Preliminary General Maths Text Book

MATHS Quest

General Mathematics PRELIMINARY COURSE

Robert Rowland

Second edition published 2008 by John Wiley & Sons Australia, Ltd 42 McDougall Street, Milton, Qld 4064 First edition published 2000 Typeset in 10.5/12.5 pt Times © John Wiley & Sons Australia, Ltd 2000, 2008 The moral rights of the author have been asserted. National Library of Australia Cataloguing-in-Publication data Rowland, Robert, 1963–. Maths quest general mathematics: preliminary course. 2nd ed. For secondary school students. ISBN 978 0 7314 0570 1 (student edition) ISBN 978 0 7314 0571 8 (teacher edition) 1. Mathematics — Textbooks. I. Title. 510

Reproduction and communication for educational purposes The Australian Copyright Act 1968 allows a maximum of one chapter or 10% of the pages of this work, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that the educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL). Reproduction and communication for other purposes Except as permitted under the Act (for example, a fair dealing for the purposes of study, research, criticism or review), no part of this book may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher. Cover photograph and internal design images: © Digital Vision Illustrated by the Wiley Art Studio Printed in China by Printplus Limited 10 9 8 7 6 5 4 3 2

Contents Introduction viii About eBookPLUS x Acknowledgements xi

CHAPTER 1 Earning money 1 Are you ready?

2 Calculating salary payments 3 Exercise 1A 4 Calculating wages 6 Exercise 1B 8 10 Quick Questions 1 11 Commission and royalties 11 Exercise 1C 14 Payment by piece 16 Exercise 1D 17 10 Quick Questions 2 18 Working overtime 19 Exercise 1E 21 Investigation — Investigating government payments 24 Additions to and deductions from gross pay 25 Exercise 1F 27 Investigation — Examining bank fees and taxes 30 10 Quick Questions 3 31 Budgeting 31 Exercise 1G 35 Summary 40 Chapter review 41 Practice examination questions 43

CHAPTER 2 Units of measurement 45 Are you ready?

46

Units of measurement 47 Exercise 2A 50 Relative error 52 Exercise 2B 54 Investigation — Measuring heights 56 10 Quick Questions 1 56 Significant figures 57 Exercise 2C 60 Rates 61 Exercise 2D 65 Percentage change 67 Exercise 2E 68

10 Quick Questions 2

69 Using ratios 69 Exercise 2F 72 Summary 74 Chapter review 75 Practice examination questions 76

CHAPTER 3 Applications of area and volume 77 Are you ready?

78 Review of area 79 Exercise 3A 81 Investigation — Maximising an area of land 84 Calculating irregular areas from a field diagram 85 Investigation — Land survey 86 Exercise 3B 87 10 Quick Questions 1 88 Solid shapes 89 Exercise 3C 91 Surface area 92 Exercise 3D 94 10 Quick Questions 2 96 Volume of a prism 97 Investigation — Exploring the volume of a prism 97 Exercise 3E 99 Volume of other solids 103 Exercise 3F 105 Summary 108 Chapter review 109 Practice examination questions 112

CHAPTER 4 Basic algebraic skills Are you ready?

113

114

General number patterns 115 Exercise 4A 117 Number pattern notation 119 Exercise 4B 122 10 Quick Questions 1 124 Adding and subtracting like terms 125 Exercise 4C 126 Substitution 127 Exercise 4D 128

iv 10 Quick Questions 2

130

Multiplication and division of algebraic expressions 131 Exercise 4E 133 Solving linear equations 134 Exercise 4F 137 Equations arising from substitution 139 Exercise 4G 141 Summary 143 Chapter review 144 Practice examination questions 146

CHAPTER 5 Statistics and society Are you ready?

147

148

Analysing data 149 Investigation — Why statistical investigation? 149 Investigation — A statistical investigation – 1 149 Statistical processes 150 Investigation — Posing questions 150 Investigation — A statistical investigation – 2 150 Exercise 5A 152 Investigation — A statistical investigation – 3 153 Exercise 5B 155 Investigation — A statistical investigation – 4 155 Exercise 5C 159 Investigation — A statistical investigation – 5 159 Investigation — A statistical investigation – 6 159 Investigation — A statistical investigation – 7 159 Quality control 160 Exercise 5D 162 Privacy and ethical issues 163 Investigation — Privacy issues 163 Investigation — Organisations that use statistics 164 Summary 165 Chapter review 166

CHAPTER 6 Data collection and sampling 167 Are you ready?

168

Target populations and sampling 169 Investigation — Gallup poll 169 Investigation — Identifying the target population 169 Exercise 6A 172 Investigation — Census or sample 174 Population characteristics 174 Investigation — Population characteristics 175 Exercise 6B 177 Investigation — Choosing a sample 179 10 Quick Questions 1 179 Bias 180 Investigation — Bias in statistics 181 Investigation — Biased sampling 182 Investigation — Spreadsheets creating misleading graphs 182 Exercise 6C 184 Investigation — Bias 185 Types of data 186 Exercise 6D 188 10 Quick Questions 2 191 Estimating populations 191 Investigation — Estimating a population 192 Exercise 6E 193 Summary 194 Chapter review 195 Practice examination questions 196

CHAPTER 7 Modelling linear relationships 199 Are you ready?

200

Graphing linear functions 201 Exercise 7A 204 Investigation — Graph of height versus age 205 Gradient and intercept 205 Exercise 7B 209 Drawing graphs using gradient and intercept 211 Exercise 7C 214 10 Quick Questions 1 215 Graphing variations 216 Exercise 7D 217 Investigation — Currency conversions 218 Step and piecewise functions 218 Exercise 7E 220 Simultaneous equations 221 Exercise 7F 222

v Summary 224 Chapter review 225 Practice examination questions 227

CHAPTER 8 Investing money 229 Are you ready?

230

Calculation of simple interest 231 Exercise 8A 234 10 Quick Questions 1 236 Graphing simple interest functions 236 Exercise 8B 239 Calculation of compound interest 241 Exercise 8C 244 10 Quick Questions 2 247 Calculating compound interest from a table of compounded values 248 Exercise 8D 251 Graphing compound interest functions 253 Exercise 8E 255 Share dividends 257 Exercise 8F 258 Graphing share performance 260 Exercise 8G 262 Investigation — Researching share prices 263 Inflation and appreciation 264 Exercise 8H 265 Summary 267 Chapter review 268 Practice examination questions 270

CHAPTER 10 Summary statistics Are you ready?

321

322

Calculating the mean 323 Investigation — Average — what does it mean? 323 Exercise 10A 328 Standard deviation 333 Exercise 10B 337 Median and mode 341 Exercise 10C 345 10 Quick Questions 1 349 Best summary statistics 350 Exercise 10D 351 Investigation — Wage rise 354 Investigation — Best summary statistics and comparison of samples 354 Summary 355 Chapter review 356 Practice examination questions 361

CHAPTER 11

CHAPTER 9 Displaying single data sets 271 Are you ready?

Stem-and-leaf plots 302 Exercise 9E 306 Five-number summaries 308 Exercise 9F 312 Summary 315 Chapter review 316 Practice examination questions 319

272

Frequency tables 273 Exercise 9A 276 Types of graphs 277 Exercise 9B 280 Investigation — Choice of graph 283 Investigation — Producing graphs using technology 283 Statistical graphs 283 Exercise 9C 287 10 Quick Questions 1 291 Range and interquartile range 292 Exercise 9D 297

Similarity of two-dimensional figures 363 Are you ready? 364

Similar figures and scale factors 365 Exercise 11A 367 Investigation — Enlarging a figure 369 Investigation — Investigating scale factors 369 Investigation — Similar triangles 370 Solving problems using similar figures 371 Exercise 11B 372 Investigation — Scale drawing of the classroom 373 House plans 374 Exercise 11C 376 Investigation — House plans 378

vi Summary 379 Chapter review 380 Practice examination questions 382

CHAPTER 12 Taxation Are you ready?

383 384

Calculating allowable deductions 385 Exercise 12A 388 Taxable income 390 Exercise 12B 392 10 Quick Questions 1 395 Medicare levy 395 Exercise 12C 397 Investigation — Medicare levy 397 Calculating tax 398 Exercise 12D 402 10 Quick Questions 2 404 Calculating GST and VAT 405 Exercise 12E 407 Graphing tax functions 409 Exercise 12F 409 Summary 411 Chapter review 412 Practice examination questions 414

CHAPTER 13 Right-angled triangles

415

Are you ready? 416 History of mathematics — Pythagoras of Samos (circa 580 BC–500 BC) 417

Pythagoras’ theorem 418 Exercise 13A 421 Calculating trigonometric ratios 423 Investigation — Looking at the tangent ratio 423 Investigation — Looking at the sine ratio 425 Investigation — Looking at the cosine ratio 426 Exercise 13B 429 10 Quick Questions 1 430 Finding an unknown side 431 Exercise 13C 435 10 Quick Questions 2 438 Finding angles 438 Exercise 13D 442 Angles of elevation and depression 445 Exercise 13E 448 Investigation — Calculation of heights 449

Proportional diagrams 450 Investigation — Checking with a proportional diagram 450 Investigation — Using proportional diagrams 450 Summary 451 Chapter review 452 Practice examination questions 454

CHAPTER 14 The language of chance Are you ready?

455

456

Informal description of chance 457 Exercise 14A 460 Investigation — Common descriptions of chance 462 Sample space 462 Exercise 14B 464 Investigation — Matching actual and expected results 465 10 Quick Questions 1 466 Tree diagrams 467 Exercise 14C 470 Investigation — Two-stage experiments 471 Equally likely outcomes 472 Exercise 14D 474 10 Quick Questions 2 475 Using the fundamental counting principle 476 Exercise 14E 479 Summary 481 Chapter review 482 Practice examination questions 484

CHAPTER 15 Relative frequency and probability 485 Are you ready?

486 Relative frequency 487 Exercise 15A 489 Investigation — Researching relative frequencies 491 Single event probability 492 Exercise 15B 494 Investigation — Comparing probabilities with actual results 497 10 Quick Questions 1 498

vii Writing probabilities as decimals and percentages 499 Exercise 15C 500 Range of probabilities 502 Exercise 15D 504 10 Quick Questions 2 506 Investigation — Graphing results 506 Complementary events 507 Exercise 15E 509 10 Quick Questions 3 511

Summary 512 Chapter review 513 Practice examination questions 514 Glossary 515 Answers 521 Index 559

Introduction Maths Quest General Mathematics — Preliminary course is the first book in a series specifically designed for the General Mathematics Stage 6 Syllabus starting in 2000. This course replaces the current syllabuses for Mathematics in Society (1981) and Mathematics in Practice (1989). There are five new areas of study: • Financial mathematics • Data analysis • Measurement • Probability • Algebraic modelling. This resource contains: • a student textbook with accompanying eBookPLUS and • a teacher edition with accompanying eGuidePLUS.

Student textbook Full colour is used throughout to produce clearer graphs and diagrams, to provide bright, stimulating photos and to make navigation through the text easier. Clear, concise theory sections contain worked examples, highlighted important text and remember boxes. Worked examples in a Think/Write format provide a clear explanation of key steps and suggest a presentation for solutions. Exercises contain many carefully graded skills and application problems, including multiple-choice questions. Cross-references to relevant worked examples appear beside the first ‘matching’ question throughout the exercises. Investigations, including spreadsheet investigations, provide further learning opportunities through discovery. Sets of 10 Quick Questions allow students to quickly review the concepts just learnt before proceeding further in the chapter. A glossary of mathematical terms is provided to assist students’ understanding of the terminology introduced in each unit of the course. Words in bold type in the theory sections of each chapter are defined in the glossary at the back of the book. Each chapter concludes with a summary and chapter review exercise, containing questions in a variety of forms (multiple-choice, short-answer and analysis) that help consolidate students’ learning of new concepts. Practice examination questions provide a ready source of problems for students to use to gain further confidence in each topic.

ix Technology is fully integrated, in line with Board of Studies recommendations. As well as graphics calculators, Maths Quest features spreadsheets, dynamic geometry software and several graphing packages. Not only does the text promote these technologies as learning tools, but demonstration versions of the programs (with the exception of Microsoft Excel) are also included, as well as hundreds of supporting files on the bonus accompanying online resources. Graphics calculator tips are incorporated throughout the text. All formulae, which are given on the HSC examination formula sheet, are marked with the symbol .

Programs included Graphmatica: an excellent graphing utility Equation grapher and regression analyser: like a graphics calculator for the PC GrafEq: graphs any relation, including complicated inequalities Poly: for visualising 3D polyhedra and their nets Tess: for producing tessellations and other symmetric planar illustrations TI Connect: calculator screen capture and program transfer CASIO Software FA-123: calculator screen capture and program transfer Cabri Geometry II: dynamic geometry program Adobe® Acrobat® Reader 4.0

Teacher edition with accompanying eGuidePLUS The teacher edition textbook contains everything in the student textbook and more. To support teachers assisting students in class, answers appear in red next to most questions in the exercises. Each exercise is annotated with relevant study design dot points. A readily accessible Work program lists all available resources and provides curriculum coverage information. The accompanying teacher eGuidePLUS contains everything in the student eBookPLUS and more. Two tests per chapter, fully worked solutions to WorkSHEETs, the work program and other curriculum advice in editable Word 2000 format are provided. Maths Quest is a rich collection of teaching and learning resources within one package. Maths Quest General Mathematics Preliminary course, Second edition, provides ample material, such as exercises, analysis questions, investigations, worksheets and technology files, from which teachers may set assessment tasks.

Next generation teaching and learning About eBookPLUS Using the JacarandaPLUS website

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To access your eBookPLUS resources, simply log on to www.jacplus.com.au. There are three easy steps for using the JacarandaPLUS system. Step 1. Create a user account The first time you use the JacarandaPLUS system, you will need to create a user account. Go to the JacarandaPLUS home page (www.jacplus.com.au) and follow the instructions on screen. Step 2. Enter your registration code Once you have created a new account and logged in, you will be prompted to enter your unique registration code for this book, which is printed on the inside front cover of your textbook.

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Acknowledgements The Maths Quest project began in 1997, and the first edition of this book was printed in 2000. In that time we believe that Maths Quest has become the best-resourced mathematical database in Australian education. I would like to thank all of those people who have supported us with our first edition. I hope that we have been able to help you in achieving your goals and have also played a part in your successes. Technology has evolved greatly since our first edition was published. The second edition has evolved from the first textbook into an interactive resource for both students and teachers. I would like to thank everyone at John Wiley & Sons Australia, Ltd for giving me the opportunity to do this. There are three people in particular whom I would like to single out for special mention: Jennifer Nolan, whose support for the Maths Quest project and for me personally has made everything possible; Ingrid Kemp, the newest addition to our team, who has brought a new set of eyes to our project and kept the ball rolling — thanks Ingrid; and finally Keith Hartmann, who has tirelessly reviewed all of the new material and has completed all of the answer checking — thanks Keith — I hope you’re enjoying retirement! Finally, and most importantly, to my family — thank you. Without your support this book and online resources would never have been completed. The author and publisher would like to thank the following copyright holders, organisations and individuals for their assistance and for permission to reproduce copyright material in this book.

Illustrative material • © AGL: p. 38 • © Blue Mountains City Council: p. 39(upper) • © Colleen Foelz: pp 77, 104 • © Corbis Corporation: pp. 27, 42, 84, 180, 190, 289(upper), 332, 383, 391, 407 • © Digital Stock/Corbis Corporation: pp. 45, 90, 271, 277(lower), 288, 301, 415, 447, 474 • © Digital Vision: pp. 12, 14, 95, 142(lower), 167, 193, 339, 345, 351 • © Fancy Images: p. 469 • © Getty Images: p. 61(lower)/AFP/Rob Elliott • © Image Addict: p. 156 • © Image 100: p. 331 • © Integral Energy: p. 37 • © Jennifer Wright: p. 102 • © John Wiley & Sons Australia: pp. 16/Narelle Kremmer; pp. 59, 163, 283, 329 & 400/Taken by Kari Ann Tapp; p. 468/Taken by Jo Patterson; p. 478/Taken by Kari-Ann Tapp © Newspix: p. 39(lower)/Samantha Studdert; pp. 321 & 337/David Crosling; p. 462/AFP/Hamish Blair • © Peter Storer: p. 304 • © Photolibrary: p. 417/SPL • © PhotoDisc, Inc: pp. 1, 3, 4, 7, 8, 19, 20, 24, 26, 30, 37(lower), 38(lower), 43, 47, 51, 56, 61(upper), 67, 68, 69, 70, 71, 73, 85, 87, 88, 91, 93, 96, 98, 107, 112, 113, 116, 118, 129, 141, 147, 154, 158, 160, 161, 177, 186, 189, 191, 197, 198, 199, 204, 205, 208, 210, 220, 222(2), 225, 229, 231, 239, 240, 243, 247, 250, 252, 253, 256, 259, 262, 265, 266, 275, 277(upper), 282(lower), 282(upper), 286, 289(lower), 290, 291, 294, 296, 303, 323, 327, 335, 336, 340(2), 346, 363, 368, 373, 381, 382, 387, 389, 401, 408(2), 430, 437, 444(2), 449, 455, 466,

xii 472, 476(2), 477, 479, 485, 490, 491, 498, 501, 505(2), 510 • © Purestock Superstock: pp. 11, 142(upper) • © Queensland Transport: p. 5 of WorkSHEET 11.2/ Reproduced with the permission of Maritime Safety Queensland. Not to be used for navigation. © Rubberball Productions: p. 6 • © Stockbyte: pp. 494, 500 • © Stockdisc: p. 314.

Software The authors and publisher would like to thank the following software providers for their assistance and for permission to use their materials. However, the use of such material does not imply that the providers endorse this product in any way. Third party software — registered full version ordering information Full versions of third party software may be obtained by contacting the companies listed below. Texas Instruments TI Connect™ and TI-GRAPHLINK software TI Connect™ and TI-GRAPHLINK software reproduced with permission of the publisher Texas Instruments Incorporated. TI Connect software available from Texas Instruments Web: http://education.ti.com/us/product/software.html Note: The TI Connectivity cable can be purchased from educational booksellers or calculator suppliers. Program-Link FA-124 Copyright © 1999–2005, CASIO COMPUTER CO., LTD. All rights reserved. fx-9860G Manager PLUS Expired (30 days) Copyright © 2006, CASIO COMPUTER CO., LTD. All rights reserved. Distributed by Shriro Australia Pty Ltd 23–27 Chaplin Drive Lane Cove NSW 2066 Web: www.casioed.net.au and find the calculator product range If you are interested in this product after expiry, please contact Shriro Australia Pty Ltd. Graphmatica Reproduced with permission of kSoft, Inc. 345 Montecillo Dr., Walnut Creek, CA 94595-2654. e-mail: [email protected] Web: http://www.graphmatica.com Software included is for evaluation purposes only. The user is expected to register share-ware if use exceeds 30 days. Order forms are available at www.graphmatica.com/register. txt

xiii Cabri Geometry™ II PLUS Reproduced with permission of Cabrilog. Cabrilog 6, Robert Schuman Place 38000 Grenoble FRANCE Web: http://www.cabri.com 1. Due to copyright restrictions, the demo version of Cabri Geometry™ II Plus must not be used in the classroom for presentation on a regular basis. 2. For site licences contact Cabrilog — Grenoble-France at «[email protected]» or www.cabri.com GrafEq and Poly Evaluation copies of GrafEq™ and Poly™ have been included with permission from Pedagoguery Software, Inc. e-mail: [email protected] Web: http://www.peda.com Microsoft® Excel, Microsoft® Word and Microsoft® PowerPoint Microsoft Excel, Microsoft Word and Microsoft PowerPoint are registered trademarks of the Microsoft Corporation in the United States and/or other countries. Screenshots reproduced throughout with permission from Microsoft. Every effort has been made to trace the ownership of copyright material. Information that will enable the publisher to trace the copyright holders or to rectify any error or omission in subsequent reprints will be welcome. In such cases, please contact the Permission Section of John Wiley & Sons Australia, who will arrange for the payment of the usual fee.

About the author Robert Rowland has been teaching Mathematics for over 20 years and currently holds the position of Head teacher, Teaching and learning at Ulladulla High School. He taught at Cabramatta High School from 1985 to 1988 before taking up his appointment at Ulladulla High School in 1989. Robert has successfully taught all levels of Mathematics to Year 12 as well as Computing Studies 7–12 and Information Processes and Technology. Robert is the coauthor of New South Wales Maths Year 9 Standard and New South Wales Maths Year 10 Standard as well as being the author of Maths Quest General Mathematics — Preliminary Course and Maths Quest General Mathematics — HSC Course.

Earning money

1 syllabus reference Financial Mathematics 1 • Earning money

In this chapter 1A Calculating salary payments 1B Calculating wages 1C Commission and royalties 1D Payment by piece 1E Working overtime 1F Additions to and deductions from gross pay 1G Budgeting

areyou

READY?

Are you ready?

Try the questions below. If you have difficulty with any of them, extra help can be obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon next to the question on the Maths Quest Preliminary Course CD-ROM or ask your teacher for a copy.

1.1

Converting units of time

1.2

Multiplying and dividing a quantity (money) by a whole number

1.3

1.4

1.5

1.6

1.8

1 Convert each of the following to the units shown in brackets. a 2 years (months) b 3 years (weeks) c 42 weeks (fortnights) d 60 months (years)

2 Calculate each of the following. a $23.50 × 26 c $528.72 × 52

b $31 432.70 ÷ 12 d $45 600 ÷ 52

Converting a percentage into a decimal

3 Convert each of the following percentages to a decimal. a 34% b 79% d 67.2% e 8.25%

c 4% f 17.5%

Finding a percentage of a quantity (money)

4 Find each of the following. a 10% of $350 c 18% of $9000

b 25% of $1424 d 12.5% of $4570

Multiplying a quantity (money) by a decimal

5 Calculate each of the following. a $8.56 × 1.5

b $12.90 × 2.5

Adding periods of time

6 Jessica has worked the following hours in one week. Thursday 6.30 pm to 9.00 pm Friday 5.45 pm to 9.00 pm Saturday 8.00 am to 2.30 pm How many hours has she worked? Expressing one quantity as a percentage of another

7 For each of the following pairs, express the first quantity as a percentage of the second quantity. a $56, $400 b $13, $20 c $125, $625

1.9

Increasing a quantity by a percentage

8 Increase each of the following by the percentage indicated. a $560 by 10% b $1120 by 5%

c $2560 by 15%

Chapter 1 Earning money

3

Calculating salary payments Methods of payment A payment received by an employee for doing a job is called income. There are many different ways people are paid for performing a job. In this section we are going to look at some of these methods of payment: salaries, wages, commission, royalties, piecework and overtime.

Salaries Many people employed in professional occupations are paid a salary. Such employees include teachers, lawyers, accountants and some doctors. A salary is a fixed amount of money that is paid to employees to do their jobs. The amount paid does not change, regardless of the number of hours worked. Salaries are usually calculated on an annual basis. A salary is therefore usually stated as an amount per annum, which means per year. Salaries are paid in weekly, fortnightly or monthly amounts. To make calculations about salaries, you will need to remember the following information. 1 year = 52 weeks = 26 fortnights = 12 months

A lecturer is paid a salary.

WORKED Example 1 Dimitri works as an accountant and receives an annual salary of $46 800. Calculate the amount that Dimitri is paid each fortnight. THINK 1 2

WRITE

There are 26 fortnights in a year, so we divide $46 800 by 26. Evaluate.

Fortnightly pay = $46 800 ÷ 26 Fortnightly Pay = $1800

We reverse this calculation when we are given the weekly, fortnightly or monthly pay of a person and are then asked to calculate the annual salary.

WORKED Example 2 Grace is a solicitor who is paid $3500 per month. Calculate Grace’s annual salary. THINK 1 2

There are 12 months in a year, so multiply $3500 (monthly pay) by 12. Evaluate.

WRITE Annual salary = $3500 × 12 Annual salary = $42 000

4

Maths Quest General Mathematics Preliminary Course

To compare a salary payment with other forms of income it may be necessary to calculate the equivalent daily or hourly payment. To do this, we need to know the number of days or hours worked per week.

WORKED Example 3 Charlotte works as a laboratory technician and is paid an annual salary of $41 560. If Charlotte works an average of 42 hours per week, calculate her equivalent hourly rate of pay. THINK 1

2

WRITE

Calculate the weekly pay by dividing the salary by 52. Calculate the hourly rate by dividing the weekly pay by 42.

Weekly pay = $41 560 ÷ 52 = $799.23 Hourly rate = $799.23 ÷ 42 = $19.03

remember 1. A salary is a fixed payment made for doing a job. 2. A salary is usually calculated on an annual basis and can be paid in weekly, fortnightly or monthly instalments. 3. To calculate information about equivalent daily or hourly rates of pay, we need information about the number of days and hours worked by the employee.

1.1

1A

SkillS

HEET


1.2

WORKED

1 Toni is paid a salary of $44 200 per annum. Calculate Toni’s fortnightly pay.

1

2 Roger is paid a salary of $49 920 per annum. Calculate Roger’s weekly pay.

Example

SkillS

HEET

Multiplying and dividing a quantity (money) by a whole number et

reads L Sp he

EXCE

Calculating salary payments

Payroll calculations

3 Frieda is paid a salary of $54 000 per annum. Calculate Frieda’s monthly pay. 4 Wendy works as an office secretary and is paid a salary of $38 740 per annum. Calculate Wendy’s pay if she is paid: a weekly b fortnightly c monthly. 5 Darren earns a salary of $43 000 per annum. Calculate Darren’s fortnightly pay, correct to the nearest cent.


5

6 Copy and complete the table below for food production employees.

Annual salary

Weekly pay

Fortnightly pay

Monthly pay

$30 000

$39 500

$42 250

$54 350

$86 475

WORKED

Example

2

7 Maxine is paid a salary. She receives $460 per week. Calculate Maxine’s annual salary. 8 Thao receives $1250 per fortnight. Calculate Thao’s annual salary. 9 Deidre is paid monthly and receives $5800. Calculate Deidre’s annual salary. 10 multiple choice Which of the following people receives the greatest salary? A Goran, who receives $530 per week. B Bryan, who receives $1075 per fortnight. C Wayne, who receives $2330 per month. D Ron, who receives $27 900 per annum.

11 Fiona receives a salary of $29 700 per annum. If Fiona works an average of 40 hours per week, calculate the equivalent hourly rate of pay. 3 12 Jade receives a salary of $33 000 per annum. a Calculate Jade’s weekly pay, correct to the nearest cent. b Jade works an average of 36 hours each week. Calculate the hourly rate to which Jade’s salary is equivalent. Give your answer correct to the nearest cent.

WORKED

Example

13 Karina is on an annual salary of $35 776. Letitia is on a wage and is paid $16.00 per hour. a Calculate Karina’s weekly pay. b If Karina works an average of 42 hours per week, calculate whether Karina or Letitia receive the better rate of pay. 14 Garry earns $42 500 per year while his friend Henry earns $18.50 per hour. Calculate the number of hours that Henry will need to work each week to earn more money than Garry does.

6


Calculating wages Most people in the workforce earn a wage. A wage is paid at an hourly rate. The hourly rate at which a person is usually paid is called an ordinary rate. The wage for each week is calculated by multiplying the ordinary rate by the number of hours worked during that week.

WORKED Example 4 Sadiq works as a mechanic and is paid $13.65 per hour. Calculate Sadiq’s wage in a week where he works 38 hours. THINK

WRITE

Multiply $13.65 (the hourly rate) by 38 (the number of hours worked).

Wage = $13.65 × 38 Wage = $518.70

To compare two people’s wages, we can’t just look at the amount of money each receives in a pay packet. We must also consider the number of hours each has worked. Wages are compared by looking at the hourly rate. To calculate the hourly rate of an employee we need to divide the wage by the number of hours worked.

WORKED Example 5 Georgina works 42 hours as a data entry operator for a computer company. Her wage for the week totalled $483.84. Calculate Georgina’s hourly rate of pay. THINK

WRITE

Divide $483.84 (the wage) by 42 (number of hours worked).

Hourly rate = $483.84 ÷ 42 Hourly rate = $11.52

Using a similar method we are able to calculate the number of hours worked by an employee, given their wage and hourly rate of pay. The number of hours worked is found by dividing the wage by the hourly rate. In some cases, wages are increased because an allowance is paid for working in unfavourable conditions. An allowance is an additional payment made when the working conditions are difficult or unpleasant.


7

For example, a road worker may be paid an allowance for working in the rain. In these cases, the allowance must be multiplied by the number of hours worked in the unfavourable conditions and this amount added to the normal pay.

WORKED Example 6 Ryan is a road worker and is paid $9.45 per hour for a 35-hour week. For working on wet days he is paid a wet weather allowance of 86c per hour. Calculate Ryan’s pay if for 12 hours of the week he works in the rain.

THINK 1

2

3

WRITE

Calculate Ryan’s normal pay by multiplying $9.45 (hourly rate) by 35 (number of hours worked). Calculate the wet weather allowance by multiplying 0.86 (the wet weather allowance) by 12 (number of hours worked in the wet). Add the normal pay to the wet weather allowance to calculate the total pay.

Normal pay = $9.45 × 35 = $330.75 Allowance = $0.86 × 12 = $10.32 Total pay = $330.75 + $10.32 = $341.07

This type of allowance is also paid to casual workers. When you are employed on a casual basis you do not receive any holiday pay and you do not get paid for days you have off because you are sick. The casual rate is a higher rate of pay to compensate for this.

remember 1. A wage is money earned at an hourly rate. 2. To calculate a wage we multiply the hourly rate by the number of hours worked during the week. 3. To calculate an hourly rate we divide the wage by the number of hours worked. 4. To calculate the number of hours worked we divide the wage by the hourly rate. 5. Allowances are paid for working under unfavourable conditions. The total allowance should be calculated and then added to the normal pay. 6. A casual rate is a higher rate of pay for casual workers to compensate them for having no holidays and receiving no sick leave.

8


1B EXCE

et

reads L Sp he

WORKED

Example

Payroll calculations

4

Calculating wages

1 Allan works in a newspaper printing mill and is paid $12.95 per hour. Calculate Allan’s wage in a week where he works 40 hours. 2 Copy and complete the table below by calculating the wage of each of the workers. Name

Hourly rate

Hours worked

A. Smith

$14.52

40

B. Brown

$16.45

38

N. Tran

$15.95

37.5

A. Milosevic

$20.10

41

L. McTavish

$18.04

36

3 Alicia is an apprentice chef. In the first year of her apprenticeship she earns $11.80 per hour. Calculate Alicia’s wage in a week where she works: a 36 hours b 48 hours c 42.5 hours. 4 Domonic is a fully qualified chef. He earns $13.50 per hour. Calculate Domonic’s wage in a week where he works: a 32 hours b 37 hours c 44.5 hours. 5 Katherine works as a casual waitress. Casual workers earn 20% more per hour than full-time workers to compensate for their lack of holidays and sick leave. a A full-time waitress earns $14.45 per hour. Calculate the casual rate earned by casual waitresses. b Calculate Katherine’s wage in a week where she works 6 hours on Saturday and 7 hours on Sunday. 6 multiple choice Which of the following workers earns the highest wage for the week? A Dylan, who works 35 hours at $13.50 per hour B Lachlan, who works 37 hours $12.93 per hour C Connor, who works 38 hours at $12.67 per hour D Cameron, who works 40 hours at $12.19 per hour

Wage


WORKED

Example

5

9

7 Calculate the hourly rate of a person who works 40 hours for a wage of $387.20. 8 Julie earns $11.42 per hour. Calculate the number of hours worked by Julie in a week where she is paid $445.38. 9 Copy and complete the table below. Name

Wage

Hours worked

Hourly rate

A. White

$416.16

36

B. Black

$538.80

40

C. Green

$369.63

37

D. Brown

$813.96

$19.38

E. Scarlet

$231.30

$15.42

F. Grey

$776.72

$20.44

10 Calculate the hourly rate of a casual worker who earns $250.80 for 20 hours work. 11 multiple choice Which of the following workers is paid at the highest hourly rate? A Melissa, who works 35 hours for $366.45 B Belinda, who works 36 hours for $376.20 C April, who works 38 hours for $399.76 D Nicole, who works 40 hours for $419.60 12 multiple choice Which of the following people worked the greatest number of hours? A Su-Li, who earned $439.66 at $11.57 per hour B Denise, who earned $576.00 at $14.40 per hour C Vera, who earned $333.20 at $9.52 per hour D Camille, who earned $707.25 at $17.25 per hour 13 Richard works as an electrical linesman and is paid $10.94 per hour for a 38-hour week. When he has to work at heights he is paid a 46c per hour ‘height allowance’. Calculate 6 Richard’s pay in a week where 15 hours are spent working at heights.

WORKED

Example

14 Ingrid works as an industrial cleaner and is paid $14.60 per hour for a 35-hour working week. When Ingrid is working with toxic substances she is paid an allowance of $1.08 per hour. Calculate Ingrid’s pay if she works with toxic substances all week. 15 Rema works as a tailor and earns $9.45 per hour. a Calculate Rema’s wage in a week where she works 37 hours. b Zhong is Rema’s assistant and earns $8.20 per hour. Find the least time Zhong must work if he is to earn more money than Rema does. 16 Tamarin works 38 hours per week at $12.40 per hour. a Calculate Tamarin’s weekly wage. b Zoe earns the same amount each week as Tamarin does, but Zoe works a 40-hour week. Calculate Zoe’s hourly rate of pay.

10


Computer Application 1 Spreadsheets Throughout this chapter we are going to develop a number of spreadsheets that will calculate wages. Work through the following steps. 1. Open a spreadsheet and enter the following information. Alternatively, access the spreadsheet (Wages_1) from the Maths Quest General Mathematics Preliminary Course CD-ROM.

EXCE

et

reads L Sp he

Wages_1

2. Enter a pay rate of $11.20 per hour for each employee. 3. Enter the hours worked as follows: Frederick Astini, 40; James Carter, 38; Kelly George, 36; Dean Jones, 15; Paul Limbrick, 45. 4. In cell E7 (in the column headed Gross Pay) enter the formula =C7*D7. This will calculate the wage for Frederick Astini (the figure 448 should appear in the cell). 5. Format cell E7 as currency (cell E7 should now show $448.00). 6. Highlight cells E7 to E11 and select the Fill Down option. The wages for each employee should now be calculated and be formatted as currency. (The entries in this column should read $448.00, $425.60, $403.20, $168.00 and $504.00.) 7. If you now change the hours worked by each employee, his or her gross pay should update automatically. 8. Choose the Save As function to save the spreadsheet as Wages_1.


11

1 1 Calculate the wage of a person who works 36 hours at a pay rate of $9.56 per hour. 2 Calculate the wage of a person who works 38 hours at $13.65 per hour. 3 Donna works 15 hours on weekends at $14.56 per hour. Calculate Donna’s wage. 4 Calculate what Stephen will earn for working 8 hours at $11.88 per hour. 5 Debbie earns $489.06 for a 38-hour working week. Calculate Debbie’s hourly rate of pay. 6 Damien earns an annual salary of $47 000 and is paid weekly. Calculate Damien’s weekly pay. 7 Simone earns an annual salary of $70 000 and is paid fortnightly. Calculate Simone’s fortnightly pay. 8 Ivan earns an annual salary of $56 480 and is paid monthly. Calculate Ivan’s monthly pay. 9 Penny earns an annual salary of $44 000 and is paid weekly. Calculate Penny’s weekly pay. 10 Penny works an average of 35 hours each week. Calculate the hourly rate to which her salary is equivalent. (Answer to the nearest cent.)

Commission and royalties Commission is a method of payment used mainly for salespeople. When paid commission, a person receives a percentage of the value of goods sold.

A royalty is a payment made to a person who owns a copyright. For example, a musician who writes a piece of music is paid royalties on sales of CDs; an author who writes a book is paid according to the number of books sold. Royalties are calculated in the same way as commission, being paid as a percentage of sales.

12


WORKED Example 7 Jack is a computer salesman who is paid a commission of 12% of all sales. Calculate the commission that Jack earns in a week if he makes sales to the value of $15 000. THINK

WRITE

Calculate 12% of $15 000.

Commission = 12% of $15 000 Commission = 12 ÷100 × $15 000 Commission = $1800

In some cases, commission may operate on a sliding scale. This means that the commission rate changes with the value of sales. This type of commission is commonly used in real estate sales. In these examples, each portion of the commission is calculated separately. The final commission is the sum of each portion.

WORKED Example 8 A real estate agent is paid commission on his sales at the following rate: • 5% on the first $75 000 • 2.5% on the balance of the sale • price. Calculate the commission earned on the sale of a property for $235 000.

THINK 1 2 3 4

Calculate 5% of $75 000. Calculate the balance of the sale. Calculate 2.5% of $160 000. Add up each portion to calculate the commission.

WRITE 5% of $75 000 = $3750 Balance = $235 000 − $75 000 Balance = $160 000 2.5% of $160 000 = $4000 Commission = $3750 + $4000 Commission = $7750


13

In some cases, people receive a fixed amount (called a retainer) as well as a commission. This is to ensure that the person earns some money even if no sales are made. To calculate this type of pay, you will need to add the retainer to the commission.

WORKED Example 9 Shelley is a furniture salesperson and is paid $250 per week plus a commission of 2% of all sales. Calculate Shelley’s pay in a week where her sales total $12 250. THINK

WRITE

1

Calculate the commission of 2% of $12 250.

2

Add the $250 to the commission to calculate her pay.

Commission = 2% of $12 250 Commission = 2 ÷ 100 × 12 250 Commission = $245 Pay = $250 + $245 Pay = $495

In some cases, the commission does not begin to be paid until sales have reached a certain point. Here the commission is calculated only on sales above this fixed amount.

WORKED Example 10 Tony is a car salesman. Tony is paid $300 per week and 2% of all sales over $50 000. Calculate Tony’s pay in a week where his sales total $84 000. THINK 1 2

3

WRITE

Calculate the amount on which commission is to be paid. Find 2% of this amount.

Add the $300 to the commission to calculate Tony’s pay.

$84 000 − $50 000 = $34 000 Commission = 2% of $34 000 Commission = 2 ÷ 100 × $34 000 Commission = $680 Pay = $300 + $680 Pay = $980

remember 1. A commission is earned when a person is paid a percentage of the value of sales made. 2. Some commissions are paid on a sliding scale. In these cases, each portion of the commission is calculated separately and then totalled at the end. 3. Some commissions are paid together with a fixed payment called a retainer. To calculate an employee’s pay, the fixed payment needs to be added to the commission. 4. In some cases where a fixed payment is made, commission may not be paid on all sales, but rather on a section of sales above a certain point.

14


1C 1.3

WORKED

Example

SkillS

HEET

7 Converting a percentage into a decimal

1.4

3 Darren’s job is to sell CDs to music stores. If Darren sells CDs to the value of $40 000, calculate his commission if it is paid at a rate of: a 1% b 3% c 3.4%.


4 Linda is a car salesperson who is paid 1.5% commission. Calculate the amount of money Linda earns in a week where her sales total $95 000.

reads L Sp he

5 Ken is an author and is paid a royalty on his book sales. The royalty is 12% of the value of all sales of his book. Calculate the value of Ken’s royalty if the value of sales totals $34 500.

et

EXCE

1 Kylie is an insurance salesperson and she is paid 8% of the value of any insurance that she sells. Calculate the amount that Kylie is paid for selling insurance to the value of $25 000. 2 Beryl sells exercise equipment and is paid a commission of 10% on all sales. Calculate Beryl’s earnings in a week where her sales total is: a $2600 b $3270 c $5687.90.

SkillS

HEET

Commission and royalties

Calculations with percentages

6 multiple choice Ursula is a computer software salesperson. Ursula’s sales total $105 000 and she is paid a commission of 0.8%. How much does Ursula receive in commission? A $105 B $840 C $8400 D $84 000 7 multiple choice Asif is a sales representative for a hardware firm. Asif earns $870 commission on sales of $17 400. What rate of commission does Asif receive? A 0.05% B 0.5% C 5% D 20% WORKED

Example

8

8 A real estate agent charges commission at the following rate: • 5% on the first $75 000 • 2.5% on the balance of the sale price. Calculate the commission charged on the sale of a property valued at $250 000. 9 Gabrielle is a fashion sales representative. Gabrielle is paid a commission of 5% on the first $3000 of sales each week and 10% commission on the balance. Calculate Gabrielle’s commission in a week where her sales total $9500.


15

10 Using the sliding scale for commission shown in question 8, calculate the commission on a property that sells for: a $90 000 b $140 000 c $600 000. 11 Stanisa is a car salesman who is paid $250 per week plus a commission of 2% of any sales he makes. Calculate Stanisa’s pay in a week where his sales total $35 000. 9 12 Daniel works as a sales representative for a car accessories firm. Daniel is paid $150 per week plus 4% of any sales. Calculate Daniel’s earnings in a week where his sales total is: a $6000 b $8500 c $12 475.

WORKED

Example

13 multiple choice A group of sales representatives each have $10 000 in sales for a week. Who earns the most money? A Averil, who is paid a commission of 8% B Bernard, who is paid $250 plus 6% commission C Cathy, who is paid $350 plus 4% commission D Darrell, who is paid $540 plus 2.5% commission 14 Fred and Gina sell life insurance. Fred is paid a commission of 8% and Gina is paid $250 plus 5% commission. a How much does Fred earn for a week in which his sales are $5000? b How much does Gina earn for a week in which her sales total $5000? c In another week Gina earns $650. What is the value of Gina’s sales? d Fred wishes to earn $650 in a week. How much should his sales be? 15 Mario is a pay television salesman. Mario earns $500 per week plus 5% commission on all sales above $5000. Calculate Mario’s pay in a week where his sales total $7500. 10 16 Neville is a door-to-door encyclopedia salesman. He is paid $300 per week plus 3% commission on all sales greater than $5000. Calculate Neville’s pay in a week where his sales total is: a $4000 b $6500 c $8560.

WORKED

Example

17 multiple choice A firm employs five sales representatives. Which representative will earn the most in a week where each of their sales totals $12 480? A Peter, who receives a commission of 4% B Richard, who receives $100 plus a commission of 3% C Susan, who is paid $280 plus a commission of 1.8% D Trevor, who is paid $300 plus a commission of 3.5% on all sales over $6000

T SHEE

Work

18 Andrew and Bonito are sales representatives. Andrew is paid $300 plus a commission of 2.5% on all sales. Bonito is paid $250 plus a 3.5% commission on all sales over $3000. a Calculate Andrew’s commission in a week where his sales total $6500. b Calculate Bonito’s commission in a week where his sales total $6500. c Who will earn the most money in a week where both Andrew and Bonito make $16 000 in sales?

1.1

16


Payment by piece Payment by piece, or piecework refers to payment for the amount of work completed. It is commonly paid for jobs such as car detailing and letterbox delivery. The amount earned is calculated by multiplying the rate of payment by the number of pieces of work completed.

A person delivering to a letterbox is paid for piecework.

WORKED Example 11 Len has a job washing cars in a car yard. He is paid $2.25 per car washed. Calculate what Len earns in an afternoon where he washes 24 cars. THINK

WRITE

Multiply the pay rate by the number of cars detailed.

Pay = $2.25 × 24 Pay = $54.00

In some cases, piecework is paid for multiples, rather than for single units. For example, for letterbox deliveries you may be paid per 1000 deliveries made.

WORKED Example 12 Holly is delivering brochures to letterboxes in her local area. She is paid $23.00 per thousand brochures delivered. Calculate what Holly will earn for a delivery of 3500 brochures. THINK 1

2

WRITE

Divide 3500 by 1000 to calculate the number of thousand brochures delivered. Multiply 3.5 by $23.00 to calculate what Holly is paid.

3500 ÷ 1000 = 3.5 Holly’s pay = 3.5 × $23.00 Holly’s pay = $80.50

There are also examples where you will be asked to compare payment by piece with other methods of earning income, in particular, wages.


17

WORKED Example 13 Tristan has a job picking apples. He is paid $4.40 per basket. a Calculate Tristan’s pay for picking 21 baskets of apples in one day. b If it takes Tristan 8 hours to pick these apples, calculate the equivalent hourly rate of pay he has earned. THINK

WRITE

a Multiply 21 (the number of baskets) by $4.40 (the pay per basket).

a Pay = 21 × $4.40 Pay = $92.40

b Divide $92.40 (total pay) by 8 (number of hours worked).

b Hourly rate = $92.40 ÷ 8 Hourly rate = $11.55

remember 1. Payment by piece is payment to an employee for the amount of work completed. 2. To calculate the amount to be paid, multiply the number of units of work completed by the amount to be paid per unit. 3. Be careful when pay is calculated for completing 100 or 1000 units of work. You will need to first divide by this amount. 4. Remember your work on other methods of payment. You will need it to compare payment by piece with them.

1D WORKED

Example

11

Payment by piece

1 Julia works after school at a car yard detailing cars. If Julia is paid $10.85 per car, calculate what she will earn in an afternoon when she details 7 cars. 2 A group of four friends take a job picking fruit over summer. They are paid $4.50 per basket of fruit picked. Calculate the earnings of each person in the group if: a Ryan picked 23 baskets b Summer picked 21 baskets c Seth picked 19 baskets d Taylor picked 18 baskets. 3 Natalie advertises that she will do ironing for $12.50 per basket. Calculate Natalie’s earnings for doing 14 baskets of ironing. 4 Matthew charges $15 to mow a lawn. Calculate Matthew’s earnings in a week if he mows 9 lawns. 5 Dean works as a house cleaner. He charges $46.50 to clean a house. If Dean cleans 7 houses, calculate his earnings.

WORKED

Example

12

6 Barbara delivers pamphlets to local letterboxes. She is paid $21.80 per thousand pamphlets delivered. Calculate what Barbara will be paid for delivering 15 000 pamphlets.

18


7 A local business employs four people to deliver advertising to letterboxes. They are paid $18.40 per 1000 deliveries. Calculate the amount each person is paid. a Jim makes 5000 deliveries. b Georgia makes 7500 deliveries. c Nicholas makes 4750 deliveries. d Claire makes 6200 deliveries. 8 Raul works in a factory assembling toys. Raul is paid $19.25 per 100 toys assembled. Calculate what Raul is paid in a day where he assembles: a 300 toys b 650 toys c 540 toys. 9 Carolina works as a bicycle courier. She charges $5.70 per kilometre for her deliveries. Calculate Carolina’s earnings for a 4 km delivery. 10 Keith is a taxi owner/driver. He is paid $3.00 plus $1.60 per kilometre. Calculate the amount Keith will earn for a journey of: a 5 km b 15.5 km c 10.2 km. 11 Denise works as a fruit picker. She is paid $4.20 for every basket of fruit picked. a Calculate the amount Denise will earn in a day during which she picks 32 baskets 13 of fruit. b If it takes Denise 8 hours to pick the fruit, calculate the equivalent hourly rate of pay.

WORKED

Example

12 Charlie works in a car yard as a detailer. Charlie is paid $11.60 per car. a What will Charlie earn in an afternoon during which he details 15 cars? b If it takes Charlie 8 hours to detail the cars, calculate his hourly rate of pay. c If Charlie could finish in 6 hours, calculate the hourly rate of pay he would earn.

2 1 Kim works a 37-hour week at a rate of $12.32 per hour. Calculate her weekly wage. 2 Viet works 35 hours a week at an hourly rate of $9.89. Calculate Viet’s weekly wage. 3 Samantha receives an annual salary of $38 500 and is paid weekly. Calculate Samantha’s weekly pay. 4 Tom receives an annual salary of $86 000 and is paid fortnightly. Calculate Tom’s fortnightly pay. 5 Celine is paid $1246.40 per fortnight. Calculate her annual salary. 6 Mick is paid 7% commission on all sales he makes. Calculate his commission for a week in which his sales total $6960. 7 Christine is paid $250 per week plus 2.5% commission on all sales. Calculate Christine’s pay for a week in which her sales total $12 800. 8 Jason has a job picking fruit and is paid $4.85 per basket. Calculate Jason’s pay for a day in which he picks 43 baskets of fruit. 9 Julia has a job delivering pamphlets to letterboxes and is paid $13.40 per 1000 pamphlets delivered. Calculate Julia’s pay for delivering 4500 pamphlets. 10 Cameron is an author who receives a royalty of 8% of the value of sales of his book. Calculate Cameron’s royalty for book sales totalling $23 000.


19

Working overtime Overtime is paid when a wage earner works more than the regular hours each week. When an employee works overtime a higher rate is paid. This higher rate of pay is called a penalty rate. The rate is normally calculated at either: time and a half, which means that the person is paid 1 1--- times the usual rate of pay, 2 or double time, which means that the person is paid twice the normal rate of pay. A person may also be paid these overtime rates for working at unfavourable times, such as at night or during weekends. To calculate the hourly rate earned when working overtime we multiply the normal hourly rate by the overtime factor, which is 1 1--- for time and a half and 2 for double time. 2

WORKED Example 14 Gustavo is paid $9.78 per hour in his job as a childcare worker. Calculate Gustavo’s hourly rate when he is being paid for overtime at time and a half.

THINK

WRITE

Multiply $9.78 (the normal hourly rate) by 1 1--- (the overtime factor for time and a half).

Time and a half rate = $9.78 × 1 1--2 Time and a half rate = $14.67

2

To calculate the pay for a period of time worked at time and a half or double time, we multiply the normal pay rate by the overtime factor (either 1 1--- or 2) and then by the 2 number of hours worked at that overtime rate.

WORKED Example 15 Adrian works as a shop assistant and his normal rate of pay is $12.84 per hour. Calculate the amount Adrian earns for 6 hours work on Saturday, when he is paid time and a half. THINK

WRITE

Multiply $12.84 (the normal pay rate) by 1 1--- (the overtime factor) and by 6 (hours 2 worked at time and a half).

Pay = $12.84 × 1 1--- × 6 2 Pay = $115.56

When we calculate the total pay for a week that involves overtime, we need to calculate the normal pay and then add the amount earned for any overtime.

20


WORKED Example 16 Natasha works as a waitress and is paid $11.80 per hour for a 38-hour week. Calculate Natasha’s pay in a week where she works 5 hours at time and a half in addition to her regular hours. THINK

WRITE

1

Calculate Natasha’s normal pay.

2

Calculate Natasha’s pay for 5 hours at time and a half. Add the normal pay and the time and a half pay together.

3

Normal pay = $11.80 × 38 = $448.40 Time and a half = $11.80 × 1 1--- × 5 2 = $88.50 Total pay = $448.40 + $88.50 = $536.90

Some examples will have more than one overtime rate to consider and some will require you to work out how many hours have been worked at each rate.

WORKED Example 17 Graeme is employed as a car assembly worker and is paid $10.40 per hour for a 36-hour week. If Graeme works overtime, the first 6 hours are paid at time and a half and the remainder at double time. Calculate Graeme’s pay in a week where he works 45 hours.

THINK 1 2

3 4 5 6

Calculate the number of hours overtime Graeme worked. Of these nine hours, calculate how much was at time and a half and how much was at double time. Calculate Graeme’s normal pay. Calculate what Graeme is paid for 6 hours at time and a half. Calculate what Graeme is paid for 3 hours at double time. Calculate Graeme’s total pay by adding the time and a half and double time payments to his normal pay.

WRITE Overtime = 45 − 36 Overtime = 9 hours Time and a half = 6 hours Double time = 3 hours Normal pay = $10.40 × 36 Normal pay = $374.40 Time and a half = $10.40 × 1 1--- × 6 2 Time and a half = $93.60 Double time = $10.40 × 2 × 3 Double time = $62.40 Total pay = $374.40 + $93.60 + $62.40 Total pay = $530.40


21

remember 1. Overtime is paid when you work more than your normal working hours in a week, and you receive a higher rate of pay for the extra hours. 2. Overtime can be paid at: (a) time and a half — 1 1--- times the normal hourly rate 2 (b) double time — twice the normal hourly rate. 3. To calculate the hourly rate when working overtime, multiply the normal hourly rate by the overtime factor. 4. To calculate the pay that is received for overtime, multiply the normal hourly rate by the overtime factor by the number of hours worked at that overtime rate. 5. To calculate the total pay for a week when overtime has been worked, calculate the normal pay and the pay for each overtime rate separately, and add them.

1E Example

14

1 Reece works in a restaurant and is paid a normal hourly rate of $11.30. Calculate the amount Reece earns each hour when he is being paid time and a half.

1.5

SkillS

HEET

WORKED

Working overtime Multiplying

2 Carmen works as a waitress and is paid $11.42 per hour. Calculate Carmen’s rate a quantity (money) by a per hour on a Sunday when she is paid double time. decimal

Example

15

Multiplying and dividing a quantity by a fraction

6 Copy and complete the table below. Ordinary rate

Overtime rate

A. Nguyen

$8.90

Time and a half

4

M. Donnell

$9.35

Double time

6

F. Milosevic

$11.56

Time and a half

7

J. Carides

$13.86

Time and a half

6.5

Y. Robinson

$22.60

Double time

5.5

Name

Hours worked

1.7

Pay

7 multiple choice Ernie works as a chef and is paid $9.95 per hour. What will Ernie’s hourly rate be when he is paid time and a half for overtime? A $11.45 B $14.92 C $14.93 D $19.90

SkillS

HEET

5 Taylor works as an usher at a concert venue. She is normally paid $13.10 per hour. Calculate Taylor’s pay for 6 hours on Sunday when she is paid double time.

SkillS

HEET

WORKED

3 Gareth works as a train driver and is normally paid $11.48 per hour. For working on 1.6 public holidays he is paid double time and a half (overtime factor = 2 1--- ). Calculate 2 Gareth’s hourly rate of pay on a public holiday. Adding periods 4 Ben works in a hotel and is paid $11.88 per hour. Calculate the total amount Ben will of time earn for an 8-hour shift on Saturday when he is paid at time and a half.

22


8 multiple choice Stephanie works in a shop and is paid $9.40 per hour. Calculate how much more Stephanie will earn in 8 hours work at time and a half than she would at ordinary rates. A $37.60 B $75.20 C $112.80 D $188.00 9 multiple choice Eric works on the wharves unloading containers and is paid $14.20 per hour. Calculate the number of hours at time and a half that Eric will have to work to earn the same amount of money that he will earn in 9 hours at ordinary rates. A 4.5 hours B 6 hours C 10.5 hours D 13.5 hours 10 Rick works 37 hours at ordinary time each week and receives $12.64 per hour. Calculate Rick’s pay in a week where, in addition to his normal hours, he works 16 4 hours overtime at time and a half.

WORKED

Example

11 Kirsty works 36 hours each week at a pay rate of $16.40 per hour. Calculate Kirsty’s pay in a week where, in addition to her ordinary hours, she works 4 hours on Sunday, when she is paid double time. 12 Grant works as a courier and is paid $13.25 per hour for a 35-hour working week. Calculate Grant’s pay for a week where he works 4 hours at time and a half and 2 hours at double time in addition to his regular hours. 13 Copy and complete the table below. Ordinary rate

Normal hours

W. Clark

$8.60

38.5

4

—

A. Hurst

$9.85

37.5

—

6.5

S. Gannon

$14.50

38.5

5

2.5

G. Dymock

$16.23

37.5

4

1.5

D. Colley

$24.90

36.5

6

8.5

Name

Time and a Double time half hours hours

Total pay

14 multiple choice Jenny is a casual worker at a motel. The normal rate of pay is $10.40 per hour. Jenny works 8 hours on Saturday for which she is paid time and a half. On Sunday she works 6 hours for which she is paid double time. Jenny’s pay is equivalent to how many hours work at the normal rate of pay? A 14 B 21 C 24 D 28 15 multiple choice Patricia works a 35-hour week and is paid $14.15 per hour. Any overtime that Patricia does is paid at time and a half. Patricia wants to work enough overtime so that she earns more than $600 each week. What is the minimum number of hours that Patricia will need to work to earn this amount of money? A 40 B 41 C 42 D 43


23

16 Steven works on a car assembly line and is paid $12.40 for a 36-hour working week. The first 4 hours overtime he works each week is paid at time and a half with the rest 17 paid at double time. Calculate Steven’s earnings for a week in which he works 43 hours.

WORKED

Example

17 Kate works as a computer technician and is paid $18.56 per hour for a 38-hour working week. For the first 4 hours overtime each week Kate is paid time and a half and the rest is paid at double time. Calculate Kate’s pay in a week where she works: a 38 hours b 41 hours c 45 hours. 18 Zac works in a supermarket. He is paid at an ordinary rate of $8.85 per hour. If Zac works more than 8 hours on any one day the first 2 hours are paid at time and a half and the rest at double time. Calculate Zac’s pay if the hours worked each day are: Monday — 8 hours Tuesday — 9 hours Wednesday — 12 hours Thursday — 7 hours Friday — 10.5 hours.

Computer Application 2 Wages 1. Load the spreadsheet Wages_1 that you started earlier in this chapter and edit it with the following information. Alternatively, access the spreadsheet Wages_2 from the Maths Quest General Mathematics Preliminary Course CD-ROM. E

Wages_2

2. In cell G7 write the formula =C7*D7 + C7*1.5*E7 + C7*2*F7. This formula will calculate the gross wage for Frederick Astini. (You should get $526.40.) 3. Highlight cells G7 to G11 and choose the Fill Down option to copy this formula to the rest of this column. (Your answers should show $526.40, $442.40, $537.60, $481.60 and $644.00.) 4. Check the functioning of your spreadsheet by changing the hours worked by Frederick Astini to 38 normal hours, 3 hours at time and a half and 4 hours at double time. You should now have $554.40 in cell G7. Now change the hours for the other employees and notice the gross pay changing. Now change the hourly rate of pay for each employee. 5. Use the Save As option to save this spreadsheet under the name Wages_2. (This will mean that you have copies of both version 1 and 2 of the spreadsheet.)

sheet

L Spre XCE ad

24


Fixed incomes Many people rely on government allowances for an income. These allowances include the youth allowance, pensions and other welfare benefits.

Investigating government payments Youth allowance 1 2 3 4

What is the youth allowance? How much is paid per week for the youth allowance? What conditions are placed on receipt of the youth allowance? What will be the total received by a person after one year of receiving the youth allowance? 5 Find out the average weekly income for an 18-year-old person. Compare this with a person who receives the youth allowance.

Unemployment benefits 6 What is the difference between unemployment benefits and the youth allowance? 7 How much is paid per week for the unemployment benefit for a: a single person? b single person with children? c married person? 8 Do you have to work to receive the unemployment benefit? 9 What conditions are placed on a person receiving unemployment benefits? (For example, must they show that they are looking for work.) 10 Compare the amount received by a person on unemployment benefits with the average weekly income for an adult in Australia.

Pensions 11 Name three different types of pension that are paid by the government. 12 What are the conditions for receiving each of these pensions? 13 How much is received per week for each of these pensions? 14 Does the amount received vary according to marital status and the number of dependants?


25

Additions to and deductions from gross pay Although we may calculate a person’s pay, this is not the amount that is actually received. The amount that we calculate, based on their wage or salary, is called gross pay or gross wage. From your gross pay, several deductions may be made for items such as tax, union fees, private health insurance, superannuation and so on. The amount left after these deductions have been taken out is called the net pay and it is this amount that you actually receive. To calculate an employee’s net pay we subtract any deductions from the gross pay.

WORKED Example 18 Robert’s gross pay is $643.60 per week. Robert has deductions for tax of $144.46, superannuation of $57.92 and union fees of $11.40. Calculate Robert’s net pay. THINK

WRITE

From $643.60 (gross pay) subtract $144.46 (tax), $57.92 (superannuation) and $11.40 (union fees).

Net pay = $643.60 − $144.46 − $57.92 − $11.40 Net pay = $429.82

In some cases, you will be required to calculate the size of a deduction based on either an annual amount or a percentage.

WORKED Example 19 Bruce is a shop assistant and he has his union fees deducted from his pay each week. If the annual union fee is $324.60, calculate the size of Bruce’s weekly union deduction. THINK 1 2

Divide $324.60 (the annual union fee) by 52. Round the answer off to the nearest cent.

WRITE Weekly deduction = $324.60 ÷ 52 Weekly deduction = $6.24

WORKED Example 20 Charissa is a salary earner and her gross fortnightly salary is $1320. Charissa pays 9% of her gross pay each fortnight in superannuation. Calculate how much is deducted from Charissa’s pay each fortnight for superannuation. THINK

WRITE

Calculate 9% of $1320 (gross pay).

Superannuation = 9% of $1320 Superannuation = 9 ÷ 100 × $1320 Superannuation = $118.80

26


When employees take annual leave, they may receive an annual leave loading. When on holidays, such employees are paid an extra 17 1--- % of their gross pay for up to 2 4 weeks.

WORKED Example 21 Russell is a newspaper printer and is paid $14.75 per hour for a 36-hour working week. a Calculate Russell’s pay for a normal working week. b Calculate Russell’s total pay for his 4 weeks annual leave if he receives a 17 1--2- % annual leave loading on the 4 weeks pay.

THINK

WRITE

a Multiply $14.75 (hourly rate) by 36 (hours worked).

a Normal pay = $14.75 × 36 Normal pay = $531.00

b

b Normal 4 weeks pay = $531.00 × 4 = $2124.00

1

2

3

Multiply $531.00 (weekly pay) by 4 to find his normal pay for 4 weeks. Calculate the annual leave loading by finding 17 1--- % of 2 $1692.

Annual leave loading = 17 1--- % of $2124.00 2

= 17 1--- ÷ 100 × $2124.00 2 = $371.70 Total holiday pay = $2124.00 + $371.70 = $2495.70

Add $371.10 (annual leave loading) to $2124 (normal 4 weeks pay).

remember 1. Gross pay refers to pay before any deductions are made. 2. Net pay refers to the pay received after deductions have been taken out. Net pay = gross pay − deductions 3. Some deductions are calculated on an annual basis and then taken out in equal weekly or fortnightly amounts. 4. Some deductions are calculated as a percentage of gross earnings. 5. When employees take their annual leave a loading is often paid. This means that they are paid an extra 17 1--- % of their gross pay. 2


1F Example

18

Additions to and deductions from gross pay

1 Trevor is a tiler and his gross pay is $532.75 per week. His weekly deductions are $106.20 for tax, $47.95 for superannuation and $17.70 for health fund contributions. Calculate Trevor’s net pay each week. 2 Copy and complete the table below.

1.4

SkillS

HEET

WORKED

27


1.8

Deductions

Net pay

$345.00

$89.45

$563.68

$165.40

$765.90

$231.85

Expressing one quantity as a percentage of another

$1175.60

$429.56

1.9

$2500.00

$765.40

SkillS

HEET

Gross pay

HEET

SkillS

3 David works in a mine and is paid a wage of $15.75 per hour for a 36-hour working week. His deductions are $118.02 for tax, $32.50 for health insurance, $51.03 for superannuation and $5.00 for the miner’s social club. Calculate David’s net pay.

Increasing a quantity by a percentage

4 Belinda is on an annual salary of $65 500. Belinda is paid fortnightly. a Calculate Belinda’s fortnightly pay. b If Belinda has fortnightly deductions of $834.92 for tax, $226.73 for superannuation and $23.50 as a contribution to a professional organisation, calculate Belinda’s net pay. WORKED

Example

19

5 Lars works as a train driver and is a member of the union. If Lars’ union fees are $394.00 per year and Lars has his fees deducted from his pay weekly, calculate the size of Lars’ weekly deduction. 6 Yasmin is a salary earner who is paid fortnightly. Yasmin has her fees for private health insurance deducted from her pay fortnightly. If the annual premium for Yasmin’s health cover is $1456.50, calculate the amount that needs to be deducted from Yasmin’s pay each fortnight. 7 Dorothy is paid a wage of $13.45 per hour for a 38-hour working week. a Calculate Dorothy’s gross weekly pay. b Dorothy pays union fees of $265.60 per annum. Calculate the amount that should be deducted from her pay each week for union fees. c Dorothy has $98.73 deducted from her pay each week for tax and union fees. Calculate Dorothy’s net pay.

28


8 Patrick is on an annual salary of $56 000 and is paid fortnightly. a Calculate Patrick’s gross fortnightly pay. b Patrick pays fortnightly into a private health fund for which the annual premium is $1165.75. Calculate the fortnightly payment. c Patrick has his health fund payment and tax (total $660.60) deducted from his fortnightly pay. Calculate Patrick’s net fortnightly pay. WORKED 9 Sabrina earns a weekly wage of $623.50. She puts 9% of this wage into a Example superannuation fund. Calculate the amount that Sabrina pays in superannuation. 20 10 Arthur earns a gross fortnightly salary of $1520.50. He pays 11% of his gross salary in superannuation. Calculate the amount that Arthur has deducted from his salary each fortnight for superannuation. 11 Rex is paid $11.12 per hour for a 38-hour working week. a Calculate Rex’s gross weekly wage. b Rex pays 10.5% of his gross weekly wage in superannuation. Calculate Rex’s weekly superannuation contribution. c Rex pays tax of $68.18 as well as his superannuation contribution. Calculate Rex’s weekly net wage. 12 Raylene is on an annual salary of $75 000 and is paid fortnightly. a Calculate Raylene’s gross fortnightly salary. b Raylene pays 12.75% of her gross salary in superannuation. Calculate the amount that is deducted from Raylene’s salary each fortnight for superannuation. c Raylene has union fees of $486.00 per annum and private health insurance of $1323.70 per annum deducted from her pay fortnightly. Calculate the amount of the deduction made for both union fees and health insurance. d If Raylene pays $1009.22 in fortnightly tax, as well as the above deductions, calculate her net weekly pay. WORKED 13 Liang-Yi earns $13.60 per hour for a 38-hour working week. Example a Calculate the amount Liang-Yi will earn in a normal working week. 21 b Calculate the total amount Liang-Yi will receive for his 4 weeks annual leave if he receives a 17 1--- % holiday loading. 2 14 Paula is paid an annual salary of $45 800. a Calculate Paula’s gross weekly salary. b Calculate the total amount Paula will receive for her 4 weeks annual leave if she is paid a 17 1--- % holiday loading. 2 15 Leon is paid $12.95 per hour for a 36-hour working week. a Calculate Leon’s weekly wage. b Leon takes one week’s holiday for which he is given a 17 1--- % loading. Calculate 2 the holiday loading. c If Leon pays $83.24 in tax, calculate his net pay for his week’s holiday. 16 Scott is paid an annual salary of $68 500. a Calculate Scott’s salary for a 4-week period. b Calculate how much holiday loading Scott will receive for this 4-week period if it is paid at 17 1--- %. 2 c Scott pays $1250 per annum in private health insurance, which is deducted from his gross salary. Calculate how much health insurance Scott must pay for a 4-week period. d If Scott pays $1779.92 in tax for this 4 weeks, calculate his net pay for the 4-week holiday.


29

Computer Application 3 Wages template 1. Load your spreadsheet Wages_2 and add the Deductions and Net Pay columns. Alternatively, accesss the spreadsheet Wages_3 from the Maths Quest General Mathematics Preliminary Course CD-ROM. E

sheet

L Spre XCE ad

Wages_3

2. In cell I7 write the formula =G7 − H7. This formula will calculate Net Pay by subtracting Deductions from Gross Pay. 3. Your spreadsheet will now calculate both a person’s Gross Pay and Net Pay. Save this as Wages_3. (You should now have three versions of the spreadsheet saved.) 4. Now clear all the data from the columns Pay Rate, Normal Hours, Time and a half Hours, Double Time Hours and Deductions. You should then have a spreadsheet set up with no data and $ - (as can be seen below) where there are formulas. E

Wages template

When a spreadsheet is in this form it is called a template. The spreadsheet is now ready to accept new data and make new calculations. Save this version as Wages template. Alternatively, download the Wages template from the Maths Quest General Mathematics Preliminary Course CD-ROM.

sheet

L Spre XCE ad

30


Costs of banking Once we have earned money, we use a bank or similar financial institution to look after it until we need it. When we deposit money in the bank the bank pays interest on that account. However, for many accounts where we need instant access to our money, the interest paid is very low and there may be fees associated with using the account.

Examining bank fees and taxes

Find three bank accounts into which your pay could be deposited electronically. Answer the following questions about them. 1 What is the interest rate payable on the account? 2 Is there a minimum balance that must be maintained in the account? 3 What are the features of this account? (For example, do you get a cheque book?) 4 Is there a monthly management fee on the account? 5 How many free transactions are you allowed each month? What are the charges for exceeding this number of transactions? 6 Are the transaction fees applied differently to deposits and withdrawals? Are they levied differently for over-the-counter and automatic teller and EFTPOS transactions?

31


3 1 Wendy works 37 1--- hours per week at a rate of $12.74 per hour. Calculate Wendy’s 2 weekly wage. 2 David is paid an annual salary of $43 240. Calculate David’s fortnightly pay. 3 Rebecca is paid a commission of 7.2% of the value of all sales she makes. Calculate Rebecca’s pay in a week where her sales total $5700. 4 Veronica works assembling radios. She is paid $5.23 for every radio assembled. Calculate Veronica’s pay in a week where she assembles 45 radios. 5 Christy is paid $34.50 per 1000 letterbox deliveries. Calculate what Christy is paid for 2200 deliveries. 6 Matthew is paid $12.68 per hour at ordinary rates. Calculate what Matthew earns per hour in overtime when he is paid at time and a half. 7 Calculate Norman’s earnings for a 6-hour shift at double time when his ordinary rate of pay is $8.45 per hour. 8 Darren is a bank teller who is paid $9.80 per hour. Calculate what Darren will earn in a week where he works 37 hours at ordinary rates as well as 5 hours at time and a half. 9 Zelko’s gross wage is $459.50 per week. He has deductions of $80.93 for tax, $13.80 for superannuation and $11.25 for union fees. Calculate Zelko’s net wage. 10 Calculate what Melissa will receive for 4 weeks holiday pay if her normal pay is $512.40 per week and she is paid a 17 1--- % holiday loading. 2

Budgeting Once we have earned money we need to allocate the money to cover our expenses; otherwise, we may spend more than we earn! Allocating money to cover expenses is called making a budget. A budget is divided into two parts: income and expenditure. A budget is balanced when income and expenditure are equal. Consider the budget below, drawn up for Tanya, who earns a net wage of $700. Income

Expenditure

Wages

$700

Rent Groceries Bills Car loan Car running costs Entertainment Credit card Savings

$150 $100 $100 $75 $50 $60 $50 $115

Total

$700

Total

$700

32


When designing a budget, it is important to look for all your expenses and set money aside for them. For example, electricity bills arrive every three months and money should be set aside each week so that when the bill does arrive you have the money to pay for it. The amount set aside should be based on the normal amount of the bill over a year, with that amount divided into weekly or fortnightly amounts. For bills such as electricity and telephone, an extra amount should be allowed, as you do not know the exact amount of the bill until it arrives. Such an allowance covers the possibilities of a price rise or increased usage. This is not necessary for bills such as council rates or insurance, as these are known in advance.

WORKED Example 22 Ben receives four electricity bills each year. For the previous year they were for $136, $187, $169 and $105. How much should Ben budget for electricity bills out of each week’s pay? We should allow an extra 10% to cover the possibility of price increases or extra usage. THINK 1 2 3 4

WRITE

Calculate the total of the previous years bills. To calculate the weekly amount, divide $597 by 52. Increase $11.48 by 10%. Make a practical approximation of the answer.

Annual total = $136 + $187 + $169 + $105 Annual total = $597 Weekly amount = $597 ÷ 52 Weekly amount = $11.48 110% of $11.48 = $12.62 Ben should budget $12.50 per week to cover the electricity.

Some bills are calculated over different lengths of time, so the simplest way to develop a budget is to calculate all bills over a year.

WORKED Example 23 Marlene has the following bills. Electricity $110 every 2 months Telephone $95 per quarter Car insurance $254 every 6 months Rates $1250 per year Calculate the total amount that Marlene should budget for all of these bills each fortnight, allowing for an extra 10% to cover possible increases. THINK 1 2 3 4

Calculate the total annual amount for electricity. Calculate the total annual amount for telephone. Calculate the total annual amount for car insurance. Calculate the total annual amount for rates.

WRITE Electricity = $110 × 6 Electricity = $660 Telephone = $95 × 4 Telephone = $380 Car insurance = $254 × 2 Car insurance = $508 Rates = $1250


THINK

WRITE

6

Find the annual total for all of these bills. Increase $2798 by 10%.

7

Divide $3077.80 by 26.

8

Round off and give a written answer.

5

33

Total = $660 + $380 + $508 + $1250 Total = $2798 110% of $2798 = 110 ÷ 100 × $2798 110% of $2798 = $3077.80 Fortnightly allowance = $3077.80 ÷ 26 Fortnightly allowance = $118.38 Marlene should allow about $118 per fortnight to cover her bills.

To bring a budget into balance, any money that is not spent can be saved. The amount saved can be calculated by subtracting the expenses to which we are committed from the total earnings.

WORKED Example 24 Peter earns $950 per fortnight. He allows $110 per fortnight for his bills, $250 per fortnight for groceries, $70 for car running costs and $80 per fortnight for entertainment. Peter also has a mortgage for which the payment is $600 per month. a Calculate the amount Peter should allocate each fortnight for his mortgage. b Calculate the amount of money Peter can save each fortnight. c Draw up a budget for Peter, showing his income and expenditure. THINK

WRITE

a

a Annual mortgage = $600 × 12 Annual mortgage = $7200 Fortnightly amount = $7200 ÷ 26 Fortnightly amount = $276.92 b Total expenses = $276.92 + $110 + $250 + $70 + $80 Total expenses = $786.92 Savings = $950 − $786.92 Savings = $163.08

1 2

b

1 2

Calculate the annual mortgage amount. Calculate the fortnightly amount by dividing by 26. Calculate total expenses. Calculate savings by subtracting all expenses from $950.

c Draw up a budget by listing income and expenses in two columns.

c

Income

Expenditure

Wages

$950

Mortgage Bills Groceries Car Savings

$276.92 $110.00 $250.00 $70.00 $163.08

Total

$950

Total

$950.00

To do work on budgeting you will need to be able to interpret the information on various household bills.

34


WORKED Example 25 Look at the extract from a sample telephone bill below.

a What is the total of the bill? b For what period are the call charges? c How much of the bill is for service and equipment? THINK

WRITE

a Look in the box labelled ‘Total amount payable’. b Look at the dates following ‘Local Calls’. c Look at the amount next to ‘Service & Equipment’.

a The total of the bill is $154.10. b The calls were for the period 5 Jan to 4 Apr. c The cost for service and equipment was $51.45.


35

remember 1. A budget is a statement of income and expenditure. 2. A budget is in balance when income and expenditure are equal. 3. When preparing a budget, you should calculate weekly or fortnightly amounts based on annual expenditure. 4. Any unspent money should be set aside as savings to bring a budget into balance. 5. To manage a budget, you will need to be able to read a variety of household bills.

1G Example

2 Christopher pays $1360 each year in council rates. Calculate how much he should budget for each fortnight for council rates.

a quantity by a percentage L Spre XCE ad

4 Tristan’s mortgage repayments are $750 per month. Calculate the amount that Tristan Budgets should budget for each fortnight to cover his mortgage bill. Example

23

5 Mr and Mrs Banks have the following bills. Electricity $130 every quarter Telephone $108 per quarter Car insurance $35 per month House insurance $29.50 per month Council rates $1100 per year Calculate the amount that Mr and Mrs Banks should budget for each week, to pay all these bills, allowing an extra 10% for extra usage or price increases. 6 Mr and Mrs Duric have the following bills. Electricity $105 every 2 months Telephone $115 per quarter Car insurance $287 every 6 months Home contents insurance $365 per year Private health insurance $1200 per year Rent $180 per week Calculate the total amount that Mr and Mrs Duric must budget for each fortnight, to cover all these bills.

WORKED

Example

24

7 Neville earns $685 per week. His expenses are $100 for rent, $90 for groceries, $75 for bills, $70 in car running costs, $60 in entertainment and $50 for miscellaneous expenses. a Calculate the amount that Neville can save each week. b Present the above information in the form of a budget for Neville.

sheet

3 Isabelle pays $34.65 per month in car insurance. Calculate the amount that she should budget each week for car insurance.

WORKED

SkillS

HEET

22

1 Vesna gets her telephone bill quarterly. Last year her four bills were $89.50, $103.40, 1.9 $110.30 and $95.00. Calculate the amount that Vesna should budget for her telephone bill each week, allowing approximately 10% to cover price increases or extra usage. Increasing

E

WORKED

Budgeting

36


8 Petria has the following bills. Electricity $120 every quarter Telephone $80 every quarter Council rates $800 per annum Water rates $700 per annum Insurance $70 per month a Calculate the amount that Petria must budget each fortnight for the above bills. b Petria has a mortgage with a monthly repayment of $900. Calculate the amount that Petria must budget each fortnight for her mortgage. c Petria has a net fortnightly pay of $1345. If Petria budgets $250 per fortnight for groceries, $80 for entertainment, $30 for medical expenses and $70 for car running costs, calculate the amount that Petria can save each fortnight. d Prepare the above information in a budget for Petria. WORKED

Example

9 Look at the extract from a sample telephone bill below.

25

a b c d

What is the total of the bill? For what period are the local calls charged? What is the charge for international calls? If four of these bills are received each year, what amount should be budgeted per week to pay them?


37

10 Look at the extracts from a sample electricity bill below.

a What is the amount due for this bill? b What was the amount charged for off-peak use on this

bill? c How many days does this bill cover? d How many kWh of power were used under the

Domestic heading? e What is the present reading of the domestic meter? f What was the previous reading of the off-peak meter?

38


11 Look at the extracts from a sample gas bill below.

Customer Number

Invoice Number Date of Issue

00001 21 / 09 / 07

AGL Retail Energy Pty Ltd, ACN 074 839 464 PO Box 944, North Sydney NSW 2059

Account Enquiries 131 606 Gas faults and emergencies 24hrs 131 808 24 Hour Emergency (Gasfitters, Electricians) 131 909 Sales 131 707 Callers Outside Victoria 1800 645 221

03A

MR BILL SAMPLE 787 SAMPLE RD SAMPLEVILLE VIC 3149

Payment Due 10 / 10 / 07 Last Bill

Payments Received

Balance

This Bill

Total Due

$55.92

$55.92

$0.00

$143.75

$143.75

$ Gas Charge

See over for details

Total Due

143.75 $143.75

Details Supply Address: 787 SAMPLE RD, SAMPLEVILLE Average MJ Per Day

This Bill Same Bill Last Year 161

Average Cost Per Day

123

$2.00

$1.50

Gas Consumption: Type

Gas

Meter Number

MS666421 / 09 / 07

Current Date

Reading

1874 11 / 07 / 07

Previous Reading

Units Consumed

1578

11593 MJ1

Date

296

Megajoules Consumed

1 - To convert Gas Units to megajoules, multiply the Units by 39.166666

Consumption Charge:

Tariff - General Domestic Rate Total for 72 days was 11593 MJ, charged at 1.2400¢ per MJ Total Gas Charge =

a b c d

What is the amount due for this bill? How many days does this bill cover? What is the cost per MJ on this bill? What is the daily gas consumption in MJ for this household?

$143.75


39

12 Look at the extract from the sample bill for council rates below.

BLUE MOUNTAINS CITY COUNCIL

RATES AND CHARGES NOTICE

GREAT WESTERN HIGHWAY, KATOOMBA P.O. Box 189, Katoomba, N.S.W., 2780 Phone: (STD 047) 82 0777 DX: 8305 Katoomba

FOR PERIOD 1 JULY, 2007 TO 30 JUNE, 2008

RATE ENQUIRIES DIRECT (047) 82 0538 Office and Cashier’s Hours: Monday to Friday 8.30 a.m. to 5.00 p.m.

RATE NOTICE Section 546 Local Government Act, 1993. As the owner, holder, tenant, or other person liable to pay rates and charges in respect of the below-mentioned land (or the agent to any such person) you are hereby notified that such land has been rated by Council as shown hereunder.

BILLING NUMBER 246810

J CITIZEN 10 BROWN ST SMITHVILLE NSW 2222

• SHOULD THE ADDRESS SHOWN ON THIS NOTICE BE INCORRECT, PLEASE ADVISE COUNCIL DIRECT IN WRITING

POSTING DATE 29/07/07 FIRST INSTALMENT OR FULL AMOUNT DUE DATE

DESCRIPTION AND SITUATION OF LAND RATED

31/08/07 GENERAL MANAGER PARTICULARS OF RATES AND CHARGES • FOR IMPORTANT INFORMATION AND PAYMENT METHODS PLEASE SEE REVERSE

Residential Faulconbridge Domestic Waste Charge

RATEABLE VALUE BASE DATE 1791

CENTS IN $

1.018000 91.50

AMOUNT

90000 1

916.20 91.50

• ACCRUAL OF INTEREST INTEREST ACCRUES ON RATES AND CHARGES THAT REMAIN UNPAID AFTER THE DUE DATE. INTEREST ACCRUES ON A DAILY BASIS. INTEREST DOES NOT ACCRUE ON INSTALMENTS NOT YET DUE. INTEREST RATE 10.5% PER ANNUM.

1st INSTALMENT

Domestic 31/08/07

2nd INSTALMENT

251.90 30/11/07

3rd INSTALMENT

251.90 28/02/08

4th INSTALMENT

251.90 31/05/08

Teller’s Stamp

FOR PAYMENT BY QUARTERLY INSTALMENTS PAY ABOVE AMOUNTS BY DUE DATES.

$1007.70 Teller’s Stamp

PLEASE DEDUCT ANY PAYMENTS MADE SINCE

22/7/07

What is the amount owed in council rates? What is the rateable value of the property? What is the domestic waste charge? The rates can be paid in how many instalments of what amount?

T SHEE

Work

a b c d

FOR PAYMENT IN FULL PAY THIS AMOUNT

1.2

40


summary Methods of payment • A salary is a fixed amount paid to an employee to do a job. This is usually based on an annual amount divided into weekly or fortnightly instalments. • A wage is an amount paid to an employee according to an hourly rate. The weekly wage is the hourly rate multiplied by the hours worked. • Commission or royalties are payments based on a percentage of sales. • Payment by piece is payment to an employee according to the amount of work completed.

Overtime • Overtime is paid when the employee works more than the regular hours each week. Usually the employee will be paid at either: time and a half — 1 1--- times the normal hourly rate, or 2 double time — twice the normal hourly rate.

Additions and deductions • • • •

Gross pay is the pay the employee receives before any deductions are taken out. Deductions are made from gross pay for tax, superannuation, union fees and so on. The amount left from gross pay after deductions are taken out is called net pay. Employees receive an extra 17 1--- % when they take their annual leave. This is called 2 the annual leave loading.

Budgeting • A budget is a list of income and expenses. • Budgets are used to allocate money to various purposes and to ensure that expenditure does not exceed income. • If income and expenses are equal the budget is said to be balanced.


41

CHAPTER review 1 Carole earns a salary of $39 600 per year and is paid weekly. Calculate her weekly pay. 2 Neil earns a salary of $67 400 per year and is paid fortnightly. Calculate his fortnightly pay. 3 Lainie earns a salary of $1326 per month. Calculate her annual salary. 4 Paul earns a salary of $51 000 per annum and works an average of 44 hours per week. Calculate the hourly rate to which Paul’s annual salary is equivalent.

1A 1A 1A 1A

5 Calculate the weekly wage of each of the following people. a Sandra, who works 36 hours at $14.50 per hour b Darren, who works 38 hours at $15.65 per hour c Melissa, who works 43 hours at $13.68 per hour

1B

6 Bartenders earn a standard rate of $12.30 per hour. Casual bartenders receive a casual rate of $13.80 per hour. a Kevin is a full-time bartender who works a 36-hour week. Calculate his weekly wage. b Len is a casual bartender who works 16 hours a week. Calculate Len’s weekly wage.

1B

7 Charlotte works 36 hours for a wage of $410.40. Calculate her hourly rate of pay. 8 Brian earns $11.83 per hour. Calculate the number of hours that Brian would need to work in a week if he wanted to earn $500.

1B 1B

9 Renee is a furniture salesperson who is paid 8% commission on all her sales. Calculate Renee’s pay in a week where her sales total $4940.

1C

10 Daryl is a car salesman who is paid $275 per week plus 1.5% commission on all sales. Calculate Daryl’s pay in a week where his sales total $34 900.

1C

11 Felicity sells cosmetics and is paid $150 per week plus 15% commission on all sales in excess of $1000. Calculate Felicity’s commission in a week where her sales total $3560.

1C

12 Hong has an after-school job detailing cars. Hong is paid $11.75 for every car that he details. Calculate what Hong is paid for detailing 29 cars.

1D

13 Svetlana delivers brochures to the local neighbourhood and is paid $17.50 for every 1000 brochures delivered. Calculate what Svetlana will earn for delivering 5600 brochures.

1D

14 Beatrice earns $14.20 per hour. Calculate what she will earn per hour: a on Saturdays, when she is paid time and a half b on Sundays, when she is paid double time.

1E

42


1E

15 Nicholas is a storeman who is paid a normal rate of $10.90 per hour. Calculate what Nicholas will earn for: a 6 hours work at time and a half b 5 hours work at double time.

1E

16 A photographic chemicals firm pays its factory workers $9.70 per hour. Calculate what each of the following employees earns in a week where: a Chao-ping works 38 normal hours b Elizabeth works 38 normal hours and 4 hours at time and a half c Phillip works 38 normal hours and 3 hours double time d Charlie works 38 normal hours, 4 hours time and a half and 3 hours double time.

1E

17 Eddie works as a shop assistant and is paid an ordinary rate of $10.54 per hour for a 36-hour working week. Eddie is paid time and a half for the first 4 hours overtime worked and double time for any hours beyond that. Calculate Eddy’s wage in a week where he works 47 hours. 18 Marella works as a seamstress and receives a gross wage of $439.00 per week. From her pay, $73.85 is deducted for tax, $4.80 for union fees, $39.51 for superannuation and $9.20 for health insurance. Calculate Marella’s net wage.

1F

19 Anne works as a shop assistant. Her annual union fees are $210.60. Anne has her union fees deducted from her pay weekly. Calculate the size of Anne’s weekly deduction.

1F

20 Harold earns a salary of $48 250 per annum and is paid fortnightly. a Calculate Harold’s fortnightly pay. b Harold pays 9.5% of his gross fortnightly pay into a superannuation fund. Calculate the size of Harold’s fortnightly superannuation contribution.

1F

21 Lance is paid $14.86 per hour and works 38 hours at normal time and 3 hours overtime for which he is paid time and a half. a Calculate Lance’s gross weekly pay. b Lance has his private health cover deducted from his gross pay. The annual contribution is $689.40. Calculate the amount deducted weekly from Lance’s pay. c Lance pays 11.5% of his gross pay into superannuation. Calculate the amount of Lance’s superannuation contribution. d If Lance also pays $140.30 in tax, calculate Lance’s net wage.

1G

22 Ruth has a net income of $700 per week. She has expenses of $190 for her mortgage, $90 for her bills, $80 for entertainment, $50 for car running costs, $125 for groceries and $30 for medical needs. Calculate the amount that Ruth can allocate for savings in her budget.


43

23 Amy has to budget for the following bills. Electricity $115 every 2 months Telephone $120 per quarter Insurance $62.50 per month Rates $1050 per year Calculate the amount that Amy should budget for each week to pay all of these bills.

Practice examination questions 1 multiple choice Which of the following is the highest salary? A $961.48 per week C $4165.00 per month

B $1923.12 per fortnight D $50 000 per annum

2 multiple choice Simone works as a florist and receives a normal hourly rate of $13.60. Simone’s pay for a Saturday night, when she works 6 hours at a rate of time and a half, is: A $20.40 B $81.60 C $122.40 D $163.20

3 multiple choice Noel sells computer software and receives a $250 per week retainer plus a commission of 5% of all sales over $10 000. In a week where Noel’s sales reach $13 460, he is paid a total of: A $17 B $423 C $673 D $923

1G

44


4 multiple choice Janelle works a 38-hour week at a rate of $14.50 per hour. When Janelle takes her 4 weeks annual leave she is paid a loading of 17 1--- %. Janelle’s weekly wage, when she takes her leave, 2 is: A $551 B $647.43 C $2204 D $2589.70 5 Ken works as a pest inspector. Ken is paid a wage of $15.40 per hour. a If Ken works a normal 36-hour week, calculate his wage. b Calculate Ken’s wage for a week if, in addition to his normal hours, he works 3 hours at time and a half and 2 hours at double time. c Ken receives an allowance of 79c per hour for working in confined spaces. Calculate Ken’s wage in a week if he works his normal 36 hours, but 23 of those hours are spent working in confined spaces. d Calculate the total amount which Ken will receive for his 4 weeks annual leave if he is paid an annual leave loading of 17 1--- %. 2

CHAPTER

test yourself

1

6 Danielle is a preschool teacher who receives a salary of $47 600 per annum. a Calculate the amount that she will receive each fortnight. b Danielle pays 9% of her gross salary in superannuation. Calculate her fortnightly superannuation contribution. c If Danielle also has $485.38 in tax, $45.80 for health insurance and $15.60 in union dues deducted from her pay, calculate her net fortnightly pay.

5_61_05706_NSW GM PC - 02 Page 45 Thursday, August 16, 2007 2:29 PM

Units of measurement

2 syllabus reference Measurement 1 • Units of measurement

In this chapter 2A 2B 2C 2D 2E 2F

Units of measurement Relative error Significant figures Rates Percentage change Using ratios


areyou

READY?

Are you ready?


2.1

2.2

Conversion of units

1 Complete each of the following conversions. a 5 m = ___ cm b 6.2 km = ___ m d 2000 g = ___ kg e 6.25 t = ___ kg


2 Complete each of the following conversions. a 48 hours = ___ days b 4 years = ___ weeks

2.3

Writing one quantity as a percentage of another

2.4

Rounding to a given number of decimal places

2.5

Increase or decrease by a percentage

2.6

c 8500 mm = ___ m f 750 mL = ___ L

c 8 hours = ___ min

3 In each of the following write the first quantity as a percentage of the second. Give your answers correct to 1 decimal place. a 1 cm; 2 m b 0.5 m; 15 m c 5 min; 9 hours

4 Round each of the following correct to the number of decimal places indicated in the brackets. a 2.186 486 [4] b 0.001 563 4 [3] c 48.8094 [2] d 118.3468 [1]

5 Calculate the following. a $750 increased by 12% c 3 kg increased by 7.5%

b $2500 decreased by 5% d 1.25 L decreased by 12.5%

Simplifying ratios

6 Simplify each of the following ratios. a 48 : 20 b 1.5 m : 45 cm

c 0.2 : 0.65

d

1 --4

: 1--6


Chapter 2 Units of measurement

47

Units of measurement We deal with measurement every day of our lives. Each time we pick up a ruler, look at our watch, or purchase a can of drink from the school canteen we are using measurement. The quantities measured most often are length, mass, capacity and time. The units used are the basis for the more complicated measurements of area and volume. The system we use for measurement is the International System of Units (SI system), more commonly known as the metric system. This system was introduced in Australia in 1972 to replace the ‘Imperial’ system. The metric system is simpler to use because it is based on powers of 10. Units of measurement can be compared under the metric system by examining the prefix.

Units of length Units of length are based on the metre. There are four units commonly used for measuring length: the millimetre (mm), centimetre (cm), metre (m) × 1000 and kilometre (km). 10 millimetres = 1 centimetre × 100 100 centimetres = 1 metre 1000 metres = 1 kilometre × 10 The flow chart at right shows how to convert units of measurement.

kilometres

÷ 1000

metres ÷ 100 centimetres ÷ 10 millimetres

WORKED Example 1 Complete each of the following. a 30 mm = cm b 4800 m =

km c 6.5 m =

cm d 8400 mm =

THINK

WRITE

a Changing millimetres to centimetres: divide by 10. b To change metres to kilometres: divide by 1000. c To change metres to centimetres: multiply by 100. d To change millimetres to metres: divide by 10 (to change to centimetres) then divide by 100 (to change to metres).

a 30 mm = 30 ÷ 10 cm 30 mm = 3 cm b 4800 m = 4800 ÷ 1000 km 4800 m = 4.8 km c 6.5 m = 6.5 × 100 cm 6.5 m = 650 cm d 8400 mm = 8400 ÷ 10 cm 8400 mm = 840 ÷ 100 m 8400 mm = 8.4 m

m


48


Units of mass The same method can be used to convert units of mass. There are three main units of mass: the gram (g), kilogram (kg) and tonne (t).

tonnes

× 1000

÷ 1000

kilograms × 1000

1000 grams = 1 kilogram 1000 kilograms = 1 tonne

÷ 1000 grams

Remembering these conversions can be aided by a flow chart.

WORKED Example 2 Complete the following. a 4000 g = kg b 9750 kg =

t c 3.2 kg =

g d 0.65 t =

THINK

WRITE

a To change grams to kilograms: divide by 1000. b To change kilograms to tonnes: divide by 1000. c To change kilograms to grams: multiply by 1000. d To change tonnes to kilograms: multiply by 1000.

a 4000 g = 4000 ÷ 1000 kg 4000 g = 4 kg b 9750 kg = 9750 ÷ 1000 t 9750 kg = 9.75 t c 3.2 kg = 3.2 × 1000 g 3.2 kg = 3200 g d 0.65 t = 0.65 × 1000 kg 0.65 t = 650 kg

kg

Units of capacity Capacity is the measure of liquid volume. The three common units used to measure capacity are: millilitres (mL), litres (L) and kilolitres (kL). 1000 millilitres = 1 litre 1000 litres = 1 kilolitre

× 1000

kilolitres

÷ 1000

litres × 1000

÷ 1000 millilitres

The flow chart for converting these units is similar to that for mass.

WORKED Example 3 Complete the following. a 6000 mL = L b 2500 L =

kL c 0.8 L =

mL d 10.5 kL =

THINK

WRITE

a To change millilitres to litres: divide by 1000. b To change litres to kilolitres: divide by 1000. c To change litres to millilitres: multiply by 1000. d To change kilolitres to litres: multiply by 1000.

a 6000 mL = 6000 ÷ 1000 L 6000 mL = 6 L b 2500 L = 2500 ÷ 1000 kL 2500 L = 2.5 kL c 0.8 L = 0.8 × 1000 mL 0.8 L = 800 mL d 10.5 kL = 10.5 × 1000 L 10.5 kL = 10 500 L

L



Units of time We use a more complicated system of units for time, but as we are so familiar with time, this is not difficult. 60 seconds (s) = 1 minute (min) 60 minutes = 1 hour (h) 24 hours = 1 day 7 days = 1 week

×7

weeks

49 ÷7

days × 24

÷ 24 hours

× 60

÷ 60 minutes

× 60

÷ 60 seconds

In each of the measures of time shown, the conversions are straightforward. There are, however, others which are not so precise. For example, there are 365 days in one year, except in a leap year when there are 366. We also say that there are 52 weeks in a year. However, there are actually 52 weeks and either 1 or 2 days. Finally, there are always 12 months in a year but the number of days in each month varies.

WORKED Example 4 Complete the following. a 3 min = s b 300 min = d 1 day = min e 2h= s

h

c 4 days =

h

THINK

WRITE

a To change minutes to seconds: multiply by 60.

a 3 min = 3 × 60 s 3 min = 180 s

b To change minutes to hours: divide by 60.

b 300 min = 300 ÷ 60 h 300 min = 5 h

c To change days to hours: multiply by 24.

c 4 days = 4 × 24 h 4 days = 96 h

d To change days to minutes: multiply by 24 (to change to hours) then multiply by 60 (to change to minutes).

d 1 day = 1 × 24 h 1 day = 24 × 60 min 1 day = 1440 min

e To change hours to seconds: multiply by 60 (to change to minutes) then multiply by 60 (to change to seconds).

e 2 hours = 2 × 60 min 2 hours = 120 × 60 s 2 hours = 7200 s

When making a measurement it is important to first determine the kind of quantity you are measuring (length, mass, capacity or time) and then to choose the most practical unit. For example, to measure the diameter of a coin you would use length and the most practical unit would be millimetres.


50


WORKED Example 5 Choose the most appropriate unit for each of the following measurements. a The thickness of a book b The amount of water in a glass c The amount of matter in a cricket ball d The length of a movie THINK

WRITE

a The thickness of a book is a measure of length. b The amount of water in a glass is a measure of capacity. c The amount of matter in a cricket ball is a measure of mass. d The length of a movie is a measure of time.

a Millimetres is the most practical unit to use. b Millilitres is the most practical unit to use. c Grams is the most practical unit to use. d Minutes is the most practical unit to use.

remember 1. Remember the unit conversions for length, mass, capacity and time. 2. When changing from a smaller unit to a larger unit, complete the conversion by dividing. (For example, 3 min = 3 ÷ 60 h = 0.05 h.) 3. When changing from a larger unit to a smaller unit, complete the conversion by multiplying. (For example, 2.5 t = 2.5 × 1000 kg = 2500 kg.) 4. Choose the most appropriate unit for a measurement by choosing the type of measurement, then the most practical unit.

2A 2.1

SkillS

HEET

WORKED

Example

1 Conversion of units

2.2

SkillS

HEET

WORKED


Example

2

EXCE

1 Copy and complete each of the following. a 70 mm = cm b 600 cm = d 9 cm = mm e 12 m = g 86 mm = cm h 9.2 km = j 6.4 cm = mm k 11.25 m =

m cm m cm

2 Copy and complete the following. a 8000 g = kg b 3000 kg = d 5 kg = g e 9500 kg = g 5.5 t = kg h 4.84 kg =

t t g

c f i l

5000 m = 9 km = 2400 m = 2.2 cm =

km m km mm

c 7t= kg f 2350 g = kg

et

reads L Sp he

Units of measurement

WORKED

Converting Example metric 3 units

3 Copy and complete the following. a 2000 mL = L b 11 000 L = d 15 L = mL e 4800 L = g 7.9 kL = L h 12.8 L =

kL kL mL

c 4 kL = L f 8650 mL =

L



WORKED

Example

4

4 Copy and complete the following. a 240 s = min b 360 min = h d 5 days = h e 7h= min g 2 years = days h 3 years = months j 36 h = days k 1 week = h

c f i l

51

72 h = days 3h= min 4 years = weeks 450 min = h

5 Richard is planning to have a garage built. The garage is 5.2 m long, 2.4 m wide and 2.5 m high. All builders, however, work in millimetres. What are the dimensions of the garage, in millimetres? 6 Peter is a truck driver. When he is passing through a small country town a detour takes him to a road that has a 4 tonne weight limit on all vehicles. Peter’s truck, including its load, is 3850 kg. How many kilograms under the weight limit is the truck? 7 A factory is producing orange juice. One kilogram of oranges will produce 400 mL of freshly squeezed juice. How many litres of orange juice can be produced from 4.5 tonnes of oranges? 8 At a car sales yard there are 1200 cars. It takes 20 minutes for a person to detail a car. If 15 people are employed to detail cars, how many hours will it take for all cars to be detailed? 9 multiple choice One litre of water has a mass of 1 kg. What would be the mass of 1 mL of water? A 1g B 10 g C 100 g D unknown 10 multiple choice The number of millimetres in 2.4 km is: A 24 000 B 240 000

C 2400 000

D 24 000 000

11 Choose the appropriate unit for each of the following measurements. a The time taken to run 100 metres 5 b The amount of petrol in a car’s petrol tank c A person’s height d The distance between two cities e The mass of a truck

WORKED

Example

12 The Schneider family purchase a backyard swimming pool. a Copy and complete the following statement using the appropriate unit of capacity. The amount of water used to fill the pool would be 150 . b The family decide to reduce household water consumption to compensate for filling the swimming pool. How much less water must be used per day to make up for this water over 1 year?


52


Relative error How far is it from your house to school? If you live very close to school you may give your answer in metres, or if you are a bit further away you would probably answer in kilometres. In either case the answer you give would not be exact. In fact, no measurement is exact. We are able to measure a quantity only to the degree of accuracy that our instruments allow. All measurements are approximations. In practice, we usually choose a degree of accuracy that is convenient. For example, you may say that it is 4 km from your home to school. In this case, the measurement would be given to the nearest kilometre. The actual distance from your home to school could be anything between 3.5 km and 4.5 km. Someone who lives close to school may say it is 600 m to school. This measurement is probably given to the nearest 100 m. The actual distance would be between 550 m and 650 m. When rounding off measurements, the maximum error possible is half the degree of accuracy stated.

WORKED Example 6 Hilary has her height measured at 164 cm. This measurement is given to the nearest centimetre. Between what values would her actual height be? THINK 1 2 3 4

The maximum error is half the degree of accuracy used (1 cm). Subtract 0.5 cm from 164 cm to find the smallest possible measurement. Add 0.5 cm to find the largest possible measurement. Give a written answer.

WRITE Maximum error = 0.5 cm Lower limit = 164 − 0.5 = 163.5 cm Upper limit = 164 + 0.5 = 164.5 cm Hilary’s height is between 163.5 cm and 164.5 cm.

WORKED Example 7 The mass of a trailer load of soil is given as 260 kg. The mass is given to the nearest 10 kg. Between what two masses would the true mass of the trailer load actually be? THINK 1 2 3 4

The maximum error is half the degree of accuracy used (10 kg). Subtract 5 kg from 260 kg to find the smallest possible measurement. Add 5 kg to 260 kg to find the largest possible measurement. Give a written answer.

WRITE Maximum error = 5 kg Lower limit = 260 − 5 = 255 kg Upper limit = 260 + 5 = 265 kg The soil’s mass is between 255 kg and 265 kg.



53

The true degree of accuracy in a measurement is found by determining the maximum percentage error. To do this we find the greatest possible error and then write that as a percentage of the measurement given. maximum error Percentage error = ------------------------------------- × 100% measurement

WORKED Example 8 A car’s fuel tank has a capacity of 65 litres. If this capacity is given to the nearest litre, find the degree of accuracy as a percentage (answer correct to 2 decimal places). THINK

WRITE

1

Find the maximum error. This is half the degree of accuracy used (1 L).

2

Write the percentage error rule.

3

Calculate the percentage error to 2 decimal places. Give a written answer.

4

Maximum error = 0.5 L 0.5 Percentage error = ------- × 100% 65 = 0.77% The degree of accuracy is 0.77%.

The degree of accuracy as a percentage shows how accurate a measurement is. No measuring instrument is perfect and different people can sometimes obtain a different reading from the same instrument. For example, one person may measure a person’s height as 162 cm while another may get an answer of 163 cm. To reduce the likelihood of error, we can average several readings of the same measurement.

WORKED Example 9 Taylor has her height measured by 8 people. They obtain the following results: 169 cm, 169 cm, 168 cm, 170 cm, 169 cm, 169 cm, 168 cm, 168 cm. What is the average result? THINK 1 2

WRITE

Find the total of the 8 readings. Divide the total by 8 to find the average.

Total = 169 + 169 + 168 + 170 + 169 + 169 + 168 + 168 Total = 1350 Average = 1350 ÷ 8 Average = 168.75 cm

remember 1. All measurements are approximations that are limited by the accuracy of the measuring instrument. 2. The maximum error of any measurement is half the degree of accuracy used. 3. The true accuracy of a measurement is found by calculating the percentage error. This is the maximum error as a percentage of the measured reading. 4. To obtain a more accurate measurement, several readings can be averaged.


54


2B 6

1 Each of the following measurements are given to the nearest centimetre. State the limits between which the true length lies. a 5 cm b 12 cm c 34 cm d 59 cm e 90 cm f 2m

Writing WORKED one Example quantity 7 as a percentage of another

2 Each of the following measurements are given to the nearest 10 m. State the limits between which the true length lies. a 40 m b 90 m c 250 m d 300 m e 1000 m f 2 km

WORKED

Example

2.3

SkillS

HEET

3 Each of the following measurements are given correct to 1 decimal place. State the limits between which the true length lies. a 5.3 cm b 9.8 m c 7.2 km d 5.0 mm e 9.9 km f 0.1 m

et

reads L Sp he

EXCE

Relative error

Calculations with percentages WORKED

Example

8

4 The distance between two towns is given as 45 km, correct to the nearest kilometre. a Between what two limits does the true distance lie? b What is the greatest possible error in the distance? c Write the degree of accuracy in this measurement as a percentage, correct to 2 decimal places. 5 For each of the following measurements, write the degree of accuracy as a percentage, correct to 2 decimal places. a A person’s mass is given as 67 kg, correct to the nearest kilogram. b The capacity of a bucket is 7 L, correct to the nearest litre. c The length of a park is said to be 180 m, correct to the nearest 10 m. d The volume of water in a tank is 38.6 L, correct to 1 decimal place. e The distance between Sydney and Melbourne is 1000 km, correct to the nearest 100 km. 6 a Measure each of the following lines to the nearest centimetre. iii iii iii b For each, find the maximum percentage error in the measurement, correct to 2 decimal places. 7 a Measure each of the lines in question 6, correct to the nearest millimetre. b When the lines are measured correct to the nearest millimetre, what is the maximum percentage error for each line? 8 Janice estimates that it takes 1 hour and 20 minutes to drive to a friend’s house. a If Janice’s estimate is to the nearest 10 minutes, between what two lengths of time does it take Janice to make the journey? b What is the maximum error in Janice’s estimate? c Find the degree of accuracy of Janice’s estimate, as a percentage, correct to 2 decimal places.



55

9 multiple choice The length of a house, correct to 2 decimal places, is given as 19.95 m. Which of the following statements is the most accurate? A The length of the house is between 19.9 m and 20 m. B The length of the house is between 19.945 m and 19.955 m. C The length of the house is between 19.94 m and 19.96 m. D The length of the house is between 19.955 m and 20 m. 10 multiple choice The world record for 100 m is 9.77 s. This is the time correct to 2 decimal places. What is the maximum possible error in this time? A 0.5 s B 0.05 s C 0.005 s D 0.1 s 11 multiple choice An aeroplane trip takes 17 hours, correct to the nearest hour. The degree of accuracy, correct to 2 decimal places, is: A 0.29% B 1.76% C 2.94% D 5.88% 12 multiple choice Which of the following four measurements has the greatest degree of accuracy? A The mass of a bag of beans is 400 g, correct to the nearest 50 g. B A water container contains 10 L of water, correct to the nearest litre. C The distance from home to school is 1.6 km, correct to 1 decimal place. D A roast is to cook for 1 hour and 30 min, correct to the nearest 5 minutes. 13 A paddock needs a fence which is 30 m long and 20 m wide, when measured to the nearest metre. a Between what two measurements does the length lie? b Between what two measurements does the width lie? c What is the smallest possible length of fencing needed? d What is the greatest possible length of fencing needed? e What is the maximum error in the length of fencing needed? 14 For the paddock in question 13, the area is found by multiplying the length by the width. a By multiplying the smallest possible length and width, find the smallest possible area. b By multiplying the largest possible length and width, find the largest possible area. c What is the maximum error in the area of the paddock? 15 The capacity of a jug is measured by 5 people to be 750 mL, 752 mL, 749 mL, 753 mL and 748 mL. Calculate the average of these 5 readings. 9

WORKED

Example

16 The distance between two towns is given on 4 different maps as 79 km, 81 km, 77 km and 80 km. Calculate the average of these 4 readings.


56


Measuring heights In your class, get each person to measure the height of one class member. Each person writes down their reading, without showing it to anyone else. When all readings are taken, answer the following questions. 1 What unit did you use to measure height? 2 What is the maximum error? 3 Calculate the percentage error using your reading. 4 What was the smallest height measured? 5 What was the greatest height measured? 6 Calculate the average reading.

1 Convert the following measurements into the units indicated. 1 450 cm (m) 2 6.8 km (m) 3 6800 g (kg) 4 9.45 L (mL) 5 4 min (s) 6 3 days (h) 7 5.6 m (cm) 8 40 000 mm (cm) 9 The mass of a person is shown on a scale as 65.7 kg, correct to 1 decimal place. Between what two measurements does the person’s mass lie? 10 The distance between two houses is given as 360 m, correct to the nearest 10 m. Find the degree of accuracy, correct to 1 decimal place.



57

Significant figures Consider each of the following measurements. • The distance from the Earth to the Sun is 149 000 000 km. • The distance between Sydney and Melbourne is 1040 km. • A circle with a radius of 5 cm has an area of 78.54 cm2. In each of the above cases the measurement is not exactly correct. As already stated, all measurements are approximations. Each of these measurements has had a sensible and practical approximation applied. • The distance from the Earth to the Sun has been given to the nearest one million kilometres. The distance to the nearest kilometre is needed only for very precise scientific work. • The distance between Sydney and Melbourne is given to the nearest kilometre. Noone travelling between these two cities would need to know the distance with any greater degree of accuracy. • Using the formula A = πr 2 the calculator gives the area of the circle as 78.539 816 34. Using 2 decimal places is usually a more practical way to answer such questions. The accuracy of every measurement taken is limited by the accuracy of the instrument used to take the measurement. The measurement is then given to the most practical degree of accuracy. Measurements are usually given to a required number of significant figures. In the examples above: • 149 000 000 km is to 3 significant figures • 1024 km is to 4 significant figures • 78.54 cm2 is also to 4 significant figures. Significant figures are the number of non-zero digits at the beginning of a number. The zeros that fill the remaining places are not significant and are there to maintain the correct place values. (Note: Zeros between 2 significant figures are taken to be significant, for example, 1024 has 4 significant figures, not 3.) Consider the situation below. A star is a distance of 68.04 light-years away from the Earth. If the speed of light is 299 792 km/s and a year is taken to be 365.26 days, what is the distance from the Earth to the star, in kilometres? A light-year is the distance that light will travel in 1 year. Therefore: 1 light-year = 299 792 × 60 × 60 × 24 × 365.26 = 9 460 975 039 488 km Distance = 68.04 × 9 460 975 039 488 = 643 724 741 686 764 km With such large numbers it is not usually necessary to be so exact. We could say that the distance was approximately 644 000 000 000 000 km. In this example we have rounded the distance off, correct to 3 significant figures. In this example, the zeros are not significant figures. When rounding off, however, we must include them, so that each significant figure has its correct place value. When rounding a number off to 3 significant figures, we cut the number off after the first three non-zero digits and round off using the same rules as for decimal places. We then fill out the remaining places with zeros.


58


WORKED Example 10 Round each of the following numbers off to the required number of significant figures. a 25 854 789 652 (2 significant figures) b 63 879 258 (1 significant figure) THINK

WRITE

a

a

1 2

b

1

2

Rounding off to 2 significant figures, so we look at the third significant figure. This digit is 8, so take the second significant figure up by 1 and fill out the remaining places with zeros. Rounding off to 1 significant figure, so we look at the second significant figure. This digit is 3, so it is ignored and the remaining places are filled out with zeros.

26 000 000 000

b

60 000 000

Significant figures can also be used to round off decimals. Care must be taken when reading a question to see if you are being asked to round off using significant figures or decimal places. Zeros at the front of a decimal are not considered to be significant figures. For the decimal 0.000 254 878 the first significant figure is the 2. If we round off to 2 significant figures 0.000 254 878 ≈ 0.000 25. When rounding off decimals to a set number of significant figures, the zeros at the front must be left in place but there is no need to fill out remaining places with zeros.

WORKED Example 11 Round each of the following numbers off to the number of significant figures indicated. a 0.005 254 8 (3 significant figures) b 0.014 725 8 (2 significant figures) THINK

WRITE

a

a

1

2

b

1

2

Rounding off to 3 significant figures, so we look at the fourth significant figure. This digit is 4, so it and the following digits are ignored. Rounding off to 2 significant figures, so we look to the third significant figure. This digit is 7, so the second significant figure must be increased by 1.

0.005 25 b

0.015



For very large or very small numbers we use significant figures together with scientific notation. If you look up in the sky at night, the closest star you can see is approximately 41 600 000 000 000 kilometres away. This measurement has been given correct to 3 significant figures. It can also be written as 4.16 × 1013 km, which is in scientific notation correct to 3 significant figures. 41 600 000 000 000 = 4.16 × 1013 and is entered as 4.16 EXP 13. The appearance of this on the calculator display will vary with different types of calculators. An example of a very small measurement is the width of a human hair. This may be 0.000 000 041 365 mm. In scientific notation, correct to 2 significant figures, we would write this as 4.1 × 10−8 mm.

WORKED Example 12 Write each of the following measurements in scientific notation, correct to 3 significant figures. a 25 473 269 000 km b 0.000 004 583 12 g c 499.85 L THINK

WRITE

a

Look at the fourth significant figure. This digit is a 7, so the third significant figure must be increased by 1. The decimal place must be moved 10 places left to be between the first 2 significant figures.

a

Look at the fourth significant figure. This digit is a 3, so it and the following digits are ignored. The decimal place must be moved 6 places right to be between the first 2 significant figures.

b

Look at the fourth significant figure. This digit is an 8, so the third significant figure must be increased by 1. As this digit is a 9, the previous digit must be increased until a number other than 9 is reached. The decimal place must be moved 2 places left to be between the first 2 significant figures.

c

1 2

3

b

1 2 3

c

1 2

3

2.55 × 1010

4.58 × 10−6

5.00 × 102

59


60


remember 1. A measurement is usually given to a required number of significant figures. 2. We round off using significant figures by counting the first non-zero digits. The same rules are used to round off significant figures as for decimal places. However, with significant figures zeros must be used to fill places to make sure all digits have the same place value. 3. Very large or very small measurements are written in scientific notation, correct to a required number of significant figures.

2C SkillS

HEET

2.4

WORKED

Example

10 Rounding to a given number WORKED Example of 11 decimal places

Significant figures

1 Round each of the following off to the number of significant figures indicated. a 24 587 258 (2) b 236 500 258 (1) c 8 782 568 (3) d 4587 (1) e 654 200 (1) f 287.35 (3) 2 Round each of the following off to the number of significant figures indicated. a 0.032 579 81 (2) b 0.003 658 (1) c 0.001 498 758 (3) d 6.256 677 158 (4) e 68.254 (3) f 0.000 201 47 (1) 3 Write the distance 146 565 992 km correct to: a 1 significant figure b 2 significant figures c 3 significant figures d 5 significant figures. 4 multiple choice When rounded to 2 significant figures, 0.035 81 is equal to: A 0.03 B 0.04 C 0.035

D 0.036

5 multiple choice The distance between two cities is 2986 km. Rounded to 2 significant figures, this distance becomes: A 29 km B 2900 km C 2986.00 km D 3000 km 6 multiple choice 45.5698 = 45.57 when it is rounded to which degree of accuracy? A 2 decimal places B 4 significant figures C Both A and B D Neither A nor B WORKED

Example

12

7 Write each of the following measurements in scientific notation, correct to 2 significant figures. a 471 591 400 km b 7 415 200 000 000 mm c 12 850 t d 0.002 369 g e 0.222 221 L f 0.002 99 s



61

8 The distance between the planet Mars and the Sun is given by the World Book Encyclopedia as 227 900 000 km. a Do you think this answer is correct to the nearest kilometre? Explain your answer. b To how many significant figures has this distance been given? c Give this distance correct to 2 significant figures.

kg kL mL

10 Scientists in a laboratory measure the diameter of a certain microbe to be 0.000 000 2 mm, and growing by 0.000 000 000 5 mm/day. What will be the diameter of the microbe in 10 days? (Answer in scientific notation.) 11 A light-year is defined as the distance travelled by light in one year. If light travels at 2.991 × 105 km/s, calculate in scientific notation the size of a light-year, correct to 3 significant figures. (1 year = 365.26 days)

Rates It is the last day of a test cricket match between Australia and India. To win the match, Australia need to make 280 runs in 80 overs. How many runs per over do they need to score?

T SHEE

Work

9 Copy and complete each of the following measurement conversions. a 2.4 × 106 mm = m b 9.1 × 108 m = km c 4.6 × 106 cm = mm 5 d 4.9 × 10 kg = t e 3.45 × 107 t = 6 f 8.11 × 10 kg = g g 8.9 × 107 L = 10 h 1.09 × 10 kL = L i 7.1 × 103 L =

2.1


62


This question requires us to work with rates. A rate is a comparison of two quantities of a different type. In this example we need to compare runs with overs.

WORKED Example 13 What quantities are being compared in each of the following rates? a 60 km/h b $2.50/kg c 1500 kL/year THINK

WRITE

a A kilometre is a measure of distance. An hour is a measure of time.

a 60 km/h compares distance with time.

b $2.50 is an amount of money. Kilograms is a measure of mass.

b $2.50/kg compares money with mass.

c A kilolitre is a measure of capacity. Years are a measure of time.

c 1500 kL/year compares capacity with time.

Rates, like ratios, often need to be simplified. To simplify a rate, we divide the first quantity by the second quantity. A rate is always simplified to a single unit.

WORKED Example 14 Simplify each of the following rates fully. a 240 km in 3 hours b $29.96 for 40 litres c 280 runs in 80 overs THINK

WRITE

a

a 240 km in 3 hours

b

c

1

Rewrite the original rate.

2

3

Divide the first quantity by the second quantity. (240 ÷ 3) Write the answer as a simplified rate.

1


2

3

Divide the first quantity by the second quantity. (29.96 ÷ 40) Write the answer as a simplified rate.

1


2

Divide the first quantity by the second quantity. (280 ÷ 80) Write the answer as a simplified rate.

3

= 80 km in 1 hour = 80 km/h b $29.96 for 40 litres = $0.749 for 1 litre = 74.9 c/L c 280 runs in 80 overs = 3.5 runs in 1 over = 3.5 runs/over



63

Once we are able to simplify rates, we can use them to solve problems. Solving problems usually involves multiplying or dividing quantities and rates. In each example, we need to carefully think about which of these we need to do and clearly set out the working steps.

WORKED Example 15 Giovanni is a plumber who charges $22.50/h for labour. What will be his labour charge for a job that takes 4 hours? THINK

WRITE

1

$22.50 for 1 hour, so multiply by 4 to calculate the labour charge.

$22.50 × 4 = $90.00

2

Give a written answer.

Giovanni charges $90.00 for 4 hours labour.

A common example of where a rate must be used is when modifying a recipe for a set number of people. A recipe may be given to serve 4 people but we may need to modify it to serve, say, 6. In such a case, each ingredient would need to be multiplied by 1 1--- . 2

WORKED Example 16 Below are the ingredients to make a banana pudding for 6 people. 4 bananas 10 mL lemon juice 2 eggs 20 g castor sugar 250 g coconut 20 g apricot jam Janice is having 9 people to dinner. Modify this recipe to serve 9. THINK

WRITE

1

Divide 9 by 6 to calculate the multiplication factor.

Multiplication factor = 9 ÷ 6 Multiplication factor = 1 1---

2

Multiply each ingredient by 1 1--- .

bananas = 4 × 1 1--2 =6 eggs = 2 × 1 1--2 =3 coconut = 250 g × 1 1--2 = 375 g

2

2

lemon juice = 10 mL × 1 1--2 = 15 mL castor sugar = 20 g × 1 1--2 = 30 g apricot jam = 20 g × 1 1--2 = 30 g

In a few examples a rate can compare two measurements of the same type. For example, a concentration of medicine may contain a mass/weight rate or a mass/ volume rate. This is where we are measuring the concentration of a certain substance. The concentration is the amount of one substance that is contained within another.


64


WORKED Example 17 The concentration of pentoxyverine citrate in a cough mixture is 15 mg/100 mL. A person should not consume more than 9 mg of pentoxyverine citrate per day. If one dose of the cough medicine is 10 mL, what is the maximum number of doses a person can have per day? THINK

WRITE

1

Calculate the amount of pentoxyverine citrate in one dose of cough mixture.

1 dose = 10 mL, so 100 mL = 10 doses 15 mg of pentoxyverine in 100 mL of cough mixture means 15 mg of pentoxyverine citrate in 10 doses or 1.5 mg of pentoxyverine citrate in 1 dose.

2

Divide the maximum amount of pentoxyverine citrate that can be consumed per day by the amount in each dose.

9 ÷ 1.5 = 6

3


A person can have a maximum of 6 doses of cough mixture per day.

We need to be able to use the conversion facts for measurement to convert between rates. We should be able to convert km/h to m/s and other similar rates. This is done by changing each unit separately at each stage of the conversion while keeping the equivalent rate.

WORKED Example 18 Convert the speed 20 m/s into km/h. THINK

WRITE

1

Convert 20 m/s to m/min by multiplying by 60.

20 m/s = 1200 m/min

2

Convert 1200 m/min to m/h by multiplying by 60.

20 m/s = 72 000 m/h

3

Convert 72 000 m/h to km/h by dividing by 1000.

20 m/s = 72 km/h

remember 1. A rate is a comparison of two quantities of a different type. 2. To simplify a rate, we compare the first quantity with one unit of the second quantity. 3. Rate questions need to be read carefully to see whether to multiply or divide to solve the problem. 4. Rates are converted by changing each unit separately, at each stage writing the equivalent rate.



2D WORKED

Example

13

WORKED

Example

14

65

Rates

1 What quantities are being compared in each of the following rates? a 80 km/h b $2.50/kg c $12.40/h d 50 g/L e 4 goals/game f 2°C/min g 5.1 m/s h 200 g/m2 i 78.9 c/L j 6 s/kg k 40 L/100 km l 2 m/year 2 Simplify each of the following rates (where necessary answer to 1 decimal place). a 270 km in 3 hours b $32 for 8 kg c 250 runs in 50 overs d 10 degrees in 2 h e $65 for 4 h f 90 m2 with 4 kg g 600 m in 80 s h $223 in 5 days i 500 km on 65 L j 23 goals in 8 games k 400 kL for 32 days l $42.68 for 55 L 3 multiple choice George buys 600 g of bacon at the delicatessen for $5.94. As a rate this is equal to: A $3.56/kg B 99 c/kg C $9.90/kg D $35.64/kg 4 Josie takes her car to the mechanic for a service. The mechanic worked on the car and charged Josie $68.50 for 2 1--- hours labour. At what rate has she been charged for labour? 2

5 After 15 minutes of hard exercise, Roula’s heart beat 520 times in the next 4 minutes. What is her heart rate in beats per minute? 6 Judy wants to leave Melbourne at 6:00 am bound for Sydney, a distance of 1040 km. She needs to be in Sydney by 8:00 pm that evening. If she allows for 2 hours as rest breaks, what speed must she average to arrive in time? (Answer to the nearest whole number.) 7 The race record for the Melbourne Cup is 3 min 16.9 s held by Kingston Rule. The Melbourne Cup is run over 3200 m. a How many seconds did Kingston Rule take? b What was the average speed of Kingston Rule in metres per second, to 1 decimal place? 8 Kristel’s car is filled with petrol. After travelling 345.6 km she found that her car had used 48 L of petrol. What was the fuel consumption of Kristel’s car in km/L? 9 More commonly, fuel consumption is expressed in L/100 km. A car travels 400 km on 48 L of petrol. What would be the fuel consumption in L/100 km? 10 multiple choice Hugo’s car used 56 litres of petrol on a trip of 400 km. Statement 1: Hugo’s fuel consumption is 7.14 km/L, correct to 2 decimal places. Statement 2: Hugo’s fuel consumption is 14 L/100 km. Which of the above statements is true? A 1 only B 2 only C Both 1 and 2 D Neither 1 nor 2 11 Jodie is paid $11.23 per hour for her job at the bank. How much does she earn in a week if she works 42 hours? 15

WORKED

Example


66


12 A patient in hospital is placed on an intravenous drip. The medication is given to the patient at a rate of 15 drips/min. Each drip is 0.25 mL. How much medication will the patient receive in 4 hours? 13 Below are the ingredients for seafood mornay. 600 g of rice 300 g of pink salmon 1 egg 16 60 g of butter 30 g of plain flour 450 mL of milk 90 g of shredded cheese 75 g of breadcrumbs This recipe serves 6 people. Modify the recipe so that it will serve 8 people.

WORKED

Example

14 The ingredients below make 15 chocolate cookies. 300 g of brown sugar 90 g of oil 120 g of self-raising flour 120 g of plain flour 60 g of white chocolate Modify the recipe to make 10 chocolate cookies.

30 g of cocoa powder 90 g of choc bits

15 A sore throat treatment contains 7.5 g/100 mL of povidone. If the intake of povidone must not exceed 3 grams per day and each dose of the sore throat treatment is 5 mL, 17 calculate the maximum number of doses a person can take each day.

WORKED

Example

16 A dietary supplement for cattle requires that the bull be fed 2.5 g/kg weight. Calculate the amount of the dietary supplement required for a 760 kg bull. 17 In his job as a sales assistant Jacob is paid $9.60 per hour. a On Saturdays he is paid at a rate of time and a half. How much does he receive each hour for working on Saturday? b How much does Jacob earn on a Saturday if he starts work at 8:00 am and finishes at 1:00 pm?

EXCE

et

reads L Sp he

Speed converter

WORKED

Example

18

18 Convert a speed of 15 m/s to km/h. 19 Convert each of the following rates. a 90 km/h to m/s c 8 mL/m to L/km

b 2.5 m/s to km/h d 8 km/L to L/100 km

20 multiple choice Which of the following is the most economical fuel consumption? A 10 km/L B 10 L/100 km

C 12 km/L

D 12 L/100 km

21 The instructions on a 1 kg bag of lawn food say to use 125 g/m2 of lawn. Nora buys 5 bags of the lawn food. Does she have enough to do a lawn that is 43 m2 in area? Explain your answer. 22 A car uses 45 L of petrol on a 432 km trip. a Write the fuel consumption in km/L. b How much fuel will the car use on a 324 km trip at the same rate?



67

Percentage change You should be familiar with increasing and decreasing an amount by a percentage. We will now examine consecutive percentage changes. Consider the case of a pair of rollerblades that is usually priced at $120. The price rises by 20% but then a discount of 20% is applied. The price does not revert to $120 because the increase and decrease are 20% of different amounts.

WORKED Example 19 The price of a pair of rollerblades is $120. The price is increased by 20% and then decreased by 20%. Calculate the new price of the rollerblades. THINK

WRITE

1

Increase $120 by 20%.

2

Decrease $144 by 20%.

120% of $120 = 120 ÷ 100 × $120 = $144 80% of $144 = 80 ÷ 100 × $144 = $115.20

remember 1. When we increase and decrease a quantity by the same percentage, the quantity does not revert to the original amount. 2. The increase and the decrease need to be calculated separately.


68


2E 2.5

1 The cost of a stereo system is $750. The price is increased by 10% and then decreased by 10%. Calculate the new price of the stereo. 19

WORKED

Example

SkillS

HEET

Percentage change


2 Calculate each of the following. a Increase 25 km by 5% and decrease the result by 5%. b Decrease 560 kg by 15% and then increase the result by 15%. c Increase 4 hours by 40% and then decrease the result by 40%. 3 The time taken to travel between two towns is 2 hours. A new section of road decreases the travelling time by 5%. a Calculate the new travelling time between the two towns. b An accident then increases the travelling time on a particular day by 20%. Calculate the time taken to travel between the two towns. 4 Increase $150 by 10% and then increase the result by 10%. 5 Decrease 250 kg by 5% and then decrease the result by 5%. 6 Are each of the following calculations equal? a 50 litres is increased by 10% and then the result is increased by 20%. b 50 litres is increased by 20% and then the result is increased by 10%. 7 A carpenter purchases $600 worth of items from a hardware store. He receives a 5% discount for paying cash and a further 2.5% trade discount. Calculate what he pays for the items. 8 The price of a $200 tool kit is increased by 25%. a Calculate the new price of the tool kit. b The price of the tool kit is then reduced back to $200 during a sale. Calculate the percentage discount that has been applied.



69

2 1 Copy and complete the following: 5.46 m =

cm.

2 The mass of a horse is found to be 725 kg, correct to the nearest kilogram. Within what range could the actual mass of the horse lie? 3 Calculate the maximum percentage error in the weight of the horse, correct to 3 decimal places.

4 Write 0.028 34 cm correct to 2 significant figures. 5 Write 178 569 543 in scientific notation correct to 3 significant figures. 6 Write 0.000 001 011 03 in scientific notation correct to 2 significant figures. 7 Simplify the rate 45 kg in 3 hours. 8 Convert the speed 56 km/h to m/s, correct to 1 decimal place. 9 Increase $280 by 10% and decrease the result by 5%. 10 Decrease $13.50 by 20% and then increase the result by 20%.

Using ratios Zhong and Hasam invest money in a business. Zhong invests $25 000 and Hasam invests $30 000. The business made a profit of $33 000 in the first year. If their profit is to be fairly shared, how much should each of the partners receive? We need a simple method for comparing the investment of each partner. We do this by using ratios. A ratio is a comparison of two or more quantities measured in the same units. For example, a ratio can be used to compare two quantities of money in dollars, two distances in kilometres or two masses in grams. However, we can’t use a ratio to compare quantities of different types. For example, a ratio can’t compare a distance with a mass.


70


A ratio can be simplified by dividing each quantity by the highest common factor (HCF). When this is done, we say the ratio has been fully simplified. The ratio is then easier to use in solving problems.

WORKED Example 20 Fully simplify the following ratios. a 24 cm:32 cm b $3.60:$2.10 c 3 m:80 cm THINK

WRITE

a

a 24:32 3:4 b $3.60:$2.10 360:210 12:7 c 3 m:80 cm 300:80 15:4

1 2

b

1 2 3

c

1 2 3

Write down the question, ignoring units. Divide both numbers by the HCF (8). Write down the question. Write both money quantities as cents. Divide by the HCF (30). Write the question. Change metres to centimetres. Divide by the HCF (20).

Once we are able to write ratios, we can use them to compare quantities. We do this by comparing the two parts of the ratio. We can consider each part of the ratio as consisting of a number of shares. If we know the value of one part of the ratio we can find the value of one share and hence find the other part of the ratio. This is known as the unitary method.

WORKED Example 21 Jane and Brooke’s heights are in the ratio 9:10. If Jane is 162 cm tall, how tall is Brooke? THINK 1

2 3 4

5

Model the problem by comparing the ratio to the known information. Compare the known part of the ratio (Jane’s height). Divide by 9 to find 1 share. Multiply 1 share by 10 to find the unknown part of the ratio. Give a written answer.

Returning to the problem at the start of this section, we can use ratios to divide a quantity into unequal parts.

WRITE Jane:Brooke 9:10 162 cm:??? 9 shares = 162 cm s1 share = 18 cm 10 shares = 180 cm

Brooke is 180 cm tall.



71

WORKED Example 22 Zhong and Hasam invest money in a business. Zhong invests $25 000 and Hasam invests $30 000. a What is the ratio of these investments? b If the business makes a profit of $33 000 in the first year, how much should each of the partners receive?

THINK

WRITE

a

1

b

1

a $25 000:$30 000 5:6 b 11 shares = $33 000

2

2 3 4 5

Write down the whole ratio. Divide by the HCF (5000). Sum the parts and make this equal to $33 000. Divide $33 000 by 11 to find 1 share. Zhong receives 5 shares so multiply 1 share by 5. Hasam receives 6 shares so multiply 1 share by 6. Give a written answer.

s1 share = $3000 5 shares = $15 000 6 shares = $18 000 Zhong’s share is $15 000 and Hasam’s share is $18 000.

remember 1. A ratio is a comparison of two quantities of the same type. 2. To be compared, both quantities need to be in the same unit. 3. Ratios can be simplified by dividing each part by the highest common factor (HCF). 4. To divide a quantity in a given ratio, we calculate the value of one share before calculating each part of the ratio. 5. To do most ratio problems you will need to know your basic measurement facts.


72


2F 2.6

SkillS

HEET

WORKED

Example

20

Simplifying ratios

EXCE

et

reads L Sp he

Ratios

Using ratios

1 Fully simplify each of the following ratios. a $20:$4 b 50 cm:45 cm d 560 km:240 km e 35 t:21 t g 15c:80c h 4 weeks:52 weeks j 1250 mL:300 mL k 80 cm:1 m m 1 kg:250 g n 400 mL:1 L p $4:20c q 750 kg:2 t s 3 min:45 s t 600 g:10 kg v $10:$6.50 w 3.6 km:800 m

c f i l o r u x

300 g:800 g 375 mL:500 mL 800 mm:550 mm $1:60c 40 min:1 h 900 L:3 kL 1.25 L:500 mL 3 t:450 kg

2 multiple choice The ratio 3 h:45 min fully simplified is: A 3:45 B 1:15

C 180:45

D 4:1

3 multiple choice The ratio 80 cm:2 km fully simplified is: A 40:1 B 1:2500 C 2:5 WORKED

Example

21

D 1:125

4 The ratio of boys to girls in a class is 5:4. If there are 15 boys in the class, how many girls are there? 5 In a school, the ratio of students to teachers is 35:2. If the school has 60 teachers, how many students attend the school? 6 The ratio of the weight of a male elephant to a female elephant is 10:9. If the male elephant weighs 1400 kg, what does the female elephant weigh? 7 In a cordial mixture, the ratio of syrup to water is 2:15. a How much water must be added to a 1 litre bottle of syrup? b How much cordial will this mixture make? 8 In a cricket match, the ratio of Australia’s score to England’s score is 5:3. If England made 192 runs, how many did Australia make? 9 In Parliament, the ratio of Liberal members to Labor members is 4:3. If there are 63 Labor members of the Parliament, how many Liberal members are there? 10 multiple choice In her yearly exams, the ratio of Rita’s Maths mark to her English mark was 8:7. If she scored 56% in English, what did she score in Maths? A 49% B 63% C 64% D 72% 11 Tom and Rachael divide $1000 in the ratio 7:3. How much should each receive? 12 Natalie and Kathy share a job in the ratio 3:2. If their job is a 35 hour per week job, how many hours does each person work?



73

13 In a game of netball a team scored 45 goals. The goals were scored by the goal shooter and the goal attack in the ratio 7:2. How many goals were scored by the goal attack? 14 A game of AFL at the Telstra Stadium attracts a crowd of 80 000 people. The ratio of Sydney supporters to Collingwood supporters is 11:5. How many Collingwood supporters are at the game? 15 multiple choice A fruit punch drink is to be made for a party. It consists of orange juice, pineapple juice and apple juice in the ratio 5:3:2. If we want to fill a 9 litre bucket with the punch, how much pineapple juice will be needed? A 0.9 L B 2.7 L C 3L D 5L 16 A radio station plays 14 songs in one hour. Of these, 4 are by Australian artists. a What is the ratio of Australian music to overseas music played during this hour? b If during a week this ratio is maintained, how many tracks by Australian artists will be played if a total of 2100 tracks are played? 17 Sandra and Kevin purchase a Lotto entry. The entry costs $24.80. Sandra puts in $15.50 and Kevin the rest. 22 a How much does Kevin put towards the cost of the Lotto entry? b What is the ratio of their contributions? c If the entry wins a prize of $640 000 and they agree to share the winnings in the same ratio as their contributions, how much should each receive?

WORKED

Example

18 A concrete mix is made from sand, cement and gravel in the ratio 3:2:1. How much of each component will be needed to make 150 kg of concrete? 19 At the Commonwealth Games, Australia won 60 gold, 75 silver and 30 bronze medals. a Simplify the ratio of gold:silver:bronze medals. b If at the Olympic Games medals were won in the same ratio, how many gold medals would be won if Australia won a total of 33 medals?

T SHEE

Work

20 Monica and Vicky share a two bedroom flat. Monica’s bedroom has an area of 15 m2, while Vicky’s has an area of 12 m2. a What is the ratio of the area of Vicky’s bedroom to Monica’s? b The rent on this flat is $180 per week and they agree that the rent should be split in the ratio of their bedroom areas. How much should they each contribute to the rent?

2.2


74


summary Units of measurement • Measures of length:10 mm = 1 cm, 100 cm = 1 m, 1000 m = 1 km • Measures of mass:1000 g = 1 kg, 1000 kg = 1 t • Measures of capacity:1000 mL = 1 L, 1000 L = 1 kL

Relative error • • • •

All measurements are approximations. Every measuring instrument is limited in the degree of accuracy that it allows. The maximum error is half the degree of accuracy used. A true gauge of the accuracy of a measurement is to calculate the maximum error as a percentage of the measurement we have taken.

Scientific notation • Scientific notation is a method for writing very large and very small measurements. • For numbers greater than one, we move the decimal point between the first two digits and multiply by the power of 10 corresponding to the number of places the decimal point has moved to the left. • For positive numbers less than one, we move the decimal point between the first two non-zero digits and multiply by the negative power of 10 corresponding to the number of places the decimal point has moved to the right.

Rates • A rate is a comparison of two quantities of different types. • To simplify a rate, we compare the first quantity with one unit of the second quantity. • Many problems involve using rates and these questions must be read carefully to determine whether to multiply or divide to solve the problem.

Percentage change • Percentage change involves increasing or decreasing an amount by a percentage. • When more than one percentage change is to be done, each must be carried out separately.

Ratios • A ratio is a comparison of two quantities of the same type. • Ratios often need to be simplified, and this is done by dividing each part by the highest common factor. • When a quantity needs to be divided in a given ratio, we add the parts of the ratio to find the total number of shares, then divide the quantity by this number to find the value of one share. Each part of the ratio can then be calculated by multiplying this by the number of shares in each part.



75

CHAPTER review 1 Copy and complete each of the following. a 90 mm = cm b 6m= cm d 4800 m = km e 6.9 cm = mm g 9000 g = kg h 9500 kg = t j 11 000 L = kL k 4550 mL = L m 300 s = min n 240 min = h p 4 days = h q 4 years = days

c f i l o r

6.7 km = m 11.25 m = cm 4.84 kg = g 12.8 L = mL 96 h = days 5 years = months

2A

2 An elevator has a capacity of 1.3 tonnes. If 18 people who each weigh an average of 66 kg are on the elevator, how much under the capacity is the total weight?

2A

3 In each of the following, a measurement and its degree of accuracy are given. State the limits between which the measurement lies. a 34 cm, correct to the nearest centimetre b 8.9 kg, correct to 1 decimal place c 500 km, correct to the nearest 100 km d 2.25 L, correct to 2 decimal places e 800 km, correct to the nearest 10 km

2B

4 For each of the measurements in question 3, find the degree of accuracy as a percentage, correct to 1 decimal place.

2B

5 Write each of the following measurements in scientific notation. a 60 000 000 km b 400 000 mm c 147 000 000 m d 643 000 t e 0.8739 t f 0.000 574 g g 0.002 874 mL h 0.005 874 g

2C

6 Copy and complete each of the following. a 5.2 × 105 cm = mm b 9.1 × 107 g = 7 Simplify each of the following rates. a $2.50 for 10 L b 80 km in 2 h d 3 h for 2 kg

kg

c 3.45 × 107 t =

kg

c $42 for 5 h

8 A car is travelling at 90 km/h. How far will it travel in 7 hours at this rate? 9 Eric earns $12.45/h. How many hours does he need to work to earn more than $400? 10 If petrol costs $1.27/L, how much petrol can be bought for $40? 11 The cost of a refrigerator is $900. The price is then increased by the manufacturer by 10%. When on sale, the refrigerator is sold at a discount of 10%. Calculate the sale price of the refrigerator. 12 Simplify each of the following ratios. a 9:12 b 64:48 d 40 min:25 min e $5:80c g 40 min:3 h h 600 g:2 kg

c 90 m:150 m f 500 m:3 km

2C 2D 2D 2D 2D 2E 2F

76


2F

13 Jane and Allan share an amount of money in the ratio 5:3. If Jane’s share of the money is $600, what is Allan’s share?

2F 2F

14 Divide $2000 in the ratio 3:7. 15 Yasmin and Carrie purchase a lottery ticket for $5. Yasmin paid $3.50 and Carrie paid $1.50 for the ticket. a What is the ratio of their investments in the ticket? b If the ticket won $250 000, how much should each receive?

Practice examination questions 1 multiple choice One litre of water has a mass of 1 kg. What would be the mass of 1 mL of water? A 1g B 10 g C 100 g D unknown 2 multiple choice A tree’s height is measured to be 17.3 m, correct to 1 decimal place. The maximum percentage error in the measurement of the tree is: A 0.3% B 1.7% C 2.9% D 17% 3 multiple choice The diameter of a human hair is 0.000 045 6 mm. In scientific notation this is equal to: A 4.56 × 10−4 B 4.56 × 10−5 C 4.56 × 104 D 4.56 × 105 4 multiple choice Which of the following is the greatest speed? A 100 km/h B 30 m/s C 1.5 km/min

D 2500 km/day

5 multiple choice The ratio of a tree’s height to its girth is 15:4. If the height of the tree is 26.25 m, the girth is: A 1.75 m B 6.5625 m C 7m D 98.4375 m 6 On a set of building plans the length of a rectangular house is given as 20 500 mm. a State this length in metres. b Daryl measures the length of the house in metres, correct to 1 decimal place. State the maximum error of his measurement in millimetres. c Calculate the maximum percentage error in Daryl’s measurement. d The width of the house is given as 8000 mm. Daryl calculates the area of the house as 164 000 000 mm2. Give this measurement in scientific notation.

CHAPTER

test yourself

2

7 At a certain point in the orbit of the planets Earth and Mars, the distance for a spacecraft to travel from Earth to Mars is 55 750 450 km. a Give this distance correct to 3 significant figures. b Calculate the percentage error when the distance is rounded to 3 significant figures. Give the percentage error correct to 1 significant figure. c The spacecraft takes 2 years to travel to Mars. Calculate the speed of the spacecraft in kilometres per hour, correct to 2 significant figures.

Applications of area and volume

3 syllabus reference Measurement 2 • Applications of area and volume

In this chapter 3A Review of area 3B Calculating irregular areas from a field diagram 3C Solid shapes 3D Surface area 3E Volume of a prism 3F Volume of other solids

areyou

READY?

Are you ready?


3.1

Area of squares, rectangles, triangles and circles

1 Find the area of each of the following. a

b 0.7 m 11.6 m

6.5 m

c

d 3.2 m 3.3 m 5.7 m

3.2

3.3

Converting units of area

2 Complete each of the following. a 2 cm2 = ___ mm2

b 300 000 cm2 = ___ m2

c 50 000 m2 = ___ ha

Surface area of cubes and rectangular prisms

3 Find the surface area of: a

b 8 cm 12 cm 19 cm

9 cm

3.5

3.6

Converting units of volume

4 Complete each of the following. a 2 cm3 = ___ mm3

b 3 000 000 cm3 = ___ m3

Volume of cubes and rectangular prisms

5 Calculate the volume of the figures drawn in question 3.

c 5000 mm3 = ___ cm3

Chapter 3 Applications of area and volume

79

Review of area Area is a measure of the amount of space km2 within a closed shape. Area is expressed in × 1 000 000 ÷ 10 000 square units. The exception to this rule is hectares the hectare (ha). × 10 000 ÷ 10 000 100 mm2 = 1 cm2 2 m 10 000 cm2 = 1 m2 × 10 000 ÷ 10 000 10 000 m2 = 1 ha cm2 1 000 000 m2 = 1 km2 = 100 ha × 100 ÷ 100 mm2 To convert between units we can use the flow chart at right. A square unit is a space equal to that of a square with that particular side length. For example, a square centimetre is the amount of space contained within a square with each side 1 cm. Most common shapes have a formula that we can use to find the area of that shape.

Square The formula for the area of a square is A = s2 where s represents the side length of the square.

WORKED Example 1 Find the area of the square at right. THINK 1 2 3

WRITE

Write the formula. Substitute the side length. Calculate the area.

A = s2 A = 8.52 A = 72.25 m2

8.5 m

Rectangles The formula for the area of a rectangle is A = l × b where l = length and b = breadth.

WORKED Example 2

19 mm

Find the area of the rectangle at right. THINK 1 2 3

62 mm

WRITE

Write the formula. Substitute the length and the breadth. Calculate the area.

A=l×b A = 62 × 19 A = 1178 mm2

Triangles The formula for the area of a triangle is A = triangle and h is the perpendicular height.

1 --2

× b × h where b is the base of the

80


WORKED Example 3 Find the area of the triangle at right. THINK

9.4 cm

WRITE

1

Write the formula.

A=

2

Substitute the base and the height.

A=

3

Calculate the area.

A = 60.16 cm2

1 --2 1 --2

×b×h

12.8 cm

× 12.8 × 9.4

Other quadrilaterals Formulas are also used to find the area of parallelograms, rhombuses and trapeziums. Area of a parallelogram A = b × h (b = base, h = height) Area of a rhombus

A=

1 --2

×D×d

(D, d = diagonals)

Area of a trapezium

A=

1 --2

× (a + b) × h

(a, b = parallel sides, h = height)

WORKED Example 4 Find the area of each of the following shapes. a b 26

m

18 m

9m

c

5.9 cm 7.2 cm

14 m 11.4 cm

THINK

WRITE

a

a A=b×h A = 14 × 9

3

Write the formula. Substitute the base and height. Calculate the area.

1

Write the formula.

b A=

1 --2

×D×d

2

Substitute the diagonal lengths.

A=

1 --2

× 18 × 26

3

Calculate the area.

A = 234 m2

1

Write the formula.

2 3

1 2

b

c

A = 126 m2

c A=

1 --2

× (a + b) × h

Substitute the sides and height.

A=

1 --2

× (5.9 + 11.4) × 7.2

Calculate the area.

A = 62.28 cm2

You should have seen all of these formulas in Years 7–10. You will be expected to know these formulas in your exams as they will not be given to you on your formula sheet.


81

remember You will need to remember each of the following area formulas. 1. Square

A = s2

2. Rectangle

A=l×b

3. Triangle

A=

4. Parallelogram

A=b×h

5. Rhombus

A=

6. Trapezium

A=

1 --2

×b×h

1 --2 1 --2

×D×d × (a + b) × h

3.1

HEET

SkillS

3A WORKED

Example

Review of area

1 Find the area of each of the squares below. a b

3.2

c

SkillS

HEET

1

Area of squares, rectangles, triangles and circles

Converting units of area

e

3.7 cm

f

sheet

d

L Spre XCE ad

3.6 km

29 mm

E

8 cm

Area converter (DIY)

12.5 cm

ry met

Cabri Geo

2.9 m

Example

3m

27 mm

9m

d

c 47 cm

38 mm 2.2 km

am progr –C

asio

2

2 Find the area of each of the rectangles below. a b

GC

WORKED

Area of a rectangle

Mensuration

62 cm

e

3.85 m

49.7 km

f –TI

program GC

6.4 m Mensuration 34 m 6.3 m

Example

3

Perimeter and area

c

Cabri Geo

6.2 cm 9.4 cm 9m

76 mm 82 mm

ry met

12 m

L Spre XCE ad

E

3 Find the area of each of the triangles below. a b

sheet

WORKED

Area of a triangle

82


d

e

6.3 km

f 3.7 m

8.4 km

4.2 m 9.7 m

Cabr

omet i Ge ry

WORKED

Example

Area of a parallelogram

4a

4 Find the area of each of the parallelograms below. a b 6m

c

36 cm

12 m

7.8 m 9.3 m

17 cm

d

e

38 mm

12.8 km

f 80 cm

87 mm

8m

16.9 km WORKED

Example

4

5k

m

e

.25

Cabr

Area of a trapezium

4c

f 20.9 m

14 .2

25

Example

8.8

m

7m

d

WORKED

m

m

m

c 15

omet i Ge ry

10.2 m

4m

12 m 9m

3m 39 cm

c

97 mm

7m

1m

m

km

6 Find the area of each of the trapeziums below. a b 26 mm 3m

d

c

7.7

m

m 31

9c

4b

5 Find the area of each of the rhombuses below. a b

5.6 m 3.2 m

58 mm

e

2.8 m 3.65 m 0.4 m

8.4 m

f

3.6 cm 5.4 cm 9.5 cm


7 Look at the figure at right. a Find the area of the outer rectangle. b Find the area of the inner rectangle. c Find the shaded area by subtracting the area of the inner rectangle from the area of the outer rectangle.

3 m 12 m

83 7m

20 m

8 Find the shaded area in each of the following. a b 14 m

8m

5 cm

10 m

5 cm 9 cm

16 m

c

d 3m 8m 5m 9m 10 m

8m 8m

12 m

12 m

9 multiple choice The area of the triangle at right is: A 36 cm2 B 54 cm2 2 C 108 cm D 1620 cm2

15 cm 12 cm

9 cm

10 multiple choice Which of the two statements is correct for the two shapes at right?

19 cm

19 cm

38 cm

38 cm

Statement 1. The rectangle and parallelogram have equal areas. Statement 2. The rectangle and parallelogram have equal perimeters. A Statement 1 B Statement 2 C Both statements D Neither statement 11 multiple choice The area of the figure at right is: A 54 m2 B 165 m2 2 C 225 m D 255 m2

17 m 15 m 7m 15 m

12 Len is having his lounge room carpeted. Carpet costs $27.80/m2. The lounge is rectangular with a length of 7.2 m and a width of 4.8 m. a Calculate the area of the lounge room. b Calculate the cost of carpeting the room.

84


13 A rectangular garden in a park is 15 m long and 12 m wide. A concrete path 1.5 m wide is to be laid around the garden. a Draw a diagram of the garden and the path. b Find the area of the garden. c What are the dimensions of the rectangle formed by the path? d Find the area of concrete needed for the path.

Maximising an area of land Farmer Brown needs to build a paddock for her sheep to graze. She has 1000 m of fencing with which to build this paddock. 1 If Farmer Brown builds the paddock 100 m long and 400 m wide, the area will be 40 000 m2. If she builds it 200 m long and 300 m wide, the area will be 60 000 m2. What dimensions should Farmer Brown build the paddock so it has the maximum possible area? 2 If one side of the paddock is a river, only three sides need to be fenced. If Farmer Brown still uses 1000 m of fencing, what dimensions should she now make the paddock to maximise the area?

It is possible for you to set up a spreadsheet that will calculate the area of a rectangle and substitute different values for the length and width of the paddock. Use the spreadsheet to find the maximum area of the paddock.


85

Calculating irregular areas from a field diagram Surveyors are often required to draw scale diagrams and to calculate the area of irregularly shaped blocks of land. This is done using a traverse survey. In this survey, a diagonal (traverse) is constructed between two corners of the block. The diagonal is then measured.

From this diagonal each other corner is sighted at right angles to the diagonal. Each of these lines, called an offset, is measured. These offsets then divide the block into triangles and quadrilaterals, hence we can calculate the area. C

100

The results of a traverse survey are displayed in a field diagram. The measurements through the centre of the field diagram are the points at which the offsets are taken. 100 metres is the length of the diagonal. At the sides are the measurements from the diagonal to the corners.

B

45

75 40 70

D

20 30 E 0 A

The field diagram can then be drawn as a scale diagram and the area calculated, as shown on the following page.

86


WORKED Example 5 Use the field diagram on the previous page to: a draw a scale diagram of the field (use 1 mm = 1 m) b calculate the area of the field. THINK

WRITE

a

a

1 2

3 4

Draw a 100 mm line. Draw in all offsets at the appropriate points on the traverse line. Join all corners of the field. Write all measurements on your diagram.

C

A2 45

B

A3

25 5

40

50

A1

A4 30

20

1

Calculate the area of the four triangles and the trapezium.

b A1 =

1 --2

×b×h

=

1 --2

× 75 × 45

= 1687.5 m

2

Add the areas together.

E A5

A

b

D

A2 =

1 --2

×b×h

=

1 --2

× 25 × 45

2

= 562.5 m

A4 =

1 --2

× (a + b) × h

=

1 --2

× (40 + 30) × 50

A3 =

1 --2

×b×h

= 1--- × 30 × 40 2

= 600 m2

2

A5 =

1 --2

×b×h

=

1 --2

× 20 × 30

= 1750 m2 = 300 m2 Area = 1687.5 + 562.5 + 600 + 1750 + 300 = 4900 m2

When you draw a scale diagram of the block of land, you can use measurement to find the perimeter.

Land survey 1 2 3 4

Find an area of land in or near your school and conduct a traverse survey of it. Draw a scale diagram of the area of land. Calculate the area of the land. Find the perimeter of the block.

remember 1. A traverse survey is used to calculate the area of irregularly shaped blocks of land. 2. A field diagram can be used to make a scale drawing of the land. 3. The land can be broken up into triangles and quadrilaterals. The area can then be calculated.

87


3B

Calculating irregular areas from a field diagram C

1 At right is a surveyor’s field diagram of a block of land.

WORKED

Example

50 B

a Draw a scale diagram of the block of land, using the scale 1 mm = 1 m. b Calculate the area of the block of land.

25 40

15 20 0

5

D

A

2 For the field diagram shown at right: a Draw a scale diagram of the block of land using the scale 1 mm = 1 m. b Calculate the area of the block of land. c Use measurement to find the perimeter.

D

80 C

42 65 40 28

B

35 10 0 A

3 Use the field diagrams below to calculate the area of each block of land. a b c C D D 70

B

40 40 30 15

D

C

75 23 70

B

40 30 20 20

C

E

B

100 30 90 50 30

E

30 40 20

F

0

0

0

A

A

A

E

88


1 1 Find the area of a square with a side length of 4.7 m. 2 Calculate the area of a rectangle with a length of 14.5 cm and a breadth of 9.7 cm. 3 Calculate the side length of a square with an area of 49 cm2. 4 A rectangle has a length of 7 m and an area of 84 m2. Calculate the breadth of the rectangle. 5 Calculate the area of a triangle with a base of 4.7 m and a height of 0.7 m. 6 Calculate the area of the parallelogram below. 1.7 m 9.9 m

7 Calculate the area of the rhombus below.

8.9

cm

4.5 cm

8 Calculate the area of the trapezium below. 17 mm 12 mm 39 mm

9 Calculate the area of a square with a side length of 10 mm. 10 Using your answer to question 9, copy and complete the following: 1 cm2 = mm2.


89

Solid shapes So far in this chapter we have dealt with 2-dimensional (plane) shapes. We also need to be able to recognise, classify and draw 3-dimensional (solid) shapes. Most of the solid shapes that we will be dealing with in this chapter can be classified as either prisms or pyramids. A prism is a solid shape with a constant cross-section, usually a polygon. This means that if the solid is sliced parallel to the base of the prism, the shape seen will be identical to the base. A prism is named according to the shape of its base.

WORKED Example 6 Name the prism at right. THINK

WRITE

The shape has a common cross-section and the shape of the base is a triangle.

Triangular prism

Pyramids have a plane shape as their base and have triangular sides that meet in an apex. A pyramid is also named according to the shape of its base.

WORKED Example 7 Name the pyramid at right. THINK

WRITE

The base shape is a square.

Square pyramid

A solid can also be identified by its net. The net of a solid shape is how the shape would look if it were unfolded and laid flat.

WORKED Example 8 Use the diagram of a cube to help you draw its net. THINK 1 2

The cube has six faces, each of which is a square. Draw the cube so that the six squares would fold up to form a cube.

WRITE

90


Imagine looking at a pair of railway tracks like those in the photograph below. As you can see, they appear to get closer together. The point on the horizon where they appear to meet is called the vanishing point.

When drawing a solid shape, the sides should slightly converge so that if they were extended they would meet at a similar vanishing point. This can best be done by using isometric paper.

remember 1. A prism is a solid shape where any cross-section parallel to the base is a polygon which is identical to that base shape. 2. A pyramid is a solid shape with a base and triangular sides that meet in an apex. 3. The net of a solid is how that shape would look if it were unfolded and flattened. 4. Solid shapes when drawn in perspective converge on a vanishing point. This point is best found by drawing the shape on isometric paper.


3C WORKED

Example

Solid shapes

1 Name the prism at right.

6

WORKED

Example

2 Name the pyramid at right.

7

3 Name each of the shapes below. a

b

c

4 Draw your own example of a prism and a pyramid. 5 Below is a diagram of a rectangular prism. Use the diagram to help you draw the 8 net of the rectangular prism.

WORKED

Example

.

6 Name each solid in the top row then match it with a net in the bottom row. a b c

ii

ii

iii

91

92


7 Draw the net of each of the following solids. a b

c

8 Identify the solids from the nets below. Draw the solid in your book. a b c

Work

T SHEE

3.1

9 Draw an example of each of the following on isometric paper and on your diagram mark the vanishing point. a cube b rectangular prism c triangular prism d square pyramid

Surface area Area usually refers to the space inside a 2-dimensional shape. Surface area refers to the total area occupied by the faces of a 3-dimensional shape. Surface area is measured in square units as are 2-dimensional area problems. In general, the surface area of a solid needs to be calculated by adding the area of each face separately. However, for some solids there is a unique formula.

Cube A cube has six identical faces, each of which is a square. Consider a cube of side length s. Each face can have its area calculated using the formula A = s2. Therefore, we have the formula for the surface area (SA) of a cube: SA = 6s2

s

WORKED Example 9 Find the surface area of the cube at right. THINK 1 2 3

WRITE

Write the formula. Substitute the side length. Calculate the surface area.

SA = 6s2 SA = 6 × 4.32 SA = 110.94 cm2

4.3 cm

Rectangular prism

h Consider a rectangular prism with a length of l, l a breadth of b and a height of h. Each pair of opposite faces are equal. Using the formula for a rectangle: Front and back A=l×h Top and bottom A=l×b Left and right A=b×h

b


93

Adding these gives the formula for the surface area of a rectangular prism: SA = 2(lh + lb + bh)

WORKED Example 10 Find the surface area of the rectangular prism at right. 9.7 m 3.2 m

THINK 1 2 3

WRITE

Write the formula. Substitute the length, breadth and height. Calculate the surface area.

4.3 m

SA = 2(lh + lb + bh) SA = 2(4.3 × 9.7 + 4.3 × 3.2 + 3.2 × 9.7) SA = 173.02 m2

For other solid shapes the surface area is found by adding the area of each face separately.

WORKED Example 11

5 cm

Find the surface area of the square pyramid at right. THINK

WRITE

1

Calculate the area of the square base.

2

Calculate the area of a triangular side. (Note: Each side is identical and the height of each triangular side is 5 cm.)

3

Calculate the total surface area. (Note: There are 4 identical triangular sides.)

4 cm

6 cm

A=s A = 62 A = 36 cm2 A = 1--- × b × h 2

A=

2 1 --2

×6×5

A = 15 cm2 SA = 36 + 4 × 15 SA = 96 cm2

You will be expected to know these formulas for surface area as well. They do not appear on your formula sheet.

94


remember 1. The surface area of a solid shape is the total area of each face of the shape. 2. The surface area of a cube or rectangular prism can be found using the formulas: Cube: SA = 6s2 Rectangular prism: SA = 2(lh × lb × bh) 3. The surface area of any other shape is found by adding the area of each face of the shape.

3D 3.3

WORKED

Example

SkillS

HEET

9

1 Find the surface area of each of the cubes below. a b

Surface area of cubes and rectangular prisms

c

5 cm 9 cm

d

3.4

SkillS

HEET

Surface area

32 cm

e

f

2.7 cm Surface area of triangular prisms

62 mm 2.8 m

WORKED

Example

10

2 Find the surface area of each of the following rectangular prisms. a b c

12 cm

3m

42 cm 7m

4m

21 cm

7.5 m 3.5 m 2.5 m

d

e

f

4 cm 4.1 cm

42 mm

20 cm

3.9 cm 13 cm 14 cm

7 mm 7 mm

3 Oliver is making a box in the shape of a rectangular prism. The box is to be 2.5 m long, 1.2 m wide and 0.8 m high. Calculate the surface area of the box. 4 Calculate the surface area of an open box in the shape of a cube, with a side length of 75 cm. (Hint: Since the box is open there are only five faces.)


95

5 A room is in the shape of a rectangular prism. The floor is 5 m long and 3.5 m wide. The room has a ceiling 2.5 m high. The floor is to be covered with slate tiles, the walls are to be painted blue and the roof is to be painted white. a Calculate the area to be tiled. b Each tile is 0.25 m2. Calculate the number of tiles needed. c Calculate the area to be painted blue. d Calculate the area to be painted white. e One litre of paint covers an area of 2 m2. How many litres of paint are needed to paint the room? WORKED

Example

6 Calculate the surface area of the square pyramid at right.

13 cm

11

10 cm

7 A triangular based pyramid has four equal sides as shown at right. Calculate the surface area.

8 multiple choice

3.5 cm

4 cm

Two cubes are drawn such that the side length on the second cube is double the side length on the first cube. The surface area of the larger cube will be: A twice the surface area of the smaller cube B four times the surface area of the small cube C six times the surface area of the small cube D eight times the surface area of the small cube 9 Calculate the surface area of the triangular prism below.

5 cm 4 cm 3 cm

2 cm

10 Calculate the surface area of the prism below. 3.2 m 1m 2m 6m

4m

96


2 1 Calculate the area of a rectangle with a length of 0.4 m and a breadth of 1.1 m. 2 Calculate the area of a triangle with a base of 12.3 m and a height of 4.8 m. 3 Calculate the area of the trapezium at right.

32 m 96 m

Name the solids below. 4

56 m

5

6

7 Find the surface area of the cube shown at left. 9c m

8 Find the surface area of a rectangular prism with a length of 8 cm, a breadth of 5 cm and a height of 6 cm. 9 Find the surface area of the triangular prism below.

8 cm

10 cm

20 cm 6 cm

10 Find the surface area of the square pyramid below. 6 cm

8 cm

97


Volume of a prism The volume of a solid shape is the amount of space within that shape. Consider the prism at right which has been built with cubes with sides of 1 cm. We can see by counting squares that the area of the base is 15 cm2. The height of the prism is 3 cm, and if we count the remaining cubes we find that the volume of the prism is 57 cm3.

Exploring the volume of a prism Build the prism that has been drawn above. Count the number of cubes that have been used to build the prism. Build other prisms and count the area of the base, the height and find the volume. Show that the volume can be found by multiplying the area of the front face (base) by the height perpendicular to the front face. When prisms are drawn, they are usually drawn lying down so that we can see the base. Hence, using the above example we can see that the volume of a prism can be calculated using the formula: V=A×h where A is the area of the base and h is the height.

WORKED Example 12 Calculate the volume of the prism at right. THINK 1 2 3

WRITE

Write the formula. Substitute the area of the base and the height. Calculate the volume.

V=A×h V = 63 × 5 V = 315 cm3

A = 63 cm2

5 cm

For some prisms we can develop a more specific formula for volume, without separately calculating the area of the base.

Cube The front face of the cube is a square of side length s and the height is s. V=A×h V = s2 × s since A = s2 for a square. V = s3 This becomes the formula used for the volume of a cube.

WORKED Example 13 Find the volume of the cube at right. THINK 1 2 3

Write the formula. Substitute the side length. Calculate the volume.

WRITE V = s3 V = 6.83 V = 314.432 cm3

6.8 cm

s

98


Rectangular prism Now consider a rectangular prism with a length of l, a breadth of b and a height of h. Substituting into the formula: V=A×h V = l × b × h since A = l × b.

h l

b

WORKED Example 14 Calculate the volume of the rectangular prism at right. THINK 1 2 3

12 mm

WRITE

Write the formula. Substitute the length, breadth and height. Calculate the volume.

47 mm

29 mm

V=l×b×h V = 47 × 29 × 12 V = 16 356 mm3

Cylinders A cylinder can be considered to be a circular prism. Consider the cylinder at right with a radius of r and a height of h. Substituting into h the formula r V=A×h V = π r 2h since for a circle A = π r 2. The symbol indicates that this formula will be on the HSC exam formula sheet. We also need to be aware of the relationship between volume and capacity. Capacity refers to the amount of liquid that a container holds. Capacity is measured in millilitres, litres and kilolitres. A volume of 1 cm3 = 1 mL and 1 m3 = 1000 L.

WORKED Example 15 Find the capacity of a cylinder with a radius of 1.3 m and a height of 7.8 m. THINK 1 2 3 4

WRITE

Write the formula. Substitute the radius and the height. Calculate the volume in m3. Calculate the capacity by multiplying the volume by 1000.

V = π r 2h V = π × (1.3)2 × 7.8 V ≈ 41.412 m3 Capacity = 41.412 × 1000 Capacity = 41 412 L

For any other prism, to calculate the volume we calculate the area of the base first and then use the formula V = A × h.


99

WORKED Example 16 Calculate the volume of the triangular prism at right. THINK 1

7.9 cm

WRITE A=

Calculate the area of the triangular base.

A=

1 --2 1 --2

×b×h

1.2 cm

5.6 cm

× 5.6 × 7.9

A = 22.12 cm2 2

Write the volume formula.

V=A×h

3

Substitute the area and the height.

A = 22.12 × 1.2

4

Calculate the volume.

A = 26.544 cm3

remember

1. The volume of a prism is found using the formula V = A × h. 2. Special volume formulas can be used for: Cube: V = s3 Rectangular prism: V = l × b × h Cylinder: V = π r 2h. 3. For other prisms, the volume is found by first calculating the area of the base, then using the formula V = A × h. 4. The capacity of a container can be calculated using: 1 cm3 = 1 mL and 1 m3 = 1000 L

Example

12

3.5

1 Calculate the volume of each of the solids below. a

b A = 24 cm2

c

5 cm

d

e

Converting units of volume A = 57 cm2 4 cm

12 cm

A = 15.93 mm2

A = 77.7 cm2

7.7 cm

L Spre XCE ad

Volume

2 A prism has a base area of 74.5 m2 and a height of 3.1 m. Calculate the volume.

sheet

A = 27.9

m2

SkillS

Volume of cubes and rectangular prisms

f 18 mm

9.2 m

3.6

HEET

A = 19 cm2

SkillS

HEET

WORKED

Volume of a prism

E

3E

100 WORKED

Example

13


3 Calculate the volume of each of the cubes below. a

b

c

5 cm 2.4 m 13 m

d

e

f

29 mm 8.2 m 5.64 m

WORKED

Example

14

4 Find the volume of each of the rectangular prisms below. a

b

c 3.7 m

3 cm 6 cm

4 cm

4.5 m

42 mm

6.3 m 13 mm

9 mm

d

e

f

20.5 m

16.5 m

12.5 m

50 mm 3.2 m 4.2 m 9 mm 9 mm

WORKED

Example

15

5 Calculate the volume of each of the cylinders below, correct to 1 decimal place. a

6 cm

b

c

27 cm 13 cm

12 m

12 cm 3m

d

15 cm

e

f 3 mm

18.5 cm

9 cm 25 cm

47 cm


WORKED

Example

16

101

6 For each of the following triangular prisms find: i the area of the front face ii the volume of the prism. a b 6 cm 8 cm 12 cm

8 cm 5 cm

3 cm

c

d 3.4 m

12.5 m 3.2 m 7.8 m

1.5 m

2.7 m

7 Find the volume of each of the following prisms by first calculating the area of the front face. a b 15 m 5m

5m

20 m

15 m

10 m

5m

c

d

3.1 cm

19 m 12 m

10 m 2.4 cm

1.7 cm

8 In each of the following, the prism’s front face is made up of a composite figure. For each: i calculate the area of the front face ii find the volume of the prism. a b 4 cm 4m 20 cm

10 cm 8m

10 cm

9m

16 cm

c

d 15 cm

3m

8 cm 12 cm 20 cm

12 m

6m 6 cm

9 multiple choice The shape at right could be described as a: A cube B square prism C rectangular prism D both B and C

18 m

12 m

102


10 multiple choice The area of the front face of a prism is 34.67 cm2, and the height is 3.6 cm. The volume of the prism is: A 38.27 cm2 B 38.27 cm3 C 124.12 cm2 D 124.812 cm3 11 multiple choice The dimensions of a rectangular prism are all doubled. The volume of the prism will increase by a factor of: A 2 B 4 C 6 D 8 12 A refrigerator is in the shape of a rectangular prism. The internal dimensions of the prism are 60 cm by 60 cm by 140 cm. a Find the volume of the refrigerator in cm3. b The capacity of a refrigerator is measured in litres. If 1 cm3 = 1 mL, find the capacity of the refrigerator in litres. 13 A semi-trailer is 15 m long, 2.5 m wide and 2.7 m high. Find the capacity of the semitrailer in m3. 14 A petrol tanker is shown at right. The tank is cylindrical in shape. The radius of the tank is 2 m and the length is 12 m. Calculate: a the volume of the tank, correct to 3 decimal places b the capacity of the tank, to the nearest 100 litres. (1 m3 = 1000 L).

2m

12 m

10 m 15 At right is a diagram of a concrete slab for a house. a Calculate the area of the slab. 2.5 m b The slab is to be 10 cm thick. Calculate the volume of 15 m concrete needed for the slab. (Hint: Write 10 cm as 0.1 m.) 10 m c Concrete costs $45.50/m3 to lay. Calculate the cost of this slab.

16 A rectangular roof is 14 m long and 8 m wide. When it rains, the water is collected in a cylindrical tank. a Calculate the volume of water collected on the roof when 25 mm of rain falls. b How many litres of water does the roof collect? c The cylindrical tank has a radius of 1.8 m and is 2.4 m high. What is the capacity of the tank, in litres? d By how much does the depth of water in the tank rise when the rain falls? Answer in centimetres, correct to 1 decimal place.


103

Volume of other solids Prisms are only one type of solid shape. In this section we find the volume of pyramids, cones and spheres.

Pyramids The volume of any pyramid is one-third of the volume of the corresponding prism. This leads us to the general formula for the volume of any pyramid: V = 1--- Ah 3

where A is the area of the base and h is the height of the pyramid.

WORKED Example 17 Find the volume of the pyramid at right. THINK

WRITE V = 1--- Ah

2

We are given A and h, so use the general formula. Substitute the value of A and h.

3

Calculate V.

V = 105 cm3

1

7 cm

Area = 45 cm2

3

V=

1 --3

× 45 × 7

In other cases we may need to calculate the area of the base before we are able to use the general formula for the volume of a pyramid.

Cones A cone is a circular pyramid. By substituting the formula for the area of a circle into the general formula for the volume of a pyramid, we find the formula for the volume of any cone. A = π r 2 when substituted into V = 1--- Ah becomes 3

A = π r when substituted into V = 1--- π r 2h 2

3

WORKED Example 18

8.5 cm

Find the volume of the cone at right, correct to 2 decimal places. THINK

WRITE

1

Write the formula.

V = 1--- π r 2h

2

Substitute the radius and height.

V=

3


V = 91.15 cm3

3 1 --3

× π × 3.22 × 8.5

3.2 cm

104


Spheres A sphere is a solid that looks like a ball. To find the volume of a sphere we need only the radius. The volume is calculated using the formula: V = 4--- π r 3 3

WORKED Example 19

Find the volume of a sphere with a radius of 9.5 cm, correct to the nearest cm3. THINK

WRITE

1

Write the formula.

V = 4--- π r 3

2

Substitute the radius.

V=

3


V = 3591 cm3

3 4 --3

× π × 9.53

remember

1. The volume of a pyramid is found using the formula V = 1--- Ah 3 the area of the base and h is the height. 2. The volume of a cone is found using the formula V = 1--- π r2h. 3

3. The volume of a sphere is found using the formula V = 4--- π r 3. 3

, where A is


3F Example

17

Volume of other solids

1 Find the volume of each of the pyramids below. a

b 8m

6m

A = 47 cm2

A = 25 cm2

c

d 2.5 m 9 cm A = 62 cm2 A = 13.5 m2

2 For each of the following pyramids, calculate the volume by first calculating the area of the base shape. a

b 8 cm

6 cm

15 cm

8 cm

14 cm

c

d 8 cm

12 m

6 cm 5 cm 6m 12 cm

10 m

3

a

Volume of a cone

b 6 cm A = 46 cm2

10 cm A = 30 cm2

c

14 cm

A = 150 cm2

d

52 mm

A = 12 mm2

sheet

L Spre XCE ad

3 Use the formula V = 1--- Ah to find the volume of the following cones.

E

WORKED

105

106 WORKED

Example

18


4 Find the volume of each of the following cones, correct to the nearest whole number. a b 10 cm

12 cm

5 cm

c

12 cm

d

42 cm

33 mm 42 cm

8 mm

5 A cone has a base with a diameter of 9 cm and a height of 12 cm. Calculate the volume of that cone, correct to 1 decimal place. WORKED

Example

19

6 Calculate the volume of each of the following spheres, correct to 1 decimal place. a b 6 cm

c

8 cm

d 3.2 m

12.5 m

7 Calculate the volume of a sphere with a diameter of 2.3 cm. Answer correct to 2 decimal places. 8 multiple choice Which of the following solids could not be described as a pyramid? A B

C

D

9 multiple choice A triangular pyramid and a square pyramid both have a base area of 20 cm2 and a height of 15 cm. Which has the greater volume? A The triangular pyramid B The square pyramid C Both have equal volume D This can’t be calculated.

107


10 multiple choice A spherical balloon has a volume of 500 cm3. It is then inflated so that the diameter of the balloon is doubled. The volume of the balloon will now be: A 1000 cm3 B 2000 cm3 C 3000 cm3 D 4000 cm3 11 Find the volume of the solid at right. Answer correct to 1 decimal place.

4 cm

12 A hollow rubber ball is to be made with a radius of 8 cm, and the rubber to be used is 1 cm thick. a What would be the radius of the hollow inside? b Calculate the volume of the ball. c Calculate the volume of space inside the ball. d Calculate the amount of rubber (in cm3) needed to make the ball. 13 The figure at right is a truncated cone, that is, a cone with the top cut off. a Calculate the volume of the cone before it was truncated. b The portion cut off was itself a cone. Calculate its volume. c Calculate the volume of the truncated cone.

12 cm

15 cm 3 cm 6 cm 6 cm

5 cm

14 Use the same method as in question 13 to find the volume of the truncated pyramid shown at right.

3 cm

1 cm

3 cm

2.5 cm

T SHEE

Work

15 The figure at right is of an ice-cream cone, containing a spherical scoop of ice-cream. a Calculate the volume of the cone. b Calculate the volume of the scoop of ice-cream. c Calculate the total volume of the shape. (Hint: Only half the sphere sits above the cone.)

8 cm

3.2

108


summary Area • Area formulas that you will need to remember are: Square A = s2 Rectangle A=l×b Triangle A = 1--- × b × h 2 Parallelogram A=b×h Rhombus A = 1--- × D × d 2 Trapezium A = 1--- × (a + b) × h 2 • Irregular areas have their area calculated using a survey. A traverse survey is done, a field diagram is drawn that will allow the shape to be divided into triangles and quadrilaterals, then the area is calculated.

Three-dimensional shapes • • • •

Prisms are solids with a constant cross-section. Pyramids have a plane shape as a base and triangular sides that meet in an apex. The net of a solid is what the shape would look like if it were unfolded and laid flat. Solid shapes can be drawn on isometric paper to locate the vanishing point.

Surface area • The surface area is the total area of all faces on a solid shape. • Surface area formulas: Cube SA = 6s2 Rectangular prism SA = 2(lh + lb + bh) • Many solid shapes have their surface area calculated by separately calculating the area of each face.

Volume • The volume is the amount of space inside a solid shape. • Volume formulas that you will need: Cube V = s3 Rectangular prism V=l×b×h Cylinder V = π r 2h Cone

V = 1--- π r 2h

Sphere

V = 4--- π r 3

3 3

• Any other prism has its volume calculated by using the formula V = A × h, where A is the area of the base and h is the height. • Any other pyramid has its volume calculated using the formula V =

1 --3

× A × h.

• Any formula without the icon you should have seen in years 7–10 and will be expected to remember, as they will not appear on your formula sheet in the HSC exam.

109


CHAPTER review 1 Find the area of each of the figures drawn below. a b

3A

c 6.3 cm

32 mm

17.9 cm 74 mm

5.2 cm

e

26

d

f

35 cm

m

mm

80 cm

m

43

15 cm

3m

70 cm

2 Find the area of each of the triangles drawn below. a b

3A

9 cm

40 cm 15.5 cm 18.2 cm

c

d 40 m 12.5 cm

35 m

12.5 cm C

3 At right is the field diagram for a block of land. a Use the scale 1 mm = 1 m to draw a scale diagram of the block of land. b Calculate the area of the block of land.

95 70 25

B

D

3B

36 30 0 A

4 Calculate the area of the block of land represented by the field diagram at right.

82 46 45

3B

32 37 15 26

5 Name each of the solids below. a b

0

c

3C

110


3C

6 Draw the net of each of the following solids. a b

3C

7 Name the solid shape for which the net is given at right.

3D

8 Find the surface area of each of the following solids. a b

c

2.1 m 0.8 m 3.9 m

4.2 cm

c

d

4.6 m 0.9 m

1.8 m

3D

9 Calculate the surface area of each of the figures below, by calculating the area of each face separately and adding them. a b c 10 cm 5m 6 cm

5m 4m

5 cm

4 cm 3m

15 m

12 cm

3.5 m

2m 3m 12 m

3D

10 Below is a diagram of an Olympic swimming pool. 50 m 22 m

1m

2m

a b c d

3D

Calculate the area of one side wall. Use the formula V = A × h, to calculate the volume of the pool. How many litres of water will it take to fill the pool? (1 m3 = 1000 L) The walls and floor of the pool need to be painted. Calculate the area to be painted.

11 At right are the plans for a garage that Rob is building. (Note: The garage has an iron roof and is closed at one end.) a Calculate the area that will need to be bricked. b If each brick is 20 cm long and 8 cm high, how many will be needed to complete the job?

2.5 m

6m 3m


111

12 Use the formulas to calculate the volume of each of the following cubes, rectangular prisms and cylinders. a

b

3E

c 3.8 m 11.6 m

29 mm

4.6 m

6.5 cm

d

e

f 18 mm 13 cm

32 mm

41 cm 3 cm

3 cm

8 cm

13 A prism has a base area of 45 cm2 and a height of 13 cm. Calculate the volume.

3E

14 Use the formula V = 1--- Ah to calculate the volume of each of the pyramids below.

3F

3

a

b

9 cm

c

19 mm

A = 126 mm2

A = 16 cm2

2.3 m

A = 6.9 m2

15 Calculate the volume of each of the pyramids, cones and spheres below. a

b 2.6 m 25 m

c

3.2 m 7.9 m

36 m

d

19.5 mm 23.5 mm

52 mm 19 mm

e

f 23 mm

70 cm

3F

112


Practice examination questions 1 multiple choice Which of the solids below is not a prism? A B C

D

2 multiple choice A cube has a side length of 4 cm. I: The surface area of the cube is 64 cm2. II: The volume of the cube is 96 cm3. Which of the above statements is correct? A I only B II only

C Both I and II

D Neither I nor II

3 multiple choice A prism and a pyramid both have a rectangular base of area 50 cm2 and have equal volumes. The height of the prism is 5 cm. The height of the pyramid is: A 5 cm B 10 cm C 15 cm D 20 cm 4 The figure at right is a square pyramid. a Calculate the area of the base. b Calculate the volume of the pyramid. c Calculate the surface area of the pyramid. d Draw the net of the pyramid.

CHAPTER

test yourself

3

13 cm

12 cm

5 A ice-cream cone has a base radius of 2.5 cm and a height of 12 cm. a Calculate the volume of the cone, correct to 1 decimal place. b Calculate the capacity of the cone in millilitres. c A scoop of ice-cream in the shape of a sphere is to sit inside the cone. Calculate the volume of the scoop. d How many scoops can be obtained from a 4 litre tub of icecream?

10 cm

Basic algebraic skills

4 syllabus reference Algebraic modelling 1 • Basic algebraic skills

In this chapter 4A General number patterns 4B Number pattern notation 4C Adding and subtracting like terms 4D Substitution 4E Multiplication and division of algebraic expressions 4F Solving linear equations 4G Equations arising from substitution

areyou

READY?

Are you ready?


4.1

Number patterns

4.2

Completing a formula for a table of values

1 Write down the next three numbers in each of the following sequences. a 14, 22, 30, 38, . . . b 12, 7, 2, –3, . . . c 3, 6, 12, 24, . . . d 16, 8, 4, 2, . . . 2 Complete the algebraic rule for each of the following tables. a

x

0

1

2

3

4

y

3

4

5

6

7

b

y = x + ___ c

4.4

4.5

4.6

4.7

0

1

2

3

4

y

0

3

6

9

12

y = ___x

x

0

1

2

3

4

y

3

7

11

15

19

d

y = 4x + ___

4.3

x

x

0

1

2

3

4

y

8

6

4

2

0

y = ___ – 2x

Adding and subtracting like terms

3 Simplify the following expressions. a 3g + 4g d 20x – 19x + 11

b y + 2y + 3y e 7g + 8g + 8 + 9

c 6gy – 3gy f 7h + 4t – 3h

Substituting into algebraic expressions

4 If x = 2 and y = 5, evaluate each of the following. a 9x b –3y c x–y e 7y – 10 f 8xy g 3x2y

d 2x + 5 h 6x – 2y

Multiplying algebraic terms

5 Simplify the following expressions. a 4x × 3 b 3a × 4b

c –5k × p

d 2mn × 3m

c 8a ÷ 2a

d –21xy ÷ 3

c 3x – 7 = 20

2x d ------ = 8 3

Dividing algebraic terms

6 Simplify the following expressions. a 12x ÷ 4 b 15y ÷ y Solving equations

7 Solve each of the following equations. a x + 28 = 67

b 6x = 102

Chapter 4 Basic algebraic skills

115

General number patterns There are many different types of number patterns; however, all number patterns (sequences) follow a rule. Consider the number pattern 6, 10, 14, 18, 22, . . . In this pattern (sequence), each number is 4 more than the previous one. The numbers in a sequence are called terms. In the sequence above, the first term is 6, the second term is 10, the third term is 14 and so on. The dots at the end of a sequence indicate that the pattern can be continued indefinitely.

WORKED Example 1 For the number pattern 9, 15, 21, 27, . . . a What number is added to one term to get the next? b Find the next three terms. THINK

WRITE

a

a 15 − 9 = 6, 21 − 15 = 6

1 2 3

Subtract consecutive terms. Each term differs by 6. Give a written answer.

b Each term is 6 more than the previous, so add 6 to each term to get the next.

Add 6 to each term to get the next term. b 33, 39, 45 are the next three terms.

Some sequences are generated by multiplication rather than addition.

WORKED Example 2 For the sequence 1, 4, 16, 64, . . . a What is each number multiplied by to get the next? b What are the next three terms in the sequence? THINK

WRITE

a

a 4 ÷ 1 = 4, 16 ÷ 4 = 4

1 2 3

Divide consecutive terms. Each term is 4 times the previous. Give a written answer.

b Each term is 4 times the previous, so multiply each term by 4 to get the next.

Multiply each number by 4 to get the next. b 256, 1024, 4096

As we go through the following exercise you will discover sequences that are formed by subtraction and division. There are also examples in which you will be asked to describe a number pattern in words. To do this, we state the first term and then describe what is done to the previous term to get the next.

116


WORKED Example 3 Describe each of the following patterns in words. a 45, 51, 57, 63, . . . b 2, −6, 18, −54, . . . THINK

WRITE

a

Check to see that each term increases by the same amount by subtracting consecutive terms. Describe the sequence by stating the first term and the rule to obtain each consecutive term.

a 51 − 45 = 6, 57 − 51 = 6

Divide consecutive terms to check that each term is multiplied by the same amount. Describe the sequence by stating the first term and the rule to get each consecutive term.

b (−6) ÷ 2 = −3, 18 ÷ (−6) = −3

1

2

b

1

2

The pattern has a first term of 45 and then each term is found by adding 6 to the previous term.

The pattern has a first term of 2 and then each term is found by multiplying the previous term by −3.

Patterns also occur in nature.

remember 1. A number pattern is a sequence of numbers that obey a certain rule. 2. If a pattern appears to be increasing (or decreasing) by a constant amount, subtract consecutive terms to find that amount. This number can then be added to each term to get the next. 3. If each term in a sequence appears to be found by multiplying or dividing the previous term by a constant amount, we find that amount by dividing consecutive terms. 4. To describe a pattern in words, state the first term and then describe what is done to the previous term to get the next term.


4A WORKED

Example

1 In each of the following number patterns, find the number that is added to one term to 4.1 get the next. a 3, 6, 9, 12, . . . b 2, 6, 10, 14, . . . c 4, 9, 14, 19, . . . Number d 9, 16, 23, 30, . . . e 0, 11, 22, 33, . . . f −4, 5, 14, 23, . . . patterns g 6, 19, 32, 45, . . . h −17, −13, −9, −5, . . . i 2, 4.5, 7, 9.5, . . . j --1- , --1- , --3- , 1, . . . 4

WORKED

Example

1b

General number patterns

2

4

2 For each of the sequences in question 1, write down the next three terms. 3 In each of the following number patterns, find the number which is subtracted from one term to get the next. a 10, 9, 8, 7, . . . b 30, 28, 26, 24, . . . c 40, 36, 32, 28, . . . d 40, 34, 28, 22, . . . e 19, 14, 9, 4, . . . f 22, 15, 8, 1, . . . g 65, 56, 47, 38, . . . h 3, −4, −11, −18, . . . i 5, 4.3, 3.6, 2.9, . . . j 2, 1 1--- , 1--- , − 1--- , . . . 4

2

4

4 For each of the sequences in question 3, write down the next three terms. WORKED

Example

2a

WORKED

Example

2b

5 In each of the following number patterns, find the multiplied by to get the next. a 2, 4, 8, 16, . . . b 1, 3, 9, 27, . . . d 2, 10, 50, 250, . . . e 2, 14, 98, 686, . . . g −3, 6, −12, 24, . . . h 0.1, 0.2, 0.4, 0.8, . . . j 4, 6, 9, 13.5, . . .

number which one term is c 5, 20, 80, 320, . . . f −2, −6, −18, −54, . . . i 3, 39, 507, 6591, . . .

6 For each of the sequences in question 5, write down the next three terms. 7 In each of the following sequences, find the number that each term is divided by to get the next term. a 1600, 800, 400, 200, . . . b 10 000, 1000, 100, 10, . . . c 1600, 400, 100, 25, . . . d 750, 150, 30, 6, . . . e 729, 243, 81, 27, . . . f 400 000, 20 000, 1000, 50, . . . g 512, 64, 8, 1, . . .

h 1, 1--- , 1--- , 1--- , . . .

i −100, 10, −1, 0.1, . . .

j −5000, −500, −50, −5, . . .

2

4

8

8 For each of the sequences in question 7, find the number which each term is multiplied by to get the next. (Hint: Dividing by 2 is the same as multiplying by 1--- .) 2

9 For each of the sequences in question 7, write down the next three terms. 10 In each of the following sequences, write down the first term, the fifth term and calculate the tenth term. a 2, 5, 8, 11, . . . b 45, 41, 37, 33, . . . c 4, 12, 36, 108, . . . d 192, 96, 48, 24, . . . e −25, −17, −9, −1, . . . f −6, −9, −12, −15, . . . g −2, −8, −32, −128, . . . h −48, 24, −12, 6, . . . i 1, 3, 6, 10, . . . j 1, 4, 9, 16, . . .

SkillS

HEET

1a

117

118 WORKED

Example

3


11 Describe the rule for each of the following number patterns. a d g j

5, 10, 15, 20, . . . 8, 4, 2, 1, . . . 2, 5, 12.5, 31.25, . . . −5, 10, −20, 40, . . .

b 100, 97, 94, 91, . . . e 8, 15, 22, 29, . . . h 343, 49, 7, 1, . . .

c 6, 12, 24, 48, . . . f 84, 71, 58, 45, . . . i −14, −11, −8, −5, . . .

12 Find the missing term in each of the following sequences. a c e g i

5, 12, ___, 26, 33, . . . 50, 44, 38, ___, 26, . . . 13, 5, −3, ___, −19, . . . −3, ___, −48, −192, −768, . . . 9, 18, ___, 36, 45, . . .

b d f h j

2, ___, 18, 54, 162, . . . 320, 160, ___, 40, 20, . . . −14, −9, ___, 1, 6, . . . ___, −3, 1, − 1--- , 1--- , . . . 3 9 9, 18, ___, 72, 144, . . .

13 Write down the first three terms of a number sequence with: a a first term of 2 and the next term found by adding 6 to the previous term b a first term of 2 and the next term found by multiplying the previous term by 5 c a first term of 45 and the next term found by subtracting 8 from the previous term d a first term of 2000 and the next term found by dividing the previous term by 5 e a first term of −12 and the next term found by adding 5 to the previous term f a first term of 15 and the next term found by subtracting 9 from the previous term g a first term of 20 and the next term found by dividing the previous term by −5 h a first term of 6 and the next term found by multiplying the previous term by −5 i a first term of 80 and the next term found by multiplying the previous term by

1 --2

j a first term of 6 and the next term found by multiplying the previous term by − 1--- . 4

14 In each of the following number patterns, calculate how many terms are in the sequence. a 6, 12, 18, 24, 30, . . ., 60 b 4, 9, 14, 19, . . ., 74 c 100, 90, 80, 70, . . ., 30 d 62, 59, 56, 53, 50, . . ., 8 e 1, 2, 4, 8, . . ., 1024 f 2, 6, 18, 54, . . ., 13 122 g 6561, 2187, 729, 243, . . ., 1 h 100, 20, 40, 0.8, . . ., 0.000 256 i −5, 10, −20, 40, . . ., 10 240 j − 1--- , − 1--- , − 1--- , −1, . . ., −128 8

4

2

15 In the photograph at left, we can see the inside walls of the spiral of a shell. Can you find a number pattern? (Hint: Measure the length of each dividing wall using a length of cotton and a ruler.)


119

Number pattern notation Usually we describe number patterns using pronumerals as this provides us with a shorter, more convenient method than using words. Two important pronumerals used in number patterns are ‘n’ and ‘Tn’. The pronumeral n usually stands for the position of the term in the sequence, while Tn is the value of that term. For example, the sequence 3, 6, 9, 12, 15, 18, . . . could be written as Tn = 3n. Terms in the sequence are found by substituting the position number for n. In this example, 3n is said to be the general term of the sequence. We could therefore also write this sequence as 3, 6, 9, 12, 15, . . . , 3n. For example, in this sequence: T1 = 3 × 1 =3

T2 = 3 × 2 =6

T3 = 3 × 3 =9

T4 = 3 × 4 = 12

T5 = 3 × 5 = 15

We could then go on to find any other term in the sequence. For example the 87th term: T87 = 3 × 87 = 261

WORKED Example 4

Find the sequence generated by the rule Tn = 7n. THINK

WRITE

1

The first term is found substituting n = 1.

2

The second term is found substituting n = 2.

3

The third term is found substituting n = 3.

4

The fourth term is found substituting n = 4.

5


T1 = 7 × 1 T1 = 7 T2 = 7 × 2 T1 = 14 T3 = 7 × 3 T1 = 21 T4 = 7 × 4 T1 = 28 The sequence is 7, 14, 21, 28, . . .

Graphics Calculator tip! Using the TABLE function You can use the table function on your Casio graphics calculator to generate the terms of a sequence once given the rule. Consider worked example 4. 1. From the MENU select TABLE.

120


2. Enter the rule for the number pattern using Y1 to represent Tn and X to represent n. i.e. Y1 = 7X. Press EXE once you have entered the rule. Note that you must use the X on the X,q,T button and NOT the red X which is accessed by pressing ALPHA +. 3. Press F5 for RANG. This shows the first and last number in your table as well as what to increase the values of X by. By having Start = 1, End = 100 and Pitch = 1 you will rarely need to change this setting. Press EXE after entering these settings to return to the previous screen. 4. Press F6 for TABL to display the table. You can then read off the result of each substitution. Use the arrow button to scroll down and see the values of X greater than 4. In many cases, the sequence will be presented in table form. This method makes the connection between the term and its position in the sequence clearer.

WORKED Example 5

Complete the table below for the sequence Tn = 5n − 3. n

1

2

3

4

5

Tn THINK

WRITE

1

The first term is found substituting n = 1.

2

The second term is found substituting n = 2.

3

The third term is found substituting n = 3.

4

The fourth term is found substituting n = 4.

5

The fifth term is found substituting n = 5.

6

Complete the table.

T1 = 5 × 1 − 3 =2 T2 = 5 × 2 − 3 =7 T3 = 5 × 3 − 3 = 12 T4 = 5 × 4 − 3 = 17 T5 = 5 × 5 − 3 = 22 n

1

2

3

4

5

Tn

2

7

12

17

22

The major advantage of writing sequences algebraically is that it allows us to calculate the value of any term in the sequence without working out all the terms that come before it. For example, if I want to calculate the 100th term of a sequence, I do not need to write out the first 99 terms. All I need to do is substitute n = 100 into the general term.


121

WORKED Example 6

Find the 1st, 35th and 100th term of the sequence Tn = 6n − 15. THINK

WRITE

1

The 1st term is found substituting n = 1.

T1 = 6 × 1 − 15 = −9

2

The 35th term is found substituting n = 35.

3

The 100th term is found substituting n = 100.

T35 = 6 × 35 − 15 = 195 T100 = 6 × 100 − 15 = 585

We can also look at sequences presented in table form and from there calculate the general term. Earlier we saw that the sequence 3, 6, 9, 12, 15, . . . could be written in the general form Tn = 3n. Each term in this sequence increases by 3. Any sequence that increases by 3 will have 3n as part of its general term.

WORKED Example 7 Find the general term from the sequence given below. n

1

2

3

4

5

Tn

3

8

13

18

23

THINK 1

WRITE

Terms are increasing by 5 so the general term includes 5n. 5 × 1 = 5 and 5 − 2 = 3. We then suspect Tn = 5n − 2. Test the other values in the table.

Testing Tn = 5n − 2

3

Substitute n = 2.

T2 = 5 × 2 − 2 T1 = 8

4

Substitute n = 3.

T3 = 5 × 3 − 2 T1 = 13

5

Substitute n = 4.

T4 = 5 × 4 − 2 T1 = 18

6

Substitute n = 5.

T5 = 5 × 5 − 2 T1 = 23

7


Tn = 5n − 2

2

122


remember 1. A sequence can be written algebraically. This is done using a general term called Tn, which is written in terms of n. In this statement, n is the position of the term in the sequence and Tn is the value of that term. 2. When a sequence is written in this form, the terms of the sequence are found by substituting values of n. For example, the first term, T1, is found by substituting n = 1, the second term, T2, by substituting n = 2 and so on. 3. When the sequence is written, the general term can be found by looking at the sequence and finding a pattern that can then be written in algebraic form.

4B WORKED

Example

4

Number pattern notation

1 Find the sequence generated by each of the following. a Tn = n b Tn = 4n d Tn = −6n e Tn = 2n − 1 g Tn = 23n + 13

h Tn = 1--- n +

j Tn = 20 − 3n

k Tn = 100 − 7n

2

i Tn = 10 − n

1 --2

l Tn = 2 − 5n

2 Find the sequence generated by each of the following. a Tn = 2 n b Tn = 3 n n d Tn = 10 e Tn = 4 n − 3 n g Tn = 3.2 h Tn = 2.3n n n j Tn = (−1) × 3 k Tn = (−4)n WORKED

4.2

SkillS

HEET

Completing a formula for a table of values

Example

5

c Tn = 9n f Tn = 4n + 3

c f i l

Tn = (−5)n Tn = 2 n + 5 Tn = (−1)n Tn = (−2)n + 3

3 In each of the following, copy and complete the table. a Tn = 7n b Tn = 4n − 3 n

1

2

3

4

5

Tn n

1

2

3

4

5

4

5

n

1

2

3

4

5

2

3

4

5

2

3

4

5

2

3

4

5

f Tn = 2 n + 2 1

2

3

4

5

Tn

n

1

Tn

g Tn = (−2)

h Tn = −(5)n

n

n

1

2

3

4

5

Tn

Tn

3

Tn n

n

n

2

d Tn = 7n + 2

Tn

i Tn = n

1

Tn

c Tn = 5n + 4

e Tn = 3

n

n

1

Tn j Tn = n(n − 3)

2

1

2

3

4

5

n Tn

1


123

4 For each of the following, copy and complete the table. a Tn = 9n b Tn = 3n + 2 n

1

4

9

10

20

Tn n

1

3

6

12

50

n

1

2

5

9

12

25

n

1

10

20

50

100

n

3

6

8

10

2

6

9

10

2

6

10

20

1

Tn

g Tn = (−3)n n

1

h Tn = (−5)n 4

7

10

15

Tn

n

1

Tn 3

j Tn = n(2 − n) 1

3

5

10

Tn

6

10

f Tn = 3 n − 1

Tn

Example

9

Tn

e Tn = 2 n

WORKED

5

d Tn = 8n + 5

Tn

n

1

Tn

c Tn = 6n − 1

i Tn = n

n

20

n

1

Tn

5 For the sequence Tn = 3n + 8, find the value of: a T1 b T10 c T38

d T50

6 For the sequence Tn = 30 − 4n, find the value of: a T1 b T12 c T46

d T10

7 For the sequence Tn = 4.2n, find the value of: a T1 b T5 c T8

d T1

8 For the sequence Tn = (−1)n4n, find the value of: a T1 b T15 c T25

d T56

n

9 For the sequence Tn = 100  1--- , find the value of: 2 a T1 b T2 c T5

d T10

10 multiple choice For the sequence Tn = 4n − 1, the value of T9 is: A 3 B 35 C 48

D 49

11 multiple choice Which of the following is not a term of the sequence Tn = n2 + n? A2 B 6 C 100 D 110 12 multiple choice In which of the following sequences is 100 not a term? A Tn = n 2 B Tn = 10n C Tn = 5n − 5n

D Tn = 2n2 + 10n

124


13 For each of the tables below try to discover the algebraic rule. a b 7 n 1 2 3 4 5 n 1 2 Tn 4 8 12 16 20 Tn 2 5

WORKED

Example

c

e

n

1

2

3

4

5

Tn

49

48

47

46

45

n

1

2

3

4

5

Tn

2

4

8

16

32

d

f

3

4

5

8

11

14

n

1

2

3

4

5

Tn

33

30

27

24

21

n

1

2

3

4

5

Tn

2

8

26

80

242

1 1 Write the next three terms of the sequence 1, −3, 5, −7, . . . 2 In the sequence 5, 13, 21, 29, . . . what do I do to a term to get the next? 3 In the sequence 40, 20, 10, 5, . . . what is each term multiplied by to obtain the next? 4 A number pattern begins with 2 and to get the next term we add −8 to the previous one. Find the first five terms. 5 How many terms are in the number sequence 1, 4, 9, 16, . . . 225? 6 Find the first five terms of the sequence generated by Tn = n2 + n. 7 In the sequence Tn = 3n + 2, find the value of T6. 8 Copy and complete the table below for the sequence Tn = 4n − 5. n

1

3

7

10

15

Tn 9 Is 98 a term of the sequence Tn = 5n + 3? If so, which term? 10 Find an algebraic rule for the table below. n

1

2

3

4

5

Tn

38

36

34

32

30


125

Adding and subtracting like terms In algebra, each pronumeral used stands in place of a number. Consider the expression: 5+5+5+5+5+5+5=7×5 We can write this addition as a multiplication because the same number is being added. We can say that ‘like terms’ are used. Similarly, we can simplify a similar expression using pronumerals: a + a + a + a + a + a + a + a = 8a Note that we do not use the multiplication sign in algebra. The multiplication sign is implied when it is not used. Now consider the expression 9+9+9+9+9+9+4+4+4+4+4 We can not write this as a single expression because the terms are not like. We can only simplify the like terms. 9+9+9+9+9+9+4+4+4+4+4=6×9+5×4 Similarly: x + x + x + x + x + y + y + y = 5x + 3y Only like terms, that is, the same pronumerals, can be added together.

WORKED Example 8 Simplify the expressions: a m+m+m+m+m+m+m b p + p + p + q + q + q + q + q + q. THINK

WRITE

a Write the repeated addition as a multiplication. b 1 Write p + p + p as a multiplication. 2 Write q + q + q + q + q + q as a multiplication.

a m + m + m + m + m + m + m = 7m b

p + p + p + q + q + q + q + q + q = 3p + 6q

We are able to add or subtract any expressions that use the same pronumerals.

WORKED Example 9

Simplify 5k + 9k − k. THINK

WRITE

Each term uses the same pronumeral so we add and subtract the coefficients. Remember that where no coefficient is written it is assumed to be 1.

5k + 9k − k = 13k

In each expression where more than one different pronumeral is used, we collect all the like terms (the same pronumeral). The addition or subtraction sign in such expressions belongs with what follows it. For example, in 4x − 6y + 2x the minus sign belongs to the 6y and the plus sign to the 2x.

126


WORKED Example 10 Simplify: a 5b + 2 + 2b − 6

b 4m − 3n + 2m − 5n.

THINK

WRITE

a

Rewrite the expression, grouping like terms. Complete each addition and subtraction separately.

a 5b + 2 + 2b − 6 = (5b + 2b) + (2 − 6)

Rewrite the expression, grouping like terms. Complete each addition and subtraction separately.

b 4m − 3n + 2m − 5n = (4m + 2m) + (−3n − 5n)

1 2

b

1 2

= 7b − 4

= 6m − 8n

remember 1. Like terms occur when the same pronumeral is used. 2. Like terms can be added or subtracted, unlike terms cannot. 3. In any expression, the sign belongs with the term that follows it.

4C 4.3

1 Simplify each of the following. a t+t 8 c w+w+w+w+w+w e q+q+q+q+q+q+q

WORKED

Example

SkillS

HEET



b r+r+r d x+x+x+x f n+n+n+n+n+n+n+n

2 Write the following algebraic terms as additions. a 2m b 6n c 9s d 8w 3 Simplify the following. a 5y + 7y 9 d 34j + 13j g 14r − 6r j −4w + 6w m 4j − 2j − j

e 3y

f 4r

WORKED

Example

b e h k n

15x + 4x 17k + 8k 9w − 8w −2s − 5s −2p − 17p + 25p

4 Simplify the following by collecting like terms. a 8x + 2x + 7 b h − 4 + 3h 10 d 5p + 9q − 2p − 2q e 5 + 6w − 2w g 9j + 6k − 5j + 2k h 16x − 15 − 13x j 4 + 5a − 12 − 2a k 7b − 4 − 2 + 10b m 4y − 4 + 6y − 2 n 8c − 3b − 5c + 2b p 15e − 8p + 4e + 4p q 16t − 12s + 4t − 11s

15e + 24e 14k − 10k m − 5m 14m + 5m + 3m 5z − 15z + 9z

c f i l o

WORKED

Example

c f i l o r

7k + 5k + 3k − 2k 2t + 7t + 4t 5b − 16 + 11b − 10 6r − 17 − 2r 9d − 15 + 4d + 7 8z − 4w − 8z + 6w


127

Substitution An algebraic expression has little or no meaning without a value being substituted for the pronumeral. An algebraic expression that is used in common calculations is called a formula. When using a formula, we substitute for one unknown to allow us to calculate the value of another. When substituting into a formula, we replace a pronumeral with a number and then calculate the value of the entire expression.

WORKED Example 11

The formula V = 4--- π r3 is used to calculate the volume of a sphere. Calculate the value of 3 V, correct to 2 decimal places, when r = 4.7. THINK

WRITE

1

Write the formula.

V = 4--- π r 3

2

Substitute 4.7 for r.

V=

3

Calculate the value of V and round off to 2 decimal places.

V = 434.89

3 4 --3

× π × (4.7)3

In many such examples you will be required to do calculations that require more than one substitution.

WORKED Example 12

In the formula v = u + at, calculate the value of v when u = 12.8, a = 9.8 and t = 5. THINK 1 2 3

WRITE

Write the formula. Substitute 12.8 for u, 9.8 for a and 5 for t. Calculate the value of v.

v = u + at v = 12.8 + 9.8 × 5 v = 61.8

Graphics Calculator tip! Substitution Your Casio graphics calculator can be used to complete substitutions. Consider worked example 12. 1. From the MENU select RUN.

2. We need to assign the values u = 12.8, a = 9.8 and t = 5. To do this, press 12.8 → ALPHA U EXE To do this, press 19.8 → ALPHA A EXE To do this, press 22.5 → ALPHA T EXE

128


3. To evaluate v, enter the expression to which v is equal, that is U + AT. Press EXE to obtain the value of the expression.

remember 1. A pronumeral stands in place of a number and so numbers can be substituted for pronumerals. 2. A formula is an algebraic rule for a calculation. 3. When calculating the value of an expression, rewrite the expression substituting the known values for the pronumerals, then calculate.

4D 4.4

WORKED

Example

SkillS

HEET

11 Substituting into algebraic expressions

EXCE

et

Substitution

e S = 4π r 2

WORKED

et

EXCE

Example

Substitution game

9C 1 The formula F = ------- + 32 converts degrees Celsius to degrees Fahrenheit. Use the 5 formula to convert the following temperatures to degrees Fahrenheit. a 20°C b 35°C c −5.3°C 2 For each of the following formulas, find the value of the subject given the variable. Where necessary, give your answer correct to 2 decimal places. a C = 6r (r = 5) b P = 4s (s = 7.3) c L = l 2 (l = 4.9) d C = 45 + 65d (d = 1.4)

reads L Sp he

reads L Sp he

Substitution

12

(r = 8.8)

p2 f Q = ------9.8

( p = 6.5)

3 In the formula A = (1 + r)n, find the value of A when r = 0.075 and n = 4. (Give your answer correct to 3 decimal places.) PRT 4 a Given that I = ----------- , find I when P = 2000, R = 6 and T = 5. 100 b Given that T = a + 8d, find T when a = 56 and d = −8. c Given that P = 2l + 2w, find P when l = 34 and w = 54. d Given that A = 1--- bh, find A when b = 9.8 and h = 6.7. 2

e Given that V = lbh, find V when l = 6.5, b = 6.5 and h = 5.6. D f Given that S = ---- , find S when D = 900 and T = 12. T g Given that C =

5 --9

(F − 32), find C when F = 212.

h Given that S = ut + 1--- at2, find S when u = 8, t = 4 and a = 6. 2

i Given that T = a + (n − 1)d, find T when a = −23, n = 27 and d = −2.4. j Given that c2 = a2 + b2, find c when a = 12 and b = 22.5.


129

5 The cost of hiring a taxi is $4.50 plus 60c per kilometre. a Write a formula for the cost of a taxi journey, C, in terms of distance travelled, d. b Use the formula to calculate the cost of a taxi journey of: i 5 km ii 20 km iii 50 km. 6 multiple choice A formula is given as c = mp. If m = 2 and p = 5 then c is equal to: A3 B 7 C 10 D 25 7 The formula V = 1--- π r 2h is used to calculate the volume of a cone, where r is the 3 radius and h is the height. Use the formula to calculate the volume of a cone, correct to 2 decimal places, where: a the radius is 4 cm and the height is 8 cm b the radius is 32 mm and the height is 17 mm c the radius is 4.6 cm and the height is 9.7 cm. m 8 In the formula B = ----2- , calculate the value of B when: h a m = 56 and h = 2 b m = 3.6 and h = 6 c m = 1.6 and h = 0.8. yA 9 In the formula D = --------------- , find the value of D when: y + 12 a y = 6 and A = 2 b y = 4.2 and A = 7 c y = 0.24 and A = 96. 10 Fried’s rule to calculate the infant dosage of a medicine is given by the mA formula D = --------- , where D is the 150 infant dosage, m is the infant’s age in months and A is the adult dosage. Calculate the dosage of medicine given to: a a nine-month-old baby, where the adult dosage is 50 mL b an 18-month-old child, where the adult dose is 30 mL c a two-year-old child, where the adult dose is 40 mL. yA 11 Young’s rule for the calculation of a child’s dose of medicine is D = --------------- . In this y + 12 formula, D is the child’s dose, y is the age of the child in years and A is the adult’s dose. Calculate the dosage for a three-year-old child taking a medicine for which the adult dose is 45 mL.

130


kA 12 Clark’s rule for calculating a dosage is D = ------ , where k is the mass of the child in 70 kilograms and A is the adult dose. Calculate the dosage required for a child who weighs 20 kg, where the adult dosage is 35 mL. 13 Gavin is eight years old and weighs 28 kg. The adult dosage of a medicine is 30 mL. Calculate the dosage of medicine that should be given to Gavin according to: a Fried’s rule b Young’s rule c Clark’s rule.

Work

T SHEE

4.1

14 The Body Mass Index, B, is a measure of how healthy a person is. The formula is m B = ----2- , where m is a person’s mass in kilograms and h is a person’s height in metres. h A person is considered to be healthy if 21 ≤ B ≤ 25. Calculate the Body Mass Index, correct to 1 decimal place, of the following people and comment on the health of each person. a Caroline, who is 71 kg and 1.7 m tall b Neil, who is 86 kg and 1.65 m tall c Bronwyn, who is 42 kg and 1.68 m tall

2 1 Write the next three terms in the sequence 3, 9, 15, 21, . . . 2 Write an algebraic rule for the sequence in question 1. 3 Simplify the expression r + r + r + r + r. 4 Simplify the expression p + p + p + p + q + q. 5 Simplify 4m + 6m − 11m. 6 Simplify 3v − 4w − 6v − 7w. 7 In the formula A = 1--- bh, find the value of A when b = 7.6 and h = 9.3. 2

8 In the formula A = P(1 + r)n, find the value of A when P = 5000, r = 0.09 and n = 5. 9 In the formula V = --4- π r 3, find the value of V correct to 1 decimal place when r = 9.1. 3

10 In the formula S = 4π r 2, find the value of S correct to 1 decimal place when r = 9.1.


131

Multiplication and division of algebraic expressions When we are multiplying and dividing algebraic expressions, the same rule about like terms applies; however, care needs to be taken with the notation of multiplication and division. Earlier, we saw the shorthand way of writing a repeated addition. We said: 4+4+4+4+4+4+4+4=8×4 For multiplication, we use indices: 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 = 48 In this expression, 4 is the base and 8 is the index. The same method applies to pronumerals, provided they are like terms. We can write: p × p × p × p × p = p5

WORKED Example 13 Simplify each of the following. a 3×3×3×3×3×3 b m×m×m×m THINK

WRITE

a 3 is shown 6 times.

a 3 × 3 × 3 × 3 × 3 × 3 = 36

b m is shown 4 times.

b m × m × m × m = m4

We can use index laws to simplify expressions already in index form. Consider: a3 × a4 = (a × a × a) × (a × a × a × a) = a7 This leads us to the first index law. Index Law 1:

ax × ay = ax + y

When using the index laws, the indices of the same base are added in turn. If there are coefficients (numbers in front of the pronumerals) in the expression, these are multiplied.

WORKED Example 14 Simplify each of the following. a t5 × t4 b 4m3 × 6m2

c 12a2b × 4a5b3

THINK

WRITE

a Add the indices.

a t5 × t4 = t9

b Multiply the coefficients and add the indices.

b 4m3 × 6m2 = 24m5

c Multiply coefficients, add the base ‘a’ indices, then add the base ‘b’ indices.

c 12a2b × 4a5b3 = 48a7b4

132


The reverse to the index law for multiplication is the index law for division. Index Law 2:

a x ÷ ay = a x − y or

ax ----- = a x – y ay

Division questions can be written as either a division or in fraction form. This may involve simplifying a fraction as well as applying this law. We need to be able to divide terms which are linear, quadratic and cubic in particular, as well as higher powers.

WORKED Example 15 Simplify each of the following. a w 8 ÷ w3 b 24d 4 ÷ 6d THINK

WRITE

a Subtract the indices. b Divide the coefficients then subtract the indices.

a w 8 ÷ w3 = w 5 b 24d 4 ÷ 6d = 4d 3

We are also expected to be able to simplify expressions that raise an index to a further index. This leads to our third index law. Consider: (a3)4 = a3 × a3 × a3 × a3 Adding indices: = a12 In these cases we are able to simply multiply the indices. Index Law 3:

(a x) y = a xy

WORKED Example 16 Simplify each of the following. a (n5)2 b (3w4)3

c (2p3q4)2

THINK

WRITE

a Multiply the indices. b Calculate 33 then multiply the indices. c Calculate 22 then calculate ( p3)2 and (q4)2 by multiplying the indices.

a (n 5)2 = n 10 b (3w4)3 = 27w12 c (2p3q4)2 = 4p6q8

These rules can also be used when required to expand brackets. In such examples every term inside brackets must be multiplied by the term immediately outside.

WORKED Example 17

Expand 2x3(4x5 − 3y2). THINK

WRITE

Multiply both terms inside the brackets by 2x3.

2x3(4x5 − 3y2) = 8x8 − 6x3y2


133

remember 1. Learn the index laws. • Index Law 1: a x × ay = a x + y ax • Index Law 2: a x ÷ ay = a x − y  -----y = a x – y a  • Index Law 3: (a x) y = a xy 2. Apply the index laws separately to each pronumeral.

4E

Multiplication and division of algebraic expressions

Example

b 7×7×7×7×7×7 d q×q×q×q×q×q×q f w×w×w

3 Simplify each of the following. a q3 × q4 b 14 d 45 × 42 e 4 2 g 3b × 2b h j 9 × 3j 6 k m 4x4y3 × 5x3y6 n p 6jk 4 × 8j3k 6 q

c 87 f j9

WORKED

Example

WORKED

Example

15

x5 × x6 a5 × a 5d 4 × 2d 4 8k 3 × k 5m2n4 × 4m3n5 10p4q3 × 5p

c f i l o r

y7 × y8 s × s2 7g4 × 5g6 4m × 6m 9a3b7 × 7a2b2 9r3s2t 4 × 6rst 6

4 Simplify each of the following. a a5 ÷ a

b b8 ÷ b4

c9 c ----7c

d e5 ÷ e

18g 5 e ----------3g 3

f 24h6 ÷ 4h2

g 56j7k 5 ÷ 7j 4k 3

64m 7 n 3 h -----------------16m 4 n 2

i 42p9q4 ÷ 6pq

72r 6 j ---------8r 3

84s 7 k ---------7s 5

l

5 Simplify each of the following. a (a3)4 b (g4)5 16 4 3 d (2k ) e (7m6)2 g (5p3q4)2 h (6r3s)5

6t 8 ---------18t 5

WORKED

Example

c (h2)2 f (8n2)3 i (5xy5)2

SkillS

Multiplying algebraic terms

4.6

SkillS

HEET

2 Write each of the following in expanded form. a 36 b 45 d m3 e y2

4.5

HEET

1 Write each of the following in index form. a 2×2×2×2 13 c 9×9 e p×p×p×p×p

WORKED

Dividing algebraic terms

134


6 Simplify each of the following. a m6 × m3 ÷ m5 b 5 2 d 14q ÷ 7q × 6q e g (4w4)2 × (2w3)6 h 4 5 12m × 4m j ---------------------------k 8m 3

(n3)5 × n4 3r3 ÷ (2r4)2 (8v5)2 ÷ (2v3)3 3y 4 ----------------------2 9y × 7y 6

7 Expand the brackets in each of the following. a m(m + 1) b 2f( f − 4) 17 d w(w3 − 4) e r2(r5 + s) g 4q(5q3 − 9) h 7y5(3y4 + z2) 6 j 3x (4 + 2x) k 5h(g − 4h4) m 3xy(2x + 2y) n 4ab(ab + 7) p 5m2n5(5m3 − 3n2) q 6r4s6(3r3s2 − 3)

c ( p3)6 ÷ p12 f (6s6)2 ÷ 9s7 i (2x5)3 × (4x4)2 ÷ x3 9z 4 × 6z 7 l --------------------( 3z 4 ) 3

WORKED

Example

c f i l o r

m(5m + 6n) p5( p5 − 5q) 9a4(a − 2b4) 9t2(2s3 + 6t4) 8pq( p2 − q) 7pq4r(2p4q − 7r5)

Solving linear equations An equation is an incomplete mathematical sentence. When we are given an equation, our task is to solve it. That is, to find a value for the pronumeral which makes the sentence true. The basic idea to follow when solving an equation is to undo those operations performed on the pronumeral. We do this by writing an equivalent equation made by using one of four possible steps. Step 1. We can add the same number to each side of an equation. Step 2. We can subtract the same number from each side of an equation. Step 3. We can multiply both sides of an equation by the same number. Step 4. We can divide both sides of an equation by the same number. The simplest type of equation is the one-step equation. The solution to these equations uses only one of the above four steps.

WORKED Example 18 Solve each of the following equations. a x + 48 = 75 c 7d = 91

b y − 43 = 56 q d ------ = 29 13

THINK

WRITE

a

a x + 48 = 75 x = 27 b y − 43 = 56 y = 99 c 7d = 91 d = 13 q ------ = 29 d 13 q = 377

2

Write the equation. Subtract 48 from both sides. Write the equation. Add 43 to both sides. Write the equation. Divide both sides by 7.

1

Write the equation.

2

Multiply both sides by 13.

1 2

b

1

c

1

2

d


135

When solving equations that involve more than one step to the solution, we must show the equivalent equation formed after using each of our chosen steps.

WORKED Example 19 Solve the equations. a 12 + 3x = 45

m b ---- − 14 = −25 7

THINK

WRITE

a

a 12 + 3x = 45 3x = 33 x = 11 m b ---- − 14 = −25 7 m ---- = −11 7 m = −77

3

Write the equation. Subtract 12 from each side. Divide both sides by 3.

1

Write the equation.

2

Add 14 to each side.

3

Multiply both sides by 7.

1 2

b

The solution to an equation can be checked by substituting the value found into the equation. For example, if we check x = 11 in 12 + 3x = 45. LHS = 12 + 3 × 11 = 45 = RHS Since x = 11 gives a true number sentence, we know the solution x = 11 is the correct solution to this equation. The substitution can be written, although this is not usually necessary. This is normally done mentally or on the calculator as a check that the value we have is correct. A more difficult type of equation is one that has the pronumeral occurring on both sides. In these examples, it is necessary to gather all the pronumerals on one side of the equation. This is done by adding or subtracting one pronumeral expression to each side of the equation.

WORKED Example 20

Solve the equation 6e − 4 = 3e − 43. THINK 1 2 3 4 5

Write the equation. Subtract 3e from both sides. Add 4 to both sides. Divide both sides by 3. Check the answer by substitution.

WRITE 6e − 4 = 3e − 43 3e − 4 = −43 3e = −39 e = −13 LHS = 6 × (−13) − 4 RHS = 3 × (−13) − 43 = −82 RHS = −82 Solution e = −13 is correct.

136


Graphics Calculator tip! Solving equations Your Casio graphics calculator can be used to solve equations using the EQUA function. Consider worked example 20. 1. From the MENU select EQUA.

2. Press F3 for Solver.

3. Delete any existing equation and enter the equation 6e – 4 = 3e – 43 by pressing 6 ALPHA E – 4 SHIFT = 3 ALPHA E – 43 then press EXE to store the equation. At this stage ignore any value of E that is shown, as it will be a previously stored value. 4. Press F6 to solve the equation. You can see the solution E = –13. The calculator also displays the checking of the solution. Lft = –82 and Rgt = –82 shows the result of substituting the solution E = –13 into both sides of the equation. Important note: Although your graphics calculator can be used to solve equations, it is important that you learn to solve equations without using the calculator as well. In your exams, it is possible that you will be given a question that asks you to explain the steps of solving an equation. If you do not understand the manual method of solution, you will be unable to answer such a question. Many equations involve the use of brackets. With such equations, we need to expand brackets before solving the equation.

WORKED Example 21

Solve 8(3x − 7) = −152. THINK

WRITE

1

Write the equation.

8(3x − 7) = −152

2

Expand the brackets.

24x − 56 = −152

3

Add 56 to each side of the equation.

56 – 24x = −96

4

Divide both sides of the equation by 24.

56 – 24x = −4


137

A similar approach must be used to solve equations that use fractions. If an equation has fractions, we multiply by the lowest common multiple of the denominators in the equation.

WORKED Example 22 k+2 3k – 8 Solve the equation ------------ = --------------- . 5 8 THINK

WRITE

1

Write the equation.

2

Multiply by the lowest common multiple of 5 and 8 (40). Expand the brackets. Subtract 15k from each side. Subtract 16 from each side. Divide both sides by 7.

3 4 5 6

k + 2 3k – 8 ------------ = --------------5 8 8(k + 2) = 5(3k − 8) 8k + 16 = 15k − 40 −7k + 16 = −40 −7k = −56 k=8

remember 1. An equation is an incomplete mathematical sentence. 2. When solving an equation, our task is to find a value for the pronumeral that will make the sentence true. 3. Remember the four steps that can be taken in solving an equation. 4. Write out the equivalent equation formed after each step in the solution. 5. Check your answer to each equation by substitution. This can be done mentally or on the calculator and does not always need to be written. 6. If an equation involves the use of brackets, expand the brackets before solving the equation. 7. When the equation involves the use of fractions, we first multiply by the lowest common multiple of the denominators in the equation. 8. You can use a graphics calculator to solve an equation but it is very important that you understand manual methods as well.

WORKED

Example

4.7

c 9q = 162

SkillS

f t − 253 = 78

Solving equations

i e + 79 = 45 l −7a = 84

L Spre XCE ad

o d + 8.5 = 13.7 r t − 2 4--- = 3 1--5

2

Equation solver

sheet

1 Solve each of the following one-step equations. a z + 24 = 67 b w − 34 = 54 y d ------ = 19 e r + 387 = 435 14 p g 17x = 306 h ------ = 851 23 j k − 56 = −34 k 15b = −240 u f m ------ = −4 n – --- = 8 45 5 v p 5c = 17 q ------- = 9.5 3.2

HEET

18

Solving linear equations

E

4F

138


2 multiple choice

EXCE

et

reads L Sp he

Which of the following is the exact solution to 7x = 23? 7 A x = ----B x = 3 2--C x = 3.28

Solving equations

23

GC p

sio

am – rogr Ca

Equations WORKED

Example

19a GC p

am – rogr TI

Equations

GC p

sio

am – rogr Ca

Expanding

3 Solve each of the following. a 5x = 23 b 7x = 45 d −3x = 20 e 13x = 45

c 6x = 37 f 9x = 2

4 Solve each of the following equations. a 5a + 11 = 41 b 2q − 9 = 25 d 9s − 14 = 22 e 7w + 74 = 193 g 5e − 9 = −19 h 8d + 45 = 29 j 5r − 14 = 44 k 7f + 6 = −14 m 12 + 6t = 48 n 35 − 5g = 50 p 4s + 8.5 = 2.3 q 8y − 3--- = 2 1---

c f i l o r

5 Solve each of the following equations. s v a --- + 7 = 12 b --- − 8 = 9 3 7 f p d --- − 1 = 12 e ------ + 3 = 11 4 11 k v g ------ + 20 = 27 h --- − 4 = 0 15 3 j p j ------ − 13 = −11 k --- + 35 = −4 10 7

r c --- + 5 = 11 8 s f --- − 10 = 2 4 g i --- − 1 = −1 8 m l ---- − 10 = −4 6

6 Solve each of the following equations. 4y 3p a ------ = 8 b ------ = 9 5 2 2s 5w d ----- = – 8 e ------- = 10 5 9

2q c ------ = 8 3 12m f ---------- = – 6 5

4

WORKED

Example

19b

GC p

am – rogr TI

Expanding

D x = 3.29

7

2

3z + 6 = 27 13x − 85 = 227 4c + 70 = 2 9v − 10 = 5 23 − 2b = −1 1 − 7h = −65

7 In each of the following equations, check by substitution if the answer given is correct. a x + 67 = 98 (x = 31) b r − 6.8 = 45.9 (r = 51.7) b 3 c 32p = −256 ( p = −8) d --- = --(b = 6 3--- ) 4 9 4 e 5t − 98 = 56 (t = 30.8) f 7y + 13 = −65 ( y = −11.1) 3w – 2 2e g ---------------- = 3 (w = 4 2--- ) h ------ + 1 = 7 (e = 9) 3 4 3 WORKED

Example

20

8 Solve each of the following equations. a 8a + 7 = 7a + 9 b 7b + 5 = 6b + 14 d 6d + 8 = 3d + 17 e 5e − 10 = e + 10 g 12g + 14 = 6g + 44 h 11h − 18 = 2h j 20 + 2j = j + 54 k 9k − 2 = 28 − k

c f i l

9c − 2 = 8c + 2 7f − 2 = 2f + 13 10i + 11 = 8i − 7 32 − 5l = 5 − 2l

9 multiple choice For which of the following equations is x = 12 not a solution? 5x x–4 A ------ + 1 = 16 B 4x − 7 = 53 − x C 4x − 12 = 48 − x D ----------- = 4 − x 4 2


139

10 Solve each of the following equations by first expanding the brackets. a 3(b + 5) = 30 b 5(n + 6) = 40 c 7(h − 5) = 56 21 d 9( p − 4) = 54 e 4(k − 8) = 72 f 3(m − 16) = 45 g 6(t + 9) = 84 h 4(2n + 5) = 52 i 9(3r − 7) = 72 j 6(6g + 5) = 210 k 4(5g − 1) = −44 l 7(3v − 11) = −161 m 6(z − 2) = 44 n 3(6y + 13) = 76 o 5(4u − 9) = 34

WORKED

Example

11 Solve each of the following equations. s s t–3 22 a --- = --- + 3 b ---------- = 3 4 5 v+5 v 9w d ----------- = 5 – --e ------- – 4 3 2 2 x+1 x 2m – 3 g ------------ = --- − 3 h ---------------5 2 3

WORKED

Example

t–2 ---------6 = 6 1 + 6m = ----------------7

2u – 1 u–1 c --------------- = -----------2 5 x+3 3x + 2 f ------------ = --------------4 8 i

p 3p 1 --- = ------ − --4 8 4

12 Two teams of people worked at two different car washes detailing vehicles. The cost of detailing each car is the same. Team A had 5 people who detailed 30 cars and received $20 in tips. They divided their money equally. Team B had 4 people who detailed 25 cars and received $4 in tips. They also divided their money equally. At the end of the day all 9 people had the same amount of money. a Write an equation for this situation. b Solve the equation to find the cost of getting one car detailed.

Equations arising from substitution Earlier in this chapter we covered substitution of values into a formula. In each of these examples we calculated the value of the subject of the formula. In many cases, after substitution we may be left with a value to calculate that is not the subject of the formula. Hence, the solution will require you to solve an equation.

WORKED Example 23 9C The formula F = ------- + 32 is used to convert degrees Celsius to degrees Fahrenheit. Use 5 the formula to convert 68° Fahrenheit to degrees Celsius. THINK 1

Write the formula.

2

Substitute 68 for F.

3

Multiply both sides of the equation by 5. Subtract 160 from each side. Divide both sides by 9.

4 5

WRITE 9C F = ------- + 32 5 9C 68 = ------- + 32 5 340 = 9C + 160 180 = 9C C = 20

140


Graphics Calculator tip! Solving an equation arising from substitution Earlier in the chapter we learned how to assign a value to a variable. We can assign the value in the RUN mode of the calculator and then switch to equation solving mode as shown below. Consider worked example 23.

1. From the MENU select RUN.

2. Assign the value F = 68 by pressing 68 → ALPHA F EXE .

3. From the MENU select EQUA.

4. Press F3 for Solver. Delete any equations on screen.

5. Enter the equation by pressing ALPHA F SHIFT = 9 ALPHA C ÷ 5 + 32 EXE . At this stage F = 68 is displayed as well as any previously stored value of C. Use the arrow keys to highlight C as shown.

6. Press F6 to solve the equation for C. Note that if you mistakenly highlight F in the previous step the equation will be solved for F using the previously stored value of C.

More difficult questions involve the substitution of more than one unknown. There must be only one unknown remaining for the equation to be solved.


141

WORKED Example 24 h The formula A = --- (a + b) is used to calculate the area of a trapezium. A trapezium with 2 parallel sides 16 cm and 9 cm has an area of 150 cm2. Calculate the height of the trapezium. THINK

WRITE

1

Write the formula.

2

Substitute A = 150, a = 16 and b = 9.

3 4

Multiply both sides of the equation by 2. Divide both sides by 25.

h A = --- (a + b) 2 h 150 = --- (16 + 9) 2 300 = 25h h = 12

remember 1. Begin each question by writing the formula. 2. Substitute each known value into the formula. 3. Solve the equation formed.

4G

Equations arising from substitution

1 The formula A = lb can be used to calculate the area of a rectangle. Calculate the value of b when A = 56 and l = 8. 23 2 In the formula A = lb, calculate: a l, when A = 437 and b = 23 b b, when A = 36.225 and l = 6.3.

WORKED

Example

3 The formula P = 2l + 2w is used to calculate the perimeter of a rectangle. Calculate the value of l when: a P = 64 and w = 18 b P = 142 and w = 17 c P = 12.4 and w = 3.4. 9C 4 The formula F = ------- + 32 converts 5 degrees Celsius to degrees Fahrenheit. In each of the following, find C given that F is equal to: a 50° b 80° c 100° d −10° e 15° f −5.3°.

142


5 Use the formula T = a + (n − 1)d to find: a a, when T = 30, n = 8 and d = 4 b a, when T = −447, n = 54 and d = −8 c n, when T = 5050, a = 1 and d = 1 d n, when T = 454, a = −56 and d = 6. PRT 6 a Given that I = ----------- , find R when I = 500, P = 2500 and T = 2. 100 24 b Given that T = a + 8d, find d when T = 59 and a = 11. c Given that P = 2l + 2w, find w when l = 34 and P = 176.

WORKED

Example

d Given that A = --1- bh, find h when A = 19.43 and b = 5.8. 2

e Given that V = lbh, find b when V = 74.375, 1 = 2.5 and h = 3.5. D f Given that S = ---- , find D when S = 90 and T = 12. T 5 ( F – 32 ) g Given that C = ------------------------ , find F when C is 15. 9 h Given that S = ut + 1--- at 2, find u when S = 120, t = 5 and a = 8. 2

i Given that T = a + (n − 1)d, find n when T = −3, a = 57 and d = −4. j Given that c2 = a2 + b2, find a when c = 17 and b = 15. 7 The cost of hiring a taxi can be found using the formula C = 4 + 2.5d, where d is the distance travelled in kilometres. Find the distance travelled if the taxi fare was: a $29.00 b $49.00 c $25.25.

Work

T SHEE

4.2

8 An operator connected phone call costs $1.50 connection fee plus $2.20 per minute. a Write a formula connecting the cost of the call, C, to the length of the call, m. b Calculate the cost of a call lasting: iii 1 minute iii 5 minutes iii 8 minutes. c Calculate the length of a call for which the charge is: iii $8.10 iii $16.90 iii $34.50.


143

summary Number patterns • A number pattern is a sequence of numbers that obey a certain rule. • A number pattern can be continued by following a rule that shows how to move from one term to the next, or by finding an algebraic rule that connects the value of a term with its position in the sequence. • When writing an algebraic rule for a sequence, n is the position of a term in the sequence and Tn is the value of that term.

Addition and subtraction of like terms • Like terms are those which use the same pronumeral. • We can only simplify expressions involving addition and subtraction that contain like terms.

Substitution • Pronumerals stand in place of numbers. A number can be substituted for a pronumeral in an expression before the expression is calculated.

Multiplication and division of algebraic expressions • When multiplying and dividing algebraic expressions we need to use the index laws. Index Law 1: a x × ay = a x + y ax Index Law 2: a x ÷ ay = a x − y or -----y = a x – y a Index Law 3: (a x) y = a xy • When multiplying or dividing expressions, we treat each pronumeral separately, applying the index laws when necessary. • When an expression involves the use of brackets, we multiply each term in the brackets by the term immediately outside.

Equations • An equation is a mathematical sentence with a missing value. The object of solving an equation is to find the missing value that makes the sentence correct. • In solving an equation, we can add, subtract, multiply or divide both sides of the equation to make the unknown value the subject of the equation. • Whatever is done to one side of an equation must be done to the other to maintain the equality. • Always begin by writing the equation, then write each step in the solution. • The answer to an equation can be checked by substituting the value found into the equation. • Equations can be formed when substituting into a formula. This occurs when the subject of the formula is not the value we need to find. • You can use a graphics calculator to find the solution to an equation, but it is very important to understand the manual methods as well.

144


CHAPTER review 4A

1 Write down the next three terms of each of the following sequences. a 4, 8, 12, 16, . . . b 6, 12, 24, 48, . . . c 12, 5, −2, −9, . . . d 800, 200, 50, 12.5, . . . e 1, 4, 9, 16, . . . f 3, 12, 48, 192, . . . g 1000, 100, 10, 1, . . . h 1, 3, 6, 10, . . . i −54, −45, −36, −27, . . . j 1, 1, 2, 3, . . .

4A 4A

2 Describe in words each of the sequences in question 1. 3 Write down the first five terms of a sequence which: a starts with 45 and each term is 8 less than the previous b starts with 6 and each term is 3 times the previous c starts with −34 and each term is 12 more than the previous d starts with 243 and each term is 2--- the previous. 3

4B

4 Write down the first five terms of each of the sequences given by the following algebraic rules. a Tn = 7n b Tn = 5 n c Tn = 8n − 3 d Tn = (−1)2n n n e Tn = 20 − 6n f Tn = 3 g Tn = 10 − 1 h Tn = (−6)n − 13

4B

5 Use the given rule to copy and complete the table in each of the following. a Tn = 9n b Tn = 2n + 9 n

1

3

4

7

10

Tn c Tn = 2 n

n

1

4

7

9

12

3

6

9

10

Tn d Tn = (−3)n

n

1

2

5

8

10

Tn

n

1

Tn

4C

6 Simplify each of the following. a y+y+y+y b 8w + 9w d 15t − 9t e 6q − 5q

c 6r + 9r − r f 9x + 6x − x

4C

7 Simplify each of the following. a 8m + 4n − 3m b 6a + 4 − 3a − 9 d 15m − 7 + m + 1 e 5x + 20 + 3x − 6

c 12k − 5l + 3l − 8k f 12m − 20 − 2m + 4

4D 4D

a 8 In the formula S = ----------- , find the value of S when a = 20 and r = 1--- . 2 1–r 3e + 18 9 The formula N = ------------------ + 70 is used to calculate the number of video-recorders, N, that 5 can be produced by ‘e’ employees. Calculate the number of video-recorders that can be produced by 89 employees.


h 10 In the formula A = --- ( a + b ) , calculate the value of A when h = 5, a = 8.5 and b = 6.2. 2 11 Given that S = ut + 1--- at2, find S when u = 9.5, t = 5 and a = 5.8. 2

12 Simplify each of the following. a w 3 × w5 b d 9 × 4q3 e g 6x2 × 8x4 h j 5ab3 × 4a3b2 k m b6 ÷ b2 n p 63y5 ÷ 7y q 64a 6 s -----------4 t 16a v (a3)4 w y k5 × k4 ÷ k3 z

a × a6 5p3 × p 4r4 × 3r4 7g3h2 × 4gh f4 ÷ f 45r5s2 ÷ 5r4s3 32m 6 ------------8m 4 (5m4)3 (8q4)2 ÷ 4q3

13 Expand each of the following. a m(m + 3) b 5p(2p − 6q) d 2w3(3w2 − 3) e 4pq(3p2 − 2q4) 14 Solve each of the following equations. a a + 98 = 165 b b − 76 = 84 d d ------ = 42 e −8e = −96 23 g g + 45 = 12 h 9h = 25 j 4j − 17 = 47 k 7k + 13 = 76 m 45 + 3m = 18

n 33 − 4n = 7

p p 8 – --- = – 5 5

5q q ------ = 15 4

15 Solve each of the following. a 4(s + 9) = 56 d 7v − 2 = 2v + 23

b 3(2t − 7) = 84 e 8w − 5 = 25 − 2w

g 7( y − 3) = 4( y − 9)

h 3(3z + 5) = 7(z − 4)

b–3 b–2 j ------------ = -----------5 6

6c k ------ + 4 = 16 4

4x5 × 6x3 5y × 6y 5x5y4 × 6x7y6 12m4n3 × 4mn3 45r6 ÷ 5r3 36s3t5 ÷ 9st 28 p u -----------412 p x (3p3q5)3 c f i l o r

c x3(4x5 − 2) f 7a6(3a8 − 9b2) c 43c = 3827 f f – --- = 19 4 i 12 − i = 23 l 5l + 43 = −2 t o --- + 9 = 17 5 3r r – ----- = – 15 7

c 7u − 8 = 6u + 34 f 35 − 6x = 8 − 3x a a i --- = --- + 1 4 5 d+4 3d – 4 l ------------ = --------------6 10

16 Solve the following equations by substituting the given values. a A = lw (A = 56, w = 7) b P = 2l + 2w (P = 94, l = 16) c C = 5--- (F − 32)

(C = 25)

a d S = ----------1–r

(S = 6, a = 3)

9

145 4D 4D 4E

4E 4E

4E

4F

146


Practice examination questions 1 multiple choice The number pattern 2, 4, 8, 16, . . . can be described by the algebraic rule: A Tn = 2n B Tn = 2 n C Tn = n + 2 D Tn = n2 2 multiple choice 7x − 5y − 6x + 4y = Ax+y

B x−y

C x + 9y

D x − 9y

B 20x10y

C 10x14y

D 20x14y

3 multiple choice 5x4y × (2x3)2 = A 10x10y 4 multiple choice The solution to the equation 3(2x + 4) = 4(2x − 7) is: A x = − 1--B x = 2 6--Cx=8 2

7

5 In the formula v = ut + 1--- at2: 2

a calculate the value of v when u = 0.8, t = 12 and a = 6 b calculate the value of u when v = 100, t = 4 and a = 1.6. 6 a Simplify 4x + 9y − 5x − 8y. 4x 3 y 2 × 6x 2 y 4 b Simplify ---------------------------------. 3xy 8

CHAPTER

test yourself

4

( 2x 3 y ) 2 -. c Simplify the expression -----------------8x 5 y 9 d Solve the equation 7x + 15 = 113 + 9x.

D x = 20

Statistics and society

5 syllabus reference Data analysis 1 • Statistics and society

In this chapter 5A 5B 5C 5D

Collecting data Organising data Displaying data Quality control

areyou

READY?

Are you ready?

Try the questions below. If you have difficulty with any of them, extra help can be obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon next to the question on the Maths Quest Preliminary Course CD-ROM or ask your teacher for a copy. Presenting data in a frequency table

5.2

Reading column graphs

1 Copy and complete the frequency table at right for the scores listed below. 1, 3, 2, 6, 4, 4, 3, 5, 2, 3, 1, 3, 4, 2, 5, 3, 6, 2, 3, 6, 1, 3, 2, 4

Score (x) 1 2 3 4 5 6

2 This is a graph of preferred leisure activities of a Year 8 class. a How many students preferred sport as a leisure activity? b How many students were in the class? c Which was the most favoured activity? d How many times more popular than reading was watching television? e Which two activities are closest in popularity?

Tally III

Number preferring (frequency)

5.1

Frequency (f) 3

Favourite leisure activity 10 8 6 4 2 0

Reading Television Sport

‘Hanging Other out’

Leisure activity

5.4 5.3

Drawing sector graphs

3 For the following table of values showing the number of kilograms of meat of various types that a butcher sold in a day: a copy and complete the table of values b construct a sector graph to display the data. You could use a spreadsheet program such as Excel to generate a sector graph Type of meat Lamb Beef Pork Chicken Turkey Rabbit Total

5.5

Amount sold (kg) 10 45 5 15 10 5 90

Fraction

Angle size (°)


4 In each of the following, write the first quantity as a percentage of the second. a 18, 500 b 4, 2500 c 245, 500 d 6, 2000

Chapter 5 Statistics and society

149

Analysing data There are many cases in society where data need to be analysed. Governments and businesses have data analysed regularly to try and make accurate predictions about future trends. Consider the case of a government department such as the Roads and Traffic Authority. This department needs to gather data about places where accidents occur. These data are analysed and decisions are then made about what areas need to have roadworks and what places need greater police supervision. Now consider a business example. A department store analyses sales figures throughout the year to determine its stock orders and staffing requirements. In sport, the gathering of statistics is used to measure player performance. These are kept as a matter of interest to followers of the sport, and are used by coaches when selecting their teams and planning tactics against opposition teams. When analysing data, trends need to be observed. The discovery of these trends allows for predictions of future outcomes.

Why statistical investigation? Below are some common examples of statistical analysis. (a) Weather records (b) Unemployment and inflation figures (c) Sales records (d) Hospital admissions (e) World records in sport For each of the above discuss: 1 the reason that such records are kept 2 the methods that are used to collect the information 3 what information could be gained from the analysis 4 what future predictions could be made as a result of the analysis.

A statistical investigation – 1 Choose an area of interest for which you would like to do a statistical analysis. For the analysis you have chosen, answer the following questions. 1 What information do you intend to collect? 2 Of what use is this information and to whom would it be useful? 3 What predictions or decisions could be made based on your analysis?

150


Statistical processes The purpose of completing a statistical inquiry is to turn raw data into meaningful information. Data are a set of facts that are collected, but limited data taken alone can have very little meaning. When lots of data are collected and presented and conclusions are drawn, the data become more useful information. When we set out to complete a statistical investigation, there are six stages that need to be completed: Stage 1. Posing questions Stage 2. Collecting data Stage 3. Organising data Stage 4. Summarising and displaying data Stage 5. Analysing data and drawing conclusions Stage 6. Writing a report. The whole process needs to be well planned as what can be achieved at the later stages of the process depends upon what has been done in the earlier stages.

Posing questions The initial stage of the statistical process is to determine the final information required. For example, suppose that you want to find out the best time of year to plan a skiing holiday and the best location to take that holiday? Questions that may need to be posed include: • When during the year do the best skiing conditions occur? • Which resort has the best skiing conditions on a regular basis? • What is the cost of accommodation, lift tickets and ski hire at each resort? • What facilities are available at each resort? When you have finished collecting data and can answer each question, you will be ready to make a conclusion.

Posing questions For each of the following, pose some questions that, if answered, would allow you to make a conclusion. 1 A department store manager wants to know the number of extra staff that should be hired to work in the weeks leading up to Christmas. 2 A local council wants to find out what new sporting facilities the community wants and needs. 3 The NRMA wants to know which model cars are the most secure against theft. 4 The Australian test cricket selectors need to choose a new opening batsman. 5 A potential investor wants to know which shares represent the best potential gains. 6 A newspaper wants to try and predict the winner of the next federal election.

A statistical investigation – 2 For the area that you are going to investigate, list the questions you will need to answer in order to draw a conclusion.


151

Collecting data Data can be collected using either external or internal sources. To collect data from an external source means that the data are available by doing some research. For example, if you were researching share prices, data would be available from the stock exchange. Other types of data can be obtained from organisations such as the CSIRO, local councils, the NRMA and the Australian Bureau of Statistics. If data are not available from an external source, you will need to generate the data yourself. This is called using internal sources. There are two methods of acquiring data internally. 1. Observation — this is used if the data collection does not require a response from people. For example, you may be surveying the number of customers that enter a certain shop during the day or the number of students at your school who are out of uniform. 2. Questioning — this is used when the data are obtained by getting a response from people. For example, you may be investigating what sporting facilities are needed in your local area. When designing a survey for your investigation, the questions need to be well organised and thought needs to go into how the data are going to be tabulated. It is easier to organise information if the questions asked are not open-ended. This means that the responses are limited. ‘A survey form’ ‘What sporting facilities do you think are needed in this area?’ This question invites a range of responses that may be difficult to tabulate. A better method Please rank the following would be to list a few options that are likely to prove sporting needs in this area. popular and ask people to rank them in order of Cricket nets priority. Tennis courts Similarly, when there could be a large range of Golf course responses to a question it is easier to group these Netball courts responses. Soccer field Other

WORKED Example 1 ‘What is your annual income?’ Redesign this question so that the results are easier to tabulate. THINK

WRITE

The range of responses could be simplified by putting them into income brackets.

Within what range does your income fall? $0 – $9999 $10 000 – $19 999 $20 000 – $29 999 $30 000 – $39 999 $40 000 – $49 999 $50 000 or more

152


remember 1. Data can be collected from internal or external sources. 2. An external source is where the data have been collected and are available by doing research. 3. An internal source is where you need to gather the data yourself. 4. Internal data gathering can be done by observation or questioning. 5. Observation is used when the data can be obtained without a response from other people. 6. Questioning is used when the data are found by getting people’s responses.

5A

Collecting data

1 For each of the following, state whether the data source would be internal or external. a The number of cars stolen in NSW each year b The rise or fall in a share price over the past year c The number of people who rode bikes to school today d The number of people who voted in the last federal election e Who people intend to vote for in the next federal election f The most popular band among Year 11 students at your school g The number of Holden cars sold each week in Australia h The batting average of each player in the Australian cricket team 2 For each of the following, state whether the data would be gathered using observation or questioning. a The number of sets of traffic lights in a country town b The number of students in Year 11 at your school who started high school at a different school c The most popular football team in Year 11 d The football team that attracts the largest crowds e The number of students in your class with a learner’s permit f The number of trees in your school grounds g The average weekly income of Year 11 students h The number of people who speed through an intersection 3 Rewrite the following open-ended questions so that the responses will be easier to tabulate. 1 a Where is your favourite holiday destination? b What is your weekly income? c How many movies have you seen at the cinema this year? d Who is your favourite singer or group? e How many hours study do you do each week?

WORKED

Example

4 Design a questionnaire that will provide the following information. a The sporting facilities that people would like to see in your area b The amount of income and source of income among Year 11 students


153

A statistical investigation – 3 For your investigation: 1 State whether your data will be obtained from external or internal sources. 2 If you are using an internal source, state if you will use observations or questioning. Regardless of what method you are using, give details on how you are going to collect your data. You are now ready to collect your data.

Organising data Once data have been collected, they need to be put into an organised form. This involves tallying the responses to a questionnaire, accurately recording your observations or tabulating the results of your research. This task is made easier if the questionnaire is designed with ease of tabulation in mind. Usually the results are first organised into a table and the number of responses in each category recorded. This is often done with tally marks and using the gatepost method.

WORKED Example 2 A survey is conducted among 24 students who were asked to name their favourite spectator sport. Their responses are recorded below. AFL Cricket Cricket Soccer Rugby League Cricket Tennis Cricket AFL Rugby League AFL AFL Rugby Union Soccer Netball Basketball Basketball Netball AFL Cricket Cricket AFL Rugby League Cricket THINK Draw a table and beside each sport put a tally mark for each response. Every fifth tally mark becomes a gatepost.

WRITE Sport

Tally

Frequency

AFL

|||| |

6

Basketball

||

2

Cricket

|||| ||

7

Netball

||

2

Rugby League

|||

3

Rugby Union

|

1

Soccer

||

2

Tennis

|

1

154


For simplicity, numerical data may be tabulated in groups.

WORKED Example 3 A Year 11 class was surveyed on their weekly income. The responses are shown below. $75 $115 $60 $54 $88 $0 $98 $102 $56 $45 $83 $71 $40 $37 $87 $117 $43 $79 $58 $89 $70 $105 $99 $55 Complete the table below. Income

Tally

Frequency

$0–$20 $21–$40 $41–$60 $61–$80 $81–$100 $101–$120

THINK

WRITE

Count the number of responses within each category and put a tally mark in the column.

Income

Tally

Frequency

$0–$20

|

1

$21–$40

||

2

$41–$60

|||| ||

7

$61–$80

||||

4

$81–$100

|||| |

6

||||

4

$101–$120

Here we have done some very simple tables. These tables will be dealt with in greater depth in chapter 9.

remember 1. When data are collected they are usually first organised into table form. 2. Data can be easily counted using a tally column and the gatepost method. 3. Sometimes numerical data are better organised into categories.


5B

155

Organising data car their family owned. Their

5.1

Toyota Toyota Toyota Holden Ford

Holden Toyota Mazda Ford Toyota

L Spre XCE ad

2 The results of a spelling test done by 30 students are shown below. 7 7 7

6 8 8

8 10 8

4 5 7

6 9 8

6 7 6

7 7 5

5 7 9

sheet

Put these results into a table. 6 5 4

SkillS

Presenting data in a frequency table

E

Example

HEET

1 A class of students was asked to identify the make of responses are shown below. 2 Holden Ford Nissan Mazda Ford Holden Ford Mitsubishi Nissan Holden Holden Ford Mazda Toyota Ford Holden Mitsubishi Toyota Holden Ford

WORKED

Frequency tables

9 6 7

Put these results into a table. 3 The marks scored on below. 3 87 44 95 66 54 60 66 69 71 83 74 81

WORKED

Example

a Maths exam, out of 100, by 25 Year 11 students are shown 78 66 69

69 77 70

66 79 57

92 66

78 71

Copy and complete the table below. Mark

Tally

Frequency

40–49 50–59 60–69 70–79 80–89 90–99 4 The data below show the number of customers that entered a shop each day in a certain month. 195 216 185 155

175 200 173 132

163 147 164 143

180 168 130 190

120 173 119 179

204 102 158 200

199 150 163

Choose suitable groupings to tabulate these data.

A statistical investigation – 4 Organise the data for your investigation into a suitable table.

T SHEE

Work

114 178 169 141

5.1

156


Displaying data The most common way for displaying data is by using a graph. Different graphs have different purposes, which we will look at in more detail in chapter 9. For now, we will look briefly at column graphs and sector graphs. A column graph is used when we wish to show a quantity. Categories are written on the horizontal axis and frequencies on the vertical axis.

WORKED Example 4 The table below shows the results of the survey on favourite sports. Sport

Frequency

AFL

6

Basketball

2

Cricket

7

Netball

2

Rugby League

3

Rugby Union

1

Soccer

2

Tennis

1

Show this information in a column graph. THINK

2 3

Draw the horizontal axis showing each sport. Draw a vertical axis to show frequencies up to 7. Draw the columns.

Frequency

1

WRITE 8 7 6 5 4 3 2 1 0

l e l t n r s FL al e al u o e ni A etb rick etb eag Uni occ Ten S sk C N y L by b ug Ba g Ru R Sport


157

Graphics Calculator tip! Drawing graphs Your Casio graphics calculator can be used to draw some types of graphs, including column graphs. In practice, it is probably easier to draw the column graph manually but knowing how to do this will be of use later in this course. Consider worked example 4. 1. From the MENU select STAT.

2. Delete any existing data. The calculator will not recognise any words so we will need to replace the names of the sports with the numbers 1–8. Write these numbers in List 1 and the Frequencies in List 2.

3. Press F1 for GRPH.

4. Press F6 for SET, arrow down to Graph Type and press F1 for Hist. Then arrow down to Xlist and press F1 for List 1, then arrow down to Frequency and press F3 for List 2. Your display should be as shown on the right. 5. Press EXE to return to the previous screen then F1 for GPH1. Enter a Start = 1 and a Pitch = 1.

6. Press F6 to draw the column graph. It will be drawn as a histogram, which has no gaps between the columns.

A sector graph is used when we want the graph to display a comparison of quantities. An angle is drawn at the centre of the circle that is the same fraction of 360°, as the fraction of people making each response.

158


WORKED Example 5 For the table in worked example 4, draw a sector graph. THINK 1

WRITE

Calculate each angle as a fraction of 360°.

AFL =

6 -----24

× 360°

Basketball =

= 90° Cricket =

7 -----24

2 -----24

× 360°

= 30°

× 360°

Netball =

= 105°

2 -----24

× 360°

= 30°

Rugby League =

× 360°

3 -----24

Rugby League = 45° Rugby Union =

1 -----24

× 360°

Rugby Union = 15° Soccer =

2 -----24

× 360°

= 30° 2

Tennis =

1 -----24

× 360°

= 15°

Draw the graph.

Sport AFL Basketball Cricket Netball Rugby League Rugby Union Soccer Tennis

Column graphs and sector graphs can also be drawn using a spreadsheet and the charting tool.

remember 1. A column graph is drawn when we want to display quantities. 2. A sector graph is drawn when we want to compare quantities.


5C WORKED

Example

159

Displaying data

1 Draw a column graph to display the data from question 1 of Exercise 5B.

5.2

2 Draw a sector graph to display the data from question 1 of Exercise 5B.

5

3 Draw a column graph to display the data from question 2 of Exercise 5B.

Reading column graphs

5.3

5 Draw a column graph to display the data from question 4 of Exercise 5B. 6 Draw a sector graph to compare the number of people in each category from question 3 of Exercise 5B.

SkillS

HEET

4 Draw a column graph to display the data from question 3 of Exercise 5B. Reading line graphs

5.4

SkillS

HEET

A statistical investigation – 5

Analysing data and drawing conclusions

Column graphs (DIY) L Spre XCE ad

GC

Sector graphs (DIY)

What conclusions and recommendations can you make based on your study?

The final stage is to collate all earlier stages into a written report. The written report should: 1. Pose the questions that the statistical analysis is examining. 2. Explain how data were collected, what type of data were collected and from what sources they were collected. 3. The tables used should be included. 4. Relevant graphs should be used to display the data. 5. The conclusions and recommendations should be written and include reasons why these conclusions were reached.

A statistical investigation – 7 Complete a written report for your statistical inquiry.

UV stats

program GC

–TI

Writing the report

am progr –C

asio

A statistical investigation – 6

sheet

Once the data have been organised and displayed, they need to be studied and conclusions drawn. It is at this stage that those making the statistical inquiry can reflect on the results and decide what the data mean. These conclusions should be written in point form and from them recommendations can be made to the relevant bodies.

L Spre XCE ad

sheet

For your investigation draw suitable graphs. You may prefer to do so using a spreadsheet, by entering your results and using the charting function to produce your graphs.

Drawing sector graphs E

Example

E

WORKED

SkillS

HEET

4

UV stats

160


Quality control Companies use statistics to ensure that their product is of a required standard. This is known as quality control. Consider the example of a company that produces matches. The cover of the matchbox says, ‘Minimum contents 50 matches’. To ensure that this statement is correct, a sample of matchboxes from the batch is tested. If more than 3% of the boxes have less than 50 matches, the batch is said to be unsatisfactory and the entire batch is rejected.

WORKED Example 6 A batch of matchboxes is tested for its contents. If more than 3% of the matchboxes have less than 50 matches in them, the batch is rejected. If 300 matchboxes are tested and 10 have less than 50 matches, is the batch accepted or rejected? THINK 1 2

Calculate 10 as a percentage of 300. Make a conclusion.

WRITE × 100% = 3 1--- % 3 The batch is rejected, as there are more than 3% of boxes with less than 50 matches. 10 --------300


161

In some cases you will need to be able to tabulate the results before making a conclusion.

WORKED Example 7 From a batch of batteries a sample is tested to see how long it will take for them to go flat. The results are shown below.

Life (hours)

Frequency

200–249

2

250–299

5

300–349

7

350–399

15

400–449

42

450–499

38

500–549

36

550–599

20

600–649

7

650–699

3

If more than 20% of batteries go flat in less than 400 hours, the batch is rejected. Determine whether this batch is accepted or rejected. THINK 1 2 3 4

WRITE

Count the number of batteries in the sample. Count the number of batteries that take less than 400 hours to go flat. Calculate the percentage that go flat in less than 400 hours. Make a conclusion about the batch.

175 batteries in the sample. 29 batteries went flat in less than 400 hours. 29 --------175

× 100% = 16.6%

The batch is accepted.

remember 1. Statistical methods are used in quality control. 2. You will need to calculate the percentage of a sample that meets the requirements. 3. You should be able to obtain this information from a table.

162


5D 5.5

1 A batch of 400 tyres is tested. If more than 5% of the tyres lose their tread in less than 15 000 km, the batch is rejected. If 17 tyres lose their tread in less than 15 000 km, cal6 culate if the batch is accepted or rejected.

WORKED

Example

SkillS

HEET

Quality control


2 One kilogram bags of sugar are tested to check their mass. A batch of 250 bags is tested. If more than 2% of the bags have a mass less than 1 kg, the batch is rejected. If 246 bags have a mass of 1 kg or more, calculate if the batch is accepted or rejected. 3 A batch of light globes is tested. A sample of globes are taken and tested to see how long 7 they will burn. To be considered satisfactory, at least 90% of the batch must burn for more than 1000 hours. The results are shown in the table at right. Determine if the batch is satisfactory.

WORKED

Example

4 The diameter of a batch of screws is given as 4 mm. A sample of the batch is tested. For the batch to be considered satisfactory, 95% of the screws must have a diameter between 3.9 mm and 4.1 mm. The results are shown in the table at right. Determine if the batch is satisfactory.

Hours

No. of globes

0–249

1

250–499

6

500–649

15

750–999

46

1000–1249

89

1250–1499

65

Diameter

No. of screws

3.7–3.8

3

3.8–3.9

14

3.9–4.0

58

4.0–4.1

46

4.1–4.2

1

5 Boxes of toothpicks contain a minimum of 100 toothpicks. From a batch, 60 boxes of toothpicks are selected and their contents counted. If more than 4% of the boxes contain less than 100 toothpicks, the batch is rejected. The contents of the boxes are given below. 102 101 100 102 98

103 102 101 99 101

105 102 101 100 101

100 101 102 103 100

101 100 103 100 103

101 100 100 101 99

Determine if the batch is satisfactory.

102 100 104 102 100

99 101 103 102 101

100 101 105 103 105

103 100 106 101 102

100 100 101 100 103

100 100 100 102 100


163

6 The label on a soft drink bottle states that it contains 600 mL. From a batch of bottles, 40 are selected and their volumes measured. To be considered satisfactory, 90% of the bottles must contain between 590 mL and 610 mL. The volumes found are shown below. 599 606

600 603

602 600

598 615

585 602

592 588

604 595

603 608

600

607

602

602

590

610

602

608

615

598

594

610

595

604

601

600

602

598

600

597

604

605

606

600 T SHEE

Work

Determine if the batch is satisfactory.

Privacy and ethical issues Many of the questions that are the subject of statistical analysis ask people to provide information that may be considered private. Examples of this are: • information on people’s income • information about people’s voting intentions. When such questions need to be asked, it is necessary for those conducting the inquiry to take steps to ensure that the privacy of the participants in the survey is maintained. This can be done by doing the survey anonymously. In this way, the participant can be sure that the information provided cannot be tracked back to them. In some cases, large organisations doing major research are unable to do anonymous studies. In this case there must be some guarantee to the participants that the information will remain confidential, and when the data have been collated that response forms will be destroyed. There are other issues associated with the use of statistical information. Ethical issues arise when organisations sell their data to other organisations. For example, a bank needs to keep information on its mortgage customers. If this information were then sold to an insurance company, which, in turn, was able to approach these people seeking their business, a major ethical breach would have occurred.

Privacy issues 1 Discuss with your family and friends the information they consider too private to reveal for the purposes of a survey. In your class make a list of all such issues. 2 Try to create a set of conditions under which people may be more willing to reveal more detailed private information. This may include anonymous surveying, privacy guarantees, or the use of grouping the data so specific details do not need to be revealed.

5.2

164


Statistical organisations Many major organisations carry out regular statistical analysis. These organisations include: Australian Bureau of Statistics; United Nations; World Health Organization; Morgan Gallup Polls; and NRMA. These organisations collect data for a wide variety of purposes. If you look at any statistical analysis in a book, magazine or newspaper, the source of the statistics will be given. From this you can find out the names of many similar organisations. In many cases the methods used by the organisation will also be explained.

Organisations that use statistics Choose an organisation that regularly collects statistical data. 1 Give an example of the type of information it aims to collect. 2 For what purpose is this information collected? 3 What conclusions do they hope to draw? 4 For whom is the information collected and who is going to act on the results?


165

summary Analysing data • Data are analysed by various groups for many reasons. In each case the aim is to provide a useful conclusion to a problem. • Facts that are gathered are called data. • When data are put into a meaningful form they become more meaningful as information.

Statistical processes There are six stages to a statistical inquiry. Stage 1. Posing questions — questions are set to find the information that will help solve the problem. Stage 2. Collecting data — this can be done: externally — this involves obtaining data from outside sources internally — this involves researchers generating information. Internal collection of data can be done by observation or questioning. Stage 3. Organising data — putting the results into an organised, readable form. Stage 4. Summarising and displaying data — the display of the tables used and the drawing of graphs. Stage 5. Analysing data and drawing conclusions — reading and interpreting the results to answer the original problem. Stage 6. Writing a report — the explanation of the above processes and how the conclusions were reached.

Quality control Statistical methods are used by companies to maintain standards in their products and to determine consumer satisfaction.

Major organisations Organisations such as the United Nations, Australian Bureau of Statistics and the World Health Organization use statistics for the benefit of the communities they serve.

166


CHAPTER review 5A

1 For each of the statistical investigations below, state whether the data source would be external or internal. a The wins recorded by a major football team b The number of brick homes in an area c The number of sales made by a department store in a month d The amount of traffic passing through an intersection in one hour

5A

2 For each of the following, state whether the data would be gathered using observation or questioning. a The number of surfers on various beaches b The brand of surfboard preferred by surfers c The average height of Year 11 students at your school d The most popular movie among Year 11 students at your school

5A

3 Rewrite each of the following questions so that the results are easier to tabulate. a What is the distance from your home to school? b Describe the type of house you live in. c What is the number of kilometres that your family car travels in one year? d What is your favourite sport?

5B

4 A survey is conducted on the number of people living in each household in a street. The results are shown below. 1 6 3

4 4 3

5 4 4

2 6 2

2 3 2

3 2

4 3

6 5

1 1

2 3

5 4

Put these results into a table.

5B

5 A group of Year 11 students were asked to state the number of CDs that they had purchased in the last year. The results are shown below. 12 9 12

1 5 6

13 11 18

20 0 7

5 14 10

22 25 9

35 3 6

12 8 23

17 10 14

20 9 19

Put the results into a table using the categories 0–4, 5–9, 10–14 etc.

5C 5C CHAPTER

test yourself

5

6 Draw a column and a sector graph to represent the results to question 4. 7 Draw a column and a sector graph to represent the results to question 5.

Data collection and sampling

6 syllabus reference Data analysis 2 • Data collection and sampling

In this chapter 6A Target populations and sampling 6B Population characteristics 6C Bias 6D Types of data 6E Estimating populations

areyou

READY?

Are you ready?


6.1

Writing one amount as a percentage of another

6.2

Unitary method of percentages

6.3

1 In each of the following write the first quantity as a percentage of the second. a 12; 200 b 54; 250 c 45; 527

2 Find the total quantity given that: a 5% of it is 230

b 15% of it is 126

c 2.4% of it is 28.8.

b 6% of 1150

c 8.25% of 1327.

Finding a percentage of a quantity

3 Find: a 15% of 900

Chapter 6 Data collection and sampling

169

Target populations and sampling In the previous chapter we looked at statistics and the role they play in society. In this chapter we investigate in more detail how data are collected. To gather data we conduct a poll. A poll is the recording of responses to a set of questions known as a questionnaire.

Gallup poll The most famous poll is named after its founder, the American statistician, George Gallup, who was born in 1901. Find out about Gallup and his work and how Gallup polls are used today. The first step in gathering the relevant data for a statistical investigation is to target the population to be investigated. This means identifying the sections of the population for whom the statistical investigation will have meaning. For example, if investigating the medical needs of a community, we would not conduct our survey at the local fitness club. For such a survey, we would survey doctors and other medical personnel, as well as a selection of patients who use the existing facilities.

Identifying the target population For each of the following statistical investigations, identify the population that you would target for a survey. 1 The school ‘End of Year’ Committee wants to find out the preferred venue, band and meals for the Year 12 farewell. 2 The local council wants to know what sporting facilities are needed in the local area. 3 A newspaper wants a survey to predict the winner of a forthcoming election. 4 A group of people planning to build a preschool would like to know what facilities attract people to a particular preschool. 5 A recording label wants to estimate the potential success of a ‘grunge’ band. When starting an investigation, we must determine the quantity of data needed for the database. Consider the case of a company calculating the TV ratings. Does the company need to find out what every household is watching? Obviously they do not; they ask a selection of homes to record their TV viewing. Now consider the case of selecting a band to play at the Year 12 farewell. In this case it is reasonable to ask every Year 12 student their opinion. Data can be collected in one of two ways: 1. Census. In a census an entire population is counted. Australians complete ‘The Census’ every five years. This is a survey of every household in the nation conducted by the Bureau of Statistics. For the purposes of most statistical investigations, a census surveys everyone in the target population, such as the Year 12 example above. 2. Sample. A sample is a more practical method for doing most surveys. Only a selection of the target population is surveyed, with the results taken to be representative of the whole group. The TV ratings example is one where a sample is used.

170


WORKED Example 1 In each of the following, state if the information was obtained by census or sample. a A school uses the roll to count the number of students absent each day. b The television ratings, in which 2000 families complete a survey on what they watch over a one week period. c A light globe manufacturer tests every hundredth light globe off the production line. d A teacher records the examination marks of her class. THINK

WRITE

a Every student is counted at roll call each morning.

a Census

b Not every family is asked to complete a ratings survey.

b Sample

c Not every light globe is tested.

c Sample

d The marks of every student are recorded.

d Census

To ensure that the results of your sample are representative of the whole population, the method of sampling is important. There are three main methods of choosing a sample: random sample, stratified sample and systematic sample. Method 1. Random sample. In a random sample those to be surveyed are selected by chance. When a random sample is conducted, every person in the target population should have an equal chance of being selected. For example, the names of the people to complete your survey may be drawn from a hat. If this method is used, you should get a good mixture of people in your survey. Suppose that we are going to survey students in No. of Year students a school. We want a mixture of students and could choose a fixed number of students from each year. Year 7 200 Suppose we decide to survey 60 students. We could select 10 from each year, but if we did this the survey Year 8 180 would not have the correct proportion of students from Year 9 190 each year. For example, 20% of the students at this 2 school are in Year 7, but only 16 --- % of the survey Year 10 185 3 participants are in Year 7. Year 11 135 If we randomly select the 60 participants, we could expect those chosen to resemble the overall Year 12 110 population. To get a genuinely random sample, you could use the random number generator on your Total 1000 calculator. The calculator will give you a random decimal number between 0 and 1. Each random number is multiplied by 1000 and then rounded up to give a whole number between 1 and 1000. We could then take the roll number of the student corresponding to this number. Repeating this 60 times will give us a genuinely random sample.


171

WORKED Example 2 Three students from a school are to be selected to participate in a statewide survey of school students. There are 750 students at the school. To choose the participants, a random number generator is used with the results 0.983, 0.911 and 0.421. What are the roll numbers of the students who should be selected? THINK 1

2

WRITE

Multiply the results of the random number generator by the size of the population. Round up to whole numbers.

0.983 × 750 = 737.25 0.911 × 750 = 683.25 0.421 × 750 = 315.75 The 738th, 684th and the 316th people on the roll would be surveyed.

Any other method may not give a truly representative sample. For example, if you survey people in the playground you may: • have a tendency to ask people you know • choose an area where a lot of students from a particular year tend to sit • choose more of one sex than the other. Method 2. Stratified sample. In this type of sample you deliberately choose people to complete your survey who are representative of the whole population. In the school survey you would need to select six strata that had the correct proportion of students from each year. In this example, 20% of the school population are in Year 7 and so 20% of your sample should be from Year 7.

WORKED Example 3 Adrian is conducting a survey of school students. At his school, 47% of the population are male and 53% are female. If Adrian decides to survey 60 students, how many students of each sex should he choose if he decides to use a stratified sample? THINK 1

2

3

WRITE

The population is 47% male and so 47% of the sample should be male. Find 47% of 60. The population is 53% female and so 53% of the sample should be female. Find 53% of 60. Make a conclusion about how many of each sex should participate in the survey, rounding off your answers.

47% of 60 = 0.47 × 60 47% of 60 = 28.2 53% of 60 = 0.53 × 60 53% of 60 = 31.8 There should be 28 males and 32 females in Adrian’s survey.

Method 3. Systematic sample. Systematic sampling is where those chosen for the sample are chosen in a systematic or organised way. This method is mostly used in quality control situations discussed in the previous chapter. For example, suppose that the quality and strength of sports shoes is being tested. The quality control department may test every 20th pair of shoes that come off the production line. In doing a survey, every 20th person on the school roll may be surveyed.

172


remember 1. Before beginning a statistical investigation it is important to identify the target population. 2. The survey can be conducted either by: (a) Census — the entire target population is surveyed, or (b) Sample — a selection is surveyed such that those selected are representative of the entire target population. 3. There are three methods for selecting a sample. Method 1. Random sample — chance is the only factor in deciding who is surveyed. This is best done using a random number generator. Method 2. Stratified sample — those sampled are chosen in proportion to the entire population. Method 3. Systematic sample — a system is used to choose those who are to be in the sample.

6A WORKED

Example

1

Target populations and sampling

1 A school conducts an election for a new school captain. Every teacher and student in the school votes. Is this an example of a census or a sample? Explain your answer. 2 A survey is conducted by a council to see what sporting facilities the community needs. If 500 people who live in the community are surveyed, is this an example of a census or a sample? 3 For each of the following surveys, state whether a census or a sample has been used. a Two hundred people in a shopping centre are asked to nominate the supermarket where they do most of their grocery shopping. b To find the most popular new car on the road, 500 new car buyers are asked what make and model car they purchased. c To find the most popular new car on the road, the make and model of every new car registered are recorded. d To find the average mark in the Mathematics half-yearly exam, every student’s mark is recorded. e To test the quality of tyres on a production line, every 100th tyre is road tested. 4 For each of the following, recommend whether you would use a census or a sample to obtain the results. a To find the most watched television program on Monday night at 7:30 pm b To find the number of cars sold during a period of one year c To find the number of cars that pass through the tollgates on the Sydney Harbour Bridge each day d To find the percentage of computers produced by a company that are defective 5 An opinion poll is conducted to try to predict the outcome of an election. Two thousand people are telephoned and asked about their voting intention. Is this an example of a census or a sample?


WORKED

Example

2

173

6 A factory has 500 employees. Each employee has an employee number between 1 and 500. Five employees are selected to participate in an Occupational Health and Safety survey. To choose the participants, a random number generator is used. The results are 0.326, 0.352, 0.762, 0.989 and 0.018. What are the employee numbers of those to participate in the survey? 7 A school has 837 students. A survey of 10 students in the school is to be conducted. A random number generator is used to select the participants. If the random numbers chosen are: 0.988 0.251 0.498 0.661 0.247 0.031 0.967 0.932 0.229 0.443 what are the roll numbers of the students who should be selected? 8 A survey is to be conducted of 20 out of 50 000 people in a country town. Those selected are to be chosen using a random number generator. a Use your calculator to generate 20 random numbers. b Calculate the electoral roll numbers of the people who should be chosen for the survey. 9 For each of the following, state whether the sample used is an example of random, stratified or systematic sampling. a Every 10th tyre coming off a production line is tested for quality. b A company employs 300 men and 450 women. The sample of employees chosen for a survey contains 20 men and 30 women. c The police breathalyse the driver of every red car. d The names of the participants in a survey are drawn from a hat. e Fans at a football match fill in a questionnaire. The ground contains 8000 grandstand seats and 20 000 general admission seats. The questionnaire is then given to 40 people in the grandstand and 100 people who paid for a general admission seat. 10 multiple choice Which of the following is an example of a systematic sample? A The first 20 students who arrive at school each day participate in the survey. B Twenty students to participate in the survey are chosen by a random number generator. C Twenty students to participate in the survey are selected in proportion to the number of students in each school year. D Ten boys and 10 girls are chosen to participate in the survey. 11 multiple choice Which of the following statistical investigations would be practical to complete by census? A A newspaper wants to know public opinion on a political issue. B A local council wants to know if a skateboard ramp would be popular with young people in the area. C An author wants a cricket player’s statistics for a book being written. D An advertising agency wants to know the most watched program on television.

12 Zara is conducting a survey of the people at work. At her work 68% of the workers are male and 32% are female. If Zara decides to survey 50 workers, how many of each sex 3 should she choose, if she chooses to use a stratified sample?

WORKED

Example

174


13 The table below shows the percentage of students that are in each year of a school.

Year

Percentage of students

7

20%

8

19%

9

21%

10

16%

11

13%

12

11%

If 40 students are to participate in a survey, find the number chosen from each year group if stratified sampling is used.

Census or sample For each of the following statistical investigations, state whether you would gather data using a census or sample. For those for which you would use a sample, state the best method for selecting the sample. 1 A company wants to test the life of its batteries. 2 A sporting club wants to elect a new club president. 3 A market research company wants to determine the most popular brand of toothpaste. 4 A theme park wants to know from which state and suburb its visitors come. 5 A Gallup poll is conducted to determine the preferred prime minister.

Population characteristics Characteristics about a population can be estimated by taking a sample of that population. For example, to estimate the average income of Australians we could conduct an anonymous survey of a sample of the Australian population. To get an accurate estimate, it is important that the sample taken has similar characteristics to the entire population. For example, the sample should have an equal number of males and females, and the ages of those in the sample should be in the same proportion as for the whole population. The area sampled should include a cross-section of people according to socio-economic level, ethnic background, religion etc., similar to that of the whole population. In such cases we use the stratified sampling technique. Characteristics (such as sex and age) of the population and the sample must also match, if we expect our sample to have incomes in the same proportions as those of the general population.


175

Population characteristics If we are trying to estimate the average income of Australian families, what would be the effect if our sample: 1 contained a majority of one sex? 2 contained only people in the 15–21 age group? 3 contained only people who live in a small country town? To calculate the number of participants in a sample from each strata of the population, we calculate each strata as a fraction of the total population and multiply this fraction by the total number to be chosen in the sample.

WORKED Example 4 The table at right shows the enrolment at a high school. Sam is conducting a music survey for the school disco. He plans to survey 60 students. Show how Sam should break down his sample in terms of year and sex.

THINK 1

Write each strata as a fraction of the total and then multiply by the 60 to be chosen in the sample.

Year 7 8 9 10 11 12 Total Grand total

Boys 96 85 92 80 71 69 493

Girls 102 87 88 74 75 65 491 984

WRITE Year 7 boys =

96 --------984

× 60

102 --------984

× 60

≈ 5.85 Year 7 girls =

≈ 6.22 Year 8 boys =

85 --------984

× 60


87 --------984

× 60

≈ 5.30 Year 9 boys =

92 --------984

× 60


88 --------984

× 60

≈ 5.37 Year 10 boys =

80 --------984

× 60

≈ 4.88 Continued over page

176


THINK

WRITE Year 10 girls =

74 --------984

× 60

≈ 4.51 Year 11 boys =

71 --------984

× 60


75 --------984

× 60

≈ 4.57 Year 12 boys =

69 --------984

× 60

≈ 4.21 Year 12 girls = 2

To complete the table, round off each of these answers to the nearest whole number.

65 --------984

× 60

≈ 3.96 Number of students to be sampled Year

Boys

Girls

7

6

6

8

5

5

9

6

5

10

5

5

11

4

5

12

4

4

30

30

Total Grand total

60

Note that on some occasions after rounding off each of the answers, the total number of people to participate in the sample may add to one more than the number that we planned to select. In such cases, the person doing the sample should include this extra person as it gives a better sample of the overall population.

remember 1. Stratified sampling is used to ensure that a sample chosen is representative of the entire population. 2. To choose the number of participants from each stratum in the population, write each stratum as a fraction of the total population, and then multiply by the number of people you intend to select in the sample. Round off each of the answers to this multiplication to the nearest whole number. 3. Sometimes, as a result of rounding off, there may be one more to be chosen in the sample than was originally intended.


6B

177

Population characteristics

1 The table below shows the number of students in each year at a school. 4

WORKED

6.1

Example

7

90

8

110

9

90

10

80

11

70

12

60

Writing one amount as a percentage of another

6.2 Unitary method of percentages

6.3

SkillS

HEET

500


If a survey is to be given to 50 students at the school, how many from each year should be chosen if a stratified sample is used?

2 A company employs 300 men and 200 women. If a survey of 60 employees using a stratified sample is completed, how many people of each sex participated? 3 A business has 400 employees of which 250 are female and 150 are male. The business intends to survey 40 of their employees. If a stratified survey is to be conducted, how many employees of each sex should be surveyed? 4 In the head office of a bank there are 250 employees. Ten of these employees are senior management, 60 are middle management and 180 are employed as clerks. A survey is to be conducted of 50 staff members. How many employees at each level should be surveyed? 5 The Department of Education wants to survey a school population. At the school there are 93 teachers and 1248 students. If the department is to survey a total of 50 people, how many teachers and how many students should participate in the survey? 6 The table below shows the age and sex of the staff of a corporation. Age

SkillS

HEET

Total

HEET

Year

No. of students

SkillS

Male

Female

20–29

61

44

30–39

40

50

40–49

74

16

50–59

5

10

A survey of 50 employees is to be done. Using a stratified survey, suggest the breakdown of people to participate in terms of age and sex.

178


7 The table below shows the number of students who are in each year level at a school. Year

Number of students

7

187

8

192

9

168

10

157

11

137

12

108

If 80 students are to be selected to participate in a survey, how many should be chosen from each year level? 8 A shopping centre has a floor area of 5000 m2. There is one major department store with an area of 1500 m2, two smaller department stores of 750 m2 and 40 small stores of 50 m2. The management of the centre assigns voting rights in the shopping centre in proportion to the floor area of each business. Given that there are 200 votes to be distributed, how many votes should each business get? 9 The table below shows the population of a school. Year

Boys

Girls

7

104

98

8

112

119

9

107

110

10

97

88

11

75

82

12

68

66

A survey of 100 students is to be conducted. Complete the table below to show the number of students from each year and sex who should participate in a stratified sample. Year 7 8 9 10 Work

T SHEE

6.1

11 12

Boys

Girls


179

Choosing a sample Consider how you would choose your sample if you wished to conduct a survey for your next school disco. Use the method in worked example 4 to select the number of boys and girls that should be chosen from each year to do your survey. Step 1. Find out the number of boys and girls enrolled in each year at your school. Step 2. Calculate the percentage of the whole school population in each year for both boys and girls. Step 3. Choose a suitable sample size and calculate the number of boys and girls needed from each year to complete your survey.

1 For each of the following, state whether a census or sample has been used. 1 A school votes to elect a school captain. 2 Five hundred drivers complete a survey on the state of a major highway. 3 Every insurance customer completes a questionnaire when renewing their policy. For each of the following, state the type of sample that has been taken. 4 A computer selects 500 phone numbers. 5 Every 1000th person in the telephone book. 6 Private and business telephone numbers are chosen in proportion to the number of private and business listings. 7 In a school there are 1000 students and 100 teachers. A sample of 20 members of the school is to participate in an Occupational Health and Safety review. If a stratified sample is chosen, how many students should participate? 8 In the Parliament there are 90 Liberals and 60 Labor members. For a committee, 3 Liberal and 3 Labor members are selected. Explain why this is not a stratified sample. 9 For question 8, calculate the correct number of people who should have been selected from each party. 10 Give an advantage that stratified sampling has over random sampling.

180


Bias No doubt you have heard the comment, ‘There are lies, damned lies and statistics’. This implies that we should be wary of statistical figures quoted. Indeed, we should always make informed decisions of our own and not simply accept the mass of statistics that bombards us through the media. Bias can be introduced into statistics by: 1. questionnaire design 2. sampling bias 3. the interpretation of results.

Bias in questionnaire design Consider a survey designed to collect data on opinions relating to culling kangaroo numbers in Australia. The questions may be designed to be emotive in nature. Respondents in these situations feel obliged to show compassion. Posing a question in the form, ‘The kangaroo is identified as a native Australian animal, not found anywhere else in the world. Would you be in favour of culling kangaroos in Australia?’, would almost certainly encourage a negative response. Using a leading question (one which leads the respondent to answer in a particular way) can cause bias to creep into responses. Rephrasing the question in the form, ‘As you know, kangaroos cause massive damage on many farming properties. You’d agree that their numbers need culling, wouldn’t you?’, would encourage a positive response. Using terminology that is unfamiliar to a large proportion of those being surveyed would certainly produce unreliable responses. ‘Do you think we need to cull herbivorous marsupial mammals in Australia?’, would cause most respondents to answer according to their understanding of the terms used. If the survey was conducted by an interviewer, the term could be explained. In the case of a self-administered survey, there would be no indication of whether the question was understood or not.

Sampling bias As discussed previously, an ideal sample should reflect the characteristics of the population. Statistical calculations performed on the sample would then be a reliable indication of the population’s features. Selecting a sample using a non-random method, as discussed earlier, generally tends to introduce an element of bias.


181

Particular responses can be selected from all those received. In collecting information on a local issue, an interviewer on a street corner may record responses from many passers-by. From all the data collected, a sample could be chosen to support the issue, or alternatively another sample could be chosen to refute the same issue. A sample may be selected under abnormal conditions. Consider a survey to determine which lemonade was more popular — Kirks or Schweppes. Collecting data one week when one of the brands was on special at half price would certainly produce misleading results. Data are often collected by radio and television stations via telephone polls. A ‘Yes’ response is recorded on a given phone-in number, while the ‘No’ respondents are asked to ring a different phone-in number. This type of sampling does not produce a representative sample of the population. Only those who are highly motivated tend to ring and there is no monitoring of the number of times a person might call, recording multiple votes. When data are collected from mailing surveys, bias results if the non-response rate is high (even if the selected sample was a random one). The responses received often represent only those with strong views on the subject, while those with more moderate views tend to lack representation in their correct proportion.

Statistical interpretation bias Once the data have been collected, collated and subjected to statistical calculations, bias may still occur in the interpretation of the results. Misleading graphs can be drawn leading to a biased interpretation of the data. Graphical representations of a set of data can give a visual impression of ‘little change’ or ‘major change’ depending on the scales used on the axes. The use of terms such as ‘majority’, ‘almost all’ and ‘most’ are open to interpretation. When we consider that 50.1% ‘for’ and 49.9% ‘against’ represents a ‘majority for’ an issue, the true figures have been hidden behind words with very broad meanings. Although we would probably not learn the real facts, we should be wary of statistical issues quoted in such terms.

Bias in statistics The aim of this investigation is to study statistical data that you suspect to be biased. Conduct a search of newspapers, magazines or any printed material to collect instances of quoted statistics that you believe to be biased. There are occasions when television advertisements quote statistical figures as a result of questionable sampling techniques. For each example, discuss: 1 the purpose of the survey 2 how the data might have been collected 3 the question(s) that may have been asked (try to pose the question(s) in a variety of ways to influence different outcomes) 4 ways in which bias might be introduced 5 variations in interpretation of the data.

182


Biased sampling Discuss the problems that would be caused by each of the following biased samples. 1 A survey is to be conducted to decide the most popular sport in a local community. A sample of 100 people was questioned at a local football match. 2 A music store situated in a shopping centre wants to know the type of music that it should stock. A sample of 100 people was surveyed. The sample was taken from people who passed by the store between 10 and 11 am on a Tuesday. 3 A newspaper conducting a Gallup poll on an election took a sample of 1000 people from Sydney.

Spreadsheets creating misleading graphs Let us practise producing misleading graphs. Consider the data in this table. Year Wages ($m) % increase in wages Profits ($m) % increase in profits

1990

1995

2000

2005

6 25 1 20

9 50 1.5 50

13 44 2.5 66

20 54 5 100

Graph 2

We shall use a spreadsheet to produce misleading graphs based on these data. Graph 1

Graph 3


183

1 Enter the data as indicated in the spreadsheet (see page 182). 2 Graph the data using the Chart Wizard. You should obtain a graph similar to Graph 1. 3 Copy and paste the graph twice within the spreadsheet. 4 Graph 2 gives the impression that the wages are a great deal higher than the profits. This effect was obtained by reducing the horizontal axis. Experiment with shortening the horizontal length and lengthening the vertical axis. 5 In Graph 3 we get the impression that the wages and profits are not very different. This effect was obtained by lengthening the horizontal axis and shortening the vertical axis. Experiment with various combinations. 6 Print out your three graphs and examine their differences. Note that all three graphs have been drawn from the same data using valid scales. A cursory glance leaves us with three different impressions. Clearly, it is important to look carefully at the scales on the axes of graphs. Another method which could be used to change the shape of a graph is to change the scale of the axes. 7 Right click on the axis value, enter the Format axis option, click on the Scale tab, then experiment with changing the scale values on both axes. Techniques such as these are used to create different visual impressions of the same data. 8 Use the data in the table to create a spreadsheet, then produce two graphs depicting the percentage increase in both wages and profits over the years giving the impression that: a the profits of the company have not grown at the expense of wage increases (the percentage increase in wages is similar to the percentage increase in profits) b the company appears to be exploiting its employees (the percentage increase in profits is greater than that for wages).

WORKED Example 5 Discuss why the following selected samples could provide bias in the statistics collected. a In order to determine the extent of unemployment in a community, a committee phoned two households (randomly selected) from each page of the local telephone book during the day. b A newspaper ran a feature article on the use of animals to test cosmetics. A form beneath the article invited responses to the article. THINK

WRITE

a

a Phoning two randomly selected households per page of the telephone directory is possibly a representative sample. However, those without a home phone and those with unlisted numbers could not form part of the sample. An unanswered call during the day would not necessarily imply that the resident was at work.

1

Consider phone book selection.

2

Consider those with no phone contact.

3

Consider the hours of contact.

Continued over page

184


THINK

WRITE

b

b Selecting a sample from a circulated newspaper excludes those who do not have access to the paper. In emotive issues such as these, only those with strong views will bother to respond, so the sample will represent extreme points of view.

1

Consider the newspaper circulation.

2

Consider the urge to respond.

remember Bias can be introduced at each of the following stages: 1. questionnaire design 2. sampling bias 3. interpretation of results.

6C

Bias

1 Rewrite the following questions, removing any elements or words that might contribute to bias in responses. a The poor homeless people, through no fault of their own, experience great hardship during the freezing winter months. Would you contribute to a fund to build a shelter to house our homeless? b Most people think that, since we’ve developed as a nation in our own right and broken many ties with Great Britain, we should adopt our own national flag. You’d agree with this, wouldn’t you? c You’d know that our Australian 50 cent coin is in the shape of a dodecagon, wouldn’t you? d Many in the workforce toil long hours for low wages. By comparison, politicians seem to get life pretty easy when you take into account that they only work for part of the year and they receive all those perks and allowances. You’d agree, wouldn’t you? 2 Rewrite parts a to d in question 1 so that the expected response is reversed. 3 What forms of sampling bias can you identify in the following samples? a Choosing a sample from students on a bus travelling to a sporting venue to answer a 5 questionnaire regarding sporting facilities at their school b Sampling using ‘phone-in’ responses to an issue viewed on a television program c Promoting the results of a mail-response survey when fewer than half the selected sample replied d Comparing the popularity of particular chocolate brands when one brand has a ‘two for the price of one’ special offer e Choosing a Year 7 class and a Year 12 class to gather data relating to the use of the athletics oval after school

WORKED

Example


185

Australian currency

4 Why does this graph produce a biased visual impression? Value of A$ compared with US $1 71c 70c 69c

9 May 11 May 12 May Date

5 Comment on the following statement: ‘University tests have demonstrated that Double-White toothpaste is consistently used by the majority of teenagers and is more effective than most other toothpastes.’ 6 Surveys are conducted on samples to determine the characteristics of the population. Discuss whether the samples selected would provide a reliable indication of the population’s characteristics. Sample Population a Year 11 students Student drivers b Year 12 students Students with part-time jobs c Residents attending a Residents of a suburb neighbourhood watch meeting d Students in the school choir Music students in the school e Cars in a shopping centre car park Models of Holden cars on the road f Males at a football match Popular TV programs g Users of the local library Popular teenage magazines

Bias It is important that a sample is chosen randomly to avoid bias. Consider the following situation. The government wants to improve sporting facilities in Sydney. They decide to survey 1000 people about what facilities they would like to see improved. To do this, they choose the first 1000 people through the gate at a football match at the Sydney Cricket Ground. In this situation it is likely that the results will be biased towards improving facilities for football. It is also unlikely that the survey will be representative of the whole population in terms of equality between men and women, age of the participants and ethnic backgrounds. Questions can also create bias. Consider asking the question, ‘Is football your favourite sport?’ The question invites the response that football is the favourite sport rather than allowing a free choice from a variety of sports by the respondent. Consider each of the following surveys and discuss: a any advantages, disadvantages and possible causes of bias b a way in which a truly representative sample could be obtained.

186


1 Surveying food product choices by interviewing customers of a large supermarket chain as they emerge from the store between 9.00 am and 2.00 pm on a Wednesday. 2 Researching the popularity of a government decision by stopping people at random in a central city mall. 3 Using a telephone survey of 500 people selected at random from the phone book to find if all Australian States should have Daylight Saving Time in summer. 4 A bookseller uses a public library database to survey for the most popular novels over the last three months. 5 An interview survey about violence in sport taken at a rugby league football venue as spectators leave.

Types of data Data can be put into two categories. 1 — Categorical data Categorical data cannot be measured; they can only be put into categories. An example of categorical data is makes of cars. The categories for the data would be all possible makes of cars such as Ford, Holden, Toyota, Mazda etc. Other questions that would lead to categorical data would be things such as: • What is your hair colour? • Who is your favourite musical performer? • What method of transport do you use to get to school? 2 — Quantitative data Quantitative data can be measured. They are data to which we can assign a numerical value. Quantitative data are collected either by measurement or by counting. For example, the data collected by measuring the heights of students are quantitative data. The data collected by counting the ages of students in years are also quantitative data.


187

WORKED Example 6 State whether the following pieces of data are categorical or quantitative. a The value of sales recorded at each branch of a fast-food outlet b The breeds of dog that appear at a dog show THINK

WRITE

a The value of sales at each branch can be measured. b The breeds of dog at a show cannot be measured.

a The value of sales are quantitative data. b The breeds of dog are categorical data.

There are two types of categorical data and two types of quantitative data. Data Categorical Data which are placed in categories; that is, non-numerical form, such as hair colour, type of vehicle, and so on.

Quantitative Data which are in numerical form; such as height, number of children in the family, and so on.

Nominal

Ordinal

Discrete

Continuous

Need sub-groups to complete the description, such as hair colour: blond, brown and so on.

Need a ranking to order the description, such as achievement levels: very high, high, satisfactory and so on.

Counted in exact values, such as goals scored in a football match, shoe size and so on. Values are often, but not always, whole numbers.

Measured in a continuous decimal scale, such as mass, temperature, length and so on.

WORKED Example 7

Classify each of the following data using two selections from the following descriptive words: categorical, quantitative, nominal, ordinal, discrete and continuous. a the number of students absent from school b the types of vehicle using a certain road c the various pizza sizes available at a local takeaway d the room temperature at various times during a particular day THINK WRITE a 1 Determine whether the data are a The data are quantitative as absences are categorical or quantitative. represented by a number. The data are discrete as the number of absences 2 Determine whether the data are discrete or continuous. can be counted and is an exact value. Continued over page

188


THINK b 1 Determine whether the data are categorical or quantitative. 2 Determine whether the data are nominal or ordinal.

WRITE b The data are categorical as the types of vehicle need to be placed in non-numerical categories. The data are nominal as there is no ranking or order involved.

c

Determine whether the data are categorical or quantitative. Determine whether the data are nominal or ordinal.

c The data are categorical as the pizza sizes need to be ranked in order ranging from small to family. The data are ordinal as pizzas are ranked in order of size.

Determine whether the data are categorical or quantitative. Determine whether the data are discrete or continuous.

d The data are quantitative as room temperature is represented by a number. The data are continuous as temperature can assume any value and measurement is involved.

1 2

d

1 2

remember 1. Data can be classified as either: (a) categorical — the data are in categories, or (b) quantitative — the data can be either measured or counted. 2. Categorical data can be either: (a) nominal — where the order of the categories is not important, or (b) ordinal — the categories have a logical order. 3. Quantitative data can be either: (a) discrete — the data can take only certain values, usually whole numbers, or (b) continuous — the data can take any value depending on the degree of accuracy.

6D WORKED

Example

6

Types of data

1 State whether the data collected in each of the following situations would be categorical or quantitative. a The number of matches in each box is counted for a large sample of boxes. b The sex of respondents to a questionnaire is recorded as either M or F. c A fisheries inspector records the lengths of 40 cod. d The occurrence of hot, warm, mild and cool weather for each day in January is recorded. e The actual temperature for each day in January is recorded. f Cinema critics are asked to judge a film by awarding it a rating from one to five stars.


189

2 State whether the categorical data formed by each of the following situations are nominal or ordinal. a On a school report students are ranked as Unsatisfactory, Satisfactory, Above average or Outstanding. b The day of the week that a business has the most customers is recorded. c Visitors to a museum are recorded as being either male or female. d The colour of each traffic light on a journey is recorded. e The make of each television in an electronics store is recorded. 3 State whether the quantitative data formed by each of the following situations are discrete or continuous. a The heights of 60 tomato plants at a plant nursery b The number of jelly beans in each of 50 packets c The time taken for each student in a class of six-year-olds to tie their shoelaces d The petrol consumption rate of a large sample of cars e The IQ (intelligence quotient) of each student in a class WORKED

Example

7

4 Classify each of the following data using two words selected from the following descriptive words: categorical, quantitative, nominal, ordinal, discrete and continuous. a The population of your town or city b The types of motorbike in a parking lot c The heights of people in an identification line-up d The masses of babies in a group e The languages spoken at home by students in your class f The time spent watching TV g The number of children in the families in your suburb h The air pressure in your car’s tyres i The number of puppies in a litter j The types of radio program listened to by teenagers k The times for swimming 50 metres l The quantity of fish caught in a net m The number of CDs you own n The types of shops in a shopping centre o The football competition ladder at the end of each round p The lifetime of torch batteries q The number of people attending a rock concert r Exam grades s The types of magazine sold at a newsagency t Hotel accommodation rating 5 For each of the following, state if the data are categorical or quantitative. If quantitative, state if the data are discrete or continuous. a The number of students in each class at your school b The teams people support at a football match c The brands of peanut butter sold at a supermarket d The heights of people in your class e The interest rate charged by each bank f A person’s pulse rate


6 An opinion poll was conducted. A thousand people were given the statement ‘Euthanasia should be legalised’. Each person was offered five responses: strongly agree, agree, unsure, disagree and strongly disagree. Describe the data type in this example. 7 A teacher marks her students’ work with a grade A, B, C, D, or E. Describe the data type used. 8 A teacher marks his students’ work using a mark out of 100. Describe the data type used. 9 multiple choice The number of people who are using a particular bus service are counted over a two week period. The data formed by this survey would be an example of: A categorical and ordinal data B cateorical and nominal data C quantitative and discrete data D quantitative and continuous data 10 The following graph shows the number of days of each weather type for the Gold Coast in January. Number of days in January

14 12 10 8 6 4 2 0 H

ot

m ild ar M W Weather

Co

ol

Describe the data in this example. 11 The graph at right shows a girl’s height each year for 10 years. Describe the data in this example.

180 Height (cm)

190

160 140 120 100 5 6 7 8 9 10 11 12 13 14 15 Age


191

2 1 ‘The rich should pay more in tax, to allow the poorer families to have access to better services, wouldn’t you agree?’ Explain why this question is biased. 2 Rewrite the above question to eliminate bias. 3 A television program has a phone-in survey asking the question, ‘Should criminals receive tougher sentences?’ Explain why telephone phone-in surveys have a sampling bias. 4 Explain a better method of getting public opinion on this issue. For each of the following, state if the data are categorical or quantitative. If categorical, state whether the data are nominal or ordinal, and if quantitative, state if they are continuous or discrete. 5 Customers in a video shop vote for their favourite movie. 6 Customers in a video shop have records kept on the number of movies they hire each year. 7 The video shop keeps records of the number of times each movie has been hired. 8 The video shop keeps records of the length of each movie. 9 Give an example of categorical data which is ordinal. 10 Give an example of quantitative data which is discrete.

Estimating populations Statistics are used to estimate populations that are too difficult to count accurately. An example of this is the number of fish in a lake. The method used to estimate the population is the ‘capture–recapture’ technique. In this technique, a sample of the population is captured, tagged and then released back into the population. A second sample is then captured. The percentage of the second sample that have been previously tagged (that is, recaptured) becomes an estimate of the percentage of the entire population that has been tagged.

192


Estimating a population Make a large collection of marbles, jelly beans, matches or toothpicks. 1 Select a sample of these items and identify/tag them with a dab of liquid paper. 2 Put those selected back in with the others and mix them thoroughly. 3 Select another sample and see how many are tagged. 4 Estimate the total number. 5 Put those selected back and repeat the process to see if you get a similar answer. 6 Count the number of items to see how accurate your estimates were.

WORKED Example 8 A group of marine biologists want to estimate the fish population in a lake. To do this, they catch 100 fish, tag them and release them back into the lake. The next day, they catch 200 fish and find that 10 of them have been previously tagged. What would be their estimate of the population? THINK 1

2

3

WRITE

Calculate the percentage of the second sample that has been tagged. Use this percentage to calculate what 100% of the population would be. Give a written answer.

10 - × 100% Percentage tagged = -------200 Percentage tagged = 5%

5% of population = 100 so 1% of population = 20 and 100% of population = 2000 Their estimate of the fish population in the lake is 2000.

remember 1. The capture–recapture technique is used to estimate populations that cannot be accurately counted with ease. 2. A sample of the population is captured, tagged and released back into the population. 3. A second sample is then captured and the percentage tagged is an estimate of the percentage of the entire population that has been tagged. 4. The unitary method of percentages is then used to estimate the entire population.


6E

193

Estimating populations

1 The fish population of a river is to be estimated. A sample of 400 fish are caught, tagged and released. The next day another sample of 400 fish are caught and 40 of them 8 have tags. Estimate the fish population of the river.

WORKED

Example

2 A colony of bats live near a school. Wildlife officers try to estimate the bat population by catching 60 bats and tagging them. These bats are then released and another 60 are caught, 9 of which had tags. Estimate the size of the bat population living near the school. 3 A river’s fish population is to be estimated. On one day 1000 fish are caught, tagged and released. The next day another 1000 fish are caught. Estimate the population of the river if in the second sample of fish: a 100 had tags b 40 had tags c 273 had tags. 4 A certain fish population is said to be endangered if the population falls below 15 000. A sample of 1000 fish are caught, tagged and released. The next day another sample of 1200 fish are caught, 60 of which had tags. Is the fish population endangered? 5 To estimate the fish population of a lake, 300 fish were caught. These 300 fish (150 trout, 100 bream and 50 perch) were tagged and released. A second sample of fish were then caught. Of 100 trout, 24 had tags; of 100 bream, 20 had tags; and of 100 perch, 8 had tags. a Estimate the number of trout in the lake. b Estimate the number of bream in the lake. c Estimate the number of perch in the lake.

T SHEE

Work

6 The kangaroo population in a national park is to be estimated. On one day, 100 kangaroos were caught and tagged before being released. (Note: For each sample taken, the kangaroos are released after the number with tags is counted.) a The next day 100 were caught, 12 of which had tags. Estimate the population. b The following day another estimate was done. This time 200 were caught and 20 had tags. Estimate the population again. c A third estimate was done by catching 150 and this time 17 had tags. What will the third estimate for the population be? d For a report, the average of the three estimates is taken. Calculate this average.

6.2

194


summary Data collection • A statistical investigation can be done by either census or sample. • A census is when an entire population takes part in the investigation. • A sample is when a small group takes part in the investigation and the results are taken to be representative of the whole group. • There are three types of sample. 1. Random sample — chance is the only factor in deciding who participates. 2. Stratified sample — the sample taken is chosen so that it has the same characteristics as the whole population. 3. Systematic sample — there is a method for deciding who participates in the sample.

Population characteristics • A stratified sample can be used to ensure that the characteristics of your sample match the characteristics of the whole population. • To select a stratified sample: 1. Write each number in each stratum as a fraction of the whole population. 2. Multiply by the size of the sample that you wish to take. 3. Round off the answer for each stratum to the nearest whole number. • Sometimes this method increases the sample size, but this may be necessary to get the best representation possible in your sample.

Bias • Bias occurs when the results of an investigation are skewed to one side. This can occur because of: 1. A poorly worded question that can lead the responder into a response favouring one side. 2. A poorly chosen sample. Those participating in the investigation may not represent the whole population and be more inclined to a certain point of view. 3. Misinterpretation of results. This can occur when a graph is drawn to give a certain impression.

Classification of data • Data can be classified as being categorical or quantitative. • Categorical data are data that are non-numerical. For example, a survey of car types is not numerical. • Quantitative data are data that can be either counted or measured. For example, a survey of the daily temperature is quantitative. • Quantitative data can be either discrete or continuous. • Discrete data can take only certain values such as whole numbers. • Continuous data can take any value within a certain range.

Estimating populations Populations that can’t be accurately counted with ease are estimated by using the capture–recapture technique.


195

CHAPTER review 1 For each of the following statistical investigations, state whether a census or a sample has been used. a The average price of petrol in Sydney was estimated by averaging the price at 40 petrol stations. b The Australian Bureau of Statistics has every household in Australia complete an information form once every five years. c The performance of a cricketer is measured by looking at his performance in every match he has played. d Public opinion on an issue is sought by a telephone poll of 2000 homes.

6A

2 Name and describe three different methods for selecting a sample.

6A

3 Which method of sampling has been used for each of the following? a The quality-control department of a tyre manufacturing company road tests every 50th tyre that comes off the production line. b To select the students to participate in a survey, a spreadsheet random number generator selects the roll numbers of 50 students. c An equal number of men and women are chosen to participate in a survey on fashion.

6A

4 Use a random number generator to select 10 numbers between 1 and 1000.

6A

5 The table at right shows the number of students in each year of school. In a survey of the school population, how many students from each year should be chosen, if a sample of 60 is selected using a stratified sample?

6B

6 Bias can be introduced into statistics through: a questionnaire design b sample selection c interpretation of statistical results. Discuss how bias could be a result of techniques in the above three areas.

Year

No. of students

7

212

8

200

9

189

10

175

11

133

12

124

7 State whether each of the following data types are categorical or quantitative. a The television program that people watch at 7:00 pm b The number of pets in each household c The amount of water consumed by athletes in a marathon run d The average distance that students live from school e The mode of transport used between home and school

6C

6D

196 6D


8 For each of the quantitative data types below, determine if the data are discrete or continuous. a The dress sizes of Year 11 girls b The volume of backyard swimming pools c The amount of water used in households d The number of viewers of a particular television program e The amount of time Year 11 students spent studying

6E

9 To estimate the fish population of a lake, 100 fish are caught, tagged and released. The next day another 100 are caught and it is noted that 5 have tags. Estimate the population of the lake.

6E

10 Kimberley has a worm farm. To estimate the population of her farm, she catches 150 worms and tags them before releasing them. The next day, she catches 120 worms and finds that 24 of them have tags. Estimate the population of the worm farm.

6E

11 A sample of 200 fish are caught, tagged and released back into the population. Later Barry, Viet and Mustafa each catch a sample of fish. Barry caught 40 fish and 3 had tags. Viet caught 75 fish and 9 had tags. Mustafa caught 55 fish and 7 had tags. a Find the estimate of the population that each would have calculated. b Give an estimate for the population, based on all three samples.

Practice examination questions 1 multiple choice Which of the following is an example of a census? A A newspaper conducts an opinion poll of 2000 people. B A product survey of 1000 homes to determine what brand of washing powder is used C Every 200th jar of Vegemite is tested to see if it is the correct mass. D A federal election 2 multiple choice Which of the following is an example of a random sample? A The first 50 students to arrive at school take a survey. B Fifty students’ names are drawn from a hat and those drawn take the survey. C Ten students from each year of the school are asked to complete a survey. D One class in the school is asked to complete the survey.


197

3 multiple choice Which of the following is an example of categorical data? A The type of car that is in each home B The number of cars in each home C The distance travelled by a person’s car in a one year period D The amount of money spent on petrol in a one year period 4 multiple choice Which of the following is an example of continuous data? A The type of car that is in each home B The number of cars in each home C The distance travelled by a person’s car in a one year period D The amount of money spent on petrol in a one year period 5 Carolyn is a marine biologist. She spends the day on a boat and 500 fish are netted. Carolyn notes the types of fish netted. There are 173 blackfish, 219 drummer and 108 mullet. a Are Carolyn’s data categorical or quantitative? b The fish are tagged and released back into the school from which they were caught. Another 250 are then caught and it is noted that 63 have tags. Estimate the population of the school.

198


6 A paint company says that 1 litre of paint can paint an area of 4 m2. To test this, every 100th can is tested to see that it will cover at least 4 m2. a Are the data categorical or quantitative? If they are quantitative, are the data discrete or continuous? b What type of sample has been used? c The paint is mixed in one of five machines. Every can tested therefore comes from the same machine. A quality control officer then says that the data are biased. What is meant by the term bias? d Explain what could be done to prevent the data from being biased.

CHAPTER

test yourself

6

Modelling linear relationships

7 syllabus reference Algebraic modelling 2 • Modelling linear relationships

In this chapter 7A Graphing linear functions 7B Gradient and intercept 7C Drawing graphs using gradient and intercept 7D Graphing variations 7E Step and piecewise functions 7F Simultaneous equations

areyou

READY?

Are you ready?


7.1

Recognising linear functions

1 Which of the functions below are linear functions? a y = x2

7.2

x c y = --2

b y = 2x

2 d y = --x

Gradient of a straight line

2 Find the gradient of each of the following. a

b

y

c

y

y 6

2

x

−1

3 x 2

−6

7.3

Substitution

7.4

Graphing linear equations

7.5

3 Complete the following. a Given that y = 5 – 2x, find the value of y when x = 6. b Given that y = 4x + 7, find the value of y when x = –3. 10 – 2x c Given that y = ------------------ , find the value of y when x = 0. 5

4 Draw the graph of: a y=x+1

b y = 2x – 4

c y = 6 – 2x.


5 Solve each of the following equations. a 2x – 8 = 0 b 12 – 4x = 0

c 5x – 2 = 0

x

Chapter 7 Modelling linear relationships

201

Graphing linear functions Imagine a car travelling at a constant speed D of 60 km/h. The graph at right compares 600 500 the distance travelled with time. 400 This graph can be given by the 300 200 relation D = 60t. This relation is an 100 0 example of a function. A function is 0 1 2 3 4 5 6 7 8 9 10 t a rule with two variables. In the above example time, t, is the independent variable. This is the variable for which we can substitute any value. Distance, D, is the dependent variable as its value depends on the value substituted for t. A linear function is a graph that, when drawn, is represented by a straight line. Linear functions are drawn from a table of values. The independent variable is graphed on the horizontal axis and the dependent variable is graphed on the vertical axis.

WORKED Example 1 The table below shows the amount of money earned by a wage earner. Hours (H)

10

20

30

40

50

Wage (W)

85

170

255

340

425

Draw the graph of wage, W, against hours, H. THINK 1

2 3

DRAW

Draw the graph with H on the horizontal axis and W on the vertical axis. Plot the points (10, 85) (20, 170) (30, 255) (40, 340) and (50, 425). Join the points with a straight line.

W 450 400 350 300 250 200 150 100 50 0 0

10 20 30 40 50 H

In many examples we are required to draw a graph from an algebraic rule. In such an example we need to create our own table. To do this, we can choose any sensible value to use for the independent variable.

WORKED Example 2 The conversion of Australian dollars, A, to US dollars, U, can be given by the rule U = 0.8A. Draw the graph of this function. THINK 1

WRITE/DRAW

Draw a table choosing several values to substitute for the independent variable, A. Continued over page

202


THINK 2

3

WRITE

Calculate the value of U for each value of A in the table.

Draw the axes, plot the points generated and join each point with a straight line, extending the line as required.

A

10

20

30

40

50

U

8

16

24

32

40

U

80 70 60 50 40 30 20 10

0 10 20 30 40 50 60 70 80 90 100 A

Graphics Calculator tip! Graphing linear equations The Casio graphics calculator can be used to draw both the table of values and the graph of a linear equation. Consider worked example 2. 1. From the MENU select GRAPH.

2. Delete any existing functions and enter the rule. The calculator always uses Y as the dependent variable and X as the independent variable. Enter Y1 = 0.8X. Be sure to use the X,q,T button for X. Finish by pressing EXE . 3. Next we need to rule up our coordinate axes. This is done using the V-Window function. Press SHIFT F3 . This sets the minimum and maximum values for both X and Y as well as the increments on each axis. Enter the settings shown on the screen at right, which replicate the axes that are drawn in the worked example. 4. Press EXE to return to the previous screen and then F6 to draw the graph.

5. To see the points on the graph use the Trace function. Press SHIFT F1 and use your arrow keys to see the points drawn.


203

When applying a function we need to understand the idea of: input

process

output.

The independent variable is the input, a calculation is made which is the process and the output is the value of the dependent variable. An independent variable is substituted (input), a calculation is made (process) according to the rule defined by the function and the dependent variable (output) is the result.

WORKED Example 3 A preschool has hired an entertainment group to entertain their children at a concert. The cost of staging the concert is given by the function C = 80 + 3n, where C is the cost and n is the number of children attending the concert. Draw the graph of this function. THINK

WRITE/DRAW

1

Draw a table choosing five values for n to substitute.

2

Calculate the values of C for each value of n chosen.

3

Draw the axes, plot the points generated and join with a straight line.

n

0

50

100

150

200

C

80

230

380

530

680

C 800 700 600 500 400 300 200 100 0 0

50 100 150 200 n

remember 1. A function is a rule for a calculation that consists of an independent and dependent variable. 2. Values are substituted for the independent variable and a value for the dependent variable is generated. 3. A linear function is represented by a straight line when graphed. 4. When graphing a linear function, the independent variable is shown on the horizontal axis and the dependent variable on the vertical axis. 5. To graph a linear function we draw up a table of values, plot the points generated by that table then join these points with a straight line.

204


7A WORKED

Example

1

SkillS

HEET

7.1

Recognising linear functions EXCE

et

reads L Sp he

Graphing linear functions

1 The table below shows the amount of money, M, earned for delivering a number of pamphlets, P, to letterboxes. P

1000

2000

3000

4000

5000

M

50

100

150

200

250

Draw the graph of this function.

Graph paper

2 Use the graph drawn in question 1 to find the amount of money earned by a person delivering: a 8000 pamphlets b 9500 pamphlets.

GC p

sio

am – rogr Ca

3 Australian dollars can be converted to Japanese yen using the algebraic rule Y = 80A. To draw a conversion graph, the table below is used.

Linear

GC p

am – rogr TI

Linear

am – rogr Ca

sio

GC p

100

200

300

400

500

Y

8000

16 000

24 000

32 000

40 000

Draw a graph converting Australian dollars to Japanese yen.

Myrule WORKED

Example

2

GC p

am – rogr TI

A

Myrule

4 The cost of an international telephone call can be given by the rule C = 1.5t, where C is the cost of the call and t is the length of the call in minutes. Draw a graph showing the cost of a telephone call. 5 Use the graph from question 4 to calculate the cost of a telephone call that lasts for: a 17 minutes b 45 minutes. 6 The distance, d, travelled by a cyclist can be given by the algebraic rule d = 15t, where t is the time in hours that the cyclist has been riding. Draw a graph showing the distance travelled by the cyclist against time.

WORKED

Example

3

7 The cost, C, of a taxi journey can be given by the rule C = 4 + 1.5d, where d is the distance of the journey in kilometres. Draw a graph showing the cost of a taxi journey. 8 A tree bought as a seedling is 80 cm tall. It then grows at an average rate of 12 cm per year. Draw a graph that will show the height of the tree each year. 9 Casey has a job that pays $10 per hour. Draw a graph that will show the money earned against hours worked. 10 It costs Bill $1850 per week to operate his business developing photographs. Bill charges $8 to develop a roll of film. Draw a graph that shows the profit or loss he makes against the number of rolls of film developed.


205

Graph of height versus age Not all graphs can be drawn as a straight line. Consider the case of height and age. 1 Find a person of each age from 1–20. Measure their height and plot their age and height as a pair of coordinates. 2 Draw a line of best fit for the points plotted. 3 The graph will flatten where people stop growing and so does not continue to rise indefinitely. Suggest a point at which this graph should stop.

Gradient and intercept Consider the graph of a car that is D travelling at 60 km/h. Earlier we drew 600 500 the graph of this as a linear function. 400 Two points on this graph are (1, 60) 300 200 and (2, 120). From the graph we can 100 0 see that for a one unit increase in the 0 1 2 3 4 5 6 7 8 9 10 t independent variable, there is a 60 unit increase in the dependent variable. For this function we can say that the gradient is 60. The gradient (m) is the rate of change in the dependent variable for a one unit increase in the independent variable. A simple formula that can be used to calculate gradient is: vertical change in position m = --------------------------------------------------------------------C = 2 + 0.5d horizontal change in position C (8, 6) 6 Using this formula, the gradient can be 5 3 units calculated by measurement from a graph 4 (2, 3) 3 by choosing any two points on the graph. 2 6 units The graph at right shows the function 1 0 C = 2 + 0.5d. 0 1 2 3 4 5 6 7 8 9 10 d On the graph, the two points (2, 3) and (8, 6) are marked. Between these two points the vertical rise = 3 and the horizontal run = 6. Using the gradient formula: gradient = 3--gradient =

6 1 --2

206


WORKED Example 4

D 20 18 16 14 12 10 8 6 4 2 0

For the linear function drawn at right, calculate the gradient.

THINK 1 2

WRITE

Choose two points on the graph: (1, 4) and (5, 20) for example. Measure the vertical rise and the horizontal run.

D = 4t

0 D 20 18 16 14 12 10 8 6 (1, 4) 4 2 0 0 1

1

2

3

4

5

6 t

D = 4t (5, 20)

16

4 2

3

4

5

6 t

3

Write the gradient formula.

4

Substitute for the rise and the run.

vertical change in position gradient = --------------------------------------------------------------------horizontal change in position -----gradient = 16

5

Calculate the gradient.

gradient = 4

4

A function with a positive gradient is called an increasing function. That means that the value of the dependent variable increases as the value of the independent variable increases. A decreasing function has a negative gradient. In such cases when calculating the gradient, we take the vertical rise to be negative. In a decreasing function, the value of the dependent variable decreases as the value of the independent variable increases.

WORKED Example 5

y

For the function drawn at right calculate the gradient.

5 4 3 2 1 –5 –4 –3 –2 –1 0 –1 –2 –3 –4 –5

THINK 1

Choose two points on the graph. In this case we choose (0, 4) and (4, 0).

WRITE

1 2 3 4 5

x

y=4–x


THINK 2

WRITE

Measure the vertical rise and the horizontal run.

y 5 (0, 4) 4 4 3 2 1 –5 –4 –3 –2 –1 0 –1 –2 –3 –4 –5

3

Write the gradient formula.

4

Substitute for the rise and the run. Calculate the gradient.

5

207

–4 (4, 0) 1 2 3 4 5

x

y=4–x

vertical change in position gradient = --------------------------------------------------------------------horizontal change in position gradient = –-----44 gradient = −1

The gradient of −1 in the example above means that for every one-unit increase in x, there is a one unit decrease in y. In the example above, the graph cuts the y-axis at 4. Therefore, for this function the y-intercept is 4. Consider worked example 3. Here the cost of hiring the C entertainment group was given by the function C = 80 + 3n. 800 700 In this example, the intercept on the vertical axis is 80; 600 500 that is, it costs $80 to hire the entertainment group without 400 any children attending the concert. The $80 is a fixed cost. 300 200 100 0

0

50 100 150 200 n

WORKED Example 6 The table below shows the cost of running an excursion for a given number of students. No. of students Cost

20

40

60

80

100

$200

$300

$400

$500

$600

a Draw a graph of the cost of this excursion. b Calculate the gradient and explain its meaning in this context. c Use your graph to find the intercept on the vertical axis and explain its meaning in this context. Continued over page

208


THINK

WRITE/DRAW

a

a

Draw a set of axes and plot the points given.

Cost ($)

1

600 550 500 450 400 350 300 250 200 150 100 50 0 0

Join with a straight line.

1

Choose two points on the graph and measure the vertical change in position and horizontal change in position.

b

Cost ($)

b

2

600 550 500 450 400 350 300 250 200 150 100 50 0

100 20

0

2

Calculate the gradient.

20 40 60 80 100 Number of students

20 40 60 80 100 Number of students

vertical change in position gradient = --------------------------------------------------------------------horizontal change in position --------gradient = 100 20

c

3

The gradient is the increased cost of the excursion per student.

1

Find the point where the graph cuts the vertical axis. The intercept is the fixed cost of running an excursion without considering the number of students.

2

gradient = 5 A gradient of 5 means that the cost of the excursion increases by $5 for every student who attends. c Intercept = 100 The excursion has a fixed cost of $100, meaning it would cost $100 even if no children attended.


209

remember 1. The gradient is the increase in the dependent variable for every one unit increase in the independent variable. 2. The gradient is denoted m, and is found using the formula: vertical change in position m = --------------------------------------------------------------------horizontal change in position 3. A positive gradient occurs when the value of the dependent variable increases as the value of the independent variable increases. 4. A negative gradient occurs when the value of the dependent variable decreases as the independent variable increases. 5. The intercept on the vertical axis gives us the value of the dependent variable when the independent variable is equal to zero.

7B

Gradient and intercept

1 For the functions below, find the gradient. a y 4

WORKED

01 23 45 6 x

5

c

d

y 10 9 8 7 6 5 4 3 2 1 0

L Spre XCE ad

Gradient 01 23 45

x Cabri Geo

y 10 9 8 7 6 5 4 3 2 1 0

Cabri Geo

ry met

01 23 4 x

Gradient

Linear graph intercepts 0 1 2 3 4 5 6 7 8 9 10 x

2 multiple choice Which of the functions below has a negative gradient? A B y y

x

Graph paper

x

sheet

L Spre XCE ad

E

Example

Gradient of a straight line

ry met

WORKED

SkillS

sheet

100 90 80 70 60 50 40 30 20 10 0

10 9 8 7 6 5 4 3 2 1 0

7.2

y

HEET

b

E

Example

210


C

D

y

y

x

WORKED

Example

x

3 The table below shows the payment made to a person on a newspaper delivery round.

6

Deliveries

200

400

600

800

1000

Payment

120

180

240

300

360

a Draw the graph of the function. b Find the gradient of the function. c Find the intercept on the vertical axis. 4 The table below shows the profit or loss made by a cinema for showing a movie. No. of people

Profit

20

−60

50

0

100

100

150

200

200

300

a Draw the graph of the function. b Find the gradient of the function. Explain the meaning of the gradient in this context. c Find the intercept on the vertical axis. Explain its meaning in this context. 5 A function is given by the rule y = 5x − 4. a Copy and complete the table below. x

0

1

2

3

y b Draw the graph of this function. c Find the gradient and intercept of this function.


211

Drawing graphs using gradient and intercept Most linear functions are represented on a number plane. Consider the graph of y = 2x + 1 drawn at right. This function has a gradient of 2. The intercept on the vertical axis (called the y-intercept) is 1. Comparing the gradient and y-intercept with the function, we can see that the number with x (called the coefficient of x) is 2 (the gradient) and we then add 1 (y-intercept) to complete the function. Any linear function can be written in the form y = mx + b , where m = gradient and b = y-intercept.

y 5 4 3 2 1 –5 –4 –3 –2 –1 0 –1 –2 –3 –4 –5

y = 2x + 1

1 2 3 4 5

x

WORKED Example 7 Find the gradient and y-intercept of: a y = 3x − 4 b y = 7 − 2x. THINK

WRITE

a

a

gradient = 3 y-intercept = −4

b

gradient = −2

1 2

b

1 2

The gradient is the coefficient of x (3). The y-intercept is the constant term (−4). The gradient is the coefficient of x (−2). The y-intercept is the constant term (7).

y-intercept = 7

We can use the gradient and y-intercept to draw the graph of a function in the form y = mx + b. By plotting the y-intercept we are able to use the gradient to plot other points. For example, a gradient of 2 means a rise of 2 units for a 1 unit increase in x. Therefore, from the y-intercept we count a rise of 2 units and a run of 1 unit to plot the next point. It is a useful check to repeat this process from the next point plotted. The points plotted can then be joined by a straight line that is the graph of the function.

WORKED Example 8

Draw the graph of y = 3x − 2. THINK 1 2 3 4

Find the gradient (3). Find the y-intercept (−2). Mark the y-intercept on the axis. Count a rise of 3 and a run of 1 to mark the point (1, 1).

WRITE gradient = 3 y-intercept = −2

Continued over page

212


THINK 5

WRITE

From (1, 1) count a rise of 3 and a run of 1 to mark the point (2, 4).

y 5 4 3 2 1 –5 –4 –3 –2 –1 0 –1 –2 –3 –4 –5

6

Join these points with a straight line.

y

1 2 3 4 5

x

y = 3x – 2

5 4 3 2 1 –5 –4 –3 –2 –1 0 –1 –2 –3 –4 –5

1 2 3 4 5

x

If the gradient is a fraction, the numerator indicates the vertical change in position and the denominator the horizontal change in position. The method of drawing the graph then remains unchanged.

WORKED Example 9

Sketch the graph of y = 2--- x − 2. 3

THINK 1 2 3 4 5

Find the gradient ( 2--- ). 3 Find the y-intercept (−2). Mark the y-intercept on the axis. Count a rise of 2 and a run of 3 to mark the point (3, 0). From (3, 0) count a rise of 2 and a run of 3 to mark the point (6, 2).

WRITE gradient = 2--3 y-intercept = −2

y 5 4 3 2 1 –4 –3 –2 –1 0 –1 –2 –3 –4 –5

1 2 3 4 5 6

x


THINK 6

213

WRITE


y 5 4 3 2 1

y = 2–3 x – 2

–4 –3 –2 –1 0 –1 –2 –3 –4 –5

1 2 3 4 5 6

x

When sketching functions with a negative gradient we need to remember to treat the rise as negative; that is, the function decreases.

WORKED Example 10

Sketch the function y = 3 − 2x. THINK 1 2 3 4 5

Find the gradient (−2). Find the y-intercept (3). Mark the y-intercept on the axis. Count a rise of −2 and a run of 1 to mark the point (1, 1). From (1, 1) count a rise of −2 and a run of 1 to mark the point (2, −1).

WRITE gradient = −2 y-intercept = 3

y 5 4 3 2 1 –5 –4 –3 –2 –1 0 –1 –2 –3 –4 –5

6


1 2 3 4 5

x

1 2 3 4 5

x

y 5 4 3 2 1 –5 –4 –3 –2 –1 0 –1 –2 –3 –4 –5

y = 3 – 2x

214


remember 1. A function is written in the form: y = mx + b where m equals the gradient and b equals the y-intercept. 2. A function can be graphed when in this form by plotting the y-intercept, then using the gradient to plot two other points, which can then be joined with a straight line.

7C EXCE

et

reads L Sp he

Equation of a straight line EXCE

et

reads L Sp he

WORKED

Example

7

WORKED

Example

8

Linear graphs WORKED

EXCE

et

reads L Sp he

Example

Cabr

2 Sketch the function y = 2x − 3. 3 Sketch the functions: a y = 2x + 1

b y = 3x − 6

c y = 5x.

4 Sketch the function y = 1--- x + 2. 2

5 Sketch the graph for each of the functions below. a y = 3--- x − 1

b y = 1--- x

4

WORKED

Example

10 Linear graphs

1 For each of the functions below, state the gradient and the y-intercept. a y = 2x + 2 b y = 3x − 8 c y = 2 − 4x x d y = 3--- x + 3 e y = --- + 1 f y = 3 − 3--- x 4 2 2

9

Graph paper omet i Ge ry

Drawing graphs using gradient and intercept

c y = 3--- x − 4

3

2

6 Sketch the function y = 4 − 3x. 7 Sketch the graphs of: a y = 6 − 3x

b y = −2x − 3

c y = − 1--- x + 4 2

8 multiple choice Which of the following could be the graph of y = − 1--- x + 1? 2 A B y y

x

C

x

D

y

x

y

x


9 multiple choice

215

y

The equation of the graph drawn at right could be: A y = 2x − 1 B y = 2x + 1 C y = --1- x − 1 D y = --1- x + 1 2

2

10 Write and draw an example of a linear function with: a a positive gradient b a negative gradient c a positive y-intercept d a y-intercept of 0 e a negative gradient and negative y-intercept f a gradient of 0 g a positive gradient and negative y-intercept. 11 Write an equation that could fit the following sketches. a b y y

x

−1

c

y

x

x

1 1 For the function y = 2x − 3, state the gradient. 2 For the function y = 2x − 3, state the y-intercept. 3 For the function y = 5 − 3x, state the gradient. 4 For the function y = 5 − 3x, state the y-intercept. 5 Write the equation of a linear function with a gradient of 3 and a y-intercept of 6. 6 Write the equation of a linear function with a gradient of −2 and a y-intercept of 4. 7 Write the equation of a linear function with a gradient of

x

1 --2

and a y-intercept of −3.

8 Draw an example of a linear function with a positive gradient. 9 Draw an example of a linear function with a negative gradient. 10 Draw an example of a linear function with a y-intercept of 0.

T SHEE

Work

2

7.1

216


Graphing variations A variation occurs when one quantity is proportional to another. Consider the following variation problem. The number of cars produced on an assembly line varies directly with the number of workers employed on the line. Twenty workers can produce 30 cars per week. From this information, we can determine that the number of cars produced each week will be 1.5 times the number of workers employed on the assembly line. Using this, we can draw the table below. No. of workers (N)

10

20

30

40

50

No. of cars produced (C)

15

30

45

60

75

These figures are plotted on the axes at right. In any example where one quantity varies directly with another, the graph that is drawn will be a linear function through the origin (0, 0). To draw the function, we need to know only one other point on the graph. This is known as a direct linear variation.

C 160 140 120 100 80 60 40 20 0 0

20 40 60 80 100 N

WORKED Example 11 The distance travelled by a car is directly proportional to the speed at which it is travelling. If the car travels 225 km in 3 hours, draw a graph of distance travelled against time. THINK

2 3

Draw a set of axes showing time on the horizontal axis and distance on the vertical axis. Plot the points (0, 0) and (3, 225). Join them with a straight line. Distance

1

WRITE

800 700 600 500 400 300 200 100 0 0 1 2 3 4 5 6 7 8 9 10 Time

If we examine the gradient of a variation function, we see that the gradient is equal to the constant of variation. For example in worked example 11, the gradient is 75. This is the speed at which the car is travelling. Any variation can be graphed using the form y = ax where a, the gradient, is also the constant of variation.


217

WORKED Example 12

It is known that y is directly proportional to x. When x = 30, y = 12. a Draw the graph of y against x. b What is the gradient of the graph? c Write an equation linking y and x. THINK

WRITE

a Draw a straight line graph through (0, 0) and (30, 12).

a y 40 35 30 25 20 15 10 5 0 0

vertical change in position Gradient = --------------------------------------------------------------------- . horizontal change in position 2 Simplify. c The equation is in the form y = ax, where a is the gradient. b

1

20 40 60 80 100

b Gradient =

x

12 -----30

Gradient = 0.4 c y = 0.4x

remember 1. When two quantities vary directly with each other, the variation can be graphed as a linear function. 2. To graph the function we need to know only one point on the graph, together with (0, 0). A straight line is then drawn through these two points. 3. The gradient of the function is the constant of variation. Hence, the variation is graphed using a linear function that can be written in the form y = ax.

7D

Graphing variations

Example

Graphing linear equations

SkillS

HEET

3 It is known that y varies directly with x. When x = 5, y = 40. Example a Draw the graph of y against x. 12 b What is the gradient of the graph? c Write an equation linking y and x. WORKED

SkillS

HEET

1 The distance travelled by a car varies directly with the time that the car has been trav- 7.3 elling. If the car travels 400 km in 5 hours, draw the graph of distance against time. 11 Substitution 2 A team of 6 people can unload 9 containers from a wharf per day. a Draw a graph showing the number of containers, n, that can be unloaded by a team of people, p. b What is the gradient of the graph drawn? 7.4 c Write an equation linking n and p.

WORKED

218


4 The distance, D, travelled by a car in a certain period of time will be directly proportional to the car’s speed, s. A car moving at 40 km/h travels 120 km. a Draw the graph of D against s. b Write an equation linking D and s. 5 The wage, W, earned by a worker is directly proportional to the hours, h, worked. A person who works 35 hours earns $306.25. Draw the graph of W against h. 6 The quantity of petrol, l, used by a car varies directly with the distance, d, travelled by the car. A car that travels 100 km uses 12.5 L of fuel. a Draw the graph of l against d. b Use the graph to find the quantity of petrol needed to travel 240 km. 7 The height of a tree, h, is directly proportional to the girth, g. A tree with a girth of 2.5 m has a height of 14 m. a Draw the graph of h against g. b Use the graph to find the height of a tree with a girth of 3 m. c Use the graph to find the girth of a tree that is 9 m tall, correct to 1 decimal place. 8 It is known that A$100 will buy US$67.50. Draw a conversion graph between Australian and US dollars.

Currency conversions Find out the current rate of conversion for each of the following foreign currencies and draw a linear function that will convert between Australian dollars and each currency. 1 US dollars 2 Euro 3 Pound Sterling 4 Japanese yen 5 New Zealand dollars

A step function is a linear function for which the rule changes as the value of the independent variable changes. Consider the case of a parking lot. The charge to park is $3 for the first 2 hours and $1 per hour after that. The graph for the parking charges is shown at right. The graph is called a step graph because it looks like a staircase.

Parking fee ($)

Step and piecewise functions 11 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 Number of hours


219

WORKED Example 13 A telephone call is charged at 75c for the first minute and 25c per minute after that. Draw a graph of the cost of the telephone call.

1 2

DRAW

Draw axes with time on the horizontal axis and cost on the vertical axis. Draw a step function at 75c with increases of 25c every minute.

Cost of telephone call ($)

THINK

3.00 2.75 2.50 2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0 0 1 2 3 4 5 6 7 8 9 10 Time (min)

A piecewise linear function is similar to a step graph. A piecewise function consists of more than one piece. In such examples we draw each linear function separately for the section of the graph to which it applies.

WORKED Example 14 A catering company charges $140 per hour for the first 2 hours and $80 per hour thereafter. Show this as a piecewise linear graph.

1 2 3

DRAW

Draw axes with time on the horizontal axis and cost on the vertical axis. For the first 2 hours draw the graph at $140 per hour. Three hours will cost $360, 4 hours $440. Plot these points and join them with a straight line.

Cost ($)

THINK

480 440 400 360 320 280 240 200 160 120 80 40 0 01 23 45 6 Time (h)

remember 1. A step function shows the increase in a quantity in steps. 2. A piecewise function follows different rules for different values of the independent variable.

220


7E

Step and piecewise functions

1 The cost of a bus fare is $1.20 for one section and 40c per section thereafter. Show this in a step graph. 13 2 The cost of posting a parcel is shown in the table below.

WORKED

Example

Mass

Cost

500 g or less

$2.50

500 g to 2 kg

$3.75

Over 2 kg

$5.50

Draw this information in a step graph. 3 A mobile telephone plan has a base charge of $25 per month, which includes 10 free calls. Every call thereafter costs $1.50. Show this information in a step graph. 4 A cyclist rides at an average 9 km/h uphill for 2 hours and then at 15 km/h for the next 14 3 hours. Draw a graph of distance travelled against time.

WORKED

Example

5 The cost of having business cards printed is $100 plus 50c each for the first 100, then 20c each thereafter. Draw a graph showing the cost of having these business cards printed. 6 Chandra earns $12 per hour for the first 35 hours worked each week. Any overtime is paid at time-and-a-half. Draw a piecewise graph that will show Chandra’s pay. 7 The PAYE tax rates in Australia are shown below. Income

Tax payable

$1 to $6000

$0

$6001 to $25 000

15% of each $1 over $6000

$25 001 to $75 000

$2850 plus 30% of each $1 over $25 000

$75 001 to $150 000

$17 850 plus 40% of each $1 over $75 000

In excess of $150 000

$47 850 plus 45% of each $1 over $150 000

Draw a piecewise function showing the amount of tax payable on income.


221

Simultaneous equations Consider the problem below. G G = 2B A class has 30 students. There are 30 twice as many girls as boys. How 27 24 many boys and girls are in this class? (10, 20) 21 We solve this problem by modelling 18 15 two linear relationships. 12 9 We can say that G = 30 − B and G = 30 – B 6 G = 2B, where G represents the 3 0 number of girls and B represents the 0 3 6 9 12 15 18 21 24 27 30 B number of boys. The solution to the problem will be the point of intersection of these linear relationships. The point of intersection on these lines is (10, 20). Therefore the solution to this problem is 10 boys and 20 girls.

Graphics Calculator tip! Finding the point of intersection Once you have graphed two functions, the point of intersection can be found using your graphics calculator. Consider the functions drawn previously. 1. From the MENU select GRAPH.

2. Delete any existing functions and enter the functions Y1 = 2X and Y2 = 30 – X.

3. Press SHIFT F3 for V-Window. Enter the settings shown at right, which match the axes drawn above.

4. Press EXE to return to the previous screen, and then press F6 to draw the graphs.

5. To find the intersection press SHIFT (a graph-solving function).

F5

for G-Solv

6. Press F5 for ISCT to find the intersection of the two graphs. This may take a moment for the calculator to find.

222


WORKED Example 15

Distance (km)

Car A is travelling at a constant speed of 60 km/h. Car B leaves 2 hours later and travels at a constant speed of 90 km/h. This is represented by the linear model below. How far from the starting point does car B overtake car A? B

600 500 400 300 200 100 0

A

0 1 2 3 4 5 6 7 8 9 10 Time (h)

THINK 1 2

WRITE

Look for the point of intersection of the two graphs. Read the distance of this point on the y-axis.

Point of intersection (6, 360). Car B overtakes car A 360 km from the starting point.

remember The point of intersection of two linear models will give the point where both conditions hold true.

7F 7.5

SkillS

HEET

Example

et

reads L Sp he

EXCE

1 At the grocery store, apples cost $5 per kg and bananas cost $2 per kg. Rhonda spends $30 on 9 kg of fruit. 15 This can be represented by the linear functions at right, where a represents the number of apples and b represents the number of bananas. Use the graph to find the mass of apples and bananas that Rhonda bought.

WORKED


Simultaneous equations

Cabr

omet i Ge ry

Simultaneous linear equations

Simultaneous equations b 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

5a + 2b = 30

a+b=9 0 1 2 3 4 5 6 7 8 9 10 a


223

2 a On the same set of axes as question 1, sketch the graphs of y = 2x + 1 and y = 7 − x. b Write down the point of intersection of the two graphs. 3 A rectangle has a length of x and a width of y. a The perimeter of the rectangle is 22 cm. This can be represented by the linear function 2x + 2y = 22. Graph this function. b The length of the rectangle is 5 cm longer than the width. This can be represented by the linear function y = x − 5. On the same set of axes graph this function. c Use the intersection of the two graphs to determine the length and width of the rectangle. 4 a Steve earns twice as much money each week as Theo. This can be represented by the linear function s = 2t, where s represents the amount of money Steve earns and t represents the amount of money Theo earns. Draw the graph of this function. b The total of Theo and Steve’s wages is $750. This can be represented by the linear function t + s = 750. Draw this function on the same set of axes. c Use the intersection of these graphs to find Theo’s wage and Steve’s wage.

T SHEE

Work

5 The sum of Tanya’s English and Maths marks is 135. a Write a linear function that represents this information and sketch the function. b Tanya’s English mark was 21 marks higher than her Maths mark. Write a linear function to represent this piece of information and draw the graph on the same set of axes. c Use the intersection of your two graphs to find Tanya’s mark in both English and Maths.

7.2

224


summary Linear functions • A function is a rule for calculation that involves an independent variable and a dependent variable. • If the function is a straight line when graphed, then the function is called a linear function. • The independent variable is shown on the horizontal axis and the dependent variable on the vertical axis. • The function can be graphed by drawing a table of values and then plotting the points generated, joining them with a straight line.

Gradient and intercept • The gradient is the increase in the dependent variable per one unit increase in the independent variable. • The gradient (m) can be found using the formula vertical change in position m = --------------------------------------------------------------------- . horizontal change in position • If the function is decreasing, then the gradient will be negative. • The intercept on the vertical axis gives the value of the dependent variable when the independent variable is equal to zero. • A function is written in the form: y = mx + b where m equals the gradient and b equals the y-intercept. • The gradient and the y-intercept can be used to help draw the graph of a function.

Variation • When two quantities vary directly with one another, the variation can be graphed as a linear function. • The variation will be in the form y = ax. • The graph is drawn from the point (0, 0) to one other point that is given. • The gradient of the linear function will equal the constant of variation.

Step and piecewise functions • A step function occurs when the value of the dependent variable increases in steps rather than continuously. • A piecewise function occurs when the function has different rules applying for different values of the independent variable.

Simultaneous equations • The point of intersection of two linear functions gives the point where both functions hold true simultaneously. • This is known as solving simultaneous equations.


225

CHAPTER review 1 The table below shows the labour charge for working on a motor vehicle. 1

2

3

4

5

55

80

105

130

155

Hours (h) Cost (C)

7A

a Draw the graph of this function. b Use the graph to find the labour charge for 8 hours work. 2 The conversion rate between Australian dollars, A, and Euro, E, can be defined by the rule E = 0.7A. Draw the linear function that will convert between the two currencies.

7A

3 The cost, C, of having a parcel delivered by courier is given by the linear function C = 20 + 3k, where k is the number of kilometres the parcel must be delivered. a Draw a graph of this function. b Use the graph to determine the cost of having the parcel delivered a distance of 12 km.

7A

4 For the functions below, find the gradient and vertical intercept. a b c y y

7B

(3, 7)

y 3

1

–4 x

x –2 3

5 The table below shows the profit or loss that would be made from a cake stall given the number of cakes sold. Number

−20

40

60

80

100

Profit

−30

20

70

120

170

a Draw the graph of this function. b State the gradient of this function. State the meaning of the gradient in this context. c State the vertical intercept for this function. State the meaning of the vertical intercept in this context.

x

7B

226 7C


6 For each of the linear functions below, state the gradient and the y-intercept. a y = 3x − 2

b y = 3--- x + 7 4

c y=5−x

7C

7 Sketch each of the functions shown below. a y = 2x − 1 c y=

1 --- x 2

b y = 6 − 3x

+3

8 The cost of a tank of petrol varies directly with the amount of petrol purchased. If 25 L of petrol costs $21.25, graph the variation.

7D 7E

9 It is known that p varies directly with q. When p = 5, q = 15. a Draw the graph of q against p. b What is the gradient of the graph? c Write an equation linking p and q. 10 The table below shows the cost per minute of a long distance telephone call. Distance of call

Cost per minute

Less than 150 km

30c

150 km – 750 km

65c

Over 750 km

90c

Show this information in a step graph.

7E

11 A tree has an initial height of 75 cm. The growth rate of the tree is approximately 75 cm per year for the first 4 years, and 40 cm per year thereafter. Show this by way of a piecewise graph.

7F

12 A rectangle has a length of l and a width of w. a The length of the rectangle is 10 cm longer than the width. This can be represented by the linear function w = l − 10. Draw this function. b The perimeter of the rectangle is 40 cm. This can be represented by a linear function. On the same set of axes draw this function. (Hint: Use the perimeter formula.) c Use the point of intersection of your two functions to find the length and the width of the rectangle.


Practice examination questions 1 multiple choice

y

Look at the linear function drawn at right. The gradient of this function is: A −2 C 1---

2

B − 1--2 D2

2

x

–4

2 multiple choice The function y = 6 − x has a gradient of: A −6

B −1

C1

D6

3 multiple choice For which of the functions drawn below is the gradient the greatest? A

B

y

y

x

C

x

D

y

y

x

x

4 multiple choice The linear function drawn at right is the graph of: A y = 1--- x − 1

B y = 1--- x + 1

Cy=1−

D y = 1--- x − 2

2

y

2

1 --- x 2

1

2

–2

5 The table below shows the values of x and y in a linear function. x

−0

−1

2

3

4

y

−3

−1

1

3

5

x

227

228 a b c d e f

CHAPTER

test yourself

7


Plot the points and draw the graph of the linear function. What is the gradient of the function? What is the y-intercept? Write the equation of this function. On the same axes, draw the graph of y = 5 − 2x. Write the solution to the pair of simultaneous equations represented on your diagram.

6 It is known that a quantity, m, varies directly with n. When m = 2.5, n = 1.5. a Draw a graph of n against m. b What is the gradient of the function? c What is the equation of this linear function?

Investing money

8 syllabus reference Financial mathematics 2 • Investing money

In this chapter 8A Calculation of simple interest 8B Graphing simple interest functions 8C Calculation of compound interest 8D Calculating compound interest from a table of compounded values 8E Graphing compound interest functions 8F Share dividends 8G Graphing share performance 8H Inflation and appreciation

areyou

READY?

Are you ready?


8.1

Converting percentages to decimals

1 Convert each of the following percentages to decimals. a 12% b 8% c 4.5%

8.2

Substitution into formulae

8.3

Graphing linear and non-linear relationships

4

2 Given: a A = 1--- bh, find A when b = 6.8 and h = 18.2 2 b T = a + (n – 1)d, find T when a = 7, n = 19 and d = 5 c I = Prn, find I when P = 12 000, r = 0.065 and n = 7.

3 Use the data in the table to draw a graph of the relationship between population and time. Time Population

0

1

2

3

10 000

12 000

14 400

17 280

8.4


8.5


8.6

d 6 1--- %

4 Convert: a 0 months into years b 15 months into years c 78 weeks into years.

5 Find: a 6% of $1200

b 8.5% of $2350


6 Write: a $56 as a percentage of $800 b $1050 as a percentage of $2800 c $625 as a percentage of $250 000.

c 0.6% of $12 500.

Chapter 8 Investing money

231

Calculation of simple interest When you deposit money in a bank, building society, or other financial institution you are actually lending them your money. Since you are lending them money you expect to receive your money back, plus an extra amount commonly known as interest. Similarly, if you borrow money from an institution, you must pay back the original sum, together with interest. A measure of the interest paid is called the interest rate. The interest rate is a percentage of the amount of money invested or borrowed and is paid each year. Even though all interest rates are expressed in the same way, interest can be calculated by using several different methods. Whether depositing or borrowing, it is important that you understand how the interest is calculated. The simplest method of interest calculation is called simple interest. Interest is calculated as a percentage of the initial deposit or borrowing (called the principal) and multiplied by the period the money was invested. The formula used to calculate simple interest is: I = Prn where: I = simple interest P = initial quantity r = percentage interest rate per period, = expressed as a decimal n = number of periods

WORKED Example 1 Calculate the simple interest earned on an investment of $5000 at 4% p.a. for 3 years. THINK 1 2

3 4

WRITE

Write down the simple interest formula. Write down the values of P, r (converting the percentage to a decimal) and n. Substitute into the formula. Calculate.

I = Prn P = $5000 r = 0.04 n=3 I = $5000 × 0.04 × 3 = $600

The total amount (A) that your deposit or debt has become after interest is added can be found using the formula: A=P+I where: A = total amount at the end of the term P = initial quantity I = simple interest

232


WORKED Example 2 $12 000 is invested for 5 years at 9.5% p.a. simple interest. Calculate the value of this investment at the end of the term. THINK 1 2 3 4 5 6 7

WRITE

Write down the formula for simple interest. Write down the value of P, r and n. Substitute the values into the given formula. Calculate the simple interest. Write down the formula for the total amount. Substitute the values into the given formula. Calculate.

I = Prn P = $12 000, r = 0.095, n = 5 I = $12 000 × 0.095 × 5 = $5700 A=P+I = $12 000 + $5700 = $17 700

Graphics Calculator tip! Calculating simple interest Your Casio graphics calculator can perform a number of financial functions by using the TVM mode. One of the options in this mode is to calculate simple interest. Examples such as worked example 2 are solved more simply by using the method above, rather than using the TVM mode of your graphics calculator. However, for some of the more complex questions later in this chapter, it is worth familiarising yourself with this method. 1. From the MENU select TVM.

2. Press F1 to select Simple Interest.

3. The calculator has two modes of calculating interest: 360 day mode or 365 day mode. You need to make sure that it is on 365 day mode. If not, press SHIFT SET UP, highlight DATE MODE and press F1 for 365.


233

4. Press EXE to return to the previous screen and enter the data for worked example 2. n = 3 × 365 (as n is in days) I% = 4 PV = –5000 (Principal or present value is entered as a negative.) 5. The calculator gives you two options. F1 : SI is simple interest. F2 : SFV is future value, in other words the principal plus interest. In this example, as we want the simple interest, we press F1 for SI. Care must be taken with simple interest questions when the length of the investment is not given in years. If the investment is given as months, it must be converted to years by writing the number of months as a fraction over 12, for example, 18 months =

18 -----12

= 1 1--- years. 2

WORKED Example 3 Calculate the simple interest earned on an investment of $7600 at 5.2% p.a. for 9 months. THINK 1 2 3 4

WRITE

Write down the formula for simple interest. Write down the value of P, r and n 9 3 - = --- year). (converting 9 months to ----12 4 Substitute the values into the given formula. Calculate the simple interest.

I = Prn P = $7600, r = 0.052, n = I = $7600 × 0.052 ×

3 --4

3 --4

I = $296.40

Some examples will ask you to calculate the length of time for which money must be invested in order to earn a given amount of interest.

WORKED Example 4 How long, to the nearest month, will it take to earn $650 simple interest if $8375 is invested at 6.25% p.a.? THINK 1 2

Write down the formula for simple interest. Write down the value of I, P and r.

WRITE I = Prn I = $650, P = $8375, r = 0.0625 Continued over page

234


THINK

WRITE

4

Substitute the values into the given formula. Simplify the RHS of the equation.

5

Make n the subject of the equation.

6

Calculate the value of n in years. Convert 0.2418 years to months by multiplying the decimal by 12. Answer the question.

3

7 8

$650 = $8375 × 0.0625 × n = $523.4375 × n $650 n = ------------------------$523.4375 n = 1.2418 years ≈ 15 months It will take approximately 1 year and 3 months to earn $650 in simple interest.

Examples of investments involving simple interest include investment bonds and debentures. Investment bonds are offered by the government, either State or federal, and larger organisations such as Telstra. Interest earned on investment bonds can be paid at varying intervals, for example monthly, quarterly, every six months (semi-annually) or yearly. Bonds are traded on financial markets. That is, they can be bought or sold prior to the term expiry date (also known as bond maturity). Debentures are similar to investment bonds but are issued by private companies to investors to raise capital. At the end of the term, the principal (or face value) is returned to the investor, while the interest earned is again paid at varying intervals.

remember 1. Interest is an amount of money paid to an investor or by a borrower for the use of money. 2. Simple interest is calculated by using only the initial investment. 3. The simple interest formula is: I = Prn where P is the initial quantity, r is the interest rate per period expressed as a decimal and n is the number of periods. 4. If using the formula to find a value other than I, substitute the known values into the formula and solve the resulting equation.

8A 8.1

SkillS

HEET

WORKED

Example

1 Converting percentages to decimals

8.2

SkillS

HEET

Substitution into formula

Calculation of simple interest

1 Veronica invests $4000 for 3 years at 5% p.a. Calculate the simple interest earned. 2 In each of the following, calculate the amount of simple interest earned. a $1200 for 1 year at 10% p.a. b $2460 for 5 years at 5% p.a. c $126 000 for 2 years at 8.5% p.a. d $9862 for 6 years at 11.25% p.a. e $1000 for 1 1--- years at 6% p.a. 2

f $1750 for 5 1--- years at 7.45% p.a. 4


235

Example

–TI

2

4 Brian has a $10 000 inheritance that he wants to invest. He invests his money in program GC government bonds for 3 years at 8% p.a. Calculate: a the simple interest earned Interest b the value of his investment on maturity.

6 Frank is 7 years old and starts a savings account with the local bank. He has $140 with which to start the account. a If the interest rate is 3 3--- %, calculate the amount of interest Frank will receive after 4 one year. b What will be the balance of Frank’s bank account after one year? 7 Loretta invests $7540 at 5.95% p.a. a Calculate the simple interest that Loretta will earn in her first year. b Loretta receives two interest payments per year. Calculate the size of each payment. c Find the total value of the investment after 4 1--- years. 2

WORKED

Example

3

8 Kath invests $9450 in a government bond that pays 6% p.a. simple interest for an 18 month investment. Calculate the simple interest earned on this investment. 9 multiple choice The simple interest paid on an investment of $5750 at 4.6% p.a. for 2 years is: A $529.00 B $541.17 C $6291.17 D $6279.00 10 multiple choice The total value of an investment of $3500 after 2 years and 6 months if simple interest is paid at the rate of 5% per annum, is: A $437.50 B $826.25 C $3937.50 D $3975.50 11 Calculate the simple interest that has to be paid, if $4650 is invested on a term deposit for 180 days at 5.75% p.a. (Hint: Write 180 days as a fraction of 1 year.)

12 How long, to the nearest month, will it take to earn $2400 simple interest, if $16 410 is invested at 9.75% p.a.? 4

WORKED

Example

13 A debenture offers to pay 8% p.a. interest on a 4 year investment. Janine wants to earn $2000 interest. What principal will Janine need to invest? 14 multiple choice What sum, to the nearest dollar, must be invested for one year at 6% per annum simple interest, in order to earn $1200 interest? A $2000 B $12 200 C $20 000 D $21 200

L Spre XCE ad

Interest

sheet

5 Karelle invests $7600 in a debenture that pays 6.9% p.a. for investments over 2 years. Calculate the total value of Karelle’s investment on maturity.

E

WORKED

asio

GC

3 Julie has $40 000 to invest. She invests it for 5 years in a State Government bond that am progr –C pays 5.6% p.a. interest. Calculate the simple interest that Julie will earn: Interest a each year b for the whole 5 years of the investment.

236


15 Sue and Harry invested $14 500 in State Government bonds at 8.65% p.a. The investment is for 10 years and the interest is paid semi-annually (that is, every six months). Calculate how much interest: a they receive every payment b will be received in total. 16 Mrs Williams invested $60 000 in government bonds at 7.5% p.a. with interest paid semi-annually (that is, every 6 months). a How much interest is she paid each 6 months? b How much interest is she paid over 3 years? c How long would the money need to be invested to earn a total of $33 750 in interest? 17 Mr and Mrs Tyquin donate money for a scholarship at the local high school. The value of the scholarship is $1500. They invest a sum of money at 8% p.a. so that each year $1500 in interest is earned. How much will Mr and Mrs Tyquin need to invest?

1 Find the simple interest on each of the following investments. 1 $4000 at 5% p.a. for 4 years 2 $9000 at 7% p.a. for 2 years 3 $15 000 at 6% p.a. for 3 years 4 $950 at 0.1% p.a. for 2 years 5 $40 000 at 3.5% p.a. for 5 years 6 $1200 at 4.6% p.a. for 2 1--- years 2

7 $5745 at 3 3--- % p.a. for 1 year 4

8 $32 500 at 4.1% p.a. for 18 months 9 $532 at 0.2% p.a. for 6 months 10 $3330 at 6.95% p.a. for 9 months

Graphing simple interest functions Suppose that we invest $10 000 at 5% p.a. simple interest. The table below shows the amount of interest that we will receive over various lengths of time. No. of years Interest

1

2

3

4

5

$500

$1000

$1500

$2000

$2500

The amount of interest earned can be graphed by the linear function at right. Note that the gradient of this graph is 500, which is the amount of one year’s interest, or 5% of the principal.

Interest ($)


237

3000 2000 1000 0 0

1

2 3 4 Years

5

WORKED Example 5 $6000 is invested at 4% p.a. a Complete the table below to calculate the interest that will have been earned over 5 years. No. of years

1

2

3

4

5

Interest b Graph the interest earned against the number of years the money is invested. THINK

WRITE/DRAW

a Use the simple interest formula to calculate the interest earned on $6000 at 4% p.a. for 1, 2, 3, 4 and 5 years.

a No. of years

b Draw the graph with Years on the horizontal axis and Interest on the vertical axis.

b

Interest ($)

Interest

1 $240

2 $480

3 $720

4 5 $960 $1200

1500 1000 500 0 0

1

2 3 4 Years

5

Graphics Calculator tip! Graphing a simple interest function By using the simple interest formula we can create a function that can be graphed using the GRAPH function on the graphics calculator. Consider worked example 5. 1. Write the simple interest formula

I = Prn

2. Substitute the known values of P = $6000 and r = 0.04. Simplify the expression.

I = 6000 × 0.04 × n I = 240n

3. From the MENU select GRAPH.

238


4. Delete any existing function and enter Y1 = 240X.

F3 for V-Window and

5. To draw up the axes press SHIFT enter the setting shown at right.

6. Press EXE to return to the previous screen and then press F6 to DRAW the graph.

We are able to compare the interest that is earned by an investment at varying interest rates by graphing the interest earned at varying rates on the one set of axes.

WORKED Example 6 Kylie has $12 000 to invest. Three different banks offer interest rates of 4%, 5% and 6%. a Complete the table below to show the interest that she would earn over 5 years. No. of years

1

2

3

4

5

Interest (4%) Interest (5%) Interest (6%) b Show this information in graph form. THINK

WRITE/DRAW

a

a

1 2 3 4 5 $480 $960 $1440 $1920 $2400 $600 $1200 $1800 $2400 $3000 $720 $1440 $2160 $2880 $3600

No. of years Interest (4%) Interest (5%) Interest (6%)

b Interest ($)

Use the simple interest formula to calculate the interest earned on $12 000 at 4% p.a. for 1, 2, 3, 4 and 5 years. 2 Use the simple interest formula to calculate the interest earned on $12 000 at 5% p.a. for 1, 2, 3, 4 and 5 years. 3 Use the simple interest formula to calculate the interest earned on $12 000 at 6% p.a. for 1, 2, 3, 4 and 5 years. b Draw a line graph for each investment. 1

4000 Interest (6%) Interest (5%) Interest (4%)

3000 2000 1000 0

0

1

2

3 4 Years

5


239

remember 1. The amount of simple interest earned on an investment for various lengths of time can be graphed as a linear function. 2. The gradient of the linear function will be equal to one year’s interest. 3. To compare the interest earned at various rates, we can graph several functions on the one set of axes.

8B WORKED

Example

1 $8000 is invested at 5% p.a. a Copy and complete the table below to calculate the interest over 5 years. No. of years

1

2

3

4

5

Interest

8.3 Graphing linear and non-linear relationships

b Draw a graph of the interest earned against the length of the investment. 2 $20 000 is to be invested at 8% p.a. a Copy and complete the table below to calculate the interest for various lengths of time. No. of years

1

2

3

4

5

Interest b Draw a graph of the interest earned against the length of the investment. c What is the gradient of the linear graph drawn? d Use your graph to find the amount of interest that would have been earned after 10 years. 3 Draw a graph to represent the interest earned by each of the following investments over 5 years. a $15 000 at 7% p.a. b $2000 at 10% p.a. c $8600 at 7.5% p.a. d $50 000 at 8.2% p.a. 4 A graph is drawn to show the interest earned on $6000 at 4.8% p.a. for various lengths of time. Without drawing the graph, state the gradient. 5 Darren invests $3200 at 2.5% p.a. for 5 years. a Graph the amount of interest that Darren would have earned at the end of each year for the 5 years. b Graph the total value of Darren’s investment at the end of each year.

SkillS

HEET

5

Graphing simple interest functions

WORKED

Example

6


6 Julieanne has $25 000 to invest at 5%, 6% or 8%. a Complete the table below to show the interest that she would earn over 5 years. 1

No. of years

2

3

4

5

Interest (5%) Interest (6%) Interest (8%) b Show this information in graph form. 7 Theo has $50 000 to invest. Theo investigates the website www.whichbank.com.au, which has an interactive component. Theo enters the figure $50 000 and the following graph is displayed.

Interest ( )

240

The Whichbank advantage

40 000 35 000 30 000 25 000 20 000 15 000 10 000 5000 0

Whichbank Eastpac NZA bank

0

1

2

3

4

5 6 Years

7

8

9 10

a Find the amount of simple interest earned after 10 years by investing with each of the three banks listed. b Use your answer to a to calculate the interest rate paid by each of the three banks. 8 Mark has $5500 to invest at 3%, 3.5% or 3.75%. a Complete the table below to show the interest that he would earn over various lengths of time. No. of years

1

2

3

4

5

Interest (3%) Interest (3.5%) Interest (3.75%) b Show this information in graph form. 9 Draw a graph to show the interest earned on an investment of $12 500 at 4.5% p.a., 5% p.a. and 5.2% p.a. Use the graph to find: a the amount of interest earned by each investment after 8 years b how much more the investment at 5.2% p.a. is worth after 10 years than the 4.5% p.a. investment. 10 Three banks offer $4000 debentures at rates of 5.2% p.a., 5.8% p.a. and 6.2% p.a. Draw a graph of the value of the debentures at maturity against the number of years of the debenture.


241

Computer Application 1 Simple interest spreadsheets Throughout this chapter we will use some spreadsheets that allow us to track the growing value of an investment over time. 1. From the Maths Quest General Mathematics Preliminary Course CD-ROM open the spreadsheet ‘Interest’. E

Interest

2. The spreadsheet ‘Simple Interest’ (Sheet 1) models an investment of $10 000 at 5% p.a. 3. Use the graphing function on your spreadsheet to draw a line graph for the amount of interest earned each year and the value of the investment after each year. 4. Change the amount of the principal and the interest rate, and note the change in the figures displayed and the chart. 5. Use this function to check your answers to Exercise 8B. 6. Save the spreadsheet as Simple Interest.

Calculation of compound interest In practice, most investments are not calculated using simple interest. If you have a bank account, you would know that when interest is paid the balance of your account grows and it is on this new balance that your next interest payment is calculated. When interest is added to the principal and this new balance is used to calculate the next interest payment, this is called compound interest. We can calculate compound interest by calculating simple interest one period at a time. The amount to which the initial investment grows is called the compounded value or future value.

sheet

L Spre XCE ad

242


WORKED Example 7 Calculate the future value of an investment of $10 000 at 10% p.a. for 3 years with interest paid at the end of each year, by calculating the simple interest for each year separately. THINK 1 2 3

4 5

6 7

WRITE Initial principal = $10 000 1st year’s interest = 10% of $10 000 1st year’s interest = $1000 2nd year’s principal = $10 000 + $1000 2nd year’s principal = $11 000

Write the initial principal. Calculate the interest for the 1st year. Calculate the 2nd year’s principal by adding the 1st year’s interest to the initial principal. Calculate the 2nd year’s interest.

2nd year’s interest = 10% of $11 000 2nd year’s interest = $1100 3rd year’s principal = $11 000 + $1100 3rd year’s principal = $12 100

Calculate the 3rd year’s principal by adding the 2nd year’s interest to the 2nd year’s principal. Calculate the 3rd year’s interest.

3rd year’s interest = 10% of $12 100 3rd year’s interest = $1210 Future value = $12 100 + $1210 Future value = $13 310

Calculate the future value of the investment by adding the 3rd year’s interest to the 3rd year’s principal.

To calculate the actual amount of interest received, we subtract the initial principal from the future value. In the example above CI = $13 310 − $10 000 = $3310 To compare this with simple interest earnings at the same rate. I = Prn = $10 000 × 0.1 × 3 = $3000 The table below shows a comparison between the value of an investment of $10 000 earning 10% p.a. at both simple interest and compound interest.

1

2

3

4

5

6

7

8

Simple interest

$1000

$2000

$3000

$4000

$5000

$6000

$7000

$8 000

Compound inteerest

$1000

$2100

$3310

$4641

$6105

$7716

$9487 $11 436

Year

We can develop a formula for the future value of an investment rather than do each example by repeated use of simple interest. Consider worked example 7. Let the compounded value after each year be An. After 1 year A1 = 10 000 × 1.1 (increasing $10 000 by 10%) After 2 years A2 = A1 × (1.1) = 10 000 × 1.1 × 1.1 (substituting the value of A1) = 10 000 × 1.12


243

A3 = A2 × 1.1 = 10 000 × 1.12 × 1.1 = 10 000 × 1.13 The pattern then continues such that the value of the investment after n years equals: $10 000 × 1.1n We can generalise this example to any investment. A = P(1 + r)n where A = final balance P = initial quantity r = percentage interest rate per compounding period, expressed as a decimal n = number of compounding periods. After 3 years

In the financial world, the terms future value (FV) and present value (PV) are sometimes used instead of amount and principal.

WORKED Example 8 Calculate the future value of an investment of $12 000 at 7% p.a. for 5 years, where interest is compounded annually. THINK 1 2 3 4

WRITE

Write down the formula for the future value. Write down the value of P, r (as a decimal) and n. Substitute into the formula. Calculate.

A = P(1 + r)n P = $12 000, r = 0.07, n = 5 A = $12 000 × 1.075 = $16 830.62

In the above example, interest is paid annually; however, this is not always the case. In many cases interest is paid more often. It may be paid six-monthly, quarterly, monthly or even daily. This is called the compounding period. If interest is paid more often than annually, the value of n is the number of compounding periods during the investment. The interest rate then needs to be converted from a rate per annum to a rate per compounding period. For example, consider an investment of $6000 at 8% p.a. for 2 years with interest compounded quarterly. Interest is paid four times per year and therefore eight times in 2 years. Therefore n = 8. The interest rate must be calculated per quarter. This is done by dividing the annual rate by four. Therefore, in this example the rate is 2% per quarter, hence r = 0.02.

244


WORKED Example 9 Calculate the future value of an investment of $6000 at 8% p.a. for 2 years with interest compounded quarterly. THINK 1 2 3 4

WRITE

Write down the formula for the future value. Write down the value of P, r (as a decimal) and n. Substitute into the formula. Calculate.

A = P(1 + r)n P = $6000, r = 0.02, n = 8 A = $6000 × 1.028 = $7029.96

Graphics Calculator tip! Compound interest warning! The TVM function on a graphics calculator can calculate compound interest. This is somewhat more complicated than using the formula A = P(1 + r)n. Further, a graphics calculator uses an imprecise method and answers may vary by up to $1 from the correct answer. For this reason we recommend not using a graphics calculator for these calculations.

remember 1. The future value of an investment under compound interest can be calculated by calculating the simple interest for each year separately. 2. The formula used to calculate the future value of an investment is: A = P(1 + r)n where A = final balance P = initial quantity n = number of compounding periods r = percentage interest rate per compounding period, expressed as a = decimal. 3. In the formula, n is the number of compounding periods over the term of the investment and r is the interest rate per compounding period. To find r, divide the interest rate per annum by the number of compounding periods per year.

8C 8.4

SkillS

HEET


Example

7

1 Ray has $5000 to invest. He invests it for 3 years at 10% p.a. with interest paid annually. Calculate the future value of the investment by calculating the simple interest on each year separately.

8.5

2 Suzanne is to invest $15 000 for 2 years at 7% p.a. with interest paid annually. Calculate the future value of the investment by calculating the simple interest for each year separately.

Finding a WORKED percentage Example of a 8 quantity

3 Kiri has $2000 to invest. She invests the money at 8% p.a. for 5 years with interest compounded annually. Use the formula A = P(1 + r)n to calculate the future value of Kiri’s investment.

SkillS

HEET

WORKED

Calculation of compound interest


245

Example

6 A passbook savings account pays interest of 0.2% p.a. Luke has $500 in such an account. Calculate the future value of the account after 2 years, if interest is compounded quarterly. 7 Noel is to invest $12 000 at 8% p.a. for 2 years with interest compounded quarterly. Calculate the future value of the investment. 8 Vicky invests $30 000 in a one-year fixed deposit at an interest rate of 6% p.a. with interest compounding monthly. a Convert the interest rate of 6% p.a. to a rate per month. b Calculate the future value of the investment upon maturity. 9 Calculate the compounded value of each of the following investments. a $960 for 1 year at 4.50% p.a. with interest compounded six-monthly. b $7500 for 3 1--- years at 5.6% p.a. with interest compounded quarterly. 2 c $152 000 for 2 1--- years at 7.2% p.a. with interest compounded six-monthly. 2 d $14 000 for 4 years at 9% p.a. with interest compounded monthly. e $120 000 for 20 years at 11.95% p.a. with interest compounded quarterly. 10 multiple choice A sum of $5000 is invested for 2 years at the rate of 4.75% p.a., compounded quarterly. The interest paid on this investment, to the nearest dollar, is: A $475 B $495 C $5475 D $5495 11 multiple choice After selling their house Mr and Mrs Dengate have $61 800. They plan to invest it at 6% p.a., with interest compounded annually. The value of their investment will first exceed $100 000 after: A 8 years B 9 years C 10 years D 11 years 12 multiple choice Warren wishes to invest $10 000 for a period of 5 years. The following investment alternatives are suggested to him. The best investment would be: A simple interest at 9% p.a. B compound interest at 8% p.a. with interest compounded annually C compound interest at 7.8% p.a. with interest compounded six-monthly D compound interest at 7.2% p.a. with interest compounded quarterly 13 multiple choice Which of the following investments, to be invested for 6 years and compounded semiannually at 8% p.a., will have a future value closest to $15 000? A $900 B $8500 C $9400 D $11 000

L Spre XCE ad

Compound interest

sheet

9

5 Carla is to invest $45 000 at 9.2% p.a. for 5 years with interest compounded sixmonthly. Calculate the future value of the investment.

E

WORKED

SkillS

HEET

4 Use the compound interest formula to calculate the future value of each of the 8.6 following investments with interest compounded annually. a $4000 at 5% p.a. for 3 years b $8000 at 3% p.a. for 5 years Writing one quantity as a c $18 000 at 8% p.a. for 4 years d $11 500 at 5.5% p.a. for 3 years percentage of e $8750 at 6.25% p.a. for 6 years another

246


14 Brittany has $13 500 to invest. An investment over a 2-year term will pay interest of 8% p.a. Calculate the compounded value of Brittany’s investment if the compounding period is: a one year b six months c three months d monthly. 15 Kerry invests $100 000 at 8% p.a. for a one-year term. For such large investments interest is compounded daily. a Calculate the daily percentage interest rate, correct to 4 decimal places. b Calculate the compounded value of Kerry’s investment on maturity. c Calculate the amount of interest paid on this investment. d Calculate the extra amount of interest earned, compared with the interest calculated at the end of the year.

Work

T SHEE

8.1

16 Simon invests $4000 for 3 years at 6% p.a. simple interest. Monica also invests $4000 for 3 years, but her interest rate is 5.6% p.a. with interest compounded quarterly. a Calculate the value of Simon’s investment on maturity. b Show that the compounded value of Monica’s investment is greater than Simon’s investment. c Explain why Monica’s investment is worth more than Simon’s, despite receiving a lower rate of interest.

Computer Application 2 Compound interest spreadsheets Earlier we wrote a spreadsheet to show the growth of an investment over a number of years. We will now write a similar spreadsheet to show the growth under compound interest. 1. From the Maths Quest General Mathematics Preliminary Course CD-ROM open the spreadsheet ‘Interest’. EXCE

et

reads L Sp he

Interest

2. Select Sheet 2, ‘Compound Interest’. This spreadsheet models a $10 000 investment at 5% p.a. interest with interest compounded annually (one compounding period per year).


247

3. Use the graphing function to draw a graph showing the growth of this investment over 10 years. Compare this graph with the graph drawn for the corresponding simple interest investment. 4. Change the number of compounding periods per year to see the change in the value of the investment. Your graph should change as you change the information. 5. Change other information, such as the principal and interest rate, to see the change in your graph. 6. Save this spreadsheet as Compound Interest.

2 1 Calculate the simple interest earned on an investment of $9240 made at 7.4% p.a. for 3 years. 2 Corey invests $14 200 for 5 years in debentures that pay 4.3% p.a. simple interest. Calculate the total value of Corey’s investment at maturity. 3 Emma invests $27 500 in investment bonds which pay 6.25% p.a. simple interest for 2 years. Calculate the interest earned in this investment. 4 The interest that Emma receives is paid in quarterly instalments. Calculate the size of each quarterly interest payment. 5 Vladimir invests $2000 at 5% p.a. for 3 years with interest compounded annually. Calculate the compounded value of Vladimir’s investment. 6 Calculate the amount of interest earned by Vladimir. 7 Calculate the compounded value of an investment of $6000 at 6.4% p.a. for 2 years with interest compounded annually. 8 Calculate the compounded value of an investment of $6000 at 6.4% p.a. for 2 years with interest compounded six-monthly. 9 Calculate the compounded value of an investment of $6000 at 6.4% p.a. for 2 years with interest compounded quarterly. 10 Calculate the compounded value of an investment of $13 200 at 7.2% p.a. for 18 months with interest compounded monthly.

248


Calculating compound interest from a table of compounded values So far we have looked at the calculation of compounded values and the amount of compound interest paid. Suppose we have $10 000 saved for a world holiday, which is going to cost $15 000. The best interest rate for investing the money is at 8%, compounded quarterly. We want to know how long we need to invest the $10 000, so that it will have a compounded value of $15 000. To solve this example we need to calculate the value of n, having been given the values of CV, PV and r. The best way to do this is to use a table showing the compound value interest factor for various investments. A compound value interest factor (CVIF) is the compounded value that $1 will amount to under a certain investment. For example, if $1 were invested at 5% p.a. for 4 years, compounded annually, its compounded value would be $1.216. We can use this to calculate the value of other amounts of money under the same investment pattern. For example, if $7600 were invested at 5% p.a. for 4 years, to calculate the compounded value of the investment we multiply $7600 by the CVIF which is 1.216. Therefore CV = $7600 × 1.216 = $9241.60 In this example $7600 is the present value (PV) of the investment and $9241.60 is the compounded value (CV). We can therefore use the formula: CV = PV × CVIF This formula simply states: compounded value = present value × interest factor. The CVIF table below shows the interest factors.

Interest rate per period Periods

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

1

1.010

1.020

1.030

1.040

1.050

1.060

1.070

1.080

1.090

1.100

2

1.020

1.040

1.061

1.082

1.103

1.124

1.145

1.166

1.188

1.210

3

1.030

1.061

1.093

1.125

1.158

1.191

1.225

1.260

1.295

1.331

4

1.041

1.082

1.126

1.170

1.216

1.262

1.311

1.360

1.412

1.464

5

1.051

1.104

1.159

1.217

1.276

1.338

1.403

1.469

1.539

1.611

6

1.062

1.126

1.194

1.265

1.340

1.419

1.501

1.587

1.677

1.772

7

1.072

1.149

1.230

1.316

1.407

1.504

1.606

1.714

1.828

1.949

8

1.083

1.172

1.267

1.369

1.477

1.594

1.718

1.851

1.993

2.144

9

1.094

1.195

1.305

1.423

1.551

1.689

1.838

1.999

2.172

2.358

10

1.105

1.219

1.344

1.480

1.629

1.791

1.967

2.159

2.367

2.594

We can now use this table to solve compound interest problems.


249

WORKED Example 10 Use the CVIF table to find the compounded value of $4560 invested at 8% p.a. for 2 years with interest compounded six-monthly. THINK

WRITE

1

Calculate the interest rate per period and number of interest periods.

Interest rate per period = 4% Interest periods = 4

2

Look up the CVIF for 4% with 4 interest periods.

CVIF = 1.170

3

Write the formula.

CV = PV × CVIF

4

Substitute the PV and the CVIF.

CV = $4560 × 1.170

5

Calculate.

CV = $5335.20

This table can also be used to help us calculate the present value of an investment that is required to produce a given compounded value. This is done using the same formula; however, you will need to solve the equation to find the value of PV.

WORKED Example 11 Liz is 16 years old. She hopes to have $3000 in 3 years to buy a used car. She finds an investment of 6% p.a. with interest compounded six-monthly. Calculate the amount of money that Liz must invest to generate a compounded value of $3000 in 3 years. THINK

WRITE

1

Calculate the interest rate per period and number of interest periods.

Interest rate per period = 3% Interest periods = 6

2

Look up the CVIF for 3% with 6 interest periods.

CVIF = 1.194

3

Write the formula.

4

Substitute for CV and CVIF.

5

Make PV the subject of the equation (by dividing by 1.194).

6

Calculate.

7


CV = PV × CVIF $3000 = PV × 1.194 $3000 PV = --------------1.194 PV = $2512.56 Liz will need to invest $2512.56 to generate $3000 in 3 years.

We can also use the table to determine the length of time that a given present value will take to reach a certain compounded value. This is done by calculating the required CVIF and looking for the first CVIF in the table, at the given interest rate, greater than that required.

250


WORKED Example 12 How long will it take $2500 to grow to $3200 when invested at 8% p.a. with interest compounded six-monthly? THINK

WRITE

4

Calculate the interest rate per period. Write the value of PV and CV. Write the formula. Substitute the values of PV and CV.

5

Make CVIF the subject of the formula.

6

Calculate the value of CVIF. Look at the 4% column of the CVIF table. The first CVIF greater than 1.28 (that is, 1.316) will be the minimum number of interest periods required to produce the required growth. Calculate the length of time for seven interest periods.

1 2 3

7

8

Interest rate per period = 4% PV = $2500, CV = $3200 CV = PV × CVIF $3200 = $2500 × CVIF $3200 CVIF = --------------$2500 = 1.28 Seven interest periods will be required.

It will take 3 1--- years for $2500 to grow to 2 $3200.

The same method can be used to find the interest rate required to achieve a certain compounded value over a fixed period of time.

remember 1. The compounded value of $1 under a particular investment is called the compounded value interest factor (CVIF). 2. The compounded value of any investment can be calculated by using a CVIF table and the formula CV = PV × CVIF. 3. We can use CVIF tables to calculate the length of time an investment will take to reach a certain compounded value, or the interest rate required to reach a certain compounded value.


8D WORKED

Example

10

251

Calculating compound interest from a table of compounded values

1 Toshika has $10 000 to invest for 4 years. The bank offers her 7% p.a. with interest compounded annually. Use the CVIF table on page 248 to calculate the compounded value of Toshika’s investment. 2 Greg has $8500 to invest for 5 years. A building society offers 8% p.a. with interest compounded twice a year. Use the CVIF table to calculate the compounded value of Greg’s investment. 3 Marlene invests $40 000 for 2 years at 8% p.a. with interest compounded quarterly. Use the CVIF table to calculate the compounded value of Marlene’s investment. 4 Roger invests $2400 for 2 years in an ‘at call’ account, which pays 4% p.a. interest with interest paid quarterly. Use the CVIF table to calculate the future value of this investment. 5 Use the CVIF table to calculate the compounded value of each of the following investments. a $5000 at 9% p.a. for 6 years with interest compounded annually b $6700 at 10% p.a. for 4 years with interest compounded six-monthly c $250 at 6% p.a. for 5 years with interest compounded six-monthly d $23 670 at 4% p.a. for 2 years with interest compounded quarterly e $13 250 at 8% p.a. for 18 months with interest compounded quarterly f $115 000 at 12% p.a. for 6 months with interest compounded monthly 6 Use the formula A = P(1 + r)n to calculate the CVIF, correct to 3 decimal places, for an investment at 2.5% for: a 1 interest period b 2 interest periods c 3 interest periods d 4 interest periods e 6 interest periods f 8 interest periods. 7 Using the CVIFs found in question 6 will allow you to calculate each of the compounded values of the following investments. a $900 at 2.5% p.a. for 3 years with interest compounded annually b $2340 at 5% p.a. for 2 years with interest compounded six-monthly c $7200 at 10% p.a. for 1 year with interest compounded quarterly d $11 000 at 10% p.a. for 2 years with interest compounded quarterly e $5750 at 10% p.a. for 1 1--- years with interest compounded quarterly 2

8 multiple choice One dollar invested at 3.5% for 5 interest periods amounts to: A 0.175 B 1.035 C 1.175 D 1.188 9 multiple choice For a certain investment the CVIF = 2.147. If the present value of the investment is $32 546, the compounded value, correct to the nearest dollar, will be: A $15 158 B $15 159 C $69 876 D $69 877

252


10 Jason wants to save for a car in 3 years. He needs to have $10 000. Use the CVIF table to calculate the amount of money that he will need to invest at 5% p.a. with interest 11 compounded annually, to have $10 000 in 3 years. Give your answer correct to the nearest dollar.

WORKED

Example

11 How long will it take $2000 to grow to $2500 when invested at 8% p.a. with interest compounded six-monthly? 12 12 Calculate the length of time that it will take: a $1000 to grow to $1100 at 10% p.a. with interest paid annually b $1000 to grow to $1500 at 7% p.a. with interest paid annually c $3000 to grow to $4000 at 6% p.a. with interest paid six-monthly d $9000 to grow to $10 000 at 8% p.a. with interest paid quarterly e $12 000 to grow to $17 500 at 10% p.a. with interest paid six-monthly.

WORKED

Example

13 Calculate the interest rate required for $1000 to grow to $1300 in 2 years, if interest is compounded quarterly. (Hint: Find the CVIF required and use the table for 8 interest periods.) 14 Use the CVIF table to calculate the interest rate, to the nearest whole number, required for each of the following investments. a $1000 to grow to $1200 in 3 years with interest compounded annually b $2000 to grow to $2600 in 4 years with interest compounded six-monthly c $500 to grow to $650 in 1 year with interest compounded quarterly d $10 000 to grow to $20 000 in 8 years with interest compounded annually e $3500 to grow to $6000 in 5 years with interest compounded six-monthly 15 Bruce, Keith and Max each have $10 000 to invest over a 5-year term. a Bruce invests at 10% p.a. simple interest. Calculate the value of Bruce’s investment at maturity. b Keith invests at 10% p.a. with interest compounded annually. Calculate the value of Keith’s investment at maturity, using the CVIF table. c Max invests at 10% p.a. with interest compounded six-monthly. Calculate the value of Max’s investment at maturity, using the CVIF table. d Calculate the total amount of interest each man received. e Write down the amount of interest each received as a percentage of their original investment.


253

Graphing compound interest functions Earlier, we drew graphs of the simple interest earned by various simple interest investments and found that these graphs were linear. This occurred because the amount of interest earned in each interest period was the same. With compound interest, the interest earned in each interest period increases, and so when we graph the future value of the investment, an exponential graph results. We can use the compounded values of $1 to complete tables that will then allow us to graph a compound interest function.

WORKED Example 13 Pierre invests $5000 at 5% p.a., with interest compounded annually. a Use the table of compounded values to complete the table below to show the future value at the end of each year. No. of years

1

2

3

4

5

Future value b Draw a graph of the future value of the investment against time. THINK

WRITE/DRAW

a Use the table of compounded values of $1 to complete the table.

a No. of years

b Draw the graph, by drawing a smooth curve between the marked points.

b Future value ($)

1 2 3 4 5 Future value $5250 $5515 $5790 $6080 $6380 7000

6000

5000 0

0

1

2

3 4 Years

5

254


a compound interest Graphics Calculator tip! Graphing function By using a graphics calculator, we can both calculate and draw a graph of the compound interest formula. 1. Write the compound interest formula.

A = P(1 + r)n

2. Substitute the known values of P = $5000 and r = 0.05. Simplify the expression.

A = 5000(1 + 0.05)n A = 5000(1.05)n

3. From the MENU select GRAPH.

4. Delete any existing function and enter Y1 = 5000(1.05)^X.

5. To draw up the axes, press SHIFT enter the setting shown at right.

F3 for V-Window and

6. Press EXE to return to the previous screen and then press F6 to DRAW the graph.

To graph the interest earned, the principal must be subtracted from the future value of the investment. As with simple interest, such graphs can be used to compare various investments.

WORKED Example 14 Amy is to invest $2000 at 5%, 6% or 7%, compounded annually. a Copy and complete the table below to find the future value of each investment at the end of each year. No. of years

1

2

3

Future value (5%) Future value (6%) Future value (7%) b Draw a graph that will allow the investments to be compared.

4

5


THINK

WRITE/DRAW

a Use the table of compounded values of $1 to complete the table.

a No. of years

b Draw each graph by joining the points with a smooth curve.

b

255

Future value ($)

1 2 3 4 5 Future value (5%) $2100 $2206 $2316 $2432 $2552 Future value (6%) $2120 $2248 $2382 $2524 $2676 Future value (7%) $2140 $2290 $2450 $2622 $2806 2800 2700 2600 2500 2400 2300 2200 2100 0

Future value (7%) Future value (6%) Future value (5%) 0

1

2 3 Years

4

5

remember 1. A table that shows the compounded value of $1, at various interest rates for various compounding periods, can be used to calculate the future value of an investment. 2. Graphing the future value of a compound interest investment results in an exponential graph. 3. To graph the amount of compound interest paid, we need to subtract the principal from the future value. 4. By drawing the graphs of several investments on one set of axes, we can compare the investments.

8E

Graphing compound interest functions

1 An amount of $8000 is invested at 5% p.a. with interest compounded annually. a Copy and complete the table below, using the formula A = P(1 + r)n, to calculate the 13 future value of the investment at the end of each year.

WORKED

Example

No. of years

1

2

3

4

5

Future value b Draw a graph of the future value against the length of the investment. 2 An amount of $12 000 is to be invested at 8% p.a. with interest compounded annually. a Copy and complete the table below to calculate the future value at the end of each year. No. of years

1

2

3

4

5

Future value b Draw a graph of the interest earned against the length of the investment. c Use your graph to find the future value of the investment after 10 years.

256


3 Draw a graph to represent the future value of the following investments against time. a $15 000 at 7% p.a. with interest compounded annually b $2000 at 10% p.a. with interest compounded annually 4 A graph is drawn to show the future value of an investment of $2000 at 6% p.a., with interest compounding six-monthly. a Complete the table below. Years

0.5

1

1.5

2

2.5

3

3.5

4

4.5

FV b Use the table to draw a graph. 5 An amount of $1200 is invested at 4% p.a. with interest compounding quarterly. a Graph the future value of the investment at the end of each year. b Graph the compound interest earned by the investment at the end of each year. 6 James has $8000 to invest at either 4%, 6% or 8%, compounding annually. 14 a Complete the table below to show the interest that he would earn over 5 years.

WORKED

Example

No. of years

1

2

3

4

5

Interest (4%) Interest (6%) Interest (8%) b Show this information in graph form. 7 Petra has $4000 to invest at 6% p.a. a Complete the table below to show the future value of the investment at the end of each year, if interest is compounded annually, or six-monthly. No. of years

1

Annually Six-monthly b Show this information in graph form.

2

3

4

5

5


257

Share dividends Investing money in banks and similar financial institutions is the most common type of investment, as it is safe and the return can be calculated in advance. An alternative to investing in a bank is to purchase shares. Shares have a risk associated with them and there is no fixed return; however, they have the potential to return more money to the investor than through a bank. When buying shares you are purchasing a share of the company. In other words, you become a part owner of that company. You can earn money from shares in two ways: 1. The profit made by a company will be paid to the company’s owners (the shareholders). That part of the profit distributed to shareholders is called a dividend. 2. The value of shares changes daily. People invest in the share market with the expectation that the value of shares will rise and they can be sold at a profit. The risk is that the shares may fall in value. Once or twice a year the directors of a company calculate the company’s profit. A certain proportion of the profit may be spent on developing the company, the remainder being distributed to the shareholders as dividends. A dividend is calculated by dividing the profit that is to be distributed by the number of shares in the company. The dividend is then declared on a per-share basis.

WORKED Example 15 A company has an after-tax profit of $34.2 million. There are 90 million shares in the company. What dividend will the company declare if all the profits are distributed to the shareholders? THINK 1 2

WRITE

The dividend is calculated by dividing the profit by the number of shares. Give a written answer.

Dividend = $34 200 000 ÷ 90 000 000 Dividend = $0.38 The dividend is 38c per share.

We can’t accurately compare the values of investments from the dividend alone. We need to consider the money that was invested in order to earn that dividend. A 38c dividend paid by a company with a share value of $12.00 is a lower return than a company that pays a 15c dividend and has a share value of $2.50. To compare the true return from any investment, we need to calculate that return as a percentage of the amount invested. For the income part of a share investment, this percentage is called the dividend yield. To calculate the dividend yield for any share, we calculate the dividend as a percentage of the share price.

WORKED Example 16 A company with a share price of $5.42 declares a dividend of 25c. Calculate the dividend yield, correct to 2 decimal places. THINK

WRITE

Write 0.25 (the dividend) over $5.42 (the share price) and multiply by 100%.

0.25 Dividend yield = ---------- × 100% 5.42 Dividend yield = 4.61%

258


remember 1. An investment in shares earns money through dividend payments and by increasing in value. 2. A dividend is a payment made to shareholders. It is calculated by dividing the profit to be distributed to shareholders by the total number of shares in the company. 3. To calculate the true worth of an investment, we calculate the dividend yield. The dividend yield is found by writing the dividend as a percentage of the share price.

8F 8.5

WORKED

Example

SkillS

HEET

Unless stated otherwise, for the calculations in this exercise, assume that companies distribute all their profits as dividends. 1 A company has issued 20 million shares and makes an after-tax profit of $5 million. Calculate the dividend to be declared by the company.


2 A company that has 2 million shares makes a profit of $3 million. Calculate the dividend that will be declared.

8.6

3 A company makes an after-tax profit of $150 000. If there are 2.5 million shares in the company, calculate the dividend that the company will declare.


4 A company with an after-tax profit of $1.2 million consists of 4.1 million shares. Calculate the dividend the company will declare, in cents, correct to 2 decimal places.

SkillS

HEET

15

Share dividends

5 A company makes a before-tax (gross) profit of $3.4 million. a If the company is taxed at the rate of 36%, calculate the amount of tax it must pay. b What will be the after-tax profit of the company? c If there are 5 million shares in the company, calculate the dividend that the company will declare. 6 A company makes a gross profit of $14.5 million and there are 8 million shares in the company. a Calculate the after-tax profit if company tax is paid at the rate of 36%. b If $3.2 million is to be reinvested in the company, calculate the amount of money that is to be distributed to the shareholders. c Calculate the dividend that this company will declare. 7 A company declares a dividend of 78c. If there are 4.2 million shares in the company, calculate the after-tax profit of the company. WORKED

Example

16

8 A company with a share price of $10.50 declares a dividend of 48c per share. Calculate the dividend yield for this company.


259

9 Copy and complete the table below. Dividend $0.56 $0.78 $1.20 $1.09 $0.04

Share price $8.40 $7.40 $23.40 $15.76 $0.76

Dividend yield

10 Hsiang purchased shares in a company for $3.78 per share. The company paid Hsiang a dividend of 11c per share. Calculate the dividend yield, correct to 2 decimal places. 11 multiple choice Which of the following companies paid the highest dividend yield? A Company A has a share value of $4.56 and pays a dividend of 35c/share. B Company B has a share value of $6.30 and pays a dividend of 62c/share. C Company C has a share value of $12.40 and pays a dividend of $1.10/share. D Company D has a share value of 85c and pays a dividend of 7.65c/share. 12 George bought $5600 worth of shares in a company. The dividend yield for that company was 6.5%. Calculate the amount that George receives in dividends. 13 Andrea bought shares in a company for $11.50 each. The company paid a dividend of 76c/share. a Calculate the dividend yield for this company. b One year later the share value is $12.12. The company then has a dividend yield of 8.75%. Calculate the dividend per share. 14 A company’s prospectus predicts that the dividend yield for the coming year will be 6.7%. Its share price is $21.50. a Calculate the dividend paid if the dividend yield in the prospectus is paid. b If there are 5.2 million shares in the company, calculate the after-tax profit of the company. 15 Janice buys shares in a company at $5.76. The company pays a dividend in July of 22.7c and a dividend in February of 26.4c. Calculate the dividend yield for the whole financial year (July to the following June). 16 The dividend paid by a company for the 2008–09 financial year was 5.6c/share, with a share price of $9.50. a Calculate the dividend yield for 2008–09. b In the 2009–10 financial year the share price rose by 12%. Calculate the share price for this year. c In 2009–10 the dividend paid to shareholders increased by 15%. Calculate the dividend paid, in cents, correct to 1 decimal place. d Calculate the dividend yield for 2009–10.

260


Graphing share performance Because shares offer no guaranteed returns, we can only use the past performance of a share to try to predict its future performance. This is done by graphing the value of the share at regular intervals and then drawing a line of best fit to try to monitor the trend. By continuing the line of best fit you can make a prediction for future share prices. This is called extrapolating information from the graph. Interpolate is the opposite of extrapolate and occurs when drawing a graph using data found at the end points.

WORKED Example 17 Share price ($)

The graph shows the share price of a company over a 3-month period. 4.40 4.20 4.00 3.80

1– Oc t 1– No v 1– De c

1– Se pt

3.60 Month

a On the graph draw a line of best fit. b Use your line of best fit to estimate the share price after another three months. THINK

WRITE

a Draw a line on the graph, which best fits between the points marked.

a Share price ($)

4.80 4.60 4.40 4.20 4.00 3.80

1– Fe b 1– M ar

1– Oc t 1– No v 1– De c 1– Ja n

1– Se pt

3.60 Month

b Extend the line of best fit for three months and read the predicted share price.

b The predicted share price is $4.80.

You should be able to produce your own graph to answer this type of question from a set of data that you have been given or have researched.


261

WORKED Example 18 Below is the share price of a company taken on the first day of the month for one year. Month

Share price

Month

Share price

January

$10.34

July

$10.98

February

$10.54

August

$11.56

March

$10.65

September

$11.34

April

$10.89

October

$11.23

May

$10.72

November

$11.48

June

$11.10

December

$11.72

a On a set of axes plot the share price for each month and draw a line of best fit. b Predict the share price in June of the following year. THINK

WRITE

a

a

Draw up a set of axes and plot the data.

12.40 12.20 12.00 11.80 11.60 11.40 11.20 11.00 10.80 10.60 10.40 10.20

1J 1- an F 1- eb M 1- ar A 1- pr M 1- ay Ju 1- n 1- Jul A 1- ug S 1- ep O 1- ct N 1- ov D 1- ec J 1- an Fe 1- b M 1- ar A 1- pr M 1- ay Ju n

Share price ($)

1

Month

12.40 12.20 12.00 11.80 11.60 11.40 11.20 11.00 10.80 10.60 10.40 10.20 1J 1- an F 1- eb M 1- ar A 1- pr M 1- ay Ju 1- n 1- Jul A 1- ug S 1- ep O 1- ct N 1- ov D 1- ec J 1- an Fe 1- b M 1- ar A 1- pr M 1- ay Ju n

Share price ($)

2

Draw a straight line on the graph that best fits in with the marked points.

Month

b

1 2

Extend the line of best fit for six months. Predict the share price by reading from the line of best fit.

b The predicted share price is $12.35.

262


remember 1. To try to predict possible future movement in share prices, we use the past performance of the share. 2. Graphing the past share price allows us to examine trends by drawing a line of best fit on the graph. 3. We can then use the line of best fit to predict the future price of a share.

Graphing share performance Share price ($)

1 The graph at right shows the movement in a share price over a 2-month period. 17 a Copy the graph into your book and on it draw a line of best fit. b Use your graph to predict the value of the share on 1 November.

WORKED

Example

6.50 6.30 6.10 5.90

1-

l Ju 1-

Ju

M ay

n

5.70 1-

8G

2 The graph at right shows the movement in a share price over a 6-month period. a Copy the graph into your book and on it draw a line of best fit. b Use your graph to predict the value of the share on 1 February.

Share price ($)

Month 4.50 4.00 3.50 3.00 A u 1- g Se pt 1O c 1- t N ov 1D ec

l

1-

1-

Ju

n Ju

1-

1-

M ay

2.50

Month

Share price ($)

1.14 1.12 1.10 1.08 1.06 1.04 1.02

1– Ja n 1– Fe 1– b M a 1– r Ap 1– r M a 1– y Ju n 1– Ju 1– l Au 1– g Se p 1– t Oc t

3 The graph at right shows the movement in a share price over a 9-month period. a Copy the graph into your book and on it draw a line of best fit. b Use your graph to predict the value of the share after a further 12 months.

Month


WORKED

Example

18

263

4 The table below shows the share price of a large multinational company over a 12-month period. Month

Share price

Month

Share price

January

$12.86

July

$13.45

February

$13.43

August

$13.86

March

$11.98

September

$14.40

April

$12.10

October

$13.65

May

$12.11

November

$13.20

June

$12.98

December

$12.86

a Plot the share prices on a set of axes and on your graph draw a line of best fit. b Use your graph to predict the value of the share after a further 6 months. 5 The table below shows the share price of BigCorp Productions Ltd over a period of one year. Month

Share price

Month

Share price

January

$12.40

July

$13.17

February

$12.82

August

$13.62

March

$12.67

September

$13.41

April

$13.05

October

$13.30

May

$13.06

November

$13.46

June

$12.89

December

$13.20

a Graph the share price for each month and show a line of best fit. b Use your line of best fit to predict the share price in December of the next year.

Researching share prices 1 Choose three companies from the business section of the newspaper. 2 Determine the movement of each share over the past year using financial journals, the newspaper or the Internet. 3 Graph the information on the share price that you have found. Include the highest and lowest point of the share price over the past year. 4 On your graph, draw a line of best fit to find the overall trend in the movement of the share price. 5 Try to predict the share price in six months from now by extending the line of best fit. 6 Find the share price each week for six months and see if your line of best fit accurately predicts the share price.

264


Inflation and appreciation One of the measures of how an economy is performing is the rate of inflation. Inflation is the rise in prices within an economy and is generally measured as a percentage. In Australia this percentage is called the Consumer Price Index (CPI). By looking at the inflation rate, we can estimate what the cost of various goods and services will be at some time in the future. To estimate the future price of an item one year ahead, we increase the price of an item by the rate of inflation.

WORKED Example 19 The cost of a new car is $35 000. If the inflation rate is 5%, estimate the price of the car after one year. THINK

WRITE

Increase $35 000 by 5%.

Future price = 105% of $35 000 Future price = 105 ÷ 100 × $35 000 Future price = $36 750

When calculating the future cost of an item several years ahead, the method of calculation is the same as for compound interest. This is because we are adding a percentage of the cost to the cost each year. Remember the compound interest formula is A = P(1 + r)n and so in these examples P is the original price, r is the inflation rate expressed as a decimal and n is the number of years.

WORKED Example 20 The cost of a television set is $800. If the average inflation rate is 4%, estimate the cost of the television set after 5 years. THINK 1 2 3 4

WRITE

Write the values of P, r and n. Write down the compound interest formula. Substitute the values of P, r and n. Calculate.

P = $800, r = 0.04, n = 5 A = P(1 + r)n A = $800 × (1.04)5 A = $973.32

A similar calculation can be made to anticipate the future value of collectable items, such as stamp collections and memorabilia from special occasions. This type of item increases in value over time if it becomes rare, and rises at a much greater rate than inflation. The amount by which an item grows in value over time is known as appreciation.


265

WORKED Example 21 Jenny purchases a rare stamp for $250. It is anticipated that the value of the stamp will rise by 20% per year. Calculate the value of the stamp after 10 years, correct to the nearest $10. THINK

WRITE

1

Write the values of P, r and n.

2

Write down the compound interest formula. Substitute the values of P, r and n. Calculate and round off to the nearest $10.

3 4

P = $250, r = 0.2, n = 10 A = P(1 + r)n = $250 × (1.2)10 = $1550

remember 1. Inflation is the measure of the rate at which prices increase. 2. The inflation rate is given as a percentage and is called the Consumer Price Index. 3. To estimate the cost of an item after one year, we increase the price by the percentage inflation rate. 4. To estimate the cost of an item after several years, we use the compound interest formula, using the inflation rate as the value of r. 5. Rare items such as collectibles and memorabilia increase in value as time goes on at a rate that is usually greater than inflation.

8H WORKED

Example

19

Inflation and appreciation

1 The cost of a motorcycle is $20 000. If the inflation rate is 4%, estimate the cost of the motorcycle after one year. 2 For each of the following, estimate the cost of the item after one year, with the given inflation rate. a An MP3 player costing $600 with an inflation rate of 3% b A toaster costing $45 with inflation at 7%

266


c A loaf of bread costing $1.80 with inflation at 6% d An airline ticket costing $560 with inflation at 3.5% e A washing machine costing $925 with inflation at 0.8% 3 An electric guitar is priced at $850 at the beginning of 2008. a If the inflation rate is 3.3% p.a., estimate the cost of the guitar at the beginning of 2009. b The government predicts inflation will fall to 2.7% in 2009. Estimate the cost of the guitar at the beginning of 2010. 4 When the Wilson family go shopping, the weekly basket of groceries costs $112.50. The inflation rate is predicted to be 4.8% for the next year. How much should the Wilson’s budget per week be for groceries for the next year? WORKED

Example

20

5 The cost of a lawnmower is $550. If the average inflation rate is predicted to be 3%, estimate the cost of the lawnmower after 4 years. 6 The cost of a litre of milk is $1.70. If the inflation rate is an average 4%, estimate the cost of a litre of milk after 10 years. 7 A daily newspaper costs $1.00. With an average inflation rate of 3.4%, estimate the cost of a newspaper after 5 years (to the nearest 5c). 8 If a basket of groceries costs $98.50 in 2008, what would the estimated cost of the groceries be in 2015 if the average inflation rate for that period is 3.2%? 9 multiple choice A bottle of soft drink costs $2.50. If the inflation rate is predicted to average 2% for the next five years, the cost of the soft drink in five years will be: A $2.60 B $2.70 C $2.75 D $2.76

10 Veronica bought a shirt signed by the Australian cricket team after it won the 2007 World Cup for $200. If the value of the shirt increases by 20% per annum for the next 21 5 years, calculate the value of the shirt (to the nearest $10).

WORKED

Example

11 Ken purchased a rare bottle of wine for $350. If the value of the wine is predicted to increase at 10% per annum, estimate the value of the wine in 20 years (to the nearest $10).

Work

T SHEE

8.2

12 The 1968 Australian 2c piece is very rare. If a coin collector purchased one in 2006 for $400 and the value of the coin increases by 15% per year, calculate its value in 2019 (to the nearest $10).


267

summary Simple interest • Simple interest is interest paid where the interest is not added to the principal before the next interest calculation. • It is calculated using the formula: I = Prn where P is the initial quantity, r is the percentage interest rate per annum expressed as a decimal and n is the number of periods. • It can be graphed as a linear function.

Compound interest • Compound interest is the interest added to the principal before the next interest calculation is made. • It can be calculated by using the formula: A = P(1 + r)n where A is the final balance, P is the initial quantity, r is the percentage interest rate per interest period expressed as a decimal and n is the number of compounding periods. • The amount of compound interest paid is found by subtracting the principal from the future value of the investment. • Compound interest can be calculated by using a table of compounded values of $1. • When a compound interest function is graphed, it is an exponential function.

Shares • When you buy shares you purchase a share in the company. There is no guaranteed return with shares, although there is a greater potential for profit than with investments such as banking and property, but with that comes a higher risk. • Profit can be made from buying shares in two ways: (a) The value of the share could rise over time. (b) The company may pay a dividend to its shareholders. The dividend when written as a percentage of the share price is called the dividend yield. • To try to predict the future movement in share prices, we can graph the past movement in the share price and draw a line of best fit on the graph. This line of best fit can be extrapolated to estimate the future price.

Inflation • The price of goods and services rise from year to year. To predict the future price of an item, we can use the compound interest formula taking the rate of inflation to be r. • The same method is used to predict the future value of collectibles and of memorabilia, which tend to rise at a rate greater than inflation.

268


CHAPTER review 8A 8A

1 Calculate the simple interest earned on an investment of $5000 at 4% p.a. for 5 years. 2 Calculate the simple interest earned on each of the following investments. a $3600 at 9% p.a. for 4 years b $23 500 at 6% p.a. for 2 years c $840 at 2.5% p.a. for 2 years d $1350 at 0.2% p.a. for 18 months e $45 820 at 4.75% p.a. for 3 1--- years 2

8A

3 Dion invests $32 500 in a debenture paying 5.6% simple interest for 4 years. a Calculate the interest earned by Dion. b Calculate the total value of Dion’s investment after 4 years. c If the debenture payed Dion in quarterly instalments, calculate the value of each interest payment.

8A

4 Bradley invests $15 000 for a period of 4 years. Calculate the simple interest rate, given that Bradley earned a total of $3900 interest.

8A

5 Kerry invests $23 500 at a simple interest rate of 4.6% p.a. If he earned $1351.25 in interest, calculate the length of time for which the money was invested.

8C

6 An amount of $7500 is to be invested at 6% p.a. a Copy and complete the table below to calculate the simple interest over 5 years. 1

No. of years

2

3

4

5

Interest b Draw a graph of the interest earned against the length of the investment. c What is the gradient of the linear graph drawn? d Use your graph to find the amount of interest that would have been earned after 10 years.

8C

7 Vicky invests $2400 at 5% p.a. for 3 years with interest compounded annually. Calculate the compounded value of the investment at the end of the term.

8C

8 Barry has an investment with a present value of $4500. The investment is made at 6% p.a. with interest compounded six-monthly. Calculate the value of the investment in 4 years.

8C

9 Calculate the compounded value of each of the following investments. a $3000 at 7% p.a. for 4 years with interest compounded annually b $9400 at 10% p.a. for 3 years with interest compounded six-monthly c $11 400 at 8% p.a. for 3 years with interest compounded quarterly d $21 450 at 7.2% p.a. for 18 months with interest compounded six-monthly e $5000 at 2.6% p.a. for 2 1--- years with interest compounded quarterly 2

8C

10 Dermott invested $11 500 at 3.2% p.a. for 2 years with interest compounded quarterly. Calculate the total amount of interest paid on this investment.


269

11 Kim and Glenn each invest $7500 for a period of 5 years. a Kim invests her money at 9.9% p.a. with interest compounded annually. Calculate the compounded value of Kim’s investment. b Glenn invests his money at 9.6% p.a. with interest compounded quarterly. Calculate the compounded value of Glenn’s investment. c Explain why Glenn’s investment has a greater compounded value than Kim’s.

8C

12 Use the table of CVIF values on page 248 to calculate the compounded value of each of the following investments. a $6000 at 7% p.a. for 4 years with interest compounded annually b $7230 at 9% p.a. for 7 years with interest compounded annually c $3695 at 6% p.a. for 3 years with interest compounded six-monthly d $12 400 at 10% p.a. for 5 years with interest compounded six-monthly e $2400 at 4% p.a. for 2 years with interest compounded quarterly

8C

13 $20 000 is to be invested at 4% p.a. with interest compounded annually. a Copy and complete the table below, using the compound interest formula to calculate the future value at the end of each year.

8E

No. of years

1

2

3

4

5

Future value b Draw a graph of the interest earned against the length of the investment. c Use your graph to find the future value of the investment after 10 years. 14 A company that has 10.9 million shares makes a profit of $21 million. If this entire amount is distributed among the shareholders, calculate the dividend that will be declared.

8F

15 A company that has an after-tax profit of $2.3 billion distributes this among its 156 million shares. Calculate the dividend that this company will declare.

8F

16 A company has a share price of $8.62. It declares a dividend of 45c per share. Calculate the dividend yield on this share.

8F

17 A company with a share price of 45c declares a dividend of 0.7c per share. Calculate the dividend yield on this investment.

8F

18 The dividend yield from a share valued at $19.48 is 4.2%. Calculate the dividend paid by the company, correct to the nearest cent.

8F

19 The table below shows the fluctuations in a share price over a period of 1 year. Month

Share price

Month

Share price

January

$15.76

July

$16.60

February

$16.04

August

$16.77

March

$16.27

September

$16.51

April

$16.12

October

$16.71

May

$16.49

November

$16.69

June

$16.39

December

$16.98

a On a set of axes plot the share price for each month. b Draw a line of best fit on your graph and use your line to predict the share price after a further year.

8G

270


8H

20 A MP3 player is currently priced at $80. If the current inflation rate is 4.3%, estimate the price of the MP3 player after one year.

8H

21 It is predicted that the average inflation rate for the next five years will be 3.7%. If a skateboard currently costs $125, estimate the cost of the skateboard after five years.

8H

22 In 1979, Cherie bought a limited edition photograph autographed by Sir Donald Bradman for $120. If the photograph appreciates in value by 15% per annum, calculate the value of the photograph in 2009 (to the nearest $100).

Practice examination questions 1 multiple choice The simple interest paid on $5600 at 5.6% for 3 years is: A $940.80 B $994.46 C $6540.80

D $6594.47

2 multiple choice The compound interest paid on $5600 at 5.6% for 3 years with interest compounded annually is: A $940.80 B $994.46 C $6540.80 D $6594.47 3 multiple choice A share is valued at $23.40. Greg buys 4000 shares and, at the end of the financial year, Greg receives a dividend of $4212. The dividend yield on Greg’s investment is: A 0.55% B 1.053% C 4.5% D 5.3% 4 multiple choice In 2008, a basket of groceries costs $67.50. If the inflation rate is predicted to be 2.9% for the next year, by how much can we expect the cost of the basket of groceries to rise? A $1.95 B $1.96 C $69.45 D $69.46 5 Jaclyn has $7500 saved for a holiday that she plans to take in two years time. a If Jaclyn invests the money in a debenture that pays 4.2% p.a. simple interest, calculate the amount of money that Jaclyn will have after two years. b An alternative investment for Jaclyn would be to invest her money at 4% p.a. for two years with interest compounding quarterly. Would this be a better investment? Explain your answer. c Jaclyn finally decided to buy 1500 shares in a company at $5.00 each. For the past year the dividend yield for this company was 5.1%. Is this a safe investment for Jaclyn? d After two years, the average dividend yield for this company was 4.8% p.a. of Jaclyn’s initial investment and the shares were valued at $5.75 each. Calculate the total value of Jaclyn’s investment. e Calculate Jaclyn’s profit as a percentage of her initial investment.

CHAPTER

test yourself

8

6 Frank has saved $30 000 to buy a new car. He decides to try to get another two years use out of his old car and in the meantime invest the money he has saved. a If Frank invests the $30 000 at 3.5% p.a. for two years with interest compounded annually, calculate the money that Frank has at the end of the investment. b Over the two years that Frank has invested his money, the inflation rate has averaged 4.2% p.a. Calculate the cost of the car at the end of this two years if the price rose at the same rate as inflation (to the nearest $100). c How much more money does Frank now need to buy the new car?

Displaying single data sets

9 syllabus reference Data analysis 3 • Displaying single data sets

In this chapter 9A 9B 9C 9D

Frequency tables Types of graphs Statistical graphs Range and interquartile range 9E Stem-and-leaf plots 9F Five-number summaries

areyou

READY?

Are you ready?


9.5

1 The line graph at right shows the height of a child (Timmy) over 5 years. a How tall was Timmy at the start of the measurement period? b How much did Timmy grow in the first year? c How much did Timmy grow over the five years? d How many years did it take for Timmy to grow 10 cm?

160 150 140 130 120 110 100 1990 1991 1992 1993 1994 1995 Years

Presenting data as a dot plot

2 Copy and complete the dot plot at right for the data shown below. 6, 4, 3, 11, 9, 12, 5, 7, 7, 6, 3, 6, 5, 6, 8, 6, 9, 5, 6, 7

3 4 5 6 7 8 9 10 11 12

Producing a frequency table from a frequency histogram

3 Copy and complete the following frequency table to show the data represented in the frequency histogram. Score (x) 20 21 22 23 24

Frequency (f) 5

9.6

Finding the median of a small data set

9.7

Presenting data as a stem-and-leaf plot

Frequency

9.4

Reading a line graph

Height (cm)

9.3

Increase in Timmy’s height between 1990 and 1995

170

8 7 6 5 4 3 2 1 0 20 21 22 23 24 Score

4 Find the median of each of the following data sets. a 2, 6, 8, 4, 5, 6, 7 b 13, 10, 15, 12, 18, 17, 12, 12 c 52, 45, 23, 83, 9, 45, 71, 84, 90, 183 d 9.2, 9.3, 9.4, 9.3, 9.4, 9.5, 9.9, 9.4, 9.7, 9.8, 9.0, 10.0, 9.4, 9.2, 9.9

5 Copy and complete the following stem-and-leaf plot for the given data. 52, 34, 38, 45, 10, 33, 27, 19, 33, 46

Key: 2 | 3 = 23 Stem Leaf 1 _9 2 _ 3 3_4_ 4 5_ 5 _

Chapter 9 Displaying single data sets

273

Frequency tables A frequency table is used to tabulate statistical data. A well-designed table allows the person doing the statistical investigation to easily tally up the data. In this chapter we are concerned with the tallying of quantitative data. When working with quantitative data, each No. of visits Frequency piece of data is known as a score. Quantitative data may be presented as either grouped or 0 6 ungrouped data. The table at right shows the number of cinema visits during a month by each 1 7 of 20 students. It is an example of ungrouped 2 4 data because each score is a separate group in itself. 3 2 Ungrouped data are suitable for discrete data 4 1 that do not have a wide range of scores. A frequency table usually has a tally column that allows us to enter a tally mark for each score as it is entered into the table. The gatepost method of tallying (the fifth tally mark crosses the previous four like a gatepost) allows us to easily count up the frequency at the end.

WORKED Example 1 Fifty people were surveyed and asked the number of videos that they had hired from a video store in the past month. The results are shown below. 2 1 3 0 0

4 3 4 1 2

5 3 4 5 0

3 0 2 5 4

3 5 0 1 4

3 2 1 2 4

3 2 4 3 2

0 1 0 5 2

5 3 3 3 4

1 1 2 4 5

Enter the information in a frequency table. THINK 1

Draw a table with three columns and with scores from 0 to 5.

2

Enter a tally mark for each score.

3

Count the tally marks for each score and enter the result in the frequency column.

WRITE/DRAW

Score 0 1 2 3 4 5

|||| |||| |||| |||| |||| ||||

Tally || || |||| |||| | |||| ||

Frequency 7 7 9 11 9 7

274


When the data are continuous or if the data are spread over a wide range, it is useful to group the scores into groups or classes. The following table contains grouped data. It shows the number of passengers on each of 20 bus trips. No. of passengers

Frequency

5–9

1

10–14

6

15–19

8

20–24

4

25–29

1

When making the decision to summarise some raw data by grouping them in a frequency table, the choice of class size (group size) is important. As a general rule, try to choose a class size so that 5 to 10 groups are formed. Choosing a suitable data label is also important. Consider the following data that give the number of nails in each of a sample of 40 boxes. 130 132 137 137

122 138 134 124

118 129 126 134

139 139 129 134

126 116 127 120

128 123 118 137

119 126 130 141

124 128 132 118

122 131 134 125

123 142 132 129

Let’s say that we wish to represent the data in a frequency table. It would be pointless trying to tally the data as individual (ungrouped) scores, as most scores would occur only once or twice in the whole distribution. It makes good sense to group the scores. A class size of five would be appropriate as this leads to the formation of six classes. With grouped data, an extra column is shown for the class centre. The class centre is the middle of each class. No. of nails

Class centre

Tally

Frequency

115–119

117

||||

5

120–124

122

|||| ||

7

125–129

127

|||| ||||

10

130–134

132

|||| ||||

10

135–139

137

|||| |

6

140–144

142

||

2


275

WORKED Example 2 The height of 40 students was measured and the results are shown below. 146 141 155 166 168 158 169 164 141 154 159 143 152 156 146 146 161 150 141 153 152 148 152 142 162 159 141 151 169 169 164 148 168 169 146 162 151 150 143 140 Put the above results into a frequency table.

THINK 1

2 3 4

5

WRITE

The data range from 140 cm to 168 cm. Choose a group of 5 cm beginning at 140 cm. Calculate the class centres. Draw a frequency table with four columns and room for six classes. Enter a tally mark in the appropriate class as each height is read. Complete the frequency table by counting the tally marks.

Height 140–144 145–149 150–154 155–159 160–164 165–169

Class centre 142 147 152 157 162 167

Tally |||| ||| |||| | |||| |||| |||| |||| |||| ||


In some examples the class groupings overlap at the boundary. For example, if we are collecting information on people’s mass, the categories used may be 50–55, 55–60, 60–65, etc. In such examples, the class centres would be 52.5, 57.5, 62.5, . . . etc. When tabulating such data, convention is that a score on the boundary is placed in the higher class. For example, a person with a mass of 60 kg would be placed in the 60–65 class.

remember 1. 2. 3. 4.

Statistical information may be tabulated using a frequency table. A frequency table has three columns for score, tally and frequency. The data can be entered as either grouped or ungrouped data. Ungrouped data are where each score becomes a group on its own and is suitable for a small range of scores for which the data are discrete. 5. Grouped data are put into classes. They are suitable for continuous data or discrete data for which there is a wide range of scores. For grouped data the class centre should be shown.

276


9A 9.1

SkillS

HEET


1 Twenty households were surveyed to find the number of people in that household. The results are shown below. 1 4 3 4 6 3 2 5 2 7 4 5 6 4 3 5 4 6 2 3 4 Use this information to complete a copy of the frequency table below.

WORKED

Example

Score

reads L Sp he

Tally

Frequency

2

et

EXCE

Frequency tables

Frequency tables

3 4 5 6 7 2 The marks of 25 students on a spelling test are shown below. 4 5 8 5 10 7 6 9 7 6 5 7 6 4 7 5 6 7 5 7 8 4 6 8 7 Put this information into a frequency table. 3 The scores of 50 professional golfers in a round of golf are shown below. 72 70 69 75 78 68 66 68 67 72 72 71 68 73 72 71 74 77 72 73 72 72 72 74 70 71 73 72 77 74 76 68 69 68 77 75 72 72 72 71 73 72 70 74 72 71 73 68 67 67 Display this information in a frequency table. 4 A class of 30 students sat for a Mathematics test. Their results out of 100 are shown below. 2 68 72 58 45 69 92 38 51 70 65 69 73 52 76 48 69 73 41 42 73 80 50 60 49 65 94 88 85 53 60 Use these results to copy and complete the frequency table below.

WORKED

Example

Score 30–39 40–49 50–59 60–69 70–79 80–89 90–99

Class centre

Tally

Frequency


277

5 A farmer measures the heights of his tomato plants. The results, in metres, are shown below. 0.93 1.21 2.03 1.40 1.17 1.53 1.82 1.77 1.65 0.63 1.24 1.99 0.80 2.14 1.53 2.07 1.96 1.05 0.94 1.23 1.72 1.34 0.75 1.17 1.50 1.41 1.74 1.86 1.55 1.42 1.52 1.39 1.76 1.67 1.28 1.43 2.13 Use the class groupings 0.6–0.8, 0.8–1.0, 1.0–1.2, . . . etc. to complete a frequency distribution table for these data. 6 The following data give the times (in seconds) taken for athletes to complete a 100 m sprint. 12.2 12.0 11.9 12.0 12.6 11.7 11.4 11.0 10.9 11.7 11.2 11.8 12.2 12.0 12.7 12.9 11.3 11.2 12.8 12.4 11.7 10.8 13.3 11.7 11.6 11.7 12.2 12.7 13.0 12.2 Construct a frequency distribution table for the data. Use a class size of 0.5 seconds.

Types of graphs Once statistical information has been tabulated, the next step is to display the data in some type of graph. The type of graph that is chosen depends on the purpose of the graph.

Dot plots A dot plot is used to display a set of scores on a number line. This graph is useful for showing a small number of scores.

WORKED Example 3 Below are the scores out of 10 achieved by 11 students on a Maths quiz. 7 4 8 7 6 7 6 5 8 9 5 Show this information on a dot plot. THINK 1 2

Draw a number line showing all numbers from 0 to 10. Place a dot on the appropriate number for each score, stacking the dots to show where more than one of the same score has occurred.

WRITE

0 1 2 3 4 5 6 7 8 9 10

278


Sector graph (pie chart) A sector graph is used to compare quantities. A circle is divided into proportions to be shown on the graph. To calculate the angles at the centre of the circle, the fraction of the total quantity is multiplied by 360°.

WORKED Example 4 A poll was conducted on an issue of justice based on the statement ‘Penalties for serious crime should be increased’. The responses were: Agree 35 Disagree 20 Unsure 5. Show this information in a sector graph. THINK

WRITE

2

Calculate the total number of responses. Calculate the angles at the centre of the circle by multiplying each fraction by 360°.

3

Draw the graph.

1

35 + 20 + 5 = 60 ------ × 360° Agree = 35 60 = 210° 5 - × 360° Unsure = ----60 = 30°

------ × 360° Disagree = 20 60 = 120°

Agree Disagree Unsure

Line graph A line graph is used to compare the change in one quantity with the change in another. It is suitable for quantitative data.

WORKED Example 5 The table below shows the temperature taken in a school playground over a day. Time Temp (°C)

9:00 am 15

10:00 am 11:00 am 12:00 pm 17 20 22

1:00 pm 23

2:00 pm 25

Show this information in the form of a line graph.

1 2 3

Show time on the horizontal axis and temperature on the vertical axis. Plot the points for the hourly temperature. Join each point with a straight line.

DRAW Temperature

THINK

25 20 15 0 m m m m m m m 0a 0a 0a 0p 0p 0p 0p 9:0 10:0 11:0 12:0 1:0 2:0 3:0 Time

3:00 pm 22


279

Bar and column graphs A bar graph or a column graph is used when we want to show a quantity and compare it to other quantities. This is particularly suitable for categorical data. In a bar graph, the categories are shown on the vertical axis and the results are shown on the horizontal axis. The bars are then drawn horizontally. These axes are reversed for a column graph.

WORKED Example 6 The information shown below represents the unemployment rates in various NSW regions. Hunter 12.9% Illawarra 11.8% Mid North Coast 16.4% Western NSW 9.1% Riverina 8.4% Show this information in a bar graph.

2

Draw the categories on the vertical axis and the percentages on the horizontal axis. Draw horizontal bars for each category.

Riverina Western NSW Mid North Coast Illawarra Hunter 0 5 10 15 20 Percentage unemployed

Graphs can be used to misrepresent information. This can be done by adjusting the scale on the horizontal axis. The graph at right shows the same information as worked example 6.

Riverina Region

1

DRAW

Region

THINK

Western NSW Mid North Coast Illawarra Hunter 7 9 11 13 15 17 Percentage unemployed

Radar charts A radar chart is similar to a line graph except it is circular. It is particularly suitable for showing data trends that repeat. The ‘radar’ is drawn with the data being measured placed in equal sectors around the circle and the results having a scale emanating from the centre. The points are then plotted and joined.

280


WORKED Example 7 The information below shows the sales in a department store over a year. Month January February March April May June

Sales ($m) 2.8 1.7 1.1 1.2 1.3 1.6

Month July August September October November December

Sales ($m) 1.8 1.1 1.6 1.9 2.5 3.4

Show these data in a radar chart. THINK 1 2 3 4

WRITE

Draw equal sectors of 30°. Draw the sales from the centre using 1 cm = $0.5 million. Plot the points. Join each point with a straight line.

December November

October

Sales ($m) January 3.5 3 2.5 2 1.5 1 0.5 0

February March

April

September

May

August

June July

remember 1. A graph is used to visually display data that have been tabulated. 2. You should choose the most appropriate graph for the purpose of the display and the data type. 3. You should be able to draw dot plots, sector graphs, line graphs, bar graphs and radar charts.

9B 9.2

SkillS

HEET

WORKED

Example

3 Reading a column graph

Types of graphs

1 Below are the scores out of 10 on a spelling test as achieved by a small class of students. Show the information on a dot plot. 4 5 3 7 8 10 9 6 7 7 8 6 8 6 7 2 Below is the maximum temperature in Sydney each day during February. 28 35 33 34 30 27 28 29 30 26 28 30 31 33 32 31 30 31 29 29 28 25 26 30 31 30 29 29 Show this information on a dot plot.

281


Example

4 Gary earns $600 per week. His weekly budget is: Rent $200 Bills $150 Groceries $120 Car running costs $70. The balance is put away for savings. Display Gary’s budget in a sector graph.

5

Reading a line graph

9.4

Year

2000

2001

2002

2003

2004

2005

2006

2007

Pop.

12 000

12 500

13 250

13 500

13 600

14 100

15 000

16 000

6 Tania has a savings account. The balance of the account at the beginning of each month for a year is shown in the table below. Balance

January

$200

July

$500

February

$270

August

$460

March

$320

September

$570

April

$260

October

$625

May

$380

November

$570

June

$430

December

$700

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Line graphs L Spre XCE ad

E

Month

Bar graphs

E

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Example

6

7 Below are the average weekly earnings of people in different categories of employment. Labourers $543 Plant operators $598 Salespersons $607 Clerks $620 Tradespersons $640 Para-professionals $780 Professionals $865 Managers $906 Show this information in a horizontal bar graph. 8 The table below shows the number of Olympic gold medals Australia has won since 1952. Year and venue 1952 Helsinki 1956 Melbourne

No. of gold medals 6 13

Year and venue

No. of gold medals

1980 Moscow

2

1984 Los Angeles

4

1960 Rome

8

1988 Seoul

3

1964 Tokyo

6

1992 Barcelona

7

1968 Mexico City

5

1996 Atlanta

9

1972 Munich

8

2000 Sydney

16

1976 Montreal

0

2004 Athens

17

Show this information in a column graph.

Sector graphs

sheet

Show this information in a line graph.

sheet

Balance

Column graphs

sheet

Month

sheet

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Show this information in a line graph.

WORKED

SkillS

Presenting data as a dot plot E

Example

5 The population over 8 years in a small town is shown in the table below.

SkillS

HEET

WORKED

9.3

HEET

4

3 During a season, a soccer team has 10 wins, 4 draws and 6 losses. Show these results in a sector graph.

E

WORKED

282 WORKED

Example

7


9 The table below shows the average monthly temperature in Sydney. Month

Temp (°C)

Month

Temp (°C)

January

28

July

15

February

30

August

16

March

26

September

20

April

24

October

22

May

20

November

25

June

17

December

27

Show this information in a radar chart.

10 The table below shows the percentage of televisions that are being watched over a 24-hour period. Time

Percentage

Time

Percentage

12:00 am

12%

12:00 pm

30%

2:00 am

1%

2:00 pm

33%

4:00 am

2%

4:00 pm

45%

6:00 am

8%

6:00 pm

60%

8:00 am

15%

8:00 pm

78%

10:00 am

24%

10:00 pm

55%

Show this information in a radar chart.


283

Choice of graph For different statistical investigations, some graphs may be more suitable than others. Depending on the information that needs to be displayed, each graph has its strengths and weaknesses. In the above exercise, you drew a dot plot, sector graph, line graph, column graph and radar chart. For each of these graph types describe: a The advantages of this type of display. b The disadvantages of this type of display. c An example of when this type of display would be most suitable.

E

Column graphs

E

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sheet

Line graphs

E

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sheet

Each of the graph types described in this section can be drawn using a spreadsheet or a graphics calculator. From the Maths Quest General Mathematics Preliminary Course CD-ROM, click on each of the spreadsheet icons in the margin to practise using a spreadsheet to produce a graph.

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sheet

Producing graphs using technology

Bar graphs

E

Sector graphs

Statistical graphs When displaying statistical, or quantitative data, the most useful graphs are a frequency histogram and frequency polygon. A histogram is similar to a column graph with the following essential features. • Gaps are never left between the columns, except for a half unit space before the first column. • If the chart is coloured or shaded then it is done all in one colour. (The columns are essentially all representing different levels of the same thing.) • Frequency is always plotted on the vertical axis. • For ungrouped data, the horizontal scale is marked so that the data labels appear under the centre of each column. For grouped data, the horizontal scale is marked so that the class centre of each class appears under the centre of the column.

sheet

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284


WORKED Example 8 The table below shows the number of people living in each house in a street. No. of people 1 2 3 4 5

Frequency 1 4 10 15 8

Show this information in a frequency histogram.

1

2

DRAW

Draw a set of axes with the number of people living in a house on the horizontal axis and frequency on the vertical axis. Draw the graph, leaving half a column width space before the first column.

Frequency

THINK

16 14 12 10 8 6 4 2 0

1 2 3 4 5 Number of people in a house

Frequency

A frequency polygon is a line graph that can be drawn by joining the centres of the tops of each column of the histogram. The polygon starts and finishes on the horizontal axis a half column width space from the group boundary of the first and last groups. The figure at right shows the frequency polygon 16 drawn on top of the histogram for the previous 14 12 worked example. 10 It is common practice to draw the histogram and 8 6 the polygon on the same set of axes. 4 2 0

1 2 3 4 5 Number of people in a house

WORKED Example 9 The frequency table below shows a class set of marks on an exam. Draw a frequency histogram and polygon on the same set of axes. Mark 51–60 61–70 71–80 81–90 91–100

Class centre 55.5 65.5 75.5 85.5 95.5

Frequency 3 5 12 7 3


1

2 3

DRAW

Draw a set of axes with the exam mark on the horizontal axis and frequency on the vertical axis. Show the class centres for the exam marks. Draw the columns, leaving a half column width space before the first column. Draw a line graph to the centre of each column.

Frequency

THINK

285

12 10 8 6 4 2 0

.5 .5 .5 .5 .5 55 65 75 85 95 Exam mark

No. of people

Frequency

Cumulative frequency

1

1

1

2

4

5

3

10

15

4

15

30

5

8

38

Once the cumulative frequency has been calculated, a cumulative frequency histogram and polygon (ogive) can be drawn. The cumulative frequency histogram is drawn without the half column space before the first column. The cumulative frequency polygon is drawn to the top right-hand corner of each column.


For further analysis of statistical data it is useful to draw cumulative frequency graphs. The cumulative frequency is a progressive total of the frequency column. Consider the table from worked example 8.

40 35 30 25 20 15 10 5 0 1 2 3 4 5 Number of people in a house

WORKED Example 10 The frequency table below shows the heights of people in a basketball squad. Height (cm) 170–174 175–179 180–184 185–189 190–194 195–199

Class centre 172 177 182 187 192 197

Frequency 3 6 12 10 8 1

a Add a cumulative frequency column to the table. b Draw a frequency histogram and polygon. Continued over page

286


THINK

WRITE/DRAW

a

a

2

b

Add a fourth column to the table.

Class centre

Frequency


170–174

172

3

3

175–179

177

6

9

180–184

182

12

21

185–189

187

10

31

190–194

192

8

39

195–199

197

1

40

Complete the column by keeping a running total of the frequencies.

1

Draw the axes with height on the horizontal axis and cumulative frequency on the vertical axis. Show the class centres for the height.

2

Draw the columns for the cumulative frequency histogram.

3

Height (cm)

Draw the cumulative frequency polygon by joining the lines to the top right-hand corner of each column.

b


1

40 35 30 25 20 15 10 5 0

172 177 182 187 192 197 Height (cm)


287

remember 1. A piece of numerical data is known as a score. 2. Quantitative data can be graphed using histograms and polygons. 3. When drawing histograms, always put frequency on the vertical axis and never leave gaps between columns. 4. If the histogram is illustrating ungrouped data, the data labels on the horizontal axis are placed under the centre of each column. 5. If the histogram is illustrating grouped data, the data labels on the horizontal axis (that is, the class centres) are placed under the centre of each column. 6. A polygon is a line graph that can be drawn by joining the centres of the tops of each column of the histogram. 7. The cumulative frequency is a running total of the frequency column. The cumulative frequency can also be graphed using a cumulative frequency histogram and polygon.

9C

No. of drivers (frequency)

0

5

1

8

2

13

3

4

4

3

5

1

9.5 Producing a frequency table from a frequency histogram

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Histogram, frequency polygon L Spre XCE ad

No. of children in a family

Frequency

1

3

2

5

3

8

4

4

5

2

6

1

Histogram, frequency polygon (DIY)

sheet

2 Each student in a class was asked the number of children in their family. The results are shown in the frequency distribution table at right. Show this information in a frequency histogram and polygon.

SkillS

sheet

No. of mistakes (score)

HEET

8

1 A survey was done on young drivers taking the written test for their licence. The number of mistakes each made was recorded and the results are shown in the frequency distribution table at right. Show this information in a frequency histogram.

E

Example

E

WORKED

Statistical graphs

288


3 The table below shows the age in years of the members of a surf club. Age

No. of members

18

3

19

5

20

8

21

13

22

15

23

10

24

8

25

5

Show this information on a frequency polygon. 4 The label on a box of matches states that the average contents of a box is 50 matches. Quality control surveyed 50 boxes for the number of matches and the results are shown below. 48 50 50 51 50 49 53 52 48 51 50 50 51 49 48 53 52 50 49 49 49 50 50 51 53 52 54 47 50 49 48 49 47 53 49 52 50 51 50 50 50 48 47 50 51 49 50 49 52 51 a Put this information into a frequency table. b Show the results on a frequency histogram and polygon. WORKED

Example

9

5 The table below shows the length of 71 fish caught in a competition. Length of fish (mm)

Class centre

Frequency

300–309

304.5

9

310–319

314.5

15

320–329

324.5

20

330–339

334.5

12

340–349

344.5

8

350–359

354.5

7

Show this information in a frequency histogram and polygon.


289

6 Sixty people were involved in a psychology experiment. The following frequency table shows the times taken for the 60 people to complete a puzzle for the experiment. Time taken (seconds)

Class centre

Frequency

6–8

1

8–10

4

10–12

16

12–14

18

14–16

12

16–18

8

18–20

2

a Copy the frequency table and complete the class centre column. b Show the information in a frequency histogram and polygon. WORKED

Example

10

7 The following data show the number of registered cars normally kept at each of 30 households. No. of cars

Frequency

0

2

1

8

2

11

3

6

4

2

5

1

a Copy the table and add a cumulative frequency column. b Draw a cumulative frequency histogram and polygon. 8 The following table shows the number of jelly beans in each of 60 packets. No. of jelly beans

Frequency

48

2

49

10

50

32

51

9

52

5

53

2

a Add a cumulative frequency column to the table. b Draw a cumulative frequency histogram and polygon.

290


9 The following frequency table gives the number of oysters of different lengths from a tray in a marine farm. Length (cm)

Class centre

Frequency

4–5

6

5–6

10

6–7

60

7–8

58

8–9

8

9–10

4


a Copy and complete the table. b Show the information in the form of a cumulative frequency histogram and polygon.

10 The following frequency table gives the results of testing the lives of 200 torch batteries.

Work

T SHEE

9.1

Lifetime (hours)

Frequency

20–25

6

25–30

25

30–35

70

35–40

61

40–45

30

45–50

8

a Redraw the table, including a column for class centre and cumulative frequency. b Draw a cumulative frequency histogram and polygon.


291

1 Class

Class centre

Frequency

1–10

5

11–20

15

21–30

29

31–40

37

41–50

11

1 Copy the frequency table above and complete the class centre column. 2 Complete the cumulative frequency column. 3 How many scores in the data set were above 30? 4 How many scores in the data set were 40 or less? 5 Is the data set an example of grouped or ungrouped data? 6 Draw a frequency histogram for the data set. 7 On your histogram, draw a polygon for this data set. 8 Draw a cumulative frequency histogram for the data set. 9 On your cumulative frequency histogram, draw a cumulative frequency polygon. 10 Copy and complete: Another term for a cumulative frequency polygon is an .


292


Range and interquartile range Once a set of scores has been collected, tabulated and graphed, we are ready to make some conclusions about the data. The range and interquartile range are used to measure the spread of a set of scores. The range is the difference between the highest and the lowest score. Range = highest score − lowest score

WORKED Example 11 There are 17 players in the squad for a State of Origin match. The number of State of Origin matches played by each member of the squad is shown below. 2 6 12 8 1 4 8 9 24 4 5 11 14 6 11 15 10 What is the range of this distribution? THINK 1 2 3

WRITE

The lowest number of matches played is 1. The highest number of matches played is 24. Calculate the range by subtracting the lowest score from the highest score.

Lowest score = 1 Highest score = 24 Range = 24 − 1 Range = 23

A smaller range will usually represent a more consistent set of scores. Exceptions to this are when one or two scores are much higher or lower than most. When we are calculating the range from a frequency distribution table, we find the highest and lowest score from the score column. We do not use any information from the frequency column in calculating the range. When the data are presented in grouped form, the range is found by taking the highest score from the highest class and the lowest score from the lowest class.

WORKED Example 12 The frequency distribution table at right shows the heights of boys competing for a place on a basketball team. Find the range of these data.

Height

Frequency

170–175

3

175–180

6

180–185

12

185–190

10

190–195

8

195–200

1


THINK 1 2 3

293

WRITE

The lowest score is at the bottom of the 170–175 class. The highest score is at the top of the 195–200 class. Range = highest score − lowest score.

Lowest score = 170 Highest score = 200 Range = 200 − 170 Range = 30

In many cases, the range is not a good indicator of the overall spread of scores. Consider the two sets of scores below showing the wages of people in two small businesses. A: $240, $240, $240, $245, $250, $250, $260, $800 B: $180, $200, $240, $290, $350, $400, $500, $600 The range for business A = $800 − $240 = $560

and for business B = $600 − $180 = $420

While the range for business A is greater, by looking at the wages in the two businesses, we can see that the wages in business B are generally more spread. The range uses only two scores in its calculation. The interquartile range is usually a better measure of dispersion (spread). The quartiles are found by dividing the data into quarters. The lower quartile is the lowest 25% of scores, the upper quartile is the highest 25% of scores. To calculate an interquartile range, we must first be able to calculate the median. To calculate the median, we must first arrange the scores in ascending order. The median is the middle score (if there is an odd number of scores) or the average of the two middle scores (if there is an even number of scores).

WORKED Example 13 Calculate the median of: a 2, 5, 8, 8, 8, 11, 12 b 45, 69, 69, 87, 88, 92, 99, 100. THINK

WRITE

a There are 7 scores so the median is the 4th score.

a Median = 8

b There are 8 scores so the median is the average of the 4th score and the 5th score.

87 + 88 b Median = -----------------2 = 87.5

The interquartile range is the difference between the upper quartile and the lower quartile. To find the lower and upper quartiles we arrange the scores in ascending order. The lower quartile is 1--- of the way through the distribution and the upper quartile is 3--4 4 of the way through the distribution.

294


To find the interquartile range we follow the steps below. 1. Arrange the data in ascending order. 2. Divide the data into two halves by finding the median. (a) If there is an odd number of scores, the median score should not be included in either half of the scores. (b) If there is an even number of scores, the middle will be halfway between two scores and this will divide the data neatly into two sets. 3. The lower quartile will be the median of the lower half of the data. 4. The upper quartile will be the median of the upper half of the data. 5. The interquartile range will be the difference between the medians of the two halves of the data.

WORKED Example 14 Find the interquartile range of the following data, which show the number of home runs scored in a series of baseball matches. 12, 9, 4, 6, 5, 8, 9, 4, 10, 2 THINK 1 2 3 4 5

Write the data in ascending order. Divide the data into two equal halves. The lower quartile is the median of the lower half. The upper quartile is the median of the upper half. The interquartile range is the upper quartile minus the lower quartile.

WRITE 2, 4, 4, 5, 6, 8, 9, 9, 10, 12 2, 4, 4, 5, 6 8, 9, 9, 10, 12 Lower quartile = 4 Upper quartile = 9 Interquartile range = 9 − 4 Interquartile range = 5


295

the median, range and Graphics Calculator tip! Finding interquartile range When data are entered into the statistics function on a graphics calculator, all essential summary statistics can be found. Consider the data in worked example 14. 1. From the MENU select STAT.

2. Delete any existing data from all lists and then enter the scores into List 1.

3. Press F2 for CALC and then F6 for SET. Enter the settings as shown at right. 1Var Xlist:List 1 shows that the scores are stored in List 1. 1Var Freq:1 shows that each score in List 1 is an individual score with a frequency of 1. 4. Press EXIT to return to the previous screen and then press F1 for 1Var and all the summary statistics will be displayed. 5. Scroll down using the arrow keys until you can see the median, which is equal to 7. 6. To find the range, we need to find the lowest and the highest score. On the previous screen you will see the lowest score denoted MinX. Scroll down further to find MaxX. The range is found by subtracting MinX from MaxX. Range = 12 – 2 Range = 10 7. On this screen you will see the value of the upper quartile Q3 and the lower quartile Q1. To find the interquartile range, subtract Q1 from Q3. IQR = 9 – 4 IQR = 5 In most cases we are asked to find the interquartile range of a grouped distribution. This requires us to draw a cumulative frequency polygon and find the 25th and 75th percentile. A percentile is a measure of where in a set of scores an individual score lies. For example, the 25th percentile has 25% of scores below it and 75% above it. To find the interquartile range, draw a second vertical axis that shows the 25th, 50th, and 75th percentile. A line is drawn from the 25th, 50th and 75th percentile to the ogive and then down to the horizontal axis. The value for the quartiles can then be calculated. The median is the score that is found at the 50th percentile.

296



WORKED Example 15 The cumulative frequency histogram and polygon at right shows the number of customers who order different volumes of concrete from a readymix concrete company during a day. Find the: a median b interquartile range for this distribution.

WRITE/DRAW

a

a

Draw a vertical axis showing the percentiles.

100% 75%

40 30 20 10 0

THINK 1

50

5 5 5 5 5 5 0.2 0.7 1.2 1.7 2.2 2.7 Volumes of concrete

50 40 30

50%

b

2

Draw a line for the 50th percentile to the ogive and estimate the median.

1

Draw a line for the 25th and 75th percentiles and estimate these values.

2

Calculate the interquartile range by subtracting the lower quartile from the upper quartile.

20

25%

10

0%

0

5 5 5 5 5 5 0.2 0.7 1.2 1.7 2.2 2.7 Volumes of concrete

Median = 0.9

b Lower quartile = 0.4 Upper quartile = 1.6 Interquartile range = 1.6 − 0.4 = 1.2

A data set can also be divided into deciles. A decile is a band of 10% of all scores. The deciles can be calculated in the same way as the quartiles, using the appropriate percentage on the vertical scale. For example, the top decile would be found using the 90th percentile.


297

remember 1. The range and interquartile range, also called measures of dispersion, are used to measure the spread of a set of scores. 2. The range is calculated by subtracting the lowest score from the highest score. 3. A single outlying score can enlarge the range. The interquartile range is therefore a better measure of dispersion. 4. The interquartile range is found by subtracting the lower quartile from the upper quartile. 5. The lower and upper quartiles are found by dividing the scores into two equal halves. The median of the lower half is the lower quartile and the median of the upper half is the upper quartile. 6. The interquartile range is estimated from a grouped distribution by drawing a cumulative frequency histogram and polygon.

9D Example

Example

12

Finding the median of a small data set am progr –C

2 Use the frequency distribution tables below to find the range for each of the following UV stats sets of scores. a

Frequency

1

b

Score

Frequency

2

38

23

2

6

39

46

3

12

40

52

4

10

41

62

5

7

42

42

43

45

Score

Frequency

89

12

90

25

91

36

92

34

93

11

94

9

95

4

program GC

–TI

c

Score

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asio

WORKED

9.6

HEET

11

1 Find the range of each of the following sets of data. a 2, 5, 4, 5, 7, 4, 3 b 103, 108, 111, 102, 111, 107, 110 c 2.5, 2.8, 3.4, 2.7, 2.6, 2.4, 2.9, 2.6, 2.5, 2.8 d 3.20, 3.90, 4.25, 7.29, 1.45, 2.77, 8.39 e 45, 23, 7, 47, 76, 89, 96, 48, 87, 76, 66

GC

WORKED

Range and interquartile range

UV stats

298


3 For the grouped dispersions below, state the range. a

c

Class

Frequency

51–60

b

Class

Frequency

2

150–155

12

61–70

8

155–160

25

71–80

15

160–165

38

81–90

7

165–170

47

91–100

1

170–175

39

175–180

20

Class

Frequency

40–43

48

44–47

112

48–51

254

52–55

297

56–59

199

60–63

84

4 The scores below show the number of points scored by two AFL teams over the first 10 games of the season. Sydney: 110 95 74 136 48 168 120 85 99 65 Collingwood: 125 112 89 111 96 113 85 90 87 92 a Calculate the range of the scores for each team. b Based on the results above, which team would you say is the more consistent? 5 Two machines are used to fill boxes with approximately 100 Smarties. A check is made on the operation of the two machines. Ten boxes filled by each machine have the number of Smarties in them counted. The results are shown below. Machine A: 100, 99, 99, 101, 100, 101, 100, 100, 101, 108 Machine B: 98, 104, 96, 97, 103, 96, 102, 100, 97, 104 a What is the range in the number of Smarties from machine A? b What is the range in the number of Smarties from machine B? c Ralph is the quality control officer and he argues that machine A is more consistent in its distribution of Smarties. Explain why.

EXCE

et

reads L Sp he

Median WORKED

Example

EXCE

et

reads L Sp he

Median (DIY)

13

6 Find the median for each of the data sets below. a 3, 4, 4, 5, 7, 9, 10 b 17, 20, 19, 25, 29, 27, 28, 25, 29 c 52, 55, 53, 53, 54, 55, 52, 53, 54, 52 d 12, 14, 15, 12, 14, 19, 17, 15, 18, 20 e 56, 75, 83, 47, 93, 35, 84, 83, 73, 20, 66, 90


WORKED

Example

299

7 For each of the data sets in question 6, calculate the interquartile range.

14

Example

15

8 The frequency histogram and polygon at right displays the results of a survey of 50 drivers who were asked about the number of speeding fines they have received. a Use the ogive to find the median of the distribution. b Find the lower quartile. c Find the upper quartile. d Calculate the interquartile range. e Calculate the top decile.


WORKED

50 45 40 35 30 25 20 15 10 5 0

0 1 2 3 4 5 No. of speeding fines received by drivers

9 The frequency distribution table below shows the result of a survey of 90 households who were asked about the number of times they had been the victim of crime.

a b c d

Score

Frequency

0

26

1

31

2

22

3

8

4

3

Add a column for cumulative frequency to the table. Draw a cumulative frequency histogram and polygon. Use your graph to find the median of the distribution. Calculate the interquartile range.

10 multiple choice For the frequency table below, what is the range?

A 4

B 5

Score

Frequency

25

14

26

12

27

19

28

25

29

19 C 6

D 17

11 multiple choice Calculate the interquartile range of the following data. 17, 18, 18, 19, 20, 21, 21, 23, 25 A 3

B 4

C 5

D 8

300


12 multiple choice The interquartile range is considered to be a better measure of the variability of a set of scores than the range because it: A takes into account more scores B is the difference between the upper and lower quartiles C is easier to calculate D is not affected by extreme values. 13 multiple choice The distribution below shows the ranges in the heights of 25 members of a football squad. Height (cm)

Class centre

Frequency


140–149

144.5

2

2

150–159

154.5

5

7

160–169

164.5

10

17

170–179

174.5

7

24

180–189

184.5

1

25

Which of the statements below is correct? A The range of the distribution is 40. B The range of the distribution is 49. C The range of the distribution is 9. D The interquartile range can only be estimated using a cumulative frequency polygon. 14 The frequency distribution table below shows the marks obtained by a group of people on an IQ test. IQ score

Frequency

75–85

12

85–95

25

95–105

50

105–115

24

115–125

13

a Redraw the frequency distribution table to include columns for class centre and cumulative frequency. b Draw a cumulative frequency histogram and polygon. c Find the range. d Use the graph to estimate the interquartile range.


301

15 The following frequency distribution table shows the distribution of daily maximum temperatures during the course of a full year. Maximum temperature (°C)

a b c d e

Number of days

0–5

4

5–10

22

10–15

95

15–20

124

20–25

94

25–30

19

30–35

5

35–40

2

Add a cumulative frequency column to the table. Draw an ogive of the data. Find the upper and lower quartiles of the data and calculate the interquartile range. Use the ogive to find the median (50th percentile of the data). Find the bottom decile of the data (the scores between which the lowest 10% of scores lie).

16 The following data give the number of fruit that have formed on each of 30 trees in an orchard.

a b c d

45

48

52

36

38

72

36

74

56

46

81

73

46

48

44

39

52

58

57

65

60

53

54

58

41

44

47

76

68

55

Complete a frequency distribution table for the data. Draw an ogive of the data. Use the ogive to find the median, lower quartile and upper quartile of the data. Find the interquartile range of the data.

302


Stem-and-leaf plots As an alternative to a frequency table, a stem-and-leaf plot may be used to group and summarise data. A stem is made using the first part of each piece of data. The second part of each piece of data forms the leaves. Consider the case below. The following data show the mass (in kg) of 20 possums trapped, weighed then released by a wildlife researcher. 1.8 0.9 0.7 1.4 1.6 2.1 2.7 2.2 1.8 2.3 2.3 1.5 1.1 2.2 3.0 2.5 2.7 3.2 1.9 1.7 The stem is made from the whole number part of the mass and the leaves are the decimal part. The first piece of data was 1.8 kg. The stem of this number could be considered to be 1 and the leaf 0.8. The second piece of data was 0.9. It has a stem of 0 and a leaf of 0.9. To compose a stem-and-leaf plot for these data, rule a vertical column of stems then enter the leaf of each piece of data in a neat row beside the appropriate stem. The first row of the stem-and-leaf plot records all data from 0.0 to 0.9. The second row records data from 1.0 to 1.9 etc. Attach a key to the plot to show the reader the meaning of each entry. It is convention to assemble the data in order of size, so this stem-and-leaf plot should be written in such a way that the numbers in each row of ‘leafs’ are in ascending order. Key: 0 | 7 = 0.7 kg Stem 0 1 2 3

Leaf 7 9 1 4 5 6 7 8 8 9 1 2 2 3 3 5 7 7 0 2

When preparing a stem-and-leaf plot, it is important to try to keep the numbers in neat vertical columns because a neat plot gives the reader an idea of the distribution of scores. The plot itself looks a bit like a histogram turned on its side.

WORKED Example 16 The information below shows the mass, in kilograms, of twenty 16-year-old boys. 65 45 56 57 58 54 61 72 70 69 61 58 49 52 64 71 66 65 66 60 Show this information in a stem-and-leaf plot. THINK 1 2 3

Make the ‘tens’ the stem and the ‘units’ the leaves. Write a key. Complete the plot.

WRITE

Key: 5 | 6 = 56 kg Stem 4 5 6 7

Leaf 5 9 2 4 6 7 8 8 0 1 1 4 5 5 6 6 9 0 1 2


303

It is also useful to be able to represent data with a class size of 5. This could be done for the stem-and-leaf plot on the opposite page by choosing stems 0*, 1, 1*, 2, 2*, 3, where the class with stem 1 contains all the data from 1.0 to 1.4 and stem 1* contains the data from 1.5 to 1.9 etc. If stems are split in this way, it is a good idea to include two entries in the key. The stem-and-leaf plot for the ‘possum’ data would appear as follows. Key: 1 | 1 = 1.1 kg 1* | 5 = 1.5 Stem 0* 1* 1* 2* 2* 3*

Leaf 7 9 1 4 5 6 7 8 8 9 1 2 2 3 3 5 7 7 0 2

A stem-and-leaf plot has the following advantages over a frequency distribution table. • The plot itself gives a graphical representation of the spread of data. (It is rather like a histogram turned on its side.) • All the original data are retained, so there is no loss of accuracy when calculating statistics such as the mean and standard deviation. In a grouped frequency distribution table some generalisations are made when these values are calculated.

WORKED Example 17 The following data give the length of gestation in days for 24 mothers. Prepare a stem-and-leaf plot of the data using a class size of 5. 280 288 292 281

287 273 288 292

285 295 279 268

276 279 281 282

266 284 270 275

292 271 278 281

THINK 1

2

WRITE

A class size of 5 is required. The smallest piece of data is 266 and the largest is 295 so make the stems: 26*, 27, 27*, 28, 28*, 29, 29*. The key should give a clear indication of the meaning of each entry. Enter the data piece by piece. Enter the leaves in pencil at first so that they can be rearranged into order of size. Check that 24 pieces of data have been entered. Continued over page

304


THINK 3

WRITE

Now arrange the leaves in order of size.

Key: 26* | 6 = 266 27 | 0 = 270 Stem Leaf 26* 6 8 27* 0 1 3 27* 5 6 8 9 9 28* 0 1 1 1 2 4 28* 5 7 8 8 29* 2 2 2 29* 5

Since all the original data are recorded on the stem-and-leaf plot and are conveniently arranged in order of size, the plot can be used to locate the upper and lower quartiles and the median. As was discussed earlier in the chapter: • the median is the middle score or the average of the two middle scores • the lower quartile is the median of the lower half of the data • the upper quartile is the median of the upper half of the data. Using the ‘possum’ weight data as an example: Key: 0 | 7 = 0.7 kg Stem 0 1 2 3

Leaf 7 9 1 4 5 6 7 8 8 9 1 2 2 3 3 5 7 7 0 2

There were 20 pieces of data so the median is the average of the 10th and 11th scores. Counting each score as it appeared in the stem-and-leaf plot, we can see that the 10th score is the number 1.9 and the 11th score is the number 2.1. 1.9 + 2.1 Median = --------------------2 Median = 2.0 The median divides the data into two halves. The lower quartile is the median of the lower half, which has ten scores in it. So the position of the lower quartile is given by the average of the 5th and 6th scores. The 5th score is the number 1.5. The 6th score is the number 1.6. 1.5 + 1.6 The lower quartile = --------------------2 The lower quartile = 1.55


305

The upper quartile is the median of the upper half, which also has ten scores in it. The 5th score in this half is the number 2.3. The 6th score is the number 2.5. 2.3 + 2.5 The upper quartile = --------------------2 = 2.4

WORKED Example 18 Find the interquartile range of the data presented in the following stem-and-leaf plot. Key: 15 | 7 = 157 kg Stem 15 16 17 18 19 20

Leaf 4 8 8 1 3 3 6 8 0 0 1 4 7 9 9 9 1 2 3 3 5 7 8 8 9 2 7 8 0 2

THINK 1

2

3

WRITE

There are 30 scores and so the median will be the average of the 15th and 16th scores. There are 15 scores in each half and so the lower and upper quartiles will be the 8th score in each half. The interquartile range is the difference between the upper and lower quartiles.

179 + 179 Median = -----------------------2 Median = 179 The lower quartile = 168 The upper quartile = 188 Interquartile range = upper quartile − lower Interquartile range = quartile Interquartile range = 188 − 168 Interquartile range = 20

remember When presenting the stem-and-leaf plots, observe the following points. 1. Always include a key to assist in the interpretation of the plot. 2. Choose a suitable class size. A class size of 5 is possible by using * notation on class stems. 3. After initially recording each score, rearrange the leaves so that they appear in ascending order.

306


9E 9.7

SkillS

HEET

WORKED

Example

16

Stem-and-leaf plots

1 The data below give the number of errors made each week by 20 machine operators. Prepare a stem-and-leaf diagram of the data using stems of 0, 10, 20, etc. 6 15 20 25 28 18 32 43 52 27 17 26 38 31 26 29 32 46 13 20

Presenting data as a stem-andleaf plot

2 The data below give the time taken for each of 40 runners on a 10 km fun run. Prepare a stem-and-leaf diagram for the data using a class size of 10 minutes. 36 66 42 71 WORKED

Example

17

42 75 58 42

52 45 40 50

38 42 41 46

47 55 47 40

59 38 53 52

72 42 68 37

68 46 43 54

57 48 39 48

82 39 48 52

3 The typing speed of 30 word processors is recorded below. Prepare a stem-and-leaf diagram of the data using a class size of 5. 96 102 92 96 95 102 95 115 110 108 88 86 107 111 107 108 103 121 107 96 124 95 98 102 108 112 120 99 121 130 4 Twenty transistors are tested by applying increasing voltage until they are destroyed. The maximum voltage that each could withstand is recorded below. Prepare a stemand-leaf plot of the data using a class size of 0.5. 14.8 15.2 13.8 14.0 14.8 15.7 15.5 15.6 14.7 14.3 14.6 15.2 15.9 15.1 14.3 14.6 13.9 14.7 14.5 14.2

WORKED

Example

18

5 The stem-and-leaf plot at right gives the exact Key: 248 | 4 = 248.4 g mass of 24 packets of biscuits. Find the interquartile Stem Leaf range of the data. 248 4 7 8 249 2 3 6 6 250 0 0 1 1 6 9 9 251 1 5 5 5 6 7 252 1 5 8 253 0 6 The time taken for a test vehicle to accelerate from Key: 7 | 2 = 7.2 s 0 to 100 km/h is recorded during 24 trials. Key: 7* | 6 = 7.6 s The results are represented by the stem-and-leaf plot Stem Leaf at right. 7* 2 4 4 a Find the median of the data. 7* 5 5 7 9 b Find the upper and lower quartiles of the data. 8* 0 0 1 2 4 4 4 c Find the interquartile range of the data. 8* 5 5 6 8 9 9* 2 2 3 9* 0 2


307

Questions 7 to 10 refer to the stem-and-leaf plot below. Key: 12 | 1 = 1210 Key: 12* | 5 = 1250 Stem 12* 12* 13* 13* 14* 14*

Leaf 1 2 4 5 7 7 0 1 1 5 6 6 0 2 3 0 1

9 9 2 3 4 4 7 9 9 4

7 multiple choice The class size used in the stem-and-leaf plot is: A 1 B 10 C 33

D 50

8 multiple choice The number of scores that have been recorded is: A 27 B 33 C 1210

D 1410

9 multiple choice The median of the data is: A 13.4 B 14

C 1335

D 1340

C 1290

D 1390

10 multiple choice The interquartile range of the data is: A 14 B 100

11 The maximum hand spans (in cm) of 20 male concert pianists is recorded as follows: 23.6 20.2 22.8 21.4 25.1 24.8 23.2 21.6 20.7 23.6 22.8 24.6 21.8 22.8 23.1 24.6 21.7 24.7 22.2 23.0 a Complete a stem-and-leaf plot to represent the data. b Find the median of the data. c Find the upper and lower quartile of the data. d Find the interquartile range of the data. 12 The heights (in cm) of a sample of 30 plants are recorded as follows. 93 88 94 99 91 85 126 107 110 111 98 96 117 101 97 92 101 132 103 82 114 84 96 103 108 115 90 110 126 85 a Complete a stem-and-leaf plot to represent the data. b Find the median of the data. c Find the upper and lower quartile of the data. d Find the interquartile range of the data.

308


Five-number summaries Once the median and quartiles have been calculated, we are able to summarise a data set using five numbers. This five-number summary consists of: • lower extreme — the lowest score in the data set • lower quartile — the score at the 25th percentile • median — the middle score • upper quartile — the score at the 75th percentile • upper extreme — the highest score in the data set.

WORKED Example 19 For the set of scores below, develop a five-number summary. 12 15 46 9 36 85 73 29 64 50 THINK

WRITE

3

Re-write the list in ascending order. Write the lowest score. Calculate the lower quartile.

4

Calculate the median.

5

Calculate the upper quartile. Write the upper extreme.

1 2

6

9 12 15 29 36 46 50 64 73 85 Lower extreme = 9 Lower quartile = 15 36 + 46 Median = -----------------2 Median = 41 Upper quartile = 64 Upper extreme = 85 Five-number summary = 9, 15, 41, 64, 85

In most cases you will need to calculate the five-number summary from an ogive.

WORKED Example 20 Cumulative frequency

The ogive below shows the number of seeds found in each of 60 pumpkins. 100%

60 50 40

50%

30 20 10 0

0% 0 10 15 20 25 30 35 40 45 50 55 60 65 70 Number of seeds

Use the ogive to develop a five-number summary.

309


THINK Draw the 25th, 50th and 75th percentiles on the ogive.

100%

60 Cumulative frequency

1

WRITE/DRAW

50 40 50%

30 20 10

0%

0 10 15 20 25 30 35 40 45 50 55 60 65 70 Number of seeds

2 3 4 5 6

Lower extreme = 10 Lower quartile = 31

Write the lower extreme. Use the ogive to estimate the lower quartile. Use the ogive to estimate the median. Use the ogive to estimate the upper quartile. State the upper extreme.

Median = 38 Upper quartile = 47 Upper extreme = 70 Five-number summary = 10, 31, 38, 47, 70

Once a five-number summary has been developed, it can be graphed using a box-andwhisker plot, a powerful way to display the spread of the data. The box-and-whisker plot consists of a central divided box with attached whiskers. The box spans the interquartile range, the vertical line inside the box marks the median and the whiskers indicate the range.

lower lower extreme quartile

median

upper upper quartile extreme

Box-and-whisker plots are always drawn to scale. This can be drawn with the fivenumber summary attached as labels:

4

15

21 23

28

or with a scale presented alongside the box-and-whisker plot.

0

5

10 15

20 25

30

Scale

310


WORKED Example 21 The box-and-whisker plot drawn below shows the marks achieved by a class on their end of year exam.

0

10

20 30

40 50

60 70

80 90 100

a State the median. b Find the interquartile range. c What was the highest mark in the class? THINK

WRITE

a The mark in the box shows the median (72).

a Median = 72

b

1

The lower end of the box shows the lower quartile (63).

b Lower quartile = 63

2

The upper end of the box shows the upper quartile (77).

Upper quartile = 77

3

Subtract the lower quartile from the upper quartile.

Interquartile range = 77 − 63 Interquartile range = 14

c The top end of the whisker gives the top mark (92).

c Top mark = 92

WORKED Example 22 After analysing the speed of motorists through a particular intersection, the following fivenumber summary was developed. The lowest score is 82. The lower quartile is 84. The median is 89. The upper quartile is 95. The highest score is 114. Show this information in a box-and-whisker plot. THINK 1

Draw a scale from 70 to 120 using 1 cm = 10 km/h.

2

Draw the box from 84 to 95.

3

Mark the median at 89.

4

Draw the whiskers to 82 and 114.

DRAW

70

80

90 100 110 120


311

Graphics Calculator tip! Drawing a box-and-whisker plot Earlier in the chapter we used a graphics calculator to find each number in a fivenumber summary, and used them to find the range and interquartile range. Once the data are entered, we can then get the graphics calculator to draw the boxand-whisker plot. Use the same data set that we used earlier, which was taken from worked example 14. The scores were 12, 9, 4, 6, 5, 8, 9, 4, 10 and 2. 1. From the MENU select STAT.

2. Delete any existing data from all lists and then enter the scores into List 1.

3. Press F1 for GRPH, then F6 for SET. Use the arrow keys to highlight Graph Type and press F2 for Box, which will make MedBox appear in the display. Make sure that Xlist is set to List 1 and Frequency is set to 1 as shown at right. 4. Press EXIT to return to the previous screen, then F1 for GPH1. The box-and-whisker plot then appears on the screen.

5. Press SHIFT F1 for Trace. You can then use the left and right arrow keys to display the values of the five-number summary used to draw the box-and-whisker plot.

remember 1. A five-number summary is a summary set for a distribution. 2. The five numbers used in a five-number summary are the lower extreme, lower quartile, median, upper quartile and upper extreme. 3. A box-and-whisker plot can be used to graph a five-number summary. 4. The box is used to show the interquartile range and the median is marked with a line in the box. The whiskers then extend to show the range of the data set.

312


9F Example

19

2 For each of the data sets below, write a five-number summary. a 23 45 92 80 84 83 43 83 b 2 6 4 2 5 7 1 c 60 75 29 38 69 63 45 20 29 93 8 29 93

GC p

am – rogr TI

UV stats

1 Write a five-number summary for the data set below. 15 17 16 8 25 18 20 15 17 14

WORKED

Example

20

3 A cumulative frequency histogram and polygon is shown at right. Write a five-number summary of the data set.

4 A cumulative frequency histogram and polygon is shown at right. Write a five-number summary of the data set.

WORKED

Example

21

100 90 80 70 60 50 40 30 20 10 0


UV stats

WORKED


GC p

sio

am – rogr Ca

Five-number summaries

5 From the five-number summary 6, 11, 13, 16, 32 find: a the median b the interquartile range

1

60 50 40 30 20 10 0

2

3

4 5 Score

6

7

45 55 65 75 85 95 Score

c the range.

6 From the five-number summary 101, 119, 122, 125, 128 find: a the median b the interquartile range c the range. WORKED

Example

22

7 A five-number summary is given below. Lower extreme = 39.2 Lower quartile = 46.5 Median = 49.0 Upper quartile = 52.3 Upper extreme = 57.8 Draw a box-and-whisker plot of the data. 8 The box-and-whisker plot at right shows the distribution of final points scored by a football 50 70 90 110 130 150 Points team over a season’s roster. a What was the team’s greatest points score? b What was the team’s smallest points score? c What was the team’s median points score? d What was the range of points scored? e What was the interquartile range of points scored?

313


9 The box-and-whisker plot at right shows the distribution of data formed by counting the number of honey bears in each of a large sample of packs. In any pack, what was: a the largest number of honey bears? b the smallest number of honey bears? c the median number of honey bears? d the range of numbers of honey bears? e the interquartile range of honey bears?

30

35

40 45

50 55 60 Scale

Questions 10, 11 and 12 refer to the box-and-whisker plot drawn below.

5

10

15 20

25 30 Scale

10 multiple choice The median of the data is: A 20 B 23

C 35

D 31

C 5

D 20 to 25

11 multiple choice The interquartile range of the data is: A 23 B 26 12 multiple choice Which of the following is not true of the data represented by the box-and-whisker plot? A One-quarter of the scores is between 5 and 20. B One-half of the scores is between 20 and 25. C The lowest quarter of the data is spread over a wide range. D Most of the data are contained between the scores of 5 and 20. 13 The data below show the number of sales made each day by a salesperson over a fortnight. 25, 31, 28, 43, 37, 43, 22, 45, 48, 33 a Write a five-number summary of the data. b Draw a box-and-whisker plot of the data. 14 The data below show monthly rainfall in millimetres. Jan.

Feb.

Mar.

Apr.

May

June

July

10

12

21

23

39

22

15

a Provide a five-number summary of the data. b Draw a box-and-whisker plot of the data.

Aug. Sept. Oct. 11

22

37

Nov. Dec. 45

30

314


15 The following data detail the number of hamburgers sold by a fast food outlet every day over a 4-week period. Mon.

Tues.

Wed.

Thur.

Fri.

Sat.

Sun.

125

144

132

148

187

172

181

134

157

152

126

155

183

188

131

121

165

129

143

182

181

152

163

150

148

152

179

181

a Draw a frequency table of the data. (Use a class size of 10.) b Draw a cumulative frequency histogram and polygon for the data. c Use the ogive to find approximations for the: i median ii lower quartile iii upper quartile. d Draw a box-and-whisker plot of the data.

Work

T SHEE

9.2

16 The following data show the ages of 30 mothers upon the birth of their first baby. 22 21 18 33 17 23 22 24 24 20 25 29 32 18 19 22 23 24 28 20 31 22 19 17 23 48 25 18 23 20 a Prepare a frequency table for the data. (Use a class size of 5.) b Draw an ogive for the data. c Draw a box-and-whisker plot of the data. d Describe the distribution in words. What does the distribution say about the age that mothers have their first baby?


315

summary Frequency tables • A frequency table is used to display a set of data in table form. • Ungrouped data are placed in a table and every score is displayed in the table. • Grouped data are used for continuous data or when the scores are spread over a large range. It is best to group data to create five to six classes. If the data are grouped, the table should display a column for class centre.

Graphs • A dot plot is used to display a set of scores on a scale. • A sector graph (pie chart) is used for a display that allows comparison of categorical data. • A line graph is used to show the way a quantity changes over time. • A bar or column graph is used to show quantities associated with categorical data. • A radar chart is a type of line graph that shows the way in which a quantity changes over time. It is most appropriate for a period of time that repeats.

Statistical graphs • Quantitative data are best displayed by a frequency histogram and polygon. • A frequency histogram is a column graph that is drawn with a 0.5 unit (half column) space before the first column and no other spaces between the columns. • A frequency polygon is drawn as a line graph from the corner of the axes to the centre of each column. • A cumulative frequency histogram and polygon graphs the cumulative frequency. There is no space before the first column and the polygon is drawn from the corner of the axes to the top right-hand corner of each column.

Range and interquartile range • The range is the difference between the highest score and the lowest score. • The interquartile range is the difference between the score at the 25th percentile and the 75th percentile. • The median is the score in the middle of the distribution (50th percentile). • The median, lower quartile and upper quartile can be calculated by using an ogive (cumulative frequency polygon).

Five-number summary • A five-number summary of a data set is the lower extreme, lower quartile, median, upper quartile and upper extreme. • A five-number summary can be graphed using a box-and-whisker plot. • A box-and-whisker plot shows the spread of a data set on a scale.

316


CHAPTER review 9A

1 A cricketer keeps track of the number of wickets he took in each innings in which he bowled during a season. The results are shown below. 0 2 3 1 0 6 4 1 1 2 0 0 4 3 2 2 3 1 0 1 1 1 2 0 1 0 Show this information in a frequency table.

9A

2 The following data give the amount of cut meat (in kg) obtained from 20 lambs. 4.5 6.2 5.8 4.7 4.0 3.9 6.2 6.8 5.5 6.1 5.9 5.8 5.0 4.3 4.0 4.6 4.8 5.3 4.2 4.8 Show these data in a frequency table using a class size of 0.5 kg.

9B

3 A survey is taken about the television stations being watched at 7:30 pm on a Monday night. ABC – 27

SBS – 5

Channel 7 – 48

Channel 9 – 72

Channel 10 – 28

Show this information using a sector graph.

9B

4 The number of absences from a school is taken over a week. Monday – 43

Tuesday – 55

Wednesday – 34

Thursday – 45

Friday – 63

Show this information using a radar chart.

9C

5 The table below shows the number of sales made each day over a month in a car yard. Number of sales

Frequency

0

2

1

7

2

12

3

6

4

2

5

0

6

1

Show this information in a frequency histogram and polygon.


317

6 The frequency table below shows the crowds at football matches for a team over a season. Class

Class centre

9C

Frequency

5000–9999

1

10 000–14 999

5

15 000–19 999

9

20 000–24 999

3

25 000–29 999

2

30 000–34 999

2

a Copy and complete the class centre column for the frequency table. b Show the information in a frequency histogram and polygon. 7 The frequency table below shows the marks achieved by Year 11 students on their English exam. Class

Class centre


Frequency

30–39

3

40–49

6

50–59

12

60–69

15

70–79

18

80–79

10

9C

a Copy and complete the frequency table. b Show the information on a cumulative frequency histogram and polygon.

9D

9 For each of the data sets in question 8 calculate: i the median iii the lower quartile iii the upper quartile iv the interquartile range.

9D

10 The cumulative frequency histogram and polygon at right shows the number of goals scored by a soccer team in each match over a season. Use the graph to calculate: a the median b the lower and upper quartiles c the interquartile range.


8 Find the range of each of the following sets of scores. a 28 24 26 24 25 29 22 27 25 b 118 2 56 45 72 43 69 84 159 0 c 1.9 0.7 0.5 0.8 1.1 1.5 1.4

45 40 35 30 25 20 15 10 5 0

9D

0 1 2 3 4 Number of goals scored in a soccer match

318

11 The cumulative frequency histogram and polygon at right shows the number of apples on each tree in an orchard. Use the graph to find: a the median b the lower and upper quartiles c the interquartile range.


9D


300 250 200 150 100 50 0

.5 .5 .5 .5 .5 .5 34 44 54 64 74 84 Number of apples on a tree

9E

12 Display the following scores in a stem-and-leaf plot.

9E

13 Use the stem-and-leaf plot drawn in the previous question to find: a the range b the median c the interquartile range.

9F

14 For the data set below, give a five-number summary.

9F

15 Use the ogive to develop a five-number summary for the data set below.

45 21 38 46 42 41 42 49 35 29 24 28 36 21 38 45 44 40 29 28 35 35 33 38 40 41 48 39 34 38 45 28 23 29 30 40


24 53 91 57 29 69 29 15 84 6

9F

9F

80 70 60 50 40 30 20 10 0

5

16 For the box-and-whisker plot drawn at right: 0 a state the median b calculate the range c calculate the interquartile range.

15

5

25 35 Score

10 15

45

20 25

30

35 40

45 50

55 60

17 The number of babies born each day at a hospital over a year is tabulated and the fivenumber summary is given below. Lower extreme = 1 Upper quartile = 16 Lower quartile = 8 Upper extreme = 18 Median = 14 Show this information in a box-and-whisker plot.

319


Practice examination questions 1 multiple choice The frequency table below shows the marks achieved on a test by a group of students. Score

Frequency

15

12

16

15

17

10

18

6

19

3

20

2

How many students received a mark higher than 17? A 3 B 10 C 11

D 18

2 multiple choice


3 multiple choice For the cumulative frequency polygon at right, the interquartile range of the data is: A 12 B 17 C 24 D 12 to 24

100%

25 20 15

50%

10 5 0

0% 0 5 10 15 20 25 30 35 40 45 Weight

Questions 4 and 5 refer to the box-and-whisker plot shown below.

10

15

20 25

30 35

40

45 Scale

4 multiple choice The upper quartile of the data is: A 24 B 28

C 36

D 42

C 14 to 42

D 24 to 36

5 multiple choice The interquartile range is: A 12 B 28

Cumulative frequency %

Which of the following would be greatly affected by the addition of an extreme score to the data set? A the median B the range C the interquartile range D all would be greatly affected

320


6 The data set below shows the number of admissions to a hospital each day over a month. 25 20 33 41 15 18 24 40 12 29 30 38 26 20 17 23 10 11 16 23 22 27 14 11 12 14 32 24 29 33 a b c d e

Using classes 10–14, 15–19, 20–24, etc., show this information in a frequency table. Draw a frequency histogram and polygon for the data. Draw a cumulative frequency histogram and polygon. State the range of the data set. Use the ogive to find: i the median ii the interquartile range.

7 The box-and-whisker plots below show the sales of two different brands of washing powder at a supermarket each day. Brand A Brand B 0

CHAPTER

test yourself

9

a b c d e

5

10

15

20 25

30 35

40

45

50

Scale

State the range for Brand A. State the interquartile range for Brand A. State the range for Brand B. State the interquartile range for Brand B. Describe the spread of the sales for each brand of washing powder.

Summary statistics

10 syllabus reference Data analysis 4 • Summary statistics

In this chapter 10A Calculating the mean 10B Standard deviation 10C Median and mode 10D Best summary statistics

areyou

READY?

Are you ready?


10.1

Finding the mean of a list of scores

10.2


10.3


10.5

Presenting data as a stem-and-leaf plot

1 Find the average of each of the following sets of scores. a 1, 3, 4, 6, 8 b 1.5, 1.2, 1.3, 1.5, 1.8, 1.1, 1.2, 1.7 c 180, 45, 92, 84, 96, 2, 104, 32, 8, 111

2 Draw a dot plot to represent the following data. 6 8 7 9 4 6 7 8 3 5 7

3 a Display the following sets of scores in a frequency table. 15 16 18 19 15 13 14 13 12 18 15 19 18 12 14 13 17 18 14 16 b Use the classes 0–9, 10–19, 20–29, … etc. to display the scores below in a frequency table. 45 13 9 12 28 19 36 37 28 42 28 18 39 28 36 40 28 37 28 48

4 Place the scores shown below in a stem-and-leaf plot. 48 31 20 20 46 20 25 41 32 49 24 31 31 28 46 48 41 46 27 46 29 24 36 44 29 40 41 20 39 41

Chapter 10 Summary statistics

323

Calculating the mean Average — what does it mean? Survey a group of people about what they believe is meant by the word ‘average’. Use their answers to describe what the word is generally understood to mean.

When looking at a set of statistics we are often asked for the average. The average is a figure that describes a typical score. In statistics, the correct term for the average is the mean. The mean is the first of three measures of central tendency that we will be studying. The others are the median and the mode. x -------- . The statistical symbol for the mean is –x . The formula for the mean is –x = ∑ n In Mathematics, the symbol Σ (sigma) means sum or total, x represents each individual score in a list and Σ x is therefore the sum of the scores. The sum is divided by n, which represents the number of scores.

WORKED Example 1 Find the mean of the scores 17, 16, 13, 15, 16, 20, 10, 15. THINK 1

Find the total of all scores.

2

Divide the total by 8 (the number of scores).

WRITE Total = 17 + 16 + 13 + 15 + 16 + 20 + 10 + 15 Total = 122 122 Mean = --------8 Mean = 15.25

324


Graphics Calculator tip! Finding the mean from a list of scores In the previous chapter we introduced putting a data set into a graphics calculator and extracting certain summary statistics. In this chapter we are going to explore that function further. Consider worked example 1. 1. From the MENU select STAT.

2. Delete any existing data and enter the scores into List 1.

3. Press F2 for CALC, then F6 for SET. For 1Var Xlist, enter List 1 by pressing F1 . This means that the scores are stored in List 1. For 1Var Freq, enter 1 by pressing F1 . This means every score entered has a frequency of 1. 4. Press EXIT to return to the previous screen. Press F1 for 1Var to display all summary statistics. The mean (x–) is the first summary statistic displayed.

As we have seen, large amounts of data are often presented in a frequency table. To calculate the mean in such a case, we need to add an extra column to the table. This column is the f × x column. In this column, we multiply each frequency by the score. We then total this column to find the total of all scores and divide this by the sum of the frequency column. Written as a formula this is: f ×x –x = ∑ -----------------∑f

WORKED Example 2 Complete the frequency table at right, then calculate the mean.

Score (x)

Frequency (f )

4

3

5

7

6

11

7

13

8

10

9

6 Σf =

f×x

Σf × x =


THINK 1 2

3

325

WRITE

Complete the f × x column by multiplying each score by the frequency. Sum the frequency and f × x columns.

Use the formula to calculate the mean.

Score (x)

Frequency (f )

f×x

4

3

12

5

7

35

6

11

66

7

13

91

8

10

80

9

6

54

Σ f = 50

Σ f × x = 338

f ×x –x = ∑ -----------------∑f 338 x = --------50 x = 6.76

the mean from Graphics Calculator tip! Finding a frequency table When data are presented in the form of a frequency table, the set up of the calculator is slightly different when finding summary statistics. This is because we enter the scores in List 1 and the frequencies in List 2. The calculator needs to be told that this is how the data are presented. Consider worked example 2. 1. From the MENU select STAT.

2. Delete any existing data and enter the scores in List 1 and the frequencies in List 2.

3. Press F2 for CALC, then F6 for SET. For 1Var Xlist, enter List 1 by pressing F1 . For 1Var Freq, enter List 2 by pressing F3 . This means the entries in List 2 are the frequencies corresponding to the entries in List 1. 4. Press EXIT to return to the previous screen. Press F1 for 1Var to display all summary statistics. The mean (x–) is the first summary statistic displayed.

326


The same method is used when the frequency table is given in terms of grouped data. In these cases, however, to calculate the f × x column we use the class centre multiplied by the frequency. In these cases, we obtain an estimate of the mean rather than an exact mean.

WORKED Example 3 Complete the frequency distribution table and use it to estimate the mean of the distribution. Class

Class centre (x)

Frequency (f )

25–29

4

30–34

9

35–39

13

40–44

12

45–50

7 Σf =

THINK 1 2

3

4

f×x

Σf × x =

WRITE

Calculate the class centres. Multiply each class centre by the frequency to complete the f × x column. Sum the frequency and the f × x column.

Use the formula to calculate the mean.

Class

Class centre (x)

Frequency (f )

f×x

25–29

27

4

108

30–34

32

9

288

35–39

37

13

481

40–44

42

12

504

45–50

47

7

329

Σ f = 45

Σ f × x = 1710

f ×x –x = ∑ -----------------∑f 1710 x = -----------45 x = 38

In most cases, when calculating the mean you will use your calculator and will need to set it to statistics mode. Once this is done, each score is entered and the M+ function pressed. When all scores are entered, the mean is found by using the x– function.


327

If the data are presented in the form of a frequency distribution table, you will need to check how to enter multiple scores. On many calculators, you press score × frequency followed by M+, but check with your teacher as to how your calculator works. For all statistical questions, when using your calculator clear the memory at the beginning of each question. Most calculators will display the number of scores you have entered after each entry. This is a useful check that you have cleared the memory and entered the data correctly.

WORKED Example 4 Use your calculator to find the mean of: a 10, 15, 47, 23, 56 b Score Frequency 67

10

68

23

69

35

70

28

71

12

THINK

WRITE

a

a

1 2 3

b

1 2 3

Put your calculator on to statistics mode and clear the memory. Press each score followed by M+. Get the mean by pressing –x . Set your calculator to statistics mode and clear the memory. Press each score × frequency then M+. Get the mean by pressing –x .

Mean = 30.2 b Mean = 69.1

remember 1. The mean is the statistical term for average. 2. The mean is calculated by adding all scores then dividing by the number of scores. 3. When calculating the mean from a frequency distribution table, a column for frequency × score ( f × x) is added. The mean is then calculated using the f ×x ------------------ . formula: –x = ∑ ∑f 4. If the frequency distribution uses grouped data, the f × x column is calculated using class centres. 5. The mean can be calculated using your calculator. To do so, set the calculator to statistics mode, enter the scores using the M+ function and make sure you know how to retrieve the mean using the x– function.

328


10A 10.1

WORKED

Example

SkillS

HEET

1 Finding the mean of a list of scores

10.2

SkillS

HEET


10.3

SkillS

HEET


10.4

SkillS

HEET

Organising data into class intervals

10.5

SkillS

HEET

3 An oil company surveys the price of petrol in eight Sydney suburbs. The results are below. Manly 132.9 c/L Lakemba 121.9 c/L Cronulla 129.9 c/L Liverpool 119.9 c/L Wentworthville 125.5 c/L Epping 128.9 c/L Campbelltown 125.9 c/L Penrith 120.9 c/L Based on these results, calculate the mean price of petrol in cents per litre in Sydney. 4 The seven players on a netball team have the following heights: 1.65 m, 1.81 m, 1.75 m, 1.78 m, 1.88 m, 1.92 m and 1.86 m. Calculate the mean height of the players on this team, correct to 2 decimal places. 5 A golf ball manufacturer randomly tests the mass of 10 golf balls from a batch. The batch will be considered satisfactory if the average mass of the balls is between 44.8 g and 45.2 g. The mass, in grams, of those tested are: 45.19, 45.06, 45.35, 44.78, 45.47, 44.68, 44.95, 45.32, 44.60, 44.95. Will the batch be passed as satisfactory? 6 The marks out of 10 on a spelling test are recorded in the frequency table below. Frequency

4

2

5

4

6

5

7

9

Mean (DIY)

8

3

9

5

10

2

et

EXCE

2 Majid sits for five tests in Mathematics. His marks on the tests were 45%, 90%, 67%, 86% and 75%. Calculate Majid’s mean mark on the five tests.

Score

reads L Sp he

Mean

reads L Sp he

EXCE

1 Calculate the mean of each of the following sets of scores. a 4, 8, 3, 5, 5 b 16, 24, 30, 35, 23, 11, 45, 28 c 65, 92, 56, 84 d 9.2, 9.7, 8.8, 8.1, 5.6, 7.5, 8.5, 6.4, 7.0, 6.4 e 356, 457, 182, 316, 432, 611, 299, 355

et

Presenting data as a stem-and- WORKED leaf plot Example 2

Calculating the mean

Σf =

f×x

Σf × x =

a Copy and complete the table. f ×x ------------------ to calculate the mean. b Use the formula: –x = ∑ ∑f


329

7 An electrical store records the number of televisions sold each week over a year. The results are shown in the table below. No. of televisions sold

No. of weeks

16

4

17

4

18

3

19

6

20

7

21

12

22

8

23

2

24

4

25

2 Σf =

f×x

Σf × x =

a Copy and complete the table. b Calculate the mean number of televisions sold each week over the year. Give your answer correct to 1 decimal place. 8 In a soccer season a team played 50 matches. The number of goals scored in each match is shown in the table below. No. of goals

0

1

2

3

4

5

No. of matches

4

9

18

10

5

4

a Redraw this table in the form of a frequency distribution table. b Use your table to calculate the mean number of goals scored each game. 9 A clothing store records the dress sizes sold during a day. The results are shown below. 12 14 10 12 8 12 16 10 8 12 10 12 18 10 12 14 16 10 12 12 12 14 18 10 14 12 12 14 14 10 a Present this information in a frequency table. b Calculate the mean dress size sold this day. 10 multiple choice There are eight players in a Rugby forward pack. The mean mass of the players is 104 kg. The total mass of the forward pack is: A 13 kg B 104 kg C 112 kg D 832 kg

330


11 multiple choice A small business employs five people on a mean wage of $580 per week. A manager is then employed and receives $700 per week. What is the mean wage of the six employees? A $580 B $600 C $680 D $3600 12 multiple choice The mean height of five starting players in a basketball match is 1.82 m. During a time out, a player who is 1.78 m tall is replaced by a player 1.88 m tall. What is the mean height of the players after the replacement has been made? A 1.78 m B 1.82 m C 1.84 m D 1.88 m WORKED

Example

3

13 The table below shows a set of class marks on a test out of 100. Class

Class centre (x)

Frequency (f )

31–40

1

41–50

3

51–60

4

61–70

7

71–80

11

81–90

2

91–100

2 Σf =

f×x

Σf × x =

a Copy and complete the frequency distribution table. b Use the table to calculate the mean class mark. 14 In the heats of the 100 m freestyle at a swimming meet, the times of the swimmers were recorded in the table below. Time

Class centre

No. of swimmers

50.01–51.00

4

51.01–52.00

12

52.01–53.00

23

53.01–54.00

38

54.01–55.00

15

55.01–56.00

3 Σf =

a Copy and complete the frequency distribution table. b Use the table to calculate the mean time.

f×x

Σf × x =


331

15 A cricketer played 50 innings in test cricket for the following scores. 23 65 8 112 54 0 84 12 21 4 25 105 74 40 1 15 33 45 21 47 16 70 22 33 21 8 34 36 5 7 69 104 57 78 158 0 51 16 6 16 0 49 0 14 28 52 21 3 3 7 a Put the above information into a frequency distribution table using appropriate groupings. b Use the table to estimate the batting average for this player.

16 Use the statistics function on your calculator to find the mean of each of the following scores, correct to 1 decimal place. 4 a 11, 15, 13, 12, 21, 19, 8, 14 b 2.8, 2.3, 3.6, 2.9, 4.5, 4.2 c 41, 41, 41, 42, 43, 45, 45, 45, 45, 46, 49, 50

WORKED

Example

17 Use your calculator to find the mean from each of the following tables. a

Score

Frequency

3

b

Score

Frequency

7

28

5

4

10

29

18

5

18

30

25

6

19

31

25

7

38

32

14

8

27

33

10

9

10

34

3

10

5

332


18 The table below shows the heights of a group of people. Height (cm)

Class centre

Frequency

150–154

152

7

155–159

157

14

160–164

162

13

165–169

167

23

170–174

172

24

175–179

177

12

Calculate the mean of this distribution. 19 Seventy students were timed on a 100 m sprint during their P.E. class. The results are shown in the table below. Time (s)

12–13

13–14

14–15

15–16

16–17

Number

13

17

25

15

10

a Calculate the class centre for each group in the distribution. b Use your calculator to find the mean of the distribution. 20 A drink machine is installed near a quiet beach. The number of cans sold each day over the first 10 weeks after its installation is shown below. 4 39 31 31 50 43 70 45 57 71 18 26 3 52 51 59 33 51 27 62 30 90 3 30 97 59 33 44 99 62 72 6 42 83 19 49 11 6 63 4 53 20 45 58 1 9 79 41 2 33 97 71 52 97 69 83 39 84 92 43 71 98 8 97 18 89 21 9 4 17 a Put this information into a frequency distribution table using the classes 1–10, 11–20, 21–30 etc. b Calculate the mean number of cans sold per day over these 10 weeks.


333

Standard deviation In the previous chapter, we discussed using the range and the interquartile range as a measure of the spread of a data set. The most commonly used measure of spread is the standard deviation. The standard deviation is a measure of how much a typical score in a data set differs from the mean. The standard deviation is found by entering a set of scores into your calculator, just as you do when you are finding the mean. Your calculator will have a function that gives the standard deviation. There are two standard deviation functions on your calculator. The first, σn, is the population standard deviation. This function is used when the statistical analysis is conducted on the entire population.

WORKED Example 5 Below are the scores out of 100 by a class of 20 students on a Science exam. Calculate the mean and the standard deviation. 87 69 95 73 88 47 95 63 91 66 59 70 67 83 71 57 82 65 84 69 THINK 1 2 3

WRITE

Enter the data set into your calculator. Retrieve the mean using the x– function. Retrieve the standard deviation using the σn function.

–x = 74.05 σn = 13.07

the population standard Graphics Calculator tip! Finding deviation The summary statistics displayed by your graphics calculator include the two standard deviation functions. Consider worked example 5. 1. From the MENU select STAT.


3. Press F2 for CALC, then F6 for SET. Set the calculator up for a list of scores as shown earlier and as shown at right.

334


4. Press EXIT to return to the previous screen, then F1 to display the summary statistics. The population standard deviation is displayed by the symbol xσn. Population standard deviation

When the statistical analysis is done using a sample of the population, a slightly different standard deviation function is used. Called the sample standard deviation, this value will be slightly higher than the population standard deviation. The sample standard deviation will be found on your calculator using the σn − 1 or the sn function.

WORKED Example 6 Ian surveys twenty Year 11 students and asks how much money they earn from part-time work each week. The results are given below. $65 $82 $47 $78 $108 $94 $60 $79 $88 $91 $50 $73 $68 $95 $83 $76 $79 $72 $69 $97 Calculate the mean and standard deviation. THINK 1 2 3

WRITE

Enter the statistics into your calculator. Retrieve the mean using the x– function. Retrieve the standard deviation using the σn − 1 function, as a sample has been used.

–x = $77.70 σn − 1 = $15.56

the sample standard Graphics Calculator tip! Finding deviation The summary statistics displayed by your graphics calculator include the sample standard deviation. Consider worked example 6. 1. From the MENU select STAT.


3. Press F2 for CALC, then F6 for SET. Set the calculator up for a list of scores as shown earlier and as shown at right.


335

4. Press EXIT to return to the previous screen, then F1 to display the summary statistics. The sample standard deviation is displayed by the symbol xσn – 1. Sample standard deviation

For most examples, you will need to read the question carefully to decide whether to use the population or the sample standard deviation. The standard deviation can also be calculated when the data are presented in table form. This is done by entering the data in the same way as they were when calculating the mean earlier in this chapter.

WORKED Example 7 The table below shows the scores of a class of thirty Year 3 students on a spelling test. Score

Frequency

4

1

5

2

6

4

7

9

8

6

9

7

10

1

Calculate the mean and standard deviation. THINK 1 2 3

WRITE

Enter the data into your calculator using score × frequency. Retrieve the mean using the x– function. Retrieve the standard deviation using the σn function, as the whole population is included in the statistics.

–x = 7.4 σn = 1.4

Once we have calculated the standard deviation, we can make conclusions about the reliability and consistency of the data set. The lower the standard deviation, the less spread out the data set is. By using the standard deviation, we can determine whether a set of scores is more or less consistent (or reliable) than another set. The standard deviation is the best measure of this because, unlike the range or interquartile range as a measure of dispersion, the standard deviation considers the distance of every score from the mean.

336


A higher standard deviation means that scores are less clustered around the mean and less dependable. For example, consider the following two students: Student A: –x = 60 σn = 5 Student B: –x = 60 σn = 15 Both students have the same mean. However, student A has a standard deviation of 5 and student B has a standard deviation of 15. Student A is far more consistent and can confidently be expected to score around 60 in any future exam. Student B is more inconsistent but is probably capable of scoring a higher mark than student A. This concept will be discussed further during the HSC course.

WORKED Example 8 Two brands of light globe are tested to see how long they will burn (in hours). Brand X:

850 950 1400 1150 1000 900

Brand Y:

975 1100 1050 950 1075 1025

875 1200 850 825 1000 950

975 900

Which of the two brands of light globe is more reliable?

THINK 1 2

3 4

WRITE

Enter both sets of data into your calculator. Choose the sample standard deviation because a sample of each light globe brand has been chosen. Write down the sample standard deviation for each brand. The brand with the lower standard deviation is the more reliable.

Brand X: σn − 1 = 190.4 Brand Y: σn − 1 = 62.4 Brand Y is the more reliable as it has a lower standard deviation.

remember 1. The standard deviation is a measure of the spread of a data set. 2. Standard deviation is found on your calculator by entering the data set using the calculator’s statistical mode. 3. The population standard deviation is used when an entire population is considered in the statistical analysis and can be found on the calculator using the σn function. 4. The sample standard deviation is used when a sample of the population is used in the analysis and can be found using the σn − 1 function.


10B WORKED

Example

5

WORKED

Example

6

337

Standard deviation

1 For each of the sets of scores below, calculate the standard deviation. Assume that the scores represent an entire population and answer correct to 2 decimal places. a 3, 5, 8, 2, 7, 1, 6, 5 b 11, 8, 7, 12, 10, 11, 14 c 25, 15, 78, 35, 56, 41, 17, 24 d 5.2, 4.7, 5.1, 12.6, 4.8 e 114, 12, 3.6, 42.8, 0.5 2 For each of the sets of scores below, calculate the sample standard deviation, correct to 2 decimal places. a 25, 36, 75, 85, 6, 49, 77, 80, 37, 66 b 4.8, 9.3, 7.1, 9.9, 7.0, 4.1, 6.2 c 112, 25, 56, 81, 0, 5, 178, 99, 41 d 0.3, 0.3, 0.3, 0.4, 0.5, 0.6, 0.8, 0.8, 0.8, 0.9, 1.0 e 56, 1, 258, 45, 23, 58, 48, 35, 246 3 For each of the following, state whether it is appropriate to use the population standard deviation or the sample standard deviation. a A quality control officer tests the life of 50 batteries from a batch of 1000. b The weight of every bag of potatoes is checked and recorded before being sold. c The number of people who attend every football match over a season is analysed. d A survey of 100 homes records the number of cars in each household. e The score of every HSC student in Mathematics is recorded. 4 The band ‘Aquatron’ is to release a new CD. The recording company needs to predict the number of copies that will be sold at various music stores throughout Australia. To do so, a sample of 10 music stores supplied information about the sales of the previous CD released by Aquatron, as shown below. 580 695 547 236 458 620 872 364 587 1207 a Calculate the mean number of sales at each store. b Should the population or sample standard deviation be used in this case? c What is the value of the appropriate standard deviation?

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5 A supermarket chain is analysing its sales over a week. The chain has 15 stores and the sales for each store for the past week were (in $million): 1.5 2.1 2.4 1.8 1.1 0.8 0.9 1.1 1.4 1.6 2.0 0.7 1.2 1.7 1.3 a Calculate the mean sales for the week. b Should the population or sample standard deviation be used in this case? c What is the value of the appropriate standard deviation? WORKED

Example

7

6 Use the statistical function on your calculator to find the mean and standard deviation (correct to 1 decimal place) for the information presented in the following tables. In each case, use the population standard deviation. a

b

Score

Frequency

12

45

4

24

5

c

Score

Frequency

1

75

22

46

16

76

17

47

47

39

77

8

6

21

48

61

78

10

7

7

49

52

79

12

50

36

80

21

81

29

Score

Frequency

3

7 Copy and complete the class centre column for each of the following distributions and use your calculator to find an estimate for the mean and standard deviation (correct to 2 decimal places). In each case use the population standard deviation. a Class

Class centre

c

Class centre

Frequency

Class

10–12

12

0–4

15

13–15

16

5–9

24

16–18

25

10–14

31

19–21

28

15–19

33

22–24

13

20–24

29

25–29

17

b Class

Class centre

Frequency

31–40

15

41–50

28

51–60

36

61–70

19

71–80

8

81–90

7

91–100

2

Frequency


WORKED

Example

8

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8 Below are the marks achieved by two students in five tests. Brianna: 75, 80, 70, 72, 78 Katie: 50, 95, 90, 80, 55 a Calculate the mean and standard deviation for each student. b Which of the two students is more consistent? Explain your answer. 9 multiple choice From Year 11, 21 students are chosen to complete a test. The scores are shown in the table below. Class

Frequency

10–20

1

20–30

6

30–40

9

40–50

4

50–60

1

When preparing an analysis of the typical performance of Year 11 students on the test, the standard deviation used is: A 9.209 B 9.437 C 21 D 34.048 10 multiple choice The results below are Ian’s marks in four exams for each subject that he studies. English: 63 85 78 50 Maths: 69 71 32 97 Biology: 45 52 60 41 Geography: 65 78 59 61 In which subject does Ian achieve the most consistent results? A English B Maths C Biology D Geography 11 The following frequency distribution gives the prices paid by a car wrecking yard for a sample of 40 car wrecks. Price ($)

Frequency

0–500

2

500–1000

4

1000–1500

8

1500–2000

10

2000–2500

7

2500–3000

6

3000–3500

3

Find the mean and standard deviation of the price paid for these wrecks.

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12 The table below shows the life of a sample of 175 household light globes. Life (hours)

Frequency

200–250

2

250–300

5

300–350

12

350–400

25

400–450

42

450–500

38

500–550

26

550–600

15

600–650

7

650–700

3

a Find the range of the data. b Use the class centres to find the mean and standard deviation in the lifetimes of this sample of light globes. 13 Crunch and Crinkle are two brands of potato crisps. Each are sold in packets nominally of the same size and for the same price. Upon investigation of a sample of packets of each, it is found that Crunch and Crinkle have the same mean mass (25 g). The standard deviation of the masses of Crunch is, however, 5 g and the standard deviation of the masses of Crinkle is 2 g. Which brand do you think represents better value for money under these circumstances? Why?

Work

T SHEE

10.1


341

Median and mode So far we have used the mean as a measure of the typical score in a data set. Consider the case of someone who is analysing the typical house price in an area. On a particular day, five houses are sold in the area for the following prices: $375 000 $349 000 $360 000 $411 000 $1 250 000 For these five houses the mean price is $549 000. The mean is much greater than most of the houses in the data set. This is because there is one score that is much greater than all the others. For such data sets, we need to use a different measure of central tendency. In the previous chapter, we introduced the median as the middle score in a data set, when all scores are arranged in order. For the above data set, the median house price is $375 000, a much better measure of the typical house price in this area.

WORKED Example 9 Calculate the median of the scores 3, 5, 8, 4, 4, 6, 9, 1, 6. THINK 1 Rewrite the scores in ascending order. 2 The median is the middle score.

WRITE 1, 3, 4, 4, 5, 6, 6, 8, 9 Median = 5

The median becomes more complicated when there is an even number of scores because there are two scores in the middle. When there is an even number of scores, the median is the average of the two middle scores.

WORKED Example 10 Find the median of the scores 13, 13, 16, 12, 19, 18, 20, 18. THINK 1 Write the scores in ascending order. 2

There is an even number (8) scores, so average the two middle scores.

WRITE 12, 13, 13, 16, 18, 18, 19, 20 16 + 18 Median = -----------------2 Median = 17

The median can also be calculated from the cumulative frequency column of a frequency table. The cumulative frequency column puts the scores into order and tells us what score is in each position. Consider the frequency distribution table below. Score

Frequency


4

1

1

The 1st score is 4.

5

6

7

The 2nd–7th scores are 5.

6

9

16

The 8th–16th scores are 6.

7

8

24


8

4

28


9

2

30

The 29th and 30th scores are 9.

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There are 30 scores in this distribution and so the middle two scores will be the 15th and 16th scores. By looking down the cumulative frequency column we can see that these scores are both 6. Therefore, 6 is the median of this distribution.

WORKED Example 11 Find the median for the frequency distribution at right.

THINK 1

2 3

Score

Frequency

34

3

35

8

36

12

37

9

38

8

39

5

WRITE

Redraw the frequency table with a cumulative frequency column.

There are 45 scores and so the middle score is the 23rd score. Look down the cumulative frequency column to see that the 23rd score is 36.

Score

Frequency


34

3

3

35

8

11

36

12

23

37

9

32

38

8

40

39

5

45

Median = 23rd score Median = 36

Graphics Calculator tip! Finding the median Once the data are entered into the graphics calculator, the display of summary statistics gives us the median. This is a function that most scientific calculators do not have. Consider worked example 11. 1. From the MENU select STAT.

2. Delete any existing data and enter the scores in List 1 and the frequencies in List 2.


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3. Press F2 for CALC, then F6 for SET. Set the calculator up for data stored in a frequency table as shown earlier in the chapter and as shown by the screen at right.

4. Press EXIT to return to the previous screen, then F1 to display the summary statistics. To see the median you will need to use the arrow keys to scroll down the screen by three lines. Median

When the frequency table presents grouped data, the median is estimated from the ogive as shown in the previous chapter. There are many examples where neither the mean nor the median is the appropriate measure of the typical score in a data set. Consider the case of a clothing store. It needs to re-order a supply of dresses. To know what sizes to order it looks at past sales of this particular style and gathers the following data: 8 12 14 12 16 10 12 14 16 18 14 12 14 12 12 8 18 16 12 14 For this data set the mean dress size is 13.2. Dresses are not sold in size 13.2, so this has very little meaning. The median is 13, which also has little meaning as dresses are sold only in even-numbered sizes. What is most important to the clothing store is the dress size that sells the most. In this case size 12 occurs most frequently. The score that has the highest frequency is called the mode.

WORKED Example 12 Find the mode of the scores below. 4, 5, 9, 4, 6, 8, 4, 8, 7, 6, 5, 4 THINK

WRITE

The score 4 occurs most often and so it is the mode.

Mode = 4

Graphics Calculator tip! Finding the mode Once the data are entered into the graphics calculator, the display of summary statistics gives us the mode. This is a function that most scientific calculators do not have. Consider worked example 12. 1. From the MENU select STAT.

344



3. Press F2 for CALC , then F6 for SET. Set the calculator up for a list of scores as shown earlier and as shown at right.

4. Press EXIT to return to the previous screen, then F1 to display the summary statistics. To see the mode you will need to use the arrow keys to scroll down to the last line of the display. Mode

When two scores occur most often an equal number of times, both scores are given as the mode. In this situation the scores are bimodal. If all scores occur an equal number of times, then the distribution has no mode. The Casio CFX-9850 shows only the highest mode. The Casio FX-9860GAU shows all modes, as well as the number of modes and the frequency of each. To find the mode from a frequency distribution table, we simply give the score that has the highest frequency.

WORKED Example 13 For the frequency distribution below, state the mode. Score Frequency

14

15

16

17

18

19

3

6

11

14

10

7

THINK

WRITE

The highest frequency is 14, which belongs to the score 17 and so 17 is the mode.

Mode = 17

When a table is presented using grouped data, we do not have a single mode. In these cases, the class with the highest frequency is called the modal class.

remember 1. The median is the middle score in a data set or the average of the two middle scores. 2. The median can be found using the cumulative frequency column of a frequency table. 3. The mode is the score that occurs the most.


Example

10

2 Below are the scores of eight people who played a round of golf. 75 80 81 76 84 83 81 82 Calculate the median for this set of scores.

E

L Spre XCE ad

Mode

L Spre XCE ad

Mode (DIY)

sheet

4 A factory has 80 employees. Over a two-week period the number of people absent from work each day was recorded and the results are shown below. 3, 1, 5, 4, 3, 25, 4, 2, 4, 5 a Calculate the median number of people absent from work each day. b Calculate the mean number of people absent from work each day. c Does the mean or the median give a better measure of the typical number of people absent from work each day? Explain your answer.

Median (DIY)

sheet

3 Find the median for each of the following sets of scores. a 3, 4, 5, 5, 5, 6, 9 b 5.6, 5.2, 5.4, 5.3, 5.8, 5.4, 5.3, 5.4 c 45, 62, 39, 88, 75 d 102, 99, 106, 108, 101, 103, 102, 105, 102, 101

L Spre XCE ad

sheet

WORKED

Median

E

9

L Spre XCE ad

E

Example

1 The scores of seven people on a spelling test are given below. 5 6 5 8 5 9 8 Calculate the median of these marks.

sheet

WORKED

Median and mode

E

10C

345

346 WORKED

Example

11


5 The table at right shows the number of cans of drink sold from a vending machine at a high school each day. a Copy and complete the frequency distribution table. b Use the table to calculate the median number of cans of drink sold each day from the vending machine.

Score

Frequency

17

4

18

9

19

6

20

12

21

8

22

5

23

4

24

2

6 The table at right shows the number of accidents a tow truck attends each day over a three-week period. Calculate the median number of accidents attended to by the tow truck each day.

7 The table at right shows the number of errors made by a machine each day over a 50-day period. Calculate the median number of errors made by the machine each day.


No. of accidents

No. of days

2

4

3

12

4

3

5

1

6

1

No. of errors per day

Frequency

0

9

1

18

2

13

3

6

4

3

5

1

8 multiple choice There are 25 scores in a distribution. The median score will be the: A 12th score B 12.5th score C 13th score D average of the 12th and 13th score.


347

9 multiple choice For the scores 4, 5, 5, 6, 7, 7, 9, 10 the median is: A 5 B 6 C 6.5 10 multiple choice

D 7 Score

Frequency

1

12

2

13

3

8

4

7

5

5

Consider the frequency table at right. The median of these scores is: A 2 B 3 C 8 D 13

11 The frequency distribution table below shows the number of sick days taken by each worker in a small business. Days sickness

Frequency

0–4

10

5–9

12

10–14

7

15–19

6

20–24

5

25–29

3

30–34

2


a Copy and complete the frequency distribution table. b Calculate the median class for this distribution. 12 For the frequency distribution table in question 11: a make a list of the class centres for the distribution b draw a cumulative frequency histogram and polygon c use the cumulative frequency polygon to estimate the median of the distribution. 13 For each of the following sets of scores find the mode. a 2, 5, 3, 4, 5 12 b 8, 10, 7, 10, 9, 8, 8 c 11, 12, 11, 15, 14, 13 d 0.5, 0.4, 0.6, 0.3, 0.2, 0.4, 0.6, 0.9, 0.4 e 110, 113, 100, 112, 110, 113, 110

WORKED

Example

14 Find the mode for each of the following. (Hint: Some are bimodal and others have no mode.) a 16, 17, 19, 15, 17, 19, 14, 16, 17 b 147, 151, 148, 150, 148, 152, 151 c 2, 3, 1, 9, 7, 6, 8 d 68, 72, 73, 72, 72, 71, 72, 68, 71, 68 e 2.6, 2.5, 2.9, 2.6, 2.4, 2.4, 2.3, 2.5, 2.6

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15 Use the tables below to state the mode of the distribution. a b c 13 Score Frequency Score Frequency

WORKED

Example

Score

Frequency

38

2

39

4

40

1

41

5

1

2

5

1

2

4

6

3

3

5

7

5

4

6

8

8

42

6

5

3

9

5

43

3

10

3

44

6

45

2

Frequency

16 Use the frequency histogram below to state the mode of the distribution. 40 35 30 25 20 15 10 5 0

12 13 14 15 16 17 18 19 20 Score

17 For each of the following grouped distributions, state the modal class. a b Class Frequency Class Frequency 1–4

6

1–7

3

5–8

12

8–14

8

9–12

30

15–21

9

13–16

23

22–28

25

17–20

46

29–35

12

21–24

27

36–42

11

25–28

9

43–49

2

18 The table at right shows the depth of snow during every day of the ski season. a Redraw the table to include the class centres and cumulative frequency. b Draw a cumulative frequency histogram and polygon. c Use the graph to estimate the median depth of snow for the ski season.

Depth (cm)

Frequency

0–50

8

50–100

9

100–150

12

150–200

15

200–250

6

250–300

4

300–350

2

350–400

2


349

19 The weekly wage (in dollars) of 40 people is shown below. 376 592 299 501 375 366 204 359 382 274 223 295 232 325 311 513 348 235 329 203 556 419 226 494 205 307 417 204 528 487 543 532 435 415 540 260 318 593 592 393 a Use the classes $200–$249, $250–$299, $300–$350 etc. to display the information in a frequency distribution table. b From your table, calculate the median class. c Draw a cumulative frequency histogram and polygon, and use it to estimate the median wage in the group.

1 For the set of scores 2, 5, 5, 5, 6, 7, 7, 9, 10 find: 1 the mean 2 the median 3 the mode 4 the population standard deviation 5 the sample standard deviation. For the table of scores below find: Score

Frequency


75

12

12

76

19

31

77

52

83

78

64

147

79

33

180

80

1

181

6 the mean 7 the median 8 the mode 9 the population standard deviation 10 the sample standard deviation.

350


Best summary statistics Having now examined all three summary statistics, it is important to recognise when it is appropriate to use each one. In some circumstances, one summary statistic may be more appropriate than the others. For example, a shoe manufacturer notes that in a new style of sporting footwear: mean size sold is 8.63 median size is 8.75 mode size is 9. In this case, the mode is the most useful measure as the manufacturer needs to know which size sells the most. The mean and median are of less use to the manufacturer.

WORKED Example 14 Below are the wages of ten employees in a small business. $420 a b c d

$430

$490

$475

$465 $450

$1700

$420

$420

$440

Calculate the mean wage. Calculate the median wage. Calculate the mode wage. Does the mean, median or mode give the best measure of a typical wage in this business?

THINK

WRITE

a

2

Total all the wages. Divide the total by 10.

a Total = $5710 Mean = $5710 ÷ 10 Mean = $571

1

Write the wages in ascending order.

2

Average the 5th and 6th score to find the median.

b $420 $420 $420 $430 $440 $450 $465 $475 $490 $1700 $440 + $450 Median = -----------------------------2 Median = $445

b

1

c $420 is the score that occurs most often and so this is the mode.

c Mode = $420

d The mean is larger than what is typical because of one very large wage: the mode is the lowest wage and so this is not typical. Therefore, the median is the best measure.

d The median is the best measure of the typical wage as the mode is the lowest score, which is not typical, and the mean is inflated by the $1700 wage.

For each of these examples you will need to think carefully about the relevance of each summary statistic in terms of the particular example.


351

remember 1. The three summary statistics are: mean — calculated by adding all scores, then dividing by the number of scores median — the middle score or average of the two middle scores mode — the score with the highest frequency. 2. Be careful when using the mean. One or two extreme scores can greatly increase or decrease its value. 3. When the mean is not a good measure of central tendency, the median is used. 4. The mode is the best measure in some examples where discrete data means that the mean and median may have very little meaning.

10D

Best summary statistics

1 There are ten houses in a street. A real-estate agent values each house with the am progr –C following results. 14 UV stats $350 000 $390000 $375000 $350000 $950000 $350000 $365000 $380000 $360000 $380000

WORKED

Calculate the mean house valuation. Calculate the median house valuation. Calculate the mode house valuation. Which of the above is the best measure of central tendency?

2 The table below shows the number of shoes of each size that were sold over a week at a shoe store.

a b c d

Size

Frequency

4

5

5

7

6

19

7

24

8

16

9

8

10

7

Calculate the mean shoe size sold. Calculate the median shoe size sold. Calculate the mode of the data set. Which measure of central tendency has the most meaning to the shoe store proprietor?

program GC

–TI

a b c d

asio

GC

Example

UV stats

352


3 The table below shows the crowds at football matches over a season.

a b c d e f

Crowd

Class centre

Frequency

10 000–20 000

15 000

95

20 000–30 000

25 000

64

30 000–40 000

35 000

22

40 000–50 000

45 000

15

50 000–60 000

55 000

3

60 000–70 000

65 000

0

70 000–80 000

75 000

1

Calculate the mean crowd over the season. Calculate the median class. Calculate the modal class. Draw a cumulative frequency histogram and polygon. Use the ogive to estimate the median. Which measure of central tendency would best describe the typical crowd at football matches over the season?

4 multiple choice Mr and Mrs Yousef research the typical price of a large family car. At one car yard they find six family cars. Five of the cars are priced between $30 000 and $40 000, while the sixth is priced at $80 000. What would be the best measure of the price of a typical family car? A Mean B Median C Mode D All are equally important. 5 Thirty men were asked to reveal the number of hours they spent doing housework each week. The results are given below. 1 5 2 12 2 6 2 8 14 18 0 1 1 8 20 25 3 0 1 2 7 10 12 1 5 1 18 0 2 2 a Represent the data in a frequency distribution table. (Use classes 0–4, 5–9, 10–14 etc.) b Find the mean number of hours that the men spend doing housework. c Find the median class for hours spent by the men at housework. d Find the modal class for hours spent by the men at housework. 6 The resting pulse rates of 20 female athletes below. 50 62 48 61 30 45 43 47 51 61 44 54

were measured. The results are shown 52 42 52 38

71 48 34 40


a b c d e f

353

Represent the data in a frequency distribution table using appropriate groupings. Find the mean of the data. Find the median class of the data. Find the modal class of the data. Draw an ogive of the data. Use the ogive to determine the median pulse rate.

7 The following data give the age of 25 patients admitted to the emergency ward of a hospital. 18 23 43 74 80

16 82 19 24 20

6 74 84 20 23

75 25 72 63 17

24 21 31 79 19

a Represent the data in a frequency distribution table. (Use classes 0–14, 15–29, 30–44, etc.) b Use the table to find: iii the mean age of patients admitted iii the median class of age of patients admitted iii the modal class for age of patients admitted. c Draw an ogive of the data. d Use the ogive to determine the median age. e Do any of your statistics (mean, median or mode) give a clear representation of the typical age of an emergency ward patient? f Give some reasons that could explain the pattern of the distribution of data in this question. 8 The batting scores for two cricket players over six innings are as follows: Player A: Player B: a b c d e

31, 34, 42, 28, 30, 41 0, 0, 1, 0, 250, 0

Find the mean score for each player. Which player appears to be better if the mean result is used? Find the median score for each player. Which player appears to be better when the decision is based on the median result? Which player do you think would be more useful to have in a cricket team and why? How can the mean result sometimes lead to a misleading conclusion?

9 The following frequency table gives the number of employees in different salary brackets for a small manufacturing plant. Position

Salary ($)

No. of employees

Machine operator

38 000

50

Machine mechanic

40 000

15

Floor steward

44 000

10

Manager

82 000

4

100 000

1

Chief Executive Officer

354 Work

T SHEE

10.2


a Workers are arguing for a pay rise, but the management of the factory claims that workers are well paid because the mean salary of the factory is $42 100. Are they being honest? b Suppose that you were representing the factory workers and had to write a short submission in support of the pay rise. How could you explain the management’s claim? Provide some other statistics to support your case.

Wage rise The workers in an office are trying to obtain a wage rise. In the previous year, the ten people who work in the office received a 2% rise while the company CEO received a 42% rise. 1 What was the mean wage rise received in the office last year? 2 What was the median wage rise received in the office last year? 3 What was the modal wage rise received in the office last year? 4 The company is trying to avoid paying the rise. What statistic do you think they would quote about last year’s wage rises? Why? 5 What statistic do you think the trade union would quote about wage rises? Why? 6 Which statistic do you think is the most ‘honest’ reflection of last year’s wage rises? Explain your answer. Quoting different averages can give different impressions about what is normal. Try the following task. 1 Visit a local real estate agent and study the properties for sale in the window. 2 Calculate the mean, median and mode price for houses in the area. 3 If you were a real estate agent and a person wanting to sell their home asked what the typical property sold for in the area, which figure would you quote? 4 Which figure would you quote to a person who wanted to buy a house in the area?

Best summary statistics and comparison of samples Examine each of the following statistics. • The typical mark in Maths among Year 11 students. • The number of attempts taken by Year 11 and 12 students to get their driver’s licence. • The typical number of days taken off school by Year 11 students so far this year. 1 For each of the above, gather your data by selecting a random sample. 2 Calculate the mean, median and mode for each topic. 3 Compare your results with other students who will have selected their samples from the same population. 4 In each case, state the best summary statistic and explain your answer.


355

summary The mean • For a small number of scores, the mean is calculated using the formula: x –x = ∑ -------n • When the data are presented in a frequency table, the mean can be calculated using the formula: f ×x –x = ∑ ------------------ . ∑f • The mean can also be calculated using the statistical function on your calculator.

Standard deviation • • • •

The standard deviation is a measure of the spread of a data set. The smaller the standard deviation, the smaller the spread of the data set. The standard deviation is found using the statistical function on your calculator. When the analysis is conducted on the entire population, the population standard deviation (σn) is used. • When the analysis is conducted on a sample of the population, the sample standard deviation (σn − 1) is used.

Median and mode • The median is the middle score of a data set, or the average of the two middle scores. • The mode is the score with the highest frequency.

Summary statistics • The summary statistics are the mean, median and mode. • Each summary statistic must be examined in the context of the statistical analysis to determine which is the most relevant.

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CHAPTER review 10A

1 Calculate the mean of each of the following sets of scores.

10A

2 Copy and complete the tables below and then use them to calculate the mean.

a b c d

a

b

4, 9, 5, 3, 5, 6, 2, 7, 1, 10 65, 67, 87, 45, 90, 92, 50, 23 7.2, 7.9, 7.0, 8.1, 7.5, 7.5, 8.7 5, 114, 23, 12, 25

Score (x)

Frequency (f )

5

11

6

15

7

24

8

21

9

9

f×x

Σf =

Σf × x =

Score (x)

Frequency (f )

f×x

9.2

36

9.3

48

9.4

74

9.5

65

9.6

51

9.7

32

9.8

14

9.9

2 Σf =

Σf × x =


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3 Complete the frequency distribution table below and use it to estimate the mean of the distribution. Class

Class centre (x)

Frequency (f )

21–24

3

25–28

9

29–32

17

33–36

31

37–40

29

41–44

25

45–48

19

49–52

10 Σf =

10A

f×x

Σf × x =

4 Use the statistics function on your calculator to find the mean of each of the following sets of scores.

10A

a 2, 18, 26, 121, 96, 32, 14, 2, 0, 0 b 2, 2, 12, 12, 12, 32, 32, 47, 58 c 0.2, 0.3, 0.6, 0.4, 0.3, 0.7, 0.8, 0.6, 0.5, 0.4, 0.1 5 Use the statistics function on your calculator to find the mean of the following distributions. Where necessary, give your answers correct to 1 decimal place. a

c

b

Score

Frequency

23

24

45

20

47

25

89

30

68

26

124

40

56

27

102

50

17

28

78

29

46

Score

Frequency

10

Class

Class centre

Frequency

10–12

11

18

13–15

14

32

16–18

17

34

19–21

20

40

22–24

23

28

25–27

26

14

28–30

29

6

10A

358 10B


6 The marks of 30 students in a Geography test are shown below. 66 47 43 80 42 92 92 90 92 77 67 87 75 72 42 60 86 53 95 78 46 87 49 70 82 92 93 71 62 67 a Calculate the mean. b Should the population or sample standard deviation be used in this case? c Write the value of the appropriate standard deviation.

10B

7 To find the number of attempts most people take to get their driver’s licence, a sample of twenty Year 12 students is chosen. The results are shown below. 1 2 3 3 1 2 1 2 4 1 1 1 2 2 2 3 1 2 2 3 a Calculate the mean. b Should the population or sample standard deviation be used in this case? c Write the value of the appropriate standard deviation.

10B

8 Use the statistics function on your calculator to find the mean and population standard deviation of each of the following distributions. Give each answer correct to 3 decimal places. a 0.7, 1.2, 0.5, 0.9, 1.3, 1.5, 0.1, 1.0, 0.4, 0.5 b 23, 254, 12, 89, 74, 15, 26, 45 c Score Frequency

d

26

12

27

25

28

29

29

28

30

14

Class

Class centre

Frequency

10–14

12

8

15–19

17

12

20–24

22

32

25–29

27

45

30–34

32

40

35–39

37

19

40–44

42

6


359

9 For each of the following sets of scores, find the median. a 25, 26, 26, 27, 27, 28, 30, 32, 35 b 4, 5, 8, 5, 8, 6, 7, 10, 4, 8, 4 c 3.2, 3.1, 3.0, 3.5, 3.2, 3.2, 3.2, 3.6 d 2, 3, 7, 4, 4, 8, 5, 7, 7, 6 e 121, 135, 111, 154, 147, 165, 101, 108 10 Copy and complete each of the following frequency tables and then use them to find the median. a Score

Frequency

0

2

1

6

2

11

3

7

4

6

5

3

Score

Frequency

54

2

55

5

56

14

57

11

58

6

59

1

60

1

Score

Frequency

66

8

67

10

68

12

69

14

70

7

71

5

72

4

b

c




10C

10C

360 10C


11 a Copy and complete the frequency distribution table below. Class

Class centre

Frequency

30–39

18

40–49

34

50–59

39

60–69

45

70–79

29

80–89

10

90–99

5


b What is the median class of this distribution? c Display these data in a cumulative frequency histogram and polygon. d Use your graph to estimate the median of the distribution.

10C

12 For each set of scores below, state the mode. a 2, 3, 6, 8, 4, 2, 4, 2, 6, 5, 2 b 23, 24, 19, 23, 27, 25, 31, 24, 23, 27, 27 c 1.2, 5.6, 4.7, 6.8, 4.5, 2.1

10C

13 For each of the frequency tables below, state the mode. a Score Frequency

b

1

23

2

35

3

21

4

19

5

8

Score

Frequency

14

9

15

15

16

8

17

12

18

15

19

7

20

1


361

14 Use the frequency table below to state the modal class.

10C

Class

Class centre

Frequency

30–33

31.5

12

34–37

35.5

26

38–41

39.5

34

42–45

43.5

45

46–49

47.5

52

50–53

51.5

23

15 Below are the number of goals scored by a netball team in ten matches in a tournament. 25 26 19 24 28 67 21 22 28 18 a b c d

Calculate the mean. Calculate the median. Calculate the mode. Which of the above is the best summary statistic? Explain your answer.

16 Give an example of a statistical analysis where the best summary statistic is: a the mean b the median c the mode.

Practice examination questions 1 multiple choice For the following data set, which of the statements is correct? 3, 4, 8, 7, 3, 6, 5, 3, 4, 7 A B C D

The mean is 5. The median is 5. The mode is 5. all of the above

2 multiple choice For which of the following data sets is the median greater than the mean? A B C D

10D

2, 6, 14, 14, 15, 16, 18 12, 13, 14, 14, 14, 18, 22 12, 15, 15, 15, 15, 18 1, 4, 9, 16, 25, 36, 49

3 multiple choice For the data set below, which statement is correct? 25, 45, 64, 48, 66, 85, 45, 27 A The mean is 50.625. B The sample standard deviation is 20.29. C The population standard deviation is 18.98. D all of the above

10D

362


4 multiple choice Tracey compiles a sample of new car prices. She selects 100 new car buyers and asks what price they paid for their car. To measure the spread of the distribution Tracey should use: A the population standard deviation B the sample standard deviation C both standard deviations D the mean 5 multiple choice For the statistical analysis in question 4 which summary statistic would be the most appropriate? A mean B median C mode D standard deviation 6 The table below shows the gross annual income for a sample of 100 company executives. Income

a b c d e f

Class centre

Frequency

$50 000–$75 000

12

$75 000–$100 000

18

$100 000–$125 000

26

$125 000–$150 000

24

$150 000–$175 000

12

$175 000–$200 000

8


Copy and complete the frequency table. Calculate the mean. Calculate the standard deviation. Calculate the median class Calculate the modal class. Which summary statistic best describes the typical income for a company executive?

7 In order to compare two textbooks, a teacher recommends one book to one class and another book to another class. At the end of the year the classes are each tested; the results are detailed below.

CHAPTER

test yourself

10

a b c d e

Text A 44 52 95 76 13 94 83 72 55 81 22 25 64 72 35 48 56 59 84 98 84 21 35 69 28 Text B 65 72 48 63 68 59 68 62 75 79 81 72 64 53 58 59 64 66 68 42 37 39 55 58 52 82 79 55 Calculate the mean and standard deviation for each class group. Which standard deviation did you use in part a? Explain why. Which class performed better? Which class had the more consistent results? Could a conclusion be drawn about the better textbook from the above information? Explain your answer.

Similarity of twodimensional figures

11 syllabus reference Measurement 3 • Similarity of twodimensional figures

In this chapter 11A Similar figures and scale factors 11B Solving problems using similar figures 11C House plans

areyou

READY?

Are you ready?


11.1

11.2

Simplifying ratios

1 Simplify each of the following ratios. a 15 : 12 b $56 : $49

c 0.8 : 1.25

d 40 cm : 2 m

Corresponding sides of congruent and similar triangles

2 Consider the similar triangles drawn at right. a Which side in ∆XYZ corresponds to BC? b Which angle in ∆ABC corresponds to ∠XZY?

X

Y

A

B

C

Z

11.3

Similar triangles

3 Find the value of the pronumeral in the figure at right. a

6 6

11.4

11.5

9

Using tests to prove similar triangles

4 Using the figure at right, prove that ∆ABX is similar to ∆ACX.

Finding the scale factor

5 Find the scale factor given that: a AB = 6 cm and A′B′ = 18 cm b AB = 3 cm and A′B′ = 27 mm.

A

B

X

C

365

C h a p t e r 1 1 S i m i l a r i t y o f t w o - d i m e n s i o n a l fi g u r e s

Similar figures and scale factors Have you ever read a road map or looked at plans for a house? The map or the plan is a scaled down version of the roads or house. When two objects are identical, except one is a reduction or an enlargement of the other, the objects are said to be similar. Maps and plans are practical examples of similarity. Maps and plans both use a scale. The scale tells us how many times larger an object is in reality compared to the plan. For example, a house plan may use a scale of 1:100. This means that if a wall is 1 cm long on the plan, it is 100 cm (or 1 m) in reality. All the angles shown on the plan are the same as in reality. If two walls meet at right angles on the plan, they meet at right angles in reality. Similar figures are in proportion and have the same shape. That is, each pair of corresponding sides are in the same ratio and each pair of corresponding angles are equal. To show that two triangles are similar, we can show that either of the above conditions is true. The symbol for similarity is three vertical lines (|||). For triangles, if two pairs of corresponding sides are in the same ratio and the angles they include are equal then they are similar.

WORKED Example 1

X

In the figure at right, show that LABC ||| LXYZ. A

1 2 3 4

Z

C Y

B

THINK

WRITE

∠BAC and ∠YXZ are equal. ∠ACB and ∠XZY are equal. ∠ABC and ∠XYZ are equal. Make a conclusion.

∠BAC = ∠YXZ ∠ACB = ∠XZY ∠ABC = ∠XYZ LABC ||| LXYZ (3 pairs of equal angles)

WORKED Example 2

P

Show that the triangles LMN and PQR are similar. L 4 cm

THINK

WRITE

1

Simplify the ratio LM:PQ.

2

Simplify the ratio MN:QR.

3

Simplify the ratio LM:PR.

4

Make a conclusion.

12 cm 5 cm

M 3 cm N

Q

15 cm

9 cm

R

LM:PQ = 4:12 = 1:3 MN:QR = 3:9 = 1:3 LN:PR = 5:15 = 1:3 LLMN ||| LPQR (3 pairs of sides in equal ratio)

To determine if other figures are similar, we need to examine the ratio of sides.

366


WORKED Example 3

P

Determine if the rectangles ABCD and PQRS are similar.

Q

B

A 4m D

THINK 1 2 3

15 m 10 m

S

WRITE

Simplify the ratio of corresponding sides AD and RS. Simplify the ratio of corresponding sides CD and QR. Make a conclusion.

C 6m

R

AD:RS = 4:6 = 2:3 CD:QR = 10:15 = 2:3 The rectangles are similar as their corresponding sides are in equal ratio.

When we examine similar figures we can state the ratio of sides between the two figures. The number by which we multiply measurements on the first figure to get the measurements on the second figure is called the scale factor. The scale factor is calculated by replacing the first part of the ratio of sides with one. The second part of the ratio is then calculated and is the scale factor.

WORKED Example 4

7.5 cm

The two figures at right are similar. a What is the ratio of their sides? b What is the scale factor?

3 cm

6 cm

THINK

WRITE

a

The 6 cm side corresponds to the 15 cm side. Write this as a ratio and simplify. The 3 cm side corresponds to the 7.5 cm side. Check that this simplifies to the same ratio. Make a conclusion.

a 6:15 = 2:5

The scale factor is written by comparing one unit on the first figure with the second. Make a conclusion.

b 2:5 = 1:2 1---

1

2

3

b

1

2

15 cm

3:7.5 = 30:75 = 2:5 The similar figures are in the ratio 2:5. 2

The scale factor is 2 1--- . 2

A special case of similarity occurs when the scale factor is 1. These shapes are identical and are called congruent figures.

367


remember 1. Similar figures have all pairs of corresponding angles equal and all pairs of corresponding sides in equal ratio. 2. To prove that triangles are similar, we need to prove only that either the corresponding angles are equal or that the corresponding sides are in equal ratio. 3. The ratio of the corresponding sides in similar figures can be used to calculate the scale factor. 4. If the scale factor is 1 then the figures are congruent.

11A WORKED

Example

Similar figures and scale factors

1 Prove that LABC ||| LZYX.

X

A

11.1 SkillS

Y

HEET

1

Simplifying ratios B

11.2 SkillS

Z

C

HEET

2 Prove that LABC ||| LEDC.

A

Corresponding sides of congruent and similar triangles

B C

WORKED

Example

HEET

11.3 SkillS D

E

3 Prove that LLMN ||| LPQR.

Similar triangles

P L

2

10 cm

8 cm

4 cm

5 cm

HEET

11.4 SkillS

M

4 Prove that LABC ||| LFED.

6 cm

12 cm

R

Using tests to prove similar triangles

D

A

16 cm 12 cm F

12 cm Finding the scale factor

16 cm

B 9 cm C

5 Prove that LLMN ||| LRST.

11.5 SkillS

HEET

12 cm

N Q

E R

Cabri Geo

ry met

L

Similar triangles M

N S

T

368


6 Prove that LVWX ||| LVYZ.

V

W

X

Y WORKED

Example

3

7 Determine if the rectangles ABCD and WXYZ are similar.

Example

4

9 The figures at right are similar. a What is the ratio of sides? b What is the scale factor?

L 3 cm K

4 cm

Z

Y 8 cm

P M

C

S

R

P 6 cm

D

T

R Q 12 cm

4 cm

10 cm

15 cm

M 6 cm N 3 cm P

O Q

12 On a set of house plans, a measurement of 5 cm represents a wall which is 10 m long. Calculate the scale factor. 13 On a map, a distance of 3 cm represents an actual distance of 60 km. Calculate the scale factor.

Work

T SHEE

11.1

S

40 cm

16 cm

6 cm

11 In the figure below, LMNO ||| LMPQ. Calculate the ratio of sides.

Q

4 cm

N

B 8 cm

X

6 cm

7 cm

A

E

10 The figures at right are similar. a What is the ratio of sides? b What is the scale factor?

15 cm

W A 5 cm B 2 cm D C

8 Determine if the rectangles KLMN and PQRS are similar.

WORKED

Z

369


Enlarging a figure We can draw similar figures using an enlargement factor. We will enlarge the triangle below by a scale factor of 2. 1 Mark a point, P, external to the figure. This point is called the centre of enlargement.

A P B

2 Measure the distance from P to the vertex, A. Mark a point twice this distance away in a straight line. Label this point A′.

C A'

A P B

3 4 Repeat step 2 for the vertices B and C.

C A'

A P B

C B'

C'

5 6 Join the points A′, B′ and C′.

A' A P B

C B'

C'

Investigating scale factors 1 Draw a figure on clear plastic so that it can be placed on an overhead projector. 2 Place the overhead projector 2 m from the screen and focus the image. Measure the lengths on the image and state the scale factor. 3 Repeat step 2, placing the overhead projector 3 m, 4 m and 5 m from the screen. 4 Determine if there is a relationship between the scale factors and the distance from the projector to the screen.

370


Similar triangles Two triangles are similar if they have the same shape but not necessarily the same size. One is an enlargement or reduction of the other. This means that the corresponding angles of the triangles have to be equal (to make them the same shape) and the ratio of their corresponding sides must be constant (making one smaller or larger than the other). As with congruent triangles, we do not need to know all the information about the three sides and three angles to determine if a pair of triangles is similar. Certain minimum information is sufficient. Let us investigate. 1 Draw the LABC shown (it is not drawn to scale). A Draw LXYZ, larger than ∆ABC with ∠X = ∠A, 70° ∠Y = ∠B and ∠Z = ∠C. Measure the lengths of the sides of the two triangles. Determine the ratios XY YZ of the lengths of the corresponding sides -------- , -------AB BC 60° 50° B C ZX and -------- . Are these ratios constant (within the limits of the accuracy of the CA constructions)? Does it appear that ∆XYZ is a true enlargement of LABC? Repeat the process, drawing LXYZ smaller than LABC. Is ∆XYZ similar to ∆ABC? 2 Construct the two triangles shown where LTVW is twice the size of LDEF. D The ratio of their corresponding sides is 1 cm E TV VW WT 2 cm constant as -------- = ---------- = --------- = 2. Measure 1.5 cm DE EF FD F their corresponding angles. Are the two triangles similar? 3 Construct LGHJ and LQRS to the measurements shown at right. Find the ratio of their corresponding sides (as in part 1) and measure all angles. What do you conclude?

2 cm V

4 cm

3 cm W Q

G

6 cm

3 cm H

4 Draw the right-angled triangles KLM and NPU to the dimensions given. Again, find the ratio of their corresponding sides (as in part 1) and measure all angles. What do you conclude? 5 Summarise the results of your investigation. What are the minimum requirements to ensure the similarity of two triangles?

T

30°

2 cm

J

30° 4 cm

R

S

N K 7.5 cm 5 cm

L

3 cm

M

P

4.5 cm

U

371


Solving problems using similar figures We can use similar figures to solve many problems. By setting up similar triangles we can calculate measurements of objects such as trees, which we are unable to physically measure. Another example is house plans. In this case, the ratio of sides becomes the scale of the plan making it similar to the house itself. Consider the case where we want to measure the height of a tree too tall for us to physically measure. Using shadows we can create two similar triangles.

WORKED Example 5 A tree casts a shadow 7.5 m long. At the same time a 1-metre ruler casts a shadow 60 cm long. Calculate the height of the tree.

h 1m

THINK 1 2

3

WRITE

The triangles are similar because all three angles are equal. Write the shadow lengths as a ratio and simplify. Write a proportion and solve to find the height of the tree.

7.5 m

60 cm

7.5 m:60 cm = 750:60 7.5 m:60 cm = 25:2 h 25 --- = -----1 2 h = 12.5 m

We use a similar method when reading maps or plans. The map is a similar figure to the place being mapped. We use the scale given on the map to calculate the distance between two places.

WORKED Example 6

The scale on a road map is given as 1 cm = 5 km. Jodie uses her ruler and finds the distance between the towns Huxley and Brownville is 6.2 cm. Calculate the distance between these two towns. THINK 1 2

WRITE

Multiply 6.2 cm by the given scale. Give a written answer.

6.2 cm × 5 cm/km = 31 km The actual distance between Huxley and Brownville is 31 km.

In the case of plans, the scale is often stated as a ratio. The method of solution is the same.

372


WORKED Example 7 The scale on a house plan is 1:150. The front of the house measures 8.5 cm on the plan. Calculate the actual length of the front of the house. THINK

WRITE

1

Multiply the measurement by the scale.

8.5 cm × 150 = 1275 cm

2

Change the units from cm to m.

8.5 cm × 150 = 12.75 m

3


The front of the house is 12.75 m long.

remember 1. We can calculate measurements of objects by setting up similar triangles. 2. Maps and plans are similar to the actual object. 3. The scale is the ratio of sides on the plan to those on the object. This scale can be used to make calculations.

11B WORKED

Example

5

Solving problems using similar figures

1 A tree casts a shadow 2.5 m long. At the same time a 1-metre ruler casts a shadow 40 cm long, as shown in the figure. Calculate the height of the tree.

1m 2.5 m

40 cm

2 A building casts a shadow 9.5 m long. At the same time a 1-metre ruler casts a shadow 25 cm long. a Draw a diagram to represent this situation. b Calculate the height of the building. 3 Kerry is 170 cm tall and her shadow measures 50 cm. At the same time a flagpole casts a shadow which is 3 m long. a Draw a diagram to represent this situation. b What is the ratio of sides in the similar triangles formed? c Calculate the height of the flagpole. 4 An artist holds his brush so that the top and bottom of the brush line up with the top and bottom of a tree. The brush is 10 cm long and is held 1 m away from the artist’s eye. The tree is 100 m away from the artist. Calculate the height of the tree.

1m

100 m


373

5 multiple choice A tree casts a 6 m shadow. At the same time a 1-metre ruler casts a shadow 1.2 m long. The ratio of sides in the similar figures formed is: A 5:6 B 5:1 C 6:1 D 10:1 WORKED

Example

6

6 A map gives the scale as being 1 cm = 10 km. Two towns are shown as being 6 cm apart on the map. What is the actual distance between the two towns? 7 On a map where the scale is given as 1 cm = 4 km, calculate the actual distance where the distance on the map is measured as: a 5 cm b 9 cm c 6.5 cm d 12.8 cm e 0.9 cm f 4 mm. 8 On a map where the scale is given as 1 cm = 5 km, what should the distance on the map be when the actual distance is: a 20 km? b 45 km? c 22 km? d 340 km? e 8 km? f 37.5 km? 9 multiple choice The scale on a map is given as 1 cm = 4 km. If the distance between two points on the map is 3.8 cm, then the actual distance between these two points is: A 15.2 cm B 3.8 km

C 4 km

D 15.2 km

10 multiple choice On a map the scale is given as 1 cm = 5 km. The distance from Freewell to Taleton is 64 km. How far apart should they be drawn on the map? A 0.078 125 cm B 12.8 cm C 320 cm D 12.8 km 11 The scale on a set of house plans is given as 1:500. Calculate the length of the house frontage if it is shown as 4 cm on the plans. 7

WORKED

Example

12 A set of house plans is to be drawn using a scale of 1:400. The side of the house is to be 16 m long. Calculate the length that this should be drawn on the plans.

Scale drawing of the classroom Draw a scale diagram showing the floor plan of the classroom you are now in. On your diagram show the location of all desks, cupboards, the blackboards and any other features of the room.

374


House plans House plans are a very common application of similar figures. As we saw in the previous section, plans are drawn using a ratio as the scale factor. Measurement enables us to calculate all dimensions within the house. Corresponding angles on similar figures are equal and so the angles on the plans will be the same as the angles in reality.

WORKED Example 8 Below is a plan for a house.

WC

Bathroom

Bed 3

Kitchen/Dining

Bed 2

Lounge Bed 1

Scale 1:100

a Calculate the dimensions of the house. b Calculate the area of the lounge room. THINK

WRITE

a

a Length of house on plan = 12 cm Width of house on plan = 10 cm

1

2

3

Measure the length and width of the house on the plan. Multiply each of these measurements by 100.

Write your answer.

Actual length of house = 12 cm × 100 = 1200 cm = 12 m Actual width = 10 cm × 100 = 1000 cm = 10 m The dimensions of the house are 12 m by 10 m.


THINK b 1 Measure the length and width of the lounge room on the plan. 2 Multiply each of these measurements by 100.

3 4

Calculate the area of the lounge room. Write your answer.

375

WRITE b Length of lounge room on plan = 6 cm Width of lounge room on plan = 6 cm Actual length of lounge room = 6 cm × 100 = 600 cm =6m Actual width of lounge room is also 6 m. A = 62 A = 36 m2 The area of the lounge room is 36 m2.

House plans are also drawn with a view of what the house will look like from the outside. These diagrams are called elevations. For example, the front elevation is what the house will look like from the front. Elevations are also drawn using a scale.

WORKED Example 9 The diagram below shows the front elevation of a house.

Scale 1:100

a Calculate the height of the eaves on the lower side of the house. b Measure the angle of the pitch of the roof. THINK a 1 Measure the height on the plan for the lower side of the house. 2 Multiply the plan measurement by 100.

3

b

1 2

Write your answer. Measure the angle that the slope of the roof makes with the horizontal. Write your answer.

WRITE a Height on the plan = 3.5 cm Actual height = 3.5 cm × 100 Actual height = 350 cm Actual height = 3.5 m The height of the eaves is 3.5 m. b Angle to horizontal = 45° The angle of the pitch of the roof is 45°.

376


remember 1. House plans are drawn with a ratio as the scale factor. 2. Using the scale factor, we can calculate the dimensions for the house from the plans. 3. An elevation is the view of a house from one side. 4. From an elevation, we can measure all lengths and angles and use the scale to calculate the actual measurements.

11C WORKED

Example

8

House plans

1 Below is the site plan for a block of land. N

Shed

House Garden bed

Garage

Driveway

Garden bed Scale 1:250

a What are the dimensions of the block of land? b What are the dimensions of the house?

377


2 A house plan is shown below.

Lounge

Bed 3

Bed 4

Family

Pantry Kitchen Bed 1

Bathroom

Bed 2 Toilet

Laundry Scale 1:150

a Calculate the dimensions of the house. b What are the dimensions of the lounge room? c Which bedroom is the largest? What are its dimensions? 3 Below is the front elevation of a house, drawn to scale. 9

WORKED

Example

Scale 1:100

a Calculate the height of the peak of the roof. b Calculate the height of the eaves. c Measure the angle of the pitch of the roof.

378


4 Trace the front elevation of the house at right into your book. On your diagram write all lengths and angles necessary for the construction of the house.

Scale 1:200

5 The following diagrams are representations of houses with a variety of roof types. Draw a plan of the south and east elevations of these houses. The direction of north is given. a b

N

N

Gable roof

Boxed gable roof

c

d

N

Work

T SHEE

N

11.2 Hip roof

Gambrel (a roof combining the addition of small gables to a hip roof)

House plans 1 Try to obtain a set of plans to a house. a What is the scale on the plans? b What are the dimensions of the house? c What is the total area of the house? 2 What symbols are used on the house plans to indicate the following: a a door b a window c cupboards d any other significant features? 3 Using a suitable scale, draw a set of plans for your house. Include a floor plan of your house and a front elevation.


379

summary Similar figures • Similar figures have all pairs of corresponding angles equal and corresponding sides in equal ratio. • To show that triangles are similar, we show either that all pairs of corresponding angles are equal or that all pairs of corresponding sides are in equal ratio. • For other figures it is necessary to show that both properties are true.

Scale factors • The scale factor allows us to solve problems using similar figures. • Heights of objects, such as trees, that are not easily measured can be determined by constructing similar triangles.

House plans • House plans are similar to the house being built. • We can measure all lengths and angles on a house plan and use the scale to calculate the actual measurements.

380


CHAPTER review 11A

1 Prove that LMNO ||| LPQO.

M 2 cm 18 cm

O

N 4 cm 9 cm Q

P

11A

2 Prove that LVWX ||| LYZX.

12 cm

V

W

8 cm X 12 cm Y

11A

3 Given that E is the midpoint of the line AC, prove that LABC ||| LEDC.

Z

18 cm

A

E

D 5 cm C

11A

4 ABCD and WXYZ are rectangles. a Prove that the two rectangles are similar. b State the ratio of sides in the two similar figures.

B

10 cm

W A 2 cm D

6 cm

X

B 15 cm C Z 5 cm Y

11B

5 The two triangles at right are similar. Use this to find the length of the side marked x. x

15 cm

9 cm

12 cm

11B

6 When a 1-metre ruler casts a shadow 75 cm long, a building casts a 15 m shadow. Calculate the height of the building.

11B

7 A 10 m ladder will reach 9 m up a wall. How high up a wall will a 25 m ladder reach, if it is placed at the same angle to the ground?

11B

8 A map has a scale of 1:40 000. Calculate the actual distance between two places if they are shown on the map as: a 1 cm apart b 6 cm apart c 8.5 cm apart.


381

9 A surveyor’s plan has a scale 1:800. The dimensions of a property are shown as 10 cm × 16 cm on the plan. What are the actual dimensions on the property? 10 The scale diagram below is a site plan for a block of land on which a house is to be built.

Garden bed

Garden bed

Swimming pool

House

Garage

Driveway N

Scale 1 cm = 4 m

a Calculate the dimensions of the block of land. b Calculate the dimensions of the house. c Calculate the area of the house in square metres.

11B 11C

382


Practice examination questions 1 multiple choice Which of the triangles below are similar? I

4 cm

A I and II

III

II 9 cm

3 cm

2 cm

B I and III

C II and III

6 cm

4 cm

D I, II and III

2 multiple choice Consider the statements below. Statement I. All rhombuses are similar. Statement II. All parallelograms are similar. Which of the above statements is correct? A Statement I B Statement II C Statements I and II

D Neither statement

3 multiple choice On a set of house plans a room that measures 4.5 m × 3.2 m is shown as 9 cm × 6.4 cm. The scale of the plans is: A 1:2 B 1:50 C 1:100 D 1:200 4 Harley is 160 cm tall. He casts a shadow 1 m long. a Calculate the height of a tree, which at the same time casts a shadow 4.8 m long. b Calculate the length of the shadow that would be cast by a building that is 20 m tall.

CHAPTER

test yourself

11

5 A rectangular block of land is 20 m wide and 32 m long. a Using a scale of 1:200, make a scale drawing of the block of land. b On the same diagram, a house is shown as 6.2 cm long and 5.9 cm wide. Calculate the actual dimensions of the house.

Taxation

12 syllabus reference Financial mathematics 3 • Taxation

In this chapter 12A Calculating allowable deductions 12B Taxable income 12C Medicare levy 12D Calculating tax 12E Calculating GST and VAT 12F Graphing tax functions

areyou

READY?

Are you ready?


12.1

12.2

12.3


1 Find: a 42% of $4900

b 48% of $52 000

c 1.5% of $67 500.


2 Complete the following. a Increase $450 by 10% c Decrease $600 by 12%

b Increase $750 by 15% d Decrease $945 by 25%


3 Write: a $45 as a percentage of $180 c $135 as a percentage of $90

12.4


12.5

Sketching straight lines

b $52.50 as a percentage of $420 d $48 as a percentage of $750

4 Find the amount given that: a 10% of the amount is $7.50 b 40% of the amount is $560 c 110% of the amount is $275.

5 Draw the graph of: a y = 2x + 1

b y = 4x – 2

c y = 5 – x.

C h a p t e r 1 2 Ta x a t i o n

385

Calculating allowable deductions The government collects taxes in order to pay for government services. There are several different ways in which the government collects these taxes. The one with which we are most familiar is income tax; however, there are several other forms of tax used by the government to collect money. In this chapter, we look at how taxes are calculated and collected. In chapter 1, we looked at earning money. We learned that the gross pay was the wage or salary paid by the employer. Before the employee receives this money, deductions are taken out. The amount actually received by the worker is called the net pay. There may be several payments taken out of a person’s gross pay, but for most people the largest deduction is income tax. Everyone who earns over a certain amount must pay income tax. Income tax is paid on an increasing scale, depending on the amount you earn. Most people pay income tax in each pay period. This is called Pay As You Earn tax (PAYE tax). At the end of each financial year, which runs from July 1 of one year to June 30 the following year, people who earn income must submit a tax return. A tax return is used to calculate the amount of tax that should have been paid and compares this with the amount of PAYE tax paid. The taxpayer then either receives a refund or must pay the amount owing. The amount of tax paid is calculated using your taxable income. Taxable income is your gross pay less any allowable tax deductions. Deductions are allowed for expenses incurred while earning an income. For example, a builder is allowed a tax deduction for the cost of tools or a bank teller who wears a uniform may be allowed a deduction for the dry-cleaning of that uniform. Deductions are also allowed for donations to charity over $2.

WORKED Example 1 A large company employs Ken as a plumber. Ken claims deductions of $1400 to buy tools, $25 for gumboots, $200 for two pairs of work overalls, $5 per week for dry-cleaning the overalls and $1.50 per week for work-related telephone calls. Calculate Ken’s total deductions. THINK

WRITE

1

Calculate Ken’s total dry-cleaning and telephone deductions.

2

Add up all of Ken’s deductions.

Dry-cleaning = $5 × 52 = $260 Telephone = $1.50 × 52 = $78 Deductions = $1400 + $25 + $200 + $260 + $78 = $1963

People who use their own car for work are entitled to claim a portion of the running costs as a deduction. The amount of the deduction is based on the size of the engine and the number of kilometres travelled. The tax deduction covers the cost of the fuel and a portion of the long-term costs of running a car such as registration, insurance, depreciation and maintenance.

386


WORKED Example 2 Raylene is a computer programmer. As part of her job she uses her own car to travel to visit clients and to attend training seminars. Raylene’s car is a 2.4 litre Mitsubishi Lancer, for which she is allowed a deduction of 51.9 c/km. Calculate the size of the tax deduction in a year where she travels 2547 km on work-related matters. THINK

WRITE

Multiply the number of kilometres (2547) by the rate per kilometre (0.519). Be sure to convert the rate in cents to dollars.

Travel deduction = 2547 × 0.519 Travel deduction = $1321.89

Tax deductions are also allowed for the depreciation of major equipment. For example, a teacher may own a home computer that is used to prepare lessons and store marks. The computer loses value as it becomes older and so a tax deduction is allowed for this.

WORKED Example 3 Trevor is an accountant who works from home. He owns a personal computer that is used as part of his job. Trevor bought a new computer on 1 July 2007 for $3200. Each year he is allowed a 33% deduction for the depreciation of the computer. Calculate the tax deduction allowed in: a the 2007–08 financial year b the 2008–09 financial year. THINK

WRITE

a The depreciation was 33% of the purchase price.

a Tax deduction = 33% of $3200 Tax deduction = 0.33 × $3200 Tax deduction = $1056

b

b Computer value = $3200 − $1056 Computer value = $2144

1

2

Calculate the value of the computer at the beginning of 2008–09, by subtracting the depreciation from the purchase price. The depreciation was 33% of its value at the end of the last financial year.

Tax deduction = 33% of $2144 Tax deduction = 0.33 × $2144 Tax deduction = $707.52

Another form of tax deduction comes for the cost of property needed while working. If you run a business from a shop or house, the cost of these premises is tax deductable. This includes expenses such as rent, interest on a loan if buying the property, rates, electricity and telephone. If the business is run from the family home, then a percentage of these expenses is allowed.


387

WORKED Example 4 Wendy runs a confectionery shop in a shopping centre. She pays rent of $400 per week, has an electricity bill of $326 per quarter and a telephone bill of $276 per month. Calculate the deduction that Wendy is entitled to. THINK 1 2 3 4

WRITE

Calculate the amount of rent that Wendy pays for a year. Calculate Wendy’s annual electricity bill. Calculate Wendy’s annual telephone bill. Add these expenses to calculate the tax deduction allowed.

Rent = $400 × 52 Rent = $20 800 Electricity = $326 × 4 Electricity = $1304 Telephone = $276 × 12 Telephone = $3312 Total tax deduction = $20 800 + $1304 + $3312 Total tax deduction = $25 416

remember 1. A tax deduction is an amount of money deducted from gross income before tax is calculated. 2. Tax deductions are allowed for work-related expenses as well as other expenses such as charity donations. 3. A tax deduction can be claimed on: (a) the cost of expenses incurred while earning an income (b) travel that is part of your job (c) depreciation of equipment used in performing your job (d) expenses associated with maintaining the premises used for your work (e) miscellaneous other costs.

388


12A 12.1

WORKED

Example

SkillS

HEET

1 Finding a percentage of a quantity


3 Kevin works as a waiter. Kevin must wear a white shirt with black pants, belt and bow tie. Kevin buys three shirts at $45.00 each, two pairs of pants at $76.90 each, a belt for $15 and a bow tie for $14.90. Kevin’s uniform must be dry-cleaned each week at a cost of $5.70. Kevin has other tax deductions of $345 for union fees, $60 for having his tax return prepared by an accountant and makes $50 in charity donations. Calculate Kevin’s total tax deductions.

12.3

SkillS

HEET


SkillS

HEET

12.4

1 Darren is a pest exterminator. He is allowed tax deductions for three sets of protective clothing at $167.50 each, two pairs of goggles at $34 each and four face masks at $13.60 each. He also uses a spray tank costing $269 and pays $5 per week to have his clothing professionally cleaned. Calculate Darren’s total tax deductions. 2 Jasmine is a dressmaker. Jasmine claims a tax deduction for the cost of her sewing machine ($560), an overlocker ($320), needles and cotton ($134.75), and $349.80 for dress patterns. Jasmine also claims to make $5 worth of work-related telephone calls per week. Calculate Jasmine’s total tax deductions.

12.2

SkillS

HEET

Calculating allowable deductions

WORKED

Example

2 Unitary method of percentages

4 Rajid uses his car as part of his job as an insurance assessor. He has a 1.6 L Nissan Tiida for which he is allowed a deduction of 45.7 c/km. Calculate the tax deduction Rajid is allowed in a year where he claims 3176 km in work-related travel. 5 The table below shows the rate per kilometre allowed as a tax deduction for travel in a private vehicle (for cars using up to 5000 km/year on work-related travel). Engine capacity

Allowable deduction

Up to 1.6 L

45.7 c/km

More than 1.6 L and up to 2.6 L

51.9 c/km

More than 2.6 L

53.8 c/km

Calculate the total tax deduction allowed for a person who claims: a 2000 km in a Mazda-2 with a 1.3 L engine b 2645 km in a Toyota Corolla with a 1.8 L engine c 1564 km in a Ford Focus with a 2.5 L engine d 2900 km in a Holden Commodore with a 3.6 L engine. WORKED

Example

3

6 Bruce is a teacher with a home computer that he purchased for $2500. If a 40% tax deduction is allowed for depreciation, calculate the tax deduction that Bruce is allowed in: a the first financial year b the second financial year c the third financial year. 7 Jeff is a builder. At the end of the 2006–07 financial year Jeff’s building equipment was valued at $12 350. If Jeff is allowed a tax deduction of 25% for depreciation of his equipment, calculate his deduction.


389

8 Mr and Mrs Williams own a farm. Their capital equipment, which includes items such as tractors, trucks etc. is valued at $75 000 at the beginning of the 2007–08 financial year. Each year they are allowed a 40% tax deduction for depreciation of capital equipment. a Calculate the tax deduction allowed for the: i 2007–08 financial year ii 2008–09 financial year iii 2009–10 financial year. b When the value of the capital equipment falls below $5000, the entire balance can be tax deducted and the equipment is said to be ‘written off’. In what financial year will this occur? 9 Catherine is a fashion designer who uses a computer to assist her with drawing. Catherine buys a new computer on 1 November for $3600. She therefore owned the 8 - of the financial year. A deduction of 40% p.a. is allowed for computer for only ----12 8 - of this. depreciation of the computer, but Catherine can claim only ----12 Calculate Catherine’s allowable tax deduction for the computer. 10 Gabrielle owns a small boutique in a shopping mall. Her operational expenses are: • $325 per week rent 4 • $280 per quarter in electricity • $185 per quarter in telephone bills. Calculate the total allowed in tax deductions for running this business.

WORKED

Example

11 Greg is a graphic designer who works from home. He has set up one room in the house as his office. In one financial year his household bills are: • $4500 in interest on the home mortgage • $1200 in council rates • electricity bills of $129, $187, $165 and $119 • telephone bills of $98.50, $110.60, $128.30 and $106.90 • $378.40 for building and contents insurance. As Greg’s office is 10% of the area of the house, he can claim 10% of all these bills as deductions. Calculate Greg’s tax deduction. 12 Henry is a motor mechanic who runs his own garage. Henry has the following work-related expenses: • $250 per week for rent on the garage • $290 per quarter for the electricity bill • $190 per quarter for the telephone bill • $25 per month for his mobile telephone • plan. Henry also has $85 000 in capital equipment that he depreciates at a rate of 27.5% p.a. He travels 2750 km on work-related trips in his van, which has a 3.0 L engine. Calculate Henry’s total tax deductions.

390


Taxable income A person’s taxable income is the income on which their tax is calculated. Most people have PAYE tax deducted from their wage or salary throughout the year. The amount of PAYE tax deducted each week or fortnight is the amount that would be paid, if this amount was earned each week for the entire financial year. When calculating the amount of PAYE tax to deduct from an employee’s pay, the employer makes no consideration of possible tax deductions or other sources of income. When a taxpayer completes a tax return at the end of the financial year, the amount of tax that should have been paid is calculated based on their taxable income. Taxable income is the gross income earned from all sources less any tax deductions. When calculating total income you must include your income from all sources. This means that you include any job for which you received payment throughout the year and other incomes such as interest, profits from shares, rental income etc.

WORKED Example 5 Michael is a carpet layer who earned a gross pay of $34 500 during the 2007–08 financial year. Michael also earned $278.50 for working as a polling officer during a State election and received $148.63 in interest from his bank accounts. Michael’s total tax deductions for the year were $1846.30. Calculate Michael’s taxable income. THINK

WRITE

1

Calculate the total gross income.

2

Calculate the taxable income by subtracting tax deductions from gross income.

Total income = $34 500 + $278.50 + $148.63 Total income = $34 927.13 Taxable income = $34 927.63 − $1846.30 Taxable income = $33 081.33

In order to calculate a person’s taxable income, you may need to calculate their income from a variety of sources and make a number of calculations about tax deductions.

WORKED Example 6 Murray works as a full-time jackeroo and plays two evenings per week in a band. Murray earns $471.52 per week from his full-time job and $118.53 per week from playing in the band. During the year Murray also earned $87.52 in interest from his bank accounts. a Calculate Murray’s total gross income. b It is 15 km from the property where Murray works to the club where he plays. When someone has two jobs, the cost of travel between jobs is tax deductable. If Murray is allowed a tax deduction of 45.7 c/km for travel, calculate the travel deduction that he is entitled to claim. c If Murray has other tax deductions of $948.50, calculate his taxable income.


391

THINK

WRITE

a

a Jackeroo earnings = $471.52 × 52 = $24 519.04 Band earnings = $118.53 × 52 = $6163.56 Total earnings = $24 519.04 + $6163.56 + $87.52 Total earnings = $30 770.12

b

1

Multiply Murray’s weekly earnings in each job by 52 to calculate the yearly total.

2

Add the total of each job with the interest earned to calculate total earnings.

1

Calculate the total number of trips made between jobs. Calculate the total kilometres travelled. Calculate the deduction by multiplying kilometres travelled by the allowable rate.

b Number of trips = 2 × 52 Number of trips = 104 Total km = 104 × 15 Total km = 1560 km Travel deduction = 1560 × 0.457 Travel deduction = $712.92

Calculate the total tax deduction. Calculate the taxable income by subtracting the tax deductions from the total income.

c Total deduction = $948.50 + $712.92 Total deduction = $1661.42 Taxable income = $30 770.12 − $1661.42 Taxable income = $29 108.70

2 3

c

1 2

remember 1. Taxable income is the income on which your tax is calculated. 2. To calculate taxable income, add income from all sources and subtract all tax deductions.

392


12B WORKED

Example

5

Taxable income

1 Fernando earns a gross salary of $45 900 per year. His tax deductions total $2145.75. Calculate Fernando’s taxable income. 2 Tony’s gross fortnightly pay is $649.20. a Calculate Tony’s gross yearly pay. b If Tony’s tax deductions total $1142.70, calculate his taxable income. 3 During the last financial year Janelle had a gross income of $45 670 from her job as a physiotherapist. Janelle also earned $238.79 in interest from her bank accounts. She also had tax deductions totalling $2340.45. Calculate Janelle’s taxable income. 4 Paula worked as a receptionist and earned a gross wage of $418.50 per week. Paula also earned $45 per week from a second job conducting telephone surveys. Paula had a bank account that paid her $117.40 in interest. a Calculate Paula’s total income. b If Paula had tax deductions totalling $1956.80, calculate her taxable income. 5 Janine has two part-time jobs. For one job she is paid $196.50 per week and for the other she is paid $395.60 per fortnight. a Calculate Janine’s gross annual income. b If Janine claims tax deductions of $428.40, calculate her taxable income. 6 Tavit is a telephone salesman. He is paid a commission of 5% of all sales. Over the year, Tavit makes sales that total $850 000. a Calculate the gross commission that Tavit earned. b Tavit makes $10 worth of phone calls that are tax deductable per day (5 days per week). Calculate the tax deduction that he will claim. c If Tavit has no other tax deductions, calculate his taxable income.

WORKED

Example

6

7 Stefan worked as a hairdresser and earned a gross wage of $537.90 per week. He also worked part-time at TAFE for a wage of $112.80 per week. Stefan’s income from various investments was $425.90 for the year. a Calculate Stefan’s gross annual income. b Stefan travelled 12 km between the hairdressing salon and TAFE, 80 times during the year. If he is allowed a tax deduction of 51.9 c/km for travel, calculate the amount that Stefan will claim. c If Stefan has other tax deductions totalling $1560, calculate Stefan’s taxable income. 8 Andrew earns a gross annual salary of $65 700. He also earns $165 per week from a rental property that he owns in Newcastle. a Calculate Andrew’s gross annual income. b Andrew travels 320 km to Newcastle and back four times a year to inspect his property and is allowed a tax deduction of 45.7 c/km. Andrew is also entitled to a deduction of $1200 for the council rates, $4325 in interest on his loan for the property, and $287.50 for insurance. Andrew claims $2340 in other deductions associated with his work. Calculate his total deductions. c Calculate Andrew’s taxable income.


393

9 Sandra works from home as an editor for a book company. Sandra is paid $986.50 per fortnight. a Calculate Sandra’s annual gross income. b Sandra has one room of her house set up as an office. This room is 15% of the area of the house. If Sandra’s total household expenses are $9800 per year, calculate the deduction that she can claim for her home office. c Sandra bought a $3850 computer. If Sandra claims 40% of this value as a tax deduction for depreciation, calculate the amount claimed. d Sandra is allowed to claim travel expenses from her home to the book company’s office once a week. If the distance is 50 km and Sandra is allowed a deduction at the rate of 51.9 c/km, calculate the amount that Sandra can claim for a travel deduction. e Calculate Sandra’s taxable income. 10 multiple choice Wayne’s gross fortnightly pay is $1156.60. He has tax deductions of $5 per week for dry-cleaning his work uniform, $50 per month in work-related travel expenses and $348 per year in union dues. Wayne’s taxable income is: A $28 863.60 B $29 653 C $29 668.60 D $60 143.20

Computer Application 1 Calculating taxable income We are going to use a prepared spreadsheet to calculate a person’s taxable income. E

Tax calculator

1. From your Maths Quest General Mathematics Preliminary Course CD-ROM, load the spreadsheet ‘Tax Calculator’. Enter the following data for income into Sheet 1, ‘Taxable Income’: salary $44 500, casual work $1258.50 and interest $258.50. You should now see a total income of $46 017.

sheet

L Spre XCE ad

394


2. The spreadsheet has a section that calculates the size of various deductions. Scroll down to row 23, where you will see calculation areas for travel, home office and depreciation.

(a) In cell B24 enter 2200 for the kilometres travelled and in B25 enter 3 for the engine capacity of the car. (b) We will now do a similar calculation to find the home office deduction. Enter the following data for home office. Home Office Calculator House area (m2) 170 Office area (m2) 17 Interest/rent $4500.00 Rates $1150.00 Telephone $ 600.00 Electricity $ 800.00 Gas Insurance $ 350.00 Other (c) The final section we will include is a depreciation calculator. Enter the following data for depreciation. Depreciation Calculator Item Value Rate Amount Computer $4000 40% Car Capital equipment $5000 25% Other $ 500 25% 3. Scroll up to the top of your spreadsheet. The results appear in the main calculation section. For Union fees enter $352, for Charity donations enter $90, and for Other enter $125. You should now see the total of all allowable deductions and the taxable income.


395

1 1 Candice has the following tax deductions: $1200 for travel, $800 for home office and $260 for depreciation. Calculate her total deductions. 2 Brendon has used his own car for a total of 3460 km on work-related matters during the financial year. Calculate the tax deduction that Brendon is entitled to if he can claim 45.7 c/km. 3 Noeline has a computer that is valued at $5000 at the beginning of the financial year. If Noeline depreciates the computer at 33% p.a., calculate the tax deduction that Noeline can claim in the financial year. 4 Calculate the tax deduction that Noeline can claim for the same computer in the next financial year. 5 How many years will it take for the value of Noeline’s computer to fall below $500? 6 Ravi has a gross annual income of $39 000. Ravi’s tax deductions total $2480. Calculate Ravi’s taxable income. 7 Indira has a gross annual salary of $43 000. She also earns $3450 per year from various investments. If Indira’s tax deductions total $4679, calculate her taxable income. 8 Vicky earns a gross wage of $482.50 per week. Calculate Vicky’s annual taxable income, if her tax deductions for the year total $2430. 9 Shoaib earns $650 per week from his day job and another $112.40 per week from a second part-time job. Calculate Shoaib’s taxable income, given that his tax deductions total $2119. 10 Megan has a taxable income of $23 900. Her tax deductions totalled $670. Calculate Megan’s gross annual income.

Medicare levy Medicare is Australia’s national health care scheme. As part of our tax, we pay the Medicare levy. In return for this, Medicare pays for basic health care services, such as visits to your local doctor, x-rays and pathology. The basic Medicare levy is 1.5% of taxable income. This is the rate that the majority of people pay. People who are on low incomes do not pay any Medicare levy or pay the levy at a reduced rate.

396


WORKED Example 7 Calculate the Medicare levy for a person with an annual taxable income of $44 300. THINK

WRITE

Calculate 1.5% of $44 300.

Medicare levy = 1.5% of $44 300 Medicare levy = 0.015 × $44 300 Medicare levy = $664.50

In Australia the government encourages people to take out private health insurance in addition to Medicare. This is to take the pressure off the public health system. They encourage people to do this in two ways. 1. The government provides a 30% rebate (refund) on the cost of the private health insurance. 2. People on higher incomes who do not have private health insurance are charged the Medicare levy surcharge. This surcharge is a further 1% of taxable income. The income threshold upon which families are charged the Medicare levy surcharge is shown by the table below. Number of dependent children

Surcharge income threshold

0–1

$100 000

2

$101 500

3

$103 000

4

$104 500

More than 4 dependent children

$104 500 plus $1500 for each additional child

For a single person the surcharge applies if their income exceeds $50 000 per annum.

WORKED Example 8 Calculate the total Medicare levy (including surcharge) paid by a person who has two children, no private health insurance and a taxable income of $164 000 per annum. THINK 1

Calculate the regular Medicare levy.

2

Decide if the person must pay the Medicare levy surcharge. Calculate the Medicare levy surcharge.

3 4

Calculate the total Medicare levy by adding the surcharge to the regular levy.

WRITE Medicare levy = 1.5% of $164 000 Medicare levy = 0.015 × $164 000 Medicare levy = $2460 This person’s income is over the $101 500 threshold for a person with two children. Surcharge = 1% of $164 000 Surcharge = $1640 Total Medicare levy = $2460 + $1640 Total Medicare levy = $4100


397

remember 1. The Medicare levy is a payment made as part of our tax that pays for basic health care services. 2. The Medicare levy is 1.5% of taxable income. 3. Allowances are made for people on low incomes who do not have to pay the levy or pay it at a reduced rate.

12C WORKED

Example

1 Calculate the Medicare levy for a person whose taxable income is $39 870. 2 Calculate the Medicare levy for a person with a taxable income of: a $43 250 b $56 745 c $94 000.


12.4 SkillS

HEET

3 Simon has a gross weekly wage of $451.75. a Calculate Simon’s gross annual wage. b Calculate the amount of Medicare levy that Simon pays annually.

12.1 SkillS

HEET

7

Medicare levy


4 multiple choice

Which of the following families do not have to pay the Medicare levy surcharge, assuming that none of them have private health insurance? A Income of $100 000 with no children B Income of $101 000 with one child C Income of $102 000 with two children D Income of $104 000 with four children 5 Mr and Mrs Wyatt have five children. What is the threshold for the Medicare levy surcharge for this family? 6 Calculate the total Medicare levy (including surcharge) paid by a person who has three children, no private health insurance and a taxable income of $184 000 per annum. 8 7 Calculate the total Medicare levy (including surcharge) paid by a person who has eight children, no private health insurance and a taxable income of $140 000 per annum.

WORKED

Example

Medicare levy 1 At the time of writing, the Medicare levy was 1.5% of gross income. This is reviewed each year in the federal budget. Find out the current Medicare levy. 2 What is the lower income threshold at which no Medicare levy is paid?

T SHEE

Work

8 James has an annual income of $250 000 and is single with no children. The cost of private health insurance for James would be $950 per year. Calculate how much James would save each year by joining a private heath fund as opposed to having to pay the Medicare levy surcharge.

12.1

398


Calculating tax The amount of PAYE tax payable is based on a table that shows the annual tax payable. This annual amount is then divided into a weekly or fortnightly amount. Taxable income is broken into five tax brackets. As you earn more money the rate of tax increases, as shown in the table below. The table is based on whole dollar amounts and so any cents earned are ignored for the purposes of calculating tax. Note that we do not round off when ignoring cents; we always round down. For example, a person who earns $35 956.90 has their tax calculated on $35 956. Taxable income

Tax payable

$1 to $6000

$0

$6001 to $30 000

15c for each dollar over $6000

$30 001 to $80 000

$3600 plus 30c for each $1 over $30 000

$80 001 to $180 000

$18 600 plus 40c for each $1 over $80 000

Over $180 000

$58 600 plus 45c for each $1 over $180 000

From the table we can see that there are five tax brackets. Bracket 1. People who earn less than $6000 per year pay no tax. Bracket 2. People who earn between $6001 and $30 000 pay 15c for every dollar over $6000. Bracket 3. People who earn between $30 001 and $80 000 pay $3600 plus 30c for every dollar over $30 000. Bracket 4. People who earn between $80 001 and $180 000 pay $18 600 plus 40c for every dollar over $80 000. Bracket 5. People who earn over $180 000 pay $58 600 plus 45c for every dollar over $180 000. To calculate the amount of PAYE tax that should be deducted from a person’s income, we need to see what tax bracket they are in and then apply the appropriate rule.

WORKED Example 9 Calculate the annual tax payable on a taxable income of $36 600. THINK 1 2 3 4 5

$36 600 is in the $30 001 to $80 000 tax bracket. Tax payable is $3600 plus 30c (0.3) for each $1 over $30 000. Calculate the amount over $30 000 by subtracting $30 000 from $36 600. Apply the rule $3600 plus 30c for each $1 over $30 000. Calculate.

WRITE

$36 600 − $30 000 = $6600 Tax payable = $3600 + 0.3 × $6600 Tax payable = $5580


399

When calculating the total amount of tax payable, this income tax must be added to the Medicare levy.

WORKED Example 10 Christian has a taxable income of $85 000 per year. a Calculate the income tax that Christian must pay. b Calculate the Medicare levy for Christian if he is in a private health fund. c Calculate Christian’s total tax for the year. THINK

WRITE

a

a

1 2 3

$85 000 is in the $80 001 to $180 000 tax bracket. Subtract $80 000 from $85 000 to calculate the amount over $80 000. Apply the rule $18 600 plus 40c for each $1 over $80 000.

$85 000 − $80 000 = $5000 Income tax = $18 600 + 0.4 × $5000 Income tax = $20 600

b Calculate the Medicare levy as 1.5% of $85 000.

b Medicare levy = 1.5% of $85 000 Medicare levy = 0.015 × $85 000 Medicare levy = $275

c Add the Medicare levy to the income tax to find the total tax payable.

c Total tax = $20 600 + $1275 Total tax = $21 875

When calculating the weekly tax payable, it is assumed that the gross amount earned that week is earned for the whole year, without consideration of tax deductions. The Medicare levy is taken out as part of PAYE tax. The yearly amount of tax is then calculated using this amount, then divided into a weekly payment.

WORKED Example 11 Trevor earns a gross wage of $772.70 per week. Calculate the amount of PAYE tax that is deducted by his employer. THINK 1

2 3 4

Calculate $772.70 per week as an annual amount by multiplying by 52 (ignore any cents). $40 180 is in the $30 001 to $80 000 tax bracket. Calculate the amount over $30 000, by subtracting $30 000 from $40 180. Apply the rule to calculate the annual tax payable.

WRITE $772.70 per week = $40 180 per year.

$40180 − $30000 = $10180 Annual tax = $3600 + 0.3 × $10180 Annual tax = $6654 Continued over page

400


THINK

WRITE

5

Calculate the Medicare levy as 1.5% of $40 180.

6

Calculate the total tax payable for the year. Divide $7256.70 into weekly instalments, by dividing by 52.

7

Medicare levy = 1.5% of $40180 Medicare levy = 0.015 × $40180 Medicare levy = $602.70 Total yearly tax = $6654 + $602.70 Total yearly tax = $7256.70 Weekly tax = $7256.70 ÷ 52 Weekly tax = $139.55

PAYE tax taken out by an employer is based on the pay being the employee’s only source of income without tax deductions. In most cases this means that the amount of tax paid by the end of the year will not be correct. For this reason, every taxpayer must complete a tax return. Before completing a tax return the taxpayer must collect a group certificate from each of their employers. A group certificate is a statement of gross earnings and the amount of PAYE tax that has been deducted from those earnings.

In a tax return, all group certificates are collected to find the total gross income and total PAYE tax already paid. All allowable deductions are then subtracted to calculate taxable income. The correct amount of tax is then calculated. Based on this calculation, the taxpayer will then either receive a refund or pay the difference.


401

WORKED Example 12 Catherine’s gross annual salary as a veterinarian’s assistant is $44 500. She has paid $8617.50 in PAYE tax. Catherine has also earned $560.40 in interest from an investment and has tax deductions totalling $2345. a Calculate Catherine’s taxable income. b Calculate the tax payable on Catherine’s taxable income, including the Medicare levy. c Calculate the amount that Catherine should receive as a tax refund or the amount of Catherine’s tax debt. THINK

WRITE

a

a Taxable income = $44 500 + $560.40 − Taxable income = $2345 Taxable income = $42 715.40 Taxable income = $42 715

1

2

b

1 2

3 4

5

c

1 2

3

Calculate taxable income by adding all incomes and subtracting any tax deductions. Ignore cents in stating the taxable income. $42 715 is in the $30 001 to $80 000 tax bracket. Subtract $30 000 from $42 715 to calculate the amount earned over $30 000. Apply the rule $3600 plus 30c for each $1 over $30 000. Calculate the Medicare levy.

Calculate the total tax payable by adding the income tax to the Medicare levy. Catherine has paid more tax than she needed to. So she gets a refund. Calculate the size of the refund by subtracting the amount she should pay ($8055.23) from the amount that she paid ($8617.50). Give a written answer.

b $42 715 − $30 000 = $12 715 Income tax = $3600 + 0.3 × $12 715 Income tax = $7414.50 Medicare levy = 1.5% of $42 715 Medicare levy = 0.015 × $42 715 Medicare levy = $640.73 Total tax = $7414.50 + $640.73 Total tax = $8055.23 c $8617.50 − $8055.23 = $562.27

Catherine receives a refund of $562.27.

402


remember 1. PAYE tax is the tax calculated based on your weekly or fortnightly earnings ignoring other income sources or tax deductions. 2. At the end of the financial year each taxpayer must complete a tax return, which compares the PAYE tax paid with the correct amount of tax. The taxpayer then either receives a refund or must pay the difference. 3. The rate of tax payable increases with income. The income tax scales are based on tax brackets. To calculate tax, use the tax table to determine the appropriate tax bracket and apply the rule for that bracket. 4. To calculate the total tax payable, the income tax component must be added to the Medicare levy. 5. A refund or tax debt is calculated by finding the difference between the tax that has been paid over the year and the tax that should be paid.

12D 12.1

WORKED

Example

SkillS

HEET

9

1 Use the income tax table on page 398 to calculate the income tax payable on an annual taxable income of $35 450. 2 Calculate the income tax payable on each of the following taxable incomes. a $5500 b $18 675 c $31 250 d $44 320 e $92 850 f $208 000


12.4

3 Julie receives a gross pay of $627.68 per week. a Calculate Julie’s gross annual pay (remember to ignore cents). b Calculate the annual amount of tax that Julie must pay, based on this amount (remember to ignore cents).

SkillS

HEET

Calculating tax


4 Gregory earns a gross pay of $1963.80 per fortnight. Calculate the annual amount of tax that Gregory must pay, based on this amount. WORKED

Example

10

5 Johann has an annual taxable income of $35 600. a Calculate the amount of income tax Johann must pay. b Johann is married with one child. Calculate the Medicare levy for Johann. c Calculate the total amount of tax that Johann must pay for the year. 6 For each of the following taxpayers, calculate the total amount of tax that they must pay (assume each must pay the 1.5% Medicare levy). a Andre, whose taxable income is $23 500 b Brianna, whose taxable income is $72 000 c Catelyn, whose taxable income is $106 000

WORKED

Example

11

7 Sandy earns a gross weekly pay of $478.60. Calculate the amount of PAYE tax deducted each week by her employer. 8 Ashley earns a gross fortnightly pay of $2174.35. Calculate the amount of PAYE tax that Ashley’s employer should deduct each fortnight.


403

9 multiple choice Frieda’s taxable income is $50 000 per year. The total amount of tax that she should pay for the year, including the Medicare levy is: A $3600 B $9600 C $10 350 D $20000 10 multiple choice Henry earns a gross pay of $1295.60 per fortnight. The amount of PAYE tax incuding the Medicare levy that will be deducted from Henry’s pay over a full year will be: A $4701.00 B $5210.78 C $10 101.00 D $33 670.00 11 multiple choice Ian receives a gross pay of $822.50 per week. The only deduction that Ian has taken from his gross pay is tax. Ian’s net weekly pay will be: A $142.90 B $155.24 C $667.26 D $679.60 12 At right is a group certificate for Wendell Hancock. GROUP CERTIFICATE Wendell has also earned $372.10 in interest from 12 an investment and has tax deductions totalling $1298. Wendell Hancock Gross income: $39 600.00 a Calculate Wendell’s taxable income. PAYE tax deducted: $7 074.00 b Calculate the tax payable on Wendell’s taxable income, including the Medicare levy. c Calculate the amount that Wendell should receive as a tax refund or what he must pay in tax.

WORKED

Example

13 Raymond earns a gross weekly pay of $1748.90. a Calculate Raymond’s gross annual pay. b Calculate the amount of PAYE tax including the Medicare levy (but not the surcharge) that would be deducted from Raymond’s pay each week. c During the year Raymond earned $45.15 in bank interest, and had tax deductions totalling $1296. Calculate the amount of tax that Raymond should pay for the year, based on his annual taxable income. d Calculate his refund or tax debt. 14 Vonda Flockhart is employed by day as a journalist and by night as a radio announcer. Her group certificates are shown at right. a Calculate Vonda’s gross annual pay from both jobs and the total amount of PAYE tax that Vonda has paid. b Vonda earned $184.40 in interest from bank accounts and had $3276 worth of tax deductions for the year. Calculate Vonda’s taxable income. c Calculate the amount of tax that Vonda should have paid throughout the year, including the Medicare levy, but not the surcharge. d Calculate the tax refund that Vonda is owed.

Job 1 GROUP CERTIFICATE

Vonda Flockhart Journalist Gross income: $35 000.00 PAYE tax deducted: $5 625.00 Job 2 GROUP CERTIFICATE

Vonda Flockhart Radio announcer Gross income: PAYE tax deducted:

$9605.00 $4466.33

15 Jelena receives a gross weekly pay of $1350.52. a Calculate the amount of PAYE tax, including the Medicare levy, that Jelena should have deducted from her pay for the year. b If at the end of the financial year Jelena earned $11 274.56 from other sources and had tax deductions totalling $3650, calculate Jelena’s tax refund or tax debt.

404


Computer Application 2 Tax calculation EXCE

et

reads L Sp he

Tax calculator

We are now going to continue our tax calculation spreadsheet. In the previous computer application we used the spreadsheet to calculate a person’s taxable income. We will now use it to calculate income tax and the Medicare levy. 1. From your Maths Quest General Mathematics Preliminary Course CD-ROM, open the spreadsheet ‘Tax Calculator’.

If you saved your information from the taxable income, then the second spreadsheet, ‘Income Tax’, reads the taxable income and does the appropriate calculations for income tax and the Medicare levy. 2. The formula in cell B7 calculates the Medicare levy by taking 1.5% of the taxable income. 3. The income tax is calculated by making a calculation for the appropriate tax bracket only. This figure is then transferred to the top of the spreadsheet and added with the Medicare levy to calculate the total tax payable.

2 Calculate the Medicare levy for a person with a taxable income of: 1 $40 000 2 $24 000 3 $29 467 4 $25 670 5 $33 508 Use the table on page 398 to calculate the income tax payable (without Medicare levy) on an annual taxable income of: 6 $5000 7 $19 560 8 $32 500 9 $93 675 10 $182 000


405

Calculating GST and VAT When you purchase most items you must pay the GST on that item. GST stands for Goods and Services Tax. The GST is a tax amounting to 10% of the purchase price of that item. There are some items that are exempt from the GST. These include fresh food, some educational costs and some medical costs. The GST is an example of an indirect tax. This is because the individual does not pay the tax directly to the government and there is no record kept of who is paying the tax. The tax is collected at the point of sale. To calculate the amount of GST payable on an item, we simply calculate 10% of the purchase price.

WORKED Example 13 A cricket bat has a pre-GST price of $127.50. Calculate the GST payable on the purchase of the bat. THINK

WRITE

Calculate 10% of $127.50.

GST payable = 10% of $127.50 GST payable = 0.1 × $127.50 GST payable = $12.75

When calculating the amount required to purchase an item, you will need to add the GST to the pre-tax price. The quickest way to do this will be to calculate 110% of the pre-tax price. By using this method we add the 10% GST to 100%, which represents the cost of the item. In this way there is only one calculation to make.

WORKED Example 14 The Besenko family goes to McDonald’s for lunch. The cost of the meal before GST is $19.80. How much will the Besenkos have to pay for the meal, including the GST? THINK

WRITE

1

Calculate 110% of $19.80.

2

Give a written answer, rounding your answer to the nearest 5c.

Total cost = 110% of $19.80 Total cost = 1.1 × $19.80 Total cost = $21.78 The cost of the meal will be $21.80.

When we are given the total cost of an item including GST, we need to reverse the above process to calculate the pre-tax price of the item. This means that we need to divide the total cost by 110%, written as a decimal.

406


WORKED Example 15 Calculate the pre-tax price of a car that costs $31 350, including GST. THINK 1 2

WRITE

Total cost is 110% of the price. Price is total cost divided by 1.1.

Price = $31 350 ÷ 1.1 Price = $28 500

Taxes similar to the GST apply in many countries. These taxes are levied at different percentages in different countries and in many cases are called value added tax (VAT). The methods used to calculate the amount of VAT are the same as for Australia’s GST; however, the rate must be checked for each question.

WORKED Example 16 New Zealand has a VAT levied at a rate of 12.5%. Vanessa goes on holidays to New Zealand and rents a car for five days at a rate of NZ$56.50 per day (before VAT). Calculate the total cost of renting the car including the VAT. THINK 1

2

WRITE

Calculate the cost of the car by multiplying the daily rate by the number of days. Add the VAT by calculating 112.5% of the cost.

Cost = $56.50 × 5 Cost = $282.50 Total cost = 112.5% of $282.50 Total cost = $317.81

Note that in other countries there may be 1c and 2c pieces and so we do not take the answer to the nearest 5c.

remember 1. The GST is a tax of 10% of the purchase price of all items other than a few exemptions. 2. To calculate the amount of GST payable, calculate 10% of the purchase price. 3. To calculate the total cost of an item including the GST, calculate 110% of the pre-tax price. 4. To calculate the price of an item when you are given the total cost including the GST, divide the total cost by 110%. 5. A value added tax (VAT) is a tax similar to the GST that is applied in many countries. The VAT rate varies between countries; however, the same methods of calculation are used as for the GST.


12E WORKED

Example

Calculating GST and VAT

1 Calculate the GST payable on a book that has a pre-tax price of $35.60.

12.1 SkillS

WORKED

Example

14

method

4 A pair of sports shoes that cost $112.50 has 10% GST added to the cost. Calculate the of percentages total cost of the sports shoes. 5 Calculate the total cost of each of the following items after the 10% GST has been added (prices given are pre-tax): a a football jersey priced at $114.90 b a CD priced at $29.90 c a bunch of flowers priced at $14.70 d a birthday card priced at $4.95 e a jar of coffee priced at $5.88. 6 Jia travels to New Zealand where the VAT is set at 12.5%. Calculate the amount of tax payable on each of the following items: a a camera priced at $240 b a bus fare for $7.50 c a whitewater rafting tour costing $376 d a ski lift ticket costing $23.50 e a new suitcase priced at $78.90.

SkillS

HEET

3 Calculate the GST payable on each of the following items (correct to the nearest Increase or cent): decrease by a a a barbecued chicken with a pre-tax price of $7.99 percentage b a tin of shoe polish with a pre-tax price of $4.81 c a tin of dog food with a pre-tax price of 93c 12.4 d a pack of toilet rolls with a pre-tax price of $6.25 e a pack of frozen pies with a pre-tax price of $3.36. Unitary

SkillS

HEET

2 Calculate the GST payable on each of the following items (prices given are pre-tax): Finding a a a bottle of dishwashing liquid at $2.30 percentage of b a basketball at $68.90 a quantity c a pair of cargo pants at $98.50 d a bus fare at $1.30 12.2 e a restaurant meal for which the bill totals $89.90.

HEET

13

407

408 WORKED

Example

15


7 A restaurant bill totals $108.35 including the 10% GST. Calculate the actual price of the meal before the GST was added.

8 A bus fare was $2.09 including the 10% GST. Calculate: a the bus fare without the GST b how much GST was paid. WORKED

Example

16

9 Austin travels to the USA. In the state of Utah a VAT is levied at 11%. Calculate what Austin will pay for four nights accommodation in a hotel that charges $78.40 per night before VAT. 10 Nancy travels to the USA. In California, the VAT is 7.5% of the price of the item. Calculate what Nancy will pay for each of the following items: a a postcard that has a ticketed price of $1 b Disneyland entry that is $75 c two nights accommodation at a hotel for $89.90 per night d a restaurant meal for which the bill totals $45.78 e a taxi fare that costs $6.46. 11 Sachin decides to purchase a new car. The pre-tax cost for the basic model of the car is $30 500. It is an extra $1200 for an automatic car, an extra $1600 for airconditioning, $1000 for power steering, $600 for a CD player and $450 for alloy wheels. Calculate the cost of each of the following cars, after the 10% GST has been added: a the basic model car b an automatic car with airconditioning c a car with a CD player and alloy wheels d a car with all of the above added extras.


409

Graphing tax functions We can draw linear graphs to display the tax payable. This is possible for both the GST and income tax.

12F

Graphing tax functions 50

12.5 SkillS

GST ($)

40

HEET

1 Draw a set of axes with the price on the horizontal axis and GST on the vertical axis, as shown. a Calculate the GST payable on items that cost: i $100 ii $200 iii $500. b Join these points with a straight line to show the GST function.

30 Sketching straight lines

20 10 0 0

50 100 150 200 250 300 350 400 450 500 Price ($)

Income tax ($' 000)

2 Draw a set of axes with ‘Income’ on the horizontal axis and ‘Income tax’ on the vertical axis, as shown below. 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0 200 195 190 185 180 175 170 165 160 155 150 145 140 135 130 125 120 115 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0 Income ($' 000)

a What is the tax payable on gross incomes up to $6000? Show this on the axes. b Calculate the tax payable on a gross income of: i $10 000 ii $30 000. This is the second tax bracket. Mark these points on the axes and join these points with a straight line. c Calculate the tax payable on a gross income of: i $30 000 ii $50 000 iii $80 000. This is the third tax bracket. Mark these points on the axes and join these points with a straight line.

410

Work

T SHEE

12.2


d Calculate the tax payable on a gross income of: i $100 000 ii $180 000. This is the fourth tax bracket. Mark these points on the axes and join these points with a straight line. e Calculate the tax payable on a gross income of: i $180 000 ii $190 000 iii $200 000. This is the fifth tax bracket. Mark these points on the axes and join these points with a straight line.


411

summary Allowable deductions • Allowable tax deductions are amounts that are deducted from gross income, as they are not taxable. • Deductions are allowed for work-related expenses and other items such as charity donations.

Taxable income • Taxable income is the income on which income tax is assessed. • Taxable income is calculated by subtracting any allowable deductions from gross income.

Medicare levy • • • •

The Medicare levy is part of the tax system that funds basic health care services. For most people the Medicare levy is 1.5% of gross income. People on low incomes either pay no Medicare levy or pay it at a reduced rate. People on high incomes with no private health insurance must pay a Medicare levy surcharge of an extra 1% of taxable income.

PAYE tax • Pay As You Earn (PAYE) tax is deducted from your gross pay each week or fortnight. • Tax is calculated on the taxpayer’s gross annual income. This is then divided into weekly or fortnightly amounts. • At the end of the financial year the taxpayer submits a tax return. • In a tax return the correct amount of tax for the year is calculated. The taxpayer then either receives a tax refund or pays a tax debt.

GST • The most common indirect tax is the GST, which began in Australia on 1 July 2000. • The GST is a 10% tax paid on the cost of all goods and services with the exception of some basic foods. • Some other countries have a value added tax (VAT), which is similar to the GST but levied at different rates.

412


CHAPTER review 12A

1 Tony is employed as a motor mechanic. Tony claims deductions of $1800 to buy tools, $225 for three pairs of work overalls, $5 per week for dry-cleaning of these overalls and $2.50 per week for work-related telephone calls. Calculate Tony’s total tax deductions.

12A

2 Catherine is a computer consultant who uses her own vehicle for work. Each week she must make several visits to businesses that use her computer systems. Catherine travelled 4523 km on work-related trips during the year. Calculate the tax deduction that Catherine can claim if she is entitled to claim at the rate of 47.5c per kilometre.

12A

3 Brian runs a small bakery and has equipment in his bakery to the value of $45 000 at the beginning of the 2006–07 financial year. For tax purposes he depreciates these items at a rate of 28% p.a. Calculate the tax deduction that Brian can claim for depreciation in: a 2007–08 b 2008–09 c 2009–10.

12A

4 If a $5000 computer can be depreciated at a rate of 33% p.a., how many years will it take for its value to fall below $500?

12A

5 Verity is employed as a vet. Verity has the following tax deductions. (a) $1500 for appropriate clothing (b) $5 per week for dry-cleaning (c) $2590 for new equipment (d) 28% depreciation on major equipment currently valued at $65 000 (e) 3287 km in travelling expenses (at a rate of 51.9 c/km) (f) $127.45 per month in telephone calls (g) $318 per quarter for electricity Calculate Verity’s total tax deductions.

12B

6 Eddie has a gross annual salary of $46 000 and has tax deductions that total $2117. Calculate Eddie’s taxable income.

12B

7 From her job as a journalist, Jana earns a gross annual salary of $72 000. Jana also earns $3540 per year from her investments. If Jana has tax deductions totalling $5120, calculate her taxable income.

12B

8 Allison’s gross weekly wage is $539.50. Allison also earned $107.40 per quarter in interest from a fixed term deposit. Calculate: a Allison’s total earnings for the year b Allison’s taxable income, given that she had deductions that total $2018.

12B

9 Raymond has two jobs. One earns him $938.50 per fortnight and the other $190.60 per week. He also earned $97.10 in interest throughout the year. a Calculate Raymond’s gross annual income. b In travelling between jobs, Raymond made 104 trips at 23 km per trip. For this he was entitled to claim a tax deduction at the rate of 45.7 c/km. Raymond had other tax deductions totalling $950. Calculate Raymond’s taxable income for the year.


413

10 The Medicare levy is paid at a rate of 1.5% of taxable income. Calculate the Medicare levy that must be paid by a person whose taxable income is $39 000 per year.

12C

11 Calculate the amount of Medicare levy that is payable by each of the people below. a Tanya has a taxable income of $15 500. b Sam has a taxable income of $29 000. c Emma has a taxable income of $47 500. d Gavin has a taxable income of $83 507 and is in a private health fund. e Holly has a taxable income of $99 000 and is in a private health fund.

12C

12 Use the table on page 396 to determine the Medicare levy paid by a family with 3 children, no private health insurance and a taxable income of: a $87 600 b $101 000 c $156 000.

12C

13 Use the tax table on page 398 to calculate the tax payable on a taxable income of $44 500.

12D 12D

14 Use the tax table on page 398 to calculate the tax payable on an income of: a $5000 b $19 357 c $35670 d $89 562 e $278 000. 15 Brett earns a gross weekly wage of $653.60. a Calculate Brett’s gross annual pay. b Calculate annual tax that would be payable on this annual amount. c Calculate the Medicare levy payable on this annual amount. d Calculate the total annual tax payable. e Calculate the weekly PAYE tax that would be deducted from Brett’s wages.

12D

16 Fiona has a gross fortnightly salary of $3367.90. Calculate the amount of PAYE tax that would be deducted from Fiona’s pay each fortnight.

12D

17 Neville has a gross annual salary of $43 750. He has tax deductions totalling $3495. During the year Neville has had $8381.25 deducted in PAYE tax. a Calculate Neville’s taxable income. b Calculate the total tax payable on this amount, including the Medicare levy. c Calculate the tax refund that Neville should receive.

12D

18 An electric guitar has a pre-tax price of $990. Calculate the amount of GST payable on the guitar.

12E

19 Calculate the amount of GST payable on each of the following items (prices given are pretax): a a takeaway meal at $11.30 b a lawnmower at $369.00 c a bus fare costing $1.20 d a hair style priced at $37.50 e a $12.50 movie ticket.

12E

20 A round of golf costs $20 before tax. Calculate the total cost of the game after the 10% GST is added.

12E

21 The total cost of a restaurant meal was $123.75, including GST. Calculate the actual cost of the meal without the tax.

12E

414


Practice examination questions 1 multiple choice Slavisa is a nurse with a gross annual income of $45 675. He has earned $136.50 in interest from his bank accounts and has allowable deductions of $680. Slavisa’s taxable income is: A $44 858.50 B $45 131.50 C $46 218.50 D $46 491.50 2 multiple choice Allan’s taxable income is $48 000. Allan’s tax for the year is (use the PAYE tax table on page 398): A $720 B $3600 C $9000 D $9720 3 multiple choice Bradley receives a bill for $489.50 for car repairs, which includes GST. The cost of the repairs without the tax was: A $440.55 B $445.00 C $489.50 D $538.45 4 Lleyton is employed as a forklift driver and receives a gross weekly wage of $970. a Calculate Lleyton’s gross annual wage. b If Lleyton has tax deductions totalling $1194 and has earned $75.80 from other sources, calculate Lleyton’s taxable income. c Calculate the Medicare levy for Lleyton. d Use the table on page 398 to calculate the income tax due for Lleyton. e If Lleyton’s employer has deducted $5083.20 in PAYE tax through the year, calculate Lleyton’s tax refund or tax debt.

CHAPTER

test yourself

12

5 Vicky goes to a travel agent to enquire about a holiday in New Zealand. The cost of accommodation in a New Zealand hotel is A$75 per night. a If Vicky books this at the travel agent she must pay the 10% GST on the cost of the accommodation. Calculate the total cost of four nights in Australian dollars. b If Vicky flies to New Zealand and books the hotel on arrival, the cost is NZ$80 per night. If there is a 12.5% VAT on the hotel room, calculate the total cost of four nights in New Zealand dollars. c A$1 = NZ$1.12. What is the cheapest way to book the accommodation and by how much is it cheaper?

Right-angled triangles

13 syllabus reference Measurement 4 • Right-angled triangles

In this chapter 13A Pythagoras’ theorem 13B Calculating trigonometric ratios 13C Finding an unknown side 13D Finding angles 13E Angles of elevation and depression

areyou

READY?

Are you ready?


13.1

Labelling sides of a right-angled triangle

1 In each of the following, name the opposite side, adjacent side and the hypotenuse. a A b c X P

C

B

Q

Y

R

Z

13.2

Using Pythagoras’ theorem

2 Find the value of x, correct to 1 decimal place. 10 cm a b x

5.9 cm

27 cm

x

3.5 cm

13.3

Rounding to a given number of decimal places

13.4

x Solving equations of the type a = -- to find x b

3 Round each of the following numbers to the number of decimal places indicated in brackets. a 4.7368 [2] b 18.539 [1] c 0.377 51 [3] d 507.182 09 [3] e 10.797 [2] f 0.764 281 [4]

4 Solve each of the following equations. x x a 0.6 = -----b 1.45 = --20 9

13.5

b Solving equations of the type a = -- to find x x

5 Solve each of the following equations. 6 2.7 a 0.25 = --b 7.5 = ------x x

13.7

x c 0.7328 = ------4.7

76.95 c 0.1425 = ------------x

Rounding angles to the nearest degree

6 Round each of the following to the nearest degree. a 23.698o b 47.215o c 27o24′34″

d 86o45′12″

Chapter 13 Right-angled triangles

417

History of mathematics P Y T H AG O R A S O F S A M O S ( c i rc a 5 8 0 B C – 5 0 0 B C ) a During his life: • Taoism is founded • Kung-Fu-tse (Confucius) is born • Buddhism is founded.

Pythagoras was a famous Greek mathematician and mystic who is now best known for his theorem about the sides of a triangle. He was born on Samos Island, near Greece. It is believed that he was born about 580 BC and died about 500 BC, but because of the way dates were recorded then, various dates are given for his life. Not much is known about his personal life, but we know that he had a wife, son and daughter. When Pythagoras was a young man, he travelled to Egypt and Babylonia (Mesopotamia) where he learned much of his mathematics and developed an interest in investigating it further. He founded a cult with the idea that ‘the essence of all things is a number’. This group believed that all nature could be expressed in terms of numbers. They found, however, that some numbers could not be expressed as rational numbers, such as 2 . They kept this information to themselves and there is a story that they killed one member who told somebody else about this problem. Pythagoras showed that musical notes had a mathematical pattern. He stretched a string tightly and found that it produced a certain sound and then found that if he halved the

length of the string, it produced a sound that was in harmony with the first. He also found that if it was not exactly half, or a multiple of a half, then it was a clashing sound. This approach is still used in musical instruments today. He is credited with the discovery now known as ‘Pythagoras’ theorem’. This states that ‘For a right-angled triangle, the square of the hypotenuse (long side) is equal to the sum of the squares of the two short sides’ and is normally written as c2 = a2 + b2. Other people knew of this idea long before he announced it, and there is a Babylonian tablet known as ‘Plimpton 322’ (believed to have been made about 1500 years before Pythagoras was born), c which has a set of these b sort of values (which are now called Pythagorean a triples). He used a string with knots in it to demonstrate and make right angles. Some examples of Pythagorean triples are: 3, 4 and 5 8, 15 and 17 and a more difficult example: 20, 99 and 101.

Questions 1. Where was Pythagoras born? 2. Where did he travel to and learn most of his mathematics? 3. What is the formula for his famous theorem? 4. What did he start to investigate about patterns in nature? 5. What is the name of the ancient tablet that contains Pythagorean triples? 6. What is a Pythagorean triple?

418


Pythagoras’ theorem Pythagoras’ theorem allows us to calculate the length of A a side of a right-angled triangle, if we know the lengths of hypotenuse the other two sides. Consider LABC at right. AB is the hypotenuse (the longest side). It is opposite the B C right angle. Note that the sides of a triangle can be named in either of two ways. 1. A side can be named by the two capital letters given to the vertices at each end. This is what has been done in the figure above to name the hypotenuse AB. 2. We can also name a side by using the lower-case letter of the opposite vertex. In the figure above, we could have named the hypotenuse ‘c’.

WORKED Example 1

P

Name the hypotenuse in the triangle at right. THINK

WRITE R

Q 1 2

The hypotenuse is opposite the right angle. The vertices at each end or the lower case of the opposite vertex can be used to name the side.

The hypotenuse is QR or p.

Pythagoras’ theorem In any right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. Writing this result as a formula we say: c2 = a2 + b2 This is the formula used to find the length of the hypotenuse in a right-angled triangle when we are given the lengths of the two shorter sides.

WORKED Example 2 Find the length of the hypotenuse in the triangle at right. THINK 1 2 3 4

Write the formula. Substitute the lengths of the shorter sides. Evaluate the expression for c2. Find the value of c by taking the square root.

WRITE 2

c 15 cm

c =a +b c2 = 82 + 152 2

8 cm

2

c2 = 64 + 225 c2 = 289 2 c = 289 c2 = 17 cm

419


In this example, the answer was a whole number because we are able to find 289 exactly. In most examples this will not be possible. In such cases, we are asked to write the answer correct to a given number of decimal places or significant figures. By rearranging Pythagoras’ theorem, we can write the formula to find the length of a shorter side of a triangle. If c2 = a2 + b2

then and

a2 = c2 − b2 b2 = c2 − a2.

The method of solving this type of question is the same as in the previous example, except we use subtraction instead of addition. For this reason, it is important to look at each question carefully to determine whether you are finding the length of the hypotenuse or one of the shorter sides.

WORKED Example 3

R

Find the length of side PQ in triangle PQR, correct to 3 significant figures. THINK 1 2 3 4

16 m P

WRITE

Write the formula. Substitute the lengths of the known sides. Evaluate the expression. Find the answer by finding the square root.

9m Q

a

a =c −b a2 = 162 − 92 2

2

2

= 256 − 81 = 175 a = 175 = 13.2 m

Pythagoras’ theorem states that in a right-angled triangle, c2 = a2 + b2. The converse of this theorem states that if c2 = a2 + b2 then the triangle is right angled. This is a method used by builders to ensure that a structure is ‘square’.

WORKED Example 4 Determine whether the triangle at right is right angled. THINK 1

2 3

Calculate c2 and a2 + b2 separately.

Write down an equality or inequality statement. Write a conclusion.

WRITE

5 cm

4 cm 7 cm

c2 = 72 = 49 2 2 a + b = 52 + 42 = 25 + 16 = 41 c2 ≠ a2 + b2 Therefore the triangle is not right angled.

420


These formulas can be used to solve more practical problems. In these cases, it is necessary to draw a diagram that will help you to see which method for finding a solution is required. The diagram simply needs to represent the triangle; it does not need to show details of the situation described.

WORKED Example 5 The fire brigade attends a blaze in a tall building. They need to rescue a person from the 6th floor of the building, which is 30 metres above ground level. Their ladder is 32 metres long and must be at least 10 metres from the foot of the building. Can the ladder be used to reach the people needing rescue? THINK 1

WRITE

Draw a diagram and show all given information.

burning building

30

c fire engine 10

2

3 4

5 6

Write the formula after deciding if you are finding the hypotenuse or a shorter side. Substitute the lengths of the known sides. Evaluate the expression.

c2 = a2 + b2

Find the answer by taking the square root. Give a written answer.

c = 1000 ≈ 31.62 m The ladder will be long enough to make the rescue.

c2 = 302 + 102 = 900 + 100 = 1000

remember 1. Make sure that you can identify the hypotenuse of a right-angled triangle. It is the side opposite the right angle. 2. If you are finding the length of the hypotenuse use c2 = a2 + b2. 3. If you are finding the length of a shorter side use either a2 = c2 − b2 or b2 = c2 − a2. 4. Read the question carefully to make sure that you give the answer in the correct form. 5. Begin written problems with a diagram and finish with an answer written in words.


13A WORKED

Example

Pythagoras’ theorem

1 Name the hypotenuse in each of the following triangles. a b c X

13.1 SkillS B

P

Labelling sides of a right-angled triangle

Y

R

Example

2

2 Find the length of the hypotenuse in each of the following triangles. a b c x

5 cm

Using Pythagoras’ theorem

z

m

80 mm

13.2 SkillS

C

11 m

12 cm

60 m

150 mm

Pythagoras

3 In each of the following, find the length of the hypotenuse, correct to 2 decimal places. a b c 4.9 m

Pythagoras

4.9 m

Example

3

GC

4 Find the length of each shorter side in the right-angled triangles below, correct to 1 decimal place. a b c 2.01 m

am progr –C

asio

6 cm WORKED

Cabri Geo

ry met

11.3 km

8.6 km

a

9 cm

sheet

L Spre XCE ad

E

WORKED

A

Z

HEET

Q

HEET

1

421

Pythagoras

–TI

program GC

6 cm

p

2.9 m

2.2 m

2 cm

4.37 m

t

Pythagoras

q

5 In each of the following right-angled triangles, find the length of the side marked with a pronumeral, correct to 1 decimal place. a b 10.5 cm

8 cm

m

24.5 cm

z

4 cm

c

33 mm

d

p 37.25 m

34 mm

a

52.75 m

422 WORKED

Example

4


6 Use the converse of Pythagoras’ theorem to determine if the following triangles are right angled. a b c 31 m 10 cm 8 cm 6 cm

41 cm

40 cm

38 m

16 m

9 cm

7 multiple choice

Z

X

The hypotenuse in LXYZ at right is: A XY B XZ C YZ D impossible to tell

Y

8 multiple choice Which of the following triangles is right angled? A B

5m

24.5 m

32.5 m

24.5 m 25 m

20.5 m

C 24.5 m

84 m 87.5 m

WORKED

Example

5

D

25.4 m 24.5 m

35.3 m

9 A television antenna is 12 m high. To support it, wires are attached to the ground 5 m from the foot of the antenna. Find the length of each wire. 10 Susie needs to clean the guttering on her roof. She places her ladder 1.2 m back from the edge of the guttering that is 3 m above the ground. How long will Susie’s ladder need to be (correct to 2 decimal places)? 11 A rectangular gate is 3.5 m long and 1.3 m wide. The gate is to be strengthened by a diagonal brace as shown at right. How long should the brace be (correct to 2 decimal places)?

3.5 m 1.3 m

12 A 2.5 m ladder learns against a brick wall. The foot of the ladder is 1.2 m from the foot of the wall. How high up the wall will the ladder reach (correct to 1 decimal place)? 13 Use the measurements in the diagram at right to find the height of the flagpole, correct to 1 decimal place. 7.9 m

2.4 m


423

14 An isosceles, right-angled triangle has a hypotenuse of 10 cm. Calculate the length of the shorter sides. (Hint: Call both shorter sides x.) C

15 A block of land was surveyed and the field diagram is shown. a Draw a scale diagram of the block of land. b Use Pythagoras’ theorem to calculate the perimeter of the block of land, correct to the nearest metre.

160 120 50 D

B 30 40 0 A

Calculating trigonometric ratios In the previous section we looked at Pythagoras’ theorem. This enabled us to find the length of one side of a right-angled triangle given the length of the other two. To use Pythagoras’ theorem, we had to recognise the hypotenuse in a right-angled triangle. In trigonometry, we need to be able to name the two hypotenuse shorter sides as well. We do this with reference to a given opposite angle, and label them opposite and adjacent. They are the θ sides opposite and adjacent to the given angle. The diagram adjacent shows the sides labelled with respect to the angle, θ.

The tangent ratio is a ratio of sides in similar I right-angled triangles, such as those in the G diagram. ∠BAC is common to each triangle E and is equal to 30°. We are going to look at the C ratio of the opposite side to the adjacent side in each triangle. You can do this either on your calculator or by completing the spreadsheet A H F D B ‘Tangent’ from the Maths Quest General Mathematics Preliminary Course CD-ROM. Complete each of the following measurements and calculations. BC 1 a BC = mm b AB = mm c -------- = AB DE 2 a DE = mm b AD = mm c --------- = AD FG 3 a FG = mm b AF = mm c -------- = AF HI 4 a HI = mm b AH = mm c --------- = AH Remember that ∠BAC is common to each triangle. In each of the above, part c is the ratio of the opposite side to the adjacent side of ∠BAC. What do you notice about each of these answers?

E

Tangent

sheet

Looking at the tangent ratio

L Spre XCE ad

424


Trigonometry uses the ratio of side lengths to calculate the lengths of sides and the size of angles. The ratio of the opposite side to the adjacent side is called the tangent ratio. This ratio is fixed for any particular angle. The tangent ratio for any angle, θ, can be found using the result: opposite side tan θ = ------------------------------adjacent side In the investigation on page 423, we found that for a 30° angle the ratio was 0.58. We can find a more accurate value for the tangent ratio on a calculator by pressing tan and entering 30. For all calculations in trigonometry you will need to make sure that your calculator is in DEGREES MODE. For most calculators you can check this by looking for a DEG in the display. When measuring angles: 1 degree = 60 minutes 1 minute = 60 seconds You need to be able to enter angles using both degrees and minutes into your calculator. Most calculators use a DMS (Degrees, Minutes, Seconds) button or a ° ’ ” button. Check with your teacher to see how to do this.

WORKED Example 6 Using your calculator, find the following, correct to 3 decimal places. 8 a tan 60° b 15 tan 75° c -----------------d tan 49°32′ tan 69∞ THINK

WRITE/DISPLAY

a Press tan and enter 60. × b Enter 15, press and enter 75. c Enter 8, press enter 69.

÷

and

d Press tan , enter 49, press enter 32, press DMS .

tan tan

, ,

DMS ,

a tan 60° = 1.732 b 15 tan 75° = 55.981 8 c ----------------- = 3.071 tan 69° d tan 49°32′ = 1.172

CASI

O

trigonometric ratios Graphics Calculator tip! Calculating involving degrees and minutes

Setting calculator to degrees mode

To calculate the value of a trigonometric ratio, put the calculator into RUN mode and, if the angle is in degrees only, the value is found in exactly the same way as if using a scientific calculator. Check you can do this by finding the value of tan 60° as shown above in worked example 6. If you get a different answer it may be because the calculator is not set to degrees. Click on the icon to see how to check if your calculator is set to degrees mode and how to change it if it is not.


425

Now consider worked example 6 d. 1. From the MENU select RUN.

2. To set the calculator up to accept degrees, minutes and even seconds, press OPTN , F6 for more options, then F5 for ANGL. The screen should appear as shown.

3. Now to find the trigonometric ratio, enter tan 49 and press F4 for degrees; enter 32 and press F4 for minutes; then press EXE .

The tangent ratio is used to solve problems involving the opposite side and the adjacent side of a right-angled triangle. The tangent ratio does not allow us to solve problems that involve the hypotenuse. The sine ratio (abbreviated to sin) is the name given to the ratio of the opposite side and the hypotenuse.

The tangent ratio is the ratio of the opposite side and the adjacent side in a rightangled triangle. The sine ratio is the ratio of the opposite side and the hypotenuse. Look back to the right-angled triangles used in the tangent investigation on page 423. Complete each of the following measurements and calculations by using your calculator or the spreadsheet ‘Sine’ from the Maths Quest General Mathematics Preliminary Course CD-ROM. As we saw earlier, ∠BAC is common to all of these similar triangles, and so in this exercise, we look at the ratio of the side opposite ∠BAC to the hypotenuse of each triangle. BC 1 a BC = mm b AC = mm c -------- = AC DE 2 a DE = mm b AE = mm c -------- = AE FG 3 a FG = mm b AG = mm c --------- = AG HI c ------ = AI In this exercise, part c is the ratio of the opposite side to ∠BAC to the hypotenuse. You should again notice that the answers are the same (or very close, allowing for measurement error). 4 a HI =

mm

b AI =

mm

E

Sine

sheet

Looking at the sine ratio

L Spre XCE ad

426


In any right-angled triangle with equal angles, the ratio of the opposite side to the hypotenuse will remain the same, regardless of the size of the triangle. The formula for the sine ratio is: opposite side sin θ = ------------------------------hypotenuse The value of the sine ratio for any angle is found using the sin function on the calculator. sin 30° = 0.5 Check this on your calculator.

WORKED Example 7 Find, correct to 3 decimal places: a sin 57°

18 c ----------------sin 44∞

b 9 sin 45°

THINK a Press

d 9.6 sin 26°12′. WRITE/DISPLAY

sin

a sin 57° = 0.839

and enter 57.

b Enter 9, press c Enter 18, press

× ÷

and and

sin sin

, enter 45.

b 9 sin 45° = 6.364

, enter 44.

18 c ----------------- = 25.912 sin 44°

× d Enter 9.6, press and sin , enter 26, press DMS , enter 12, press DMS .

d 9.6 sin 26°12′ = 4.238

A third trigonometric ratio is the cosine ratio. This ratio compares the length of the adjacent side and the hypotenuse. EXCE

et

reads L Sp he

Cosine

Looking at the cosine ratio Look back to the right-angled triangles used in the tangent investigation on page 423. Complete each of the following measurements and calculations. You may do so by using the spreadsheet ‘Cosine’ from the Maths Quest General Mathematics Preliminary Course CD-ROM. AB 1 a AB = mm b AC = mm c -------- = AC AD 2 a AD = mm b AE = mm c --------- = AE AF 3 a AF = mm b AG = mm c --------- = AG AH 4 a AH = mm b AI = mm c --------- = AI Again for part c, you should get the same answer for each triangle. In each case, this is the cosine ratio of the common angle BAC.


427

The cosine ratio is found using the formula: adjacent side cos θ = ------------------------------hypotenuse To calculate the cosine ratio for a given angle on your calculator, use the cos function. On your calculator check the calculation: cos 30° = 0.866

WORKED Example 8 Find, correct to 3 decimal places: a cos 27°

21.3 c -----------------cos 74∞

b 6 cos 55°

THINK a Press

4.5 d -------------------------- . cos 82∞46¢ WRITE/DISPLAY

cos

b Enter 6, press c Enter 21.3, press 74. d Enter 4.5, press enter 82, press DMS .

a cos 27° = 0.891

and enter 27. ×

and ÷

×

cos

and

and

, enter 55.

cos

cos

, enter

,

DMS , enter 46, press

b 6 cos 55° = 3.441 21.3 c ------------------ = 77.275 cos 74° 4.5 d -------------------------- = 35.740 cos 82°46′

Similarly, if we are given the sin, cos or tan of an angle, we are able to calculate the size of that angle using the calculator. We do this using the inverse functions. On most calculators these are the 2nd function of the sin, cos and tan functions and are denoted sin−1, cos−1 and tan−1.

WORKED Example 9

Find θ, correct to the nearest degree, given that sin θ = 0.738. THINK 1 2

WRITE/DISPLAY –1

Press 2nd F [sin ] and enter 0.738. Round your answer to the nearest degree.

θ = 48°

So far, we have dealt only with angles that are whole degrees. You need to be able to make calculations using minutes as well. On most calculators, you will use the DMS (Degrees, Minutes, Seconds) function or the ° ’ ” function.

428


WORKED Example 10

Given that tan θ = 1.647, calculate θ to the nearest minute. THINK 1 2

WRITE/DISPLAY

Press 2nd F [tan–1] and enter 1.647. Convert your answer to degrees and minutes by pressing DMS .

θ = 58°44′

an answer in degrees, Graphics Calculator tip! Displaying minutes and seconds Consider worked example 10. 1. From the MENU select RUN.

2. As with a scientific calculator, press SHIFT [tan-1] and enter 1.647, then press EXE .

3. Display the angle options by pressing OPTN , F6 for more choices, and then F5 for ANGL.

4. The function for getting the answer displayed in degrees, minutes and seconds is accessed by pressing F5 .

remember 1. The tangent ratio is the ratio of the opposite side and the adjacent side. opposite side tan θ = ------------------------------adjacent side 2. The sine ratio is the ratio of the opposite side and the hypotenuse. opposite side sin θ = ------------------------------hypotenuse 3. The cosine ratio is the ratio of the adjacent side and the hypotenuse. adjacent side cos θ = ------------------------------hypotenuse 4. The value of the trigonometric ratios can be found using the sin, cos and tan functions on your calculator. 5. The angle can be found when given the trigonometric ratio using the sin−1, cos−1 and tan−1 functions on your calculator.


13B WORKED

Example

WORKED

Example

7

Example

8

1 Calculate the value of each of the following, correct to 3 decimal places. a tan 57°

b 9 tan 63°

c

8.6 ----------------tan 12°

d tan 33°19′

2 Calculate the value of each of the following, correct to 3 decimal places. a sin 37°

b 9.3 sin 13°

c

14.5 ----------------sin 72°

48 d ------------------------sin 67°40′

3 Calculate the value of each of the following, correct to 3 decimal places. a cos 45°

b 0.25 cos 9°

c

6 -----------------cos 24°

d 5.9 cos 2°3′

4 Calculate the value of each of the following, correct to 4 significant figures. a sin 30° b cos 15° c tan 45° d 48 tan 85° e 128 cos 60° f 9.35 sin 8° 4.5 0.5 g -----------------h ----------------cos 32° tan 20° 15 i ----------------sin 72° 5 Calculate the value of each of the following, correct to 2 decimal places. a sin 24°38′ b tan 57°21′ c cos 84°40′ d 9 cos 55°30′ e 4.9 sin 35°50′ f 2.39 tan 8°59′ 19 49.6 g ------------------------h -------------------------tan 67°45′ cos 47°25′ 0.84 i ---------------------sin 75°5′ WORKED

Example

6 Find θ, correct to the nearest degree, given that sin θ = 0.167.

9

7 Find θ, correct to the nearest degree, given that: a sin θ = 0.698 b cos θ = 0.173 WORKED

Example

c

tan θ = 1.517.

8 Find θ, correct to the nearest minute, given that cos θ = 0.058.

10

9 Find θ, correct to the nearest minute, given that: a tan θ = 0.931 b cos θ = 0.854

c

sin θ = 0.277.

13.3 SkillS Rounding to a given number of decimal places Cabri Geo

ry met

WORKED

Calculating trigonometric ratios HEET

6

429

Sin–cos–tan ratios

430


1 1 Find the length of the hypotenuse in the triangle at right. 5 cm 12 cm

2 Find x, correct to 1 decimal place. x

12 cm

12 cm

3 Find y, correct to 3 significant figures.

y

20 cm 30 cm

4 Find z, correct to 4 significant figures.

3.7 m 7.4 m

5 Calculate sin 56°, correct to 4 decimal places. 6 Calculate 9.2 tan 50°, correct to 3 significant figures. 132 7 Calculate ----------------------- , cos 8°45′ correct to 2 decimal places. 8 Find θ, given that sin θ = 0.5. 9 Find θ to the nearest degree, given that cos θ = 0.299. 10 Find θ to the nearest minute, given that tan θ = 2.

z


431

Finding an unknown side We can use the trigonometric ratios to find the length of one hyp x opp side of a right-angled triangle if we know the length of another 30° side and an angle. Consider the triangle at right. 14 cm In this triangle we are asked to find the length of the opposite adj side and have been given the length of the adjacent side. opposite x We know from the formula that: tan θ = -------------------- . In this example, tan 30° = ------ . From adjacent 14 our calculator we know that tan 30° = 0.577. We can set up an equation that will allow us to find the value of x. opp tan θ = --------adj x tan 30° = -----14 x = 14 tan 30° = 8.083 cm

WORKED Example 11 Use the tangent ratio to find the value of h in the triangle at right, correct to 2 decimal places. THINK 1

hyp

55° 17 m

WRITE

Label the sides of the triangle opp, adj and hyp.

h

h opp

55° 17 cm adj 2 3 4 5

Write the tangent formula. Substitute for θ (55°) and the adjacent side (17 m). Make h the subject of the equation. Calculate.

opp tan θ = --------adj h tan 55° = -----17 h = 17 tan 55° = 24.28 cm

In the example above, we were told to use the tangent ratio. In practice, we need to be able to look at a problem and then decide if the solution is found using the sin, cos or tan ratio. To do this we need to examine the three formulas. opposite side tan θ = ------------------------------adjacent side We use the tan ratio when we are finding either the length of the opposite or adjacent side and are given the length of the other.

432


opposite side sin θ = ------------------------------hypotenuse The sin ratio is used when we are finding the length of the opposite side or the hypotenuse and are given the length of the other. adjacent side cos θ = ------------------------------hypotenuse The cos ratio is for problems where we are finding the length of the adjacent side or the hypotenuse and are given the length of the other. To make the decision we need to label the sides of the triangle and make a decision based on these labels.

WORKED Example 12 Find the length of the side marked x, correct to 2 decimal places. THINK 1

24 m

WRITE

x

50°

Label the sides of the triangle. hyp 24 m

opp x

50° adj 2

3 4 5

x is the opposite side and 24 m is the hypotenuse, therefore use the sin formula. Substitute for θ and the hypotenuse. Make x the subject of the equation. Calculate.

opp sin θ = --------hyp x sin 50° = -----24 x = 24 sin 50° x = 18.39 m

Graphics Calculator tip! Using the equation solver In each example when finding the length of a side, after substitution you will be left with an equation to solve to obtain your final answer. The solution steps can be cut out by using the equation solver. x Consider worked example 12. At step 3 of the solution we are left with sin 50° = ------ , 24 which can be solved as shown below. 1. From the MENU select EQUA.



433

x 3. Delete any existing equation and enter sin 50° = ------ by 24 pressing sin 50 SHIFT = X ÷ 24 EXE . Do not worry about a different value of X in the display at this stage as it is a previously stored value. 4. Press F6 for SOLV to solve this equation.

To remember each of the formulas more easily, we can use this acronym: SOHCAHTOA We pronounce this acronym as ‘Sock ca toe her’. The initials of the acronym represent the three trigonometric formulas. opp adj opp sin θ = --------cos θ = --------tan θ = --------hyp hyp adj Care needs to be taken at the substitution stage. In the above examples, the unknown side was the numerator in the fraction, hence we multiplied to find the answer. If after substitution, the unknown side is in the denominator, the final step is done by division.

WORKED Example 13 Find the length of the side marked z in the triangle at right. THINK 1

23°15'

WRITE

Label the sides opp, adj and hyp. 23°15'

hyp z

z 12.5 m

opp

12.5 m adj 2

Choose the cosine ratio because we are finding the hypotenuse and have been given the adjacent side.

3

Write the formula.

4

Substitute for θ and the adjacent side.

5

Make z the subject of the equation.

6

Calculate.

adj cos θ = --------hyp 12.5 cos 23°15′ = ---------z z cos 23°15′ = 12.5 12.5 z = -------------------------cos 23°15′ = 13.60 m

If you are using the graphics calculator to solve an equation that involves the use of degrees and minutes, the angle needs to be entered as a fraction. In worked example 13 ------ as the equation solver does not allow above, cos 23o15′ needs to be entered as cos 23 15 60 access to the degrees, minutes and seconds function.

434


Trigonometry is used to solve many practical problems. In these cases, it is necessary to draw a diagram to represent the problem and then use trigonometry to solve the problem. With written problems that require you to draw the diagram, it is necessary to give the answer in words.

WORKED Example 14 A flying fox is used in an army training camp. The flying fox is supported by a cable that runs from the top of a cliff face to a point 100 m from the base of the cliff. The cable makes a 15° angle with the horizontal. Find the length of the cable used to support the flying fox. THINK 1

WRITE

Draw a diagram and show information.

f 15° 100 m

3

Label the sides of the triangle opp, adj and hyp. Choose the cosine ratio because we are finding the hypotenuse and have been given the adjacent side.

4

Write the formula.

5


6

Make f the subject of the equation.

7

Calculate. Give a written answer.

2

8

adj cos θ = --------hyp 100 cos 15° = --------f f cos 15° = 100 100 f = -----------------cos 15° = 103.5 m The cable is approximately 103.5 m long.

remember 1. Trigonometry can be used to find the length of a side in a right-angled triangle when we are given the length of one side and the size of an angle. 2. The trigonometric formulas are: opp adj opp sin θ = --------cos θ = --------tan θ = --------hyp hyp adj 3. Take care to choose the correct trigonometric ratio for each question. 4. Substitute carefully and note the change in the calculation, depending upon whether the unknown side is in the numerator or denominator. 5. Before using your calculator, check that it is in degrees mode. 6. Be sure that you know how to enter degrees and minutes into your calculator. 7. Worded problems will require you to draw a diagram and give a written answer.

435


13C

Finding an unknown side

α

θ

Example

11

equations mmmm of themmym type x a = --b to find x

γ

13.5 SkillS

2 Use the tangent ratio to find the length of the side marked x (correct to 1 decimal place).

HEET

WORKED

x

Solving equations 71° of the type b 51 mm a = --x to find x

3 Use the sine ratio to find the length of the side marked a (correct to 2 decimal places).

13 m

a

Rearranging formulas

4 Use the cosine ratio to find the length of the side marked d (correct to 3 significant figures). d

5 The following questions use the tan, sin or cos ratios in their solution. Find the size of the side marked with the pronumeral, correct to 3 significant figures. a

b

Cosine ratio Cabri Geo

ry met

13 cm

Cabri Geo

c 49°

x

Sine ratio

ry met

12

31°

Cabri Geo

ry met

35 cm

Example

13.6 SkillS

HEET

23°

WORKED

SkillS

HEET

1 Label the sides of each of the following triangles, with respect to the angle marked 13.4 13.1 with the pronumeral. a b c Solving mmm

12.5 km

48 m

Tangent ratio

41°

68°

z

y

WORKED

Example

13

6 Find the length of the side marked with the pronumeral in each of the following Sin–cos–tan (correct to 1 decimal place). a

b

c p

21°

4.8 m

t

87 mm

77°

q 36° 8.2 m

ry met

Cabri Geo

436


7 Find the length of the side marked with the pronumeral in each of the following (correct to 3 significant figures). a b c x 23° 76°

b

a

0.85 km 8.5 km

39° 2.3 m

d

9°

e

f

d

f

64.75 m

m 116 mm

13°

x 16.75 cm 11°

g

h

i

g 20° 15.75 km

44.3 m

m 84°9' 2.34 m

83° x

j

k

q 60°32'

l t

75°19'

r

21.4 m 26.8 cm

84.6 km 29°32'

8 multiple choice Look at the diagram at right and state which of the following is correct. 9.2 A x = 9.2 sin 69° B x = ----------------sin 69° 9.2 C x = 9.2 cos 69° D x = -----------------cos 69°

x 69° 9.2

9 multiple choice Study the triangle at right and state which of the following is correct. A tan φ =

8 -----15

B tan φ =

15 -----8

C sin φ =

15 -----17

10 multiple choice Which of the statements below is not correct? A The value of tan θ can never be greater than 1. B The value of sin θ can never be greater than 1. C The value of cos θ can never be greater than 1. D tan 45° = 1

D cos φ =

φ 8 -----17

17

15 8


437

11 multiple choice Study the diagram at right and state which of the statements is correct. 22 A w = 22 cos 36° B w = ----------------22 mm sin 36° w C w = 22 cos 54° D w = 22 sin 54° 36°

12 A tree casts a 3.6 m shadow when the sun’s angle of elevation is 59°. Calculate the Example height of the tree, correct to the nearest metre. 14 13 A 10 m ladder just reaches to the top of a wall when it is leaning at 65° to the ground. How far from the foot of the wall is the ladder (correct to 1 decimal place)? WORKED

14 The diagram at right shows the paths of two ships, A and B, after they have left port. If ship B sends a distress signal, how far must ship A sail to give assistance (to the nearest kilometre)?

Port

A 60°

23 km B

15 A rectangle 13.5 cm wide has a diagonal that makes a 24° angle with the horizontal. a Draw a diagram of this situation. b Calculate the length of the rectangle, correct to 1 decimal place. 16 A wooden gate has a diagonal brace built in for support. The gate stands 1.4 m high and the diagonal makes a 60° angle with the horizontal. a Draw a diagram of the gate. b Calculate the length that the diagonal brace needs to be. 17 The wire support for a flagpole makes a 70° angle with the ground. If the support is 3.3 m from the base of the flagpole, calculate the length of the wire support (correct to 2 decimal places). T SHEE

Work

18 A ship drops anchor vertically with an anchor line 60 m long. After one hour the anchor line makes a 15° angle with the vertical. a Draw a diagram of this situation. b Calculate the depth of water, correct to the nearest metre. c Calculate the distance that the ship has drifted, correct to 1 decimal place.

13.1

438


2 Find the length of the side marked with the pronumeral in each of the following. Where necessary, give your answer correct to 1 decimal place. 1

2

20 m

10 m a

15 cm

b

8 cm

3

30 km

4

c

21° d

40 km

5

23 m

6

f

e 42.1 m

40° 39.2 cm

7

37°

8

14.9 mm

50°

119 mm

g 42°

9

10

h

17 m j

93.2 m 17°

i

45°

Finding angles In this chapter so far, we have concerned ourselves with finding side lengths. We are also able to use trigonometry to find the sizes of angles when we have been given side lengths. We need to reverse our previous processes. Consider the triangle at right. We want to find the size 10 cm of the angle marked θ. 5 cm θ


439

opp Using the formula sin θ = --------- we know that in this triangle hyp sin θ = =

5 -----10 1 --2

= 0.5 We then calculate sin−1 (0.5) to find that θ = 30°. As with all trigonometry it is important that you have your calculator set to degrees mode.

WORKED Example 15

Find the size of angle θ, correct to the nearest degree, in the triangle at right. THINK 1

4.3 m

θ 6.5 m

WRITE

Label the sides of the triangle and choose the tan ratio. hyp

4.3 opp

θ 6.5 adj

2

3 4

Substitute for the opposite and adjacent sides in the triangle and simplify. Make θ the subject of the equation. Calculate.

opp tan θ = --------adj 4.3 = ------6.5 = 0.6615 θ = tan−1(0.6615) = 33° (to the nearest degree)

Graphics Calculator tip! Using equation solver to find an angle The equation solver can be used to find an angle in the same way that it can be used to find the length of a side. Consider worked example 15. 1. From the MENU select EQUA.


440


4.3 3. Delete any existing equation and enter tan x = ------- by 6.5 pressing

tan

X

SHIFT [=] 4.3

÷

6.5 EXE .

4. Press F6 for SOLV to solve this equation.

In many cases we will need to calculate the size of an angle, correct to the nearest minute. The same method for finding the solution is used; however, you will need to use your calculator to convert to degrees and minutes.

WORKED Example 16

4.6 cm

Find the size of the angle θ, correct to the nearest minute. THINK 1

WRITE

Label the sides of the triangle and choose the sin ratio.

θ

3 4

Substitute for the opposite side and adjacent in the triangle and simplify. Make θ the subject of the equation. Calculate and convert your answer to degrees and minutes.

7.1 cm

opp 4.6 cm adj

2

θ

hyp 7.1 cm

opp sin θ = --------hyp 4.6 = ------7.1 = 0.6479 θ = sin−1(0.6479) = 40°23′ (to the nearest minute)

a solution in degrees and Graphics Calculator tip! Finding minutes If using the equation solver, the solution can only be displayed in degrees as a decimal. Consider worked example 16. To obtain the solution in degrees and minutes, use the steps shown here. 1. Solve the equation as shown previously, which gives an answer in degrees as a decimal.


441

2. Press MENU and select RUN.

3. Press X equation.

EXE

to recall the value of X from the

4. To access the angle functions, press OPTN F6 , F5 for ANGL, and F5 again to convert to degrees, minutes and seconds.

The same methods can be used to solve problems. As with finding sides, we set the question up by drawing a diagram of the situation.

WORKED Example 17 A ladder is leant against a wall. The foot of the ladder is 4 m from the base of the wall and the ladder reaches 10 m up the wall. Calculate the angle that the ladder makes with the ground. THINK 1

WRITE

Draw a diagram and label the sides. opp 10 m

hyp

θ 4m adj 2

Choose the tangent ratio and write the formula.

3

Substitute for the opposite and adjacent side, then simplify.

4

Make θ the subject of the equation.

5

Calculate.

6


opp tan θ = --------adj 10 = -----4 = 2.5

θ = tan−1(2.5) = 68°12′ The ladder makes an angle of 68°12′ with the ground.

442


remember 1. Make sure that the calculator is in degrees mode. 2. To find an angle given the trigonometric ratio, press SHIFT and then the appropriate ratio button. 3. Be sure to know how to get your calculator to display an answer in degrees and minutes. When rounding off minutes, check if the number of seconds is greater than 30. 4. When solving triangles remember the SOHCAHTOA rule to choose the correct formula. 5. In worded problems draw a diagram and give an answer written in words.

13D 13.7

SkillS

HEET

Rounding angles to the nearest degree

1 In each of the following, use the tangent ratio to find the size of the angle marked with the pronumeral, correct to the nearest degree. a

b

θ

13.8 Rounding angles to the nearest minute

13.9

SkillS

HEET

c

25 mm γ

162 mm

7m 11 m

12 m

SkillS

HEET

Finding angles

φ 3m

2 In each of the following, use the sine ratio to find the size of the angle marked with the pronumeral, correct to the nearest minute. a

Rounding angles to the nearest second

24 m

13 m

b

θ

c

4.6 m

6.5 m

9.7 km

5.6 km

θ

Cabr

omet i Ge ry

α

Sine ratio

Cabr

omet i Ge ry

Cosine ratio

3 In each of the following, use the cosine ratio to find the size of the angle marked with the pronumeral, correct to the nearest minute. a

b 15 cm

θ 9 cm

c 4.6 m

2.6 m

27.8 cm

β 19.5 cm

443


WORKED

15

4 In the following triangles, you will need to use all three trigonometric ratios. Find the Cabri Geo size of the angle marked θ, correct to the nearest degree. Tangent a b c 14 cm

ry met

Example

ratio

θ

θ

15 cm

11 cm

9 cm

ry met

Cabri Geo

θ

d

Sin–cos–tan

8 cm

7 cm 3.6 m

e

f θ

196 mm

32 mm 14.9 m

26.8 m

9.2 m

θ WORKED

Example

16

5 In each of the following, find the size of the angle marked θ, correct to the nearest minute. a

b

c

θ 30 m

θ

63 cm

θ 10 cm

19.2 m

d

2.5 m

0.6 m

3.5 m

e

f

θ

8.3 m

18.5 m

θ

6.3 m

16.3 m 18.9 m

θ

6 multiple choice A

Look at the triangle drawn at right. Which of the statements below is correct? A ∠ABC = 30° B ∠ABC = 60° C ∠CAB = 30° D ∠ABC = 45°

10 cm

5 cm

θ

C

B

7 multiple choice Consider the triangle drawn at right. θ is closest to: A 41°55′ B 41°56′ C 48°4′

θ

D 48°5′

9.3 m

8 multiple choice 3 The exact value of sin θ = ------- . The angle θ = 2 A 30° B 45° C 60°

D 90°

12.5 m

444 WORKED

Example

17


9 A 10 m ladder leans against a 6 m high wall. Find the angle that the ladder makes with the horizontal, correct to the nearest degree. 10 A kite is flying on a 40 m string. The kite is flying 10 m away from the vertical as shown in the figure at right. Find the angle the string makes with the horizontal, correct to the nearest minute.

kite

40 m

11 A ship’s compass shows a course due east of the port from which it sails. After sailing 10 nautical miles, it is found that the ship is 1.5 nautical miles off course as shown in the figure below. 1.5 nm

10 nm

Find the error in the compass reading, correct to the nearest minute. 12 The diagram below shows a footballer’s shot at goal. 7m

30 m

By dividing the isosceles triangle in half, calculate, to the nearest degree, the angle within which the footballer must kick to get the ball to go between the posts.

13 A golfer hits the ball 250 m, but 20 m off centre. Calculate the angle at which the ball deviated from a straight line, correct to the nearest minute.

10 m


445

Angles of elevation and depression The angle of elevation is measured upwards from a horizontal and refers to the angle at which we need to look up to see an object. Similarly, the angle of depression is the angle at which we need to look down from the horizontal to see an object. We are able to use the angles of elevation and depression to calculate the heights and distances of objects that would otherwise be difficult to measure.

WORKED Example 18 From a point 50 m from the foot of a building, the angle of elevation to the top of the building is measured as 40°. Calculate the height, h, of the building, correct to the nearest metre. THINK 1 2

3 4 5 6 7

WRITE

Label the sides of the triangle opp, adj and hyp. Choose the tangent ratio because we are finding the length of the opposite side and have been given the length of the adjacent side. Write the formula. Substitute for θ and the adjacent side. Make h the subject of the equation. Calculate. Give a written answer.

h 40° 50 m

h opp

hyp 40° 50 m adj

opp tan θ = --------adj h tan 40° = -----50 h = 50 tan 40° = 42 m The height of the building is approximately 42 m.

In practical situations, the angle of elevation is measured using a clinometer. Therefore, the angle of elevation is measured from a person’s height at eye level. For this reason, the height at eye level must be added to the calculated answer.

WORKED Example 19 Bryan measures the angle of elevation to the top of a tree as 64°, from a point 10 m from the foot of the tree. If the height of Bryan’s eyes is 1.6 m, calculate the height of the tree, correct to 1 decimal place. THINK 1 2

Label the sides opp, adj and hyp. Choose the tangent ratio because we are finding the length of the opposite side and have been given the length of the adjacent side.

WRITE

h opp

64° 10 m

hyp

64° 10 m adj

Continued over page

1.6 m

446


THINK

WRITE opp tan θ = --------adj

3

Write the formula.

4


5

Make h the subject of the equation.

6

Calculate h.

7

Add the eye height.

20.5 + 1.6 = 22.1

8


The height of the tree is approximately 22.1 m.

h tan 64° = -----10 h = 10 tan 64° = 20.5 m

A similar method for finding the solution is used for problems that involve an angle of depression.

WORKED Example 20 When an aeroplane is 2 km from a runway, the angle of depression to the runway is 10°. Calculate the altitude of the aeroplane, correct to the nearest metre.

THINK

2 km 10° h

WRITE

1

Label the sides of the triangle opp, adj and hyp.

2

Choose the tan ratio, because we are finding the length of the opposite side given the length of the adjacent side.

3

Write the formula.

4

Substitute for θ and the adjacent side, converting 2 km to metres.

5

Make h the subject of the equation.

6

Calculate.

7


adj 2 km 10° opp h

hyp

opp tan θ = --------adj h tan 10° = -----------2000 h = 2000 tan 10° = 353 m The altitude of the aeroplane is approximately 353 m.

Angles of elevation and depression can also be calculated by using known measurements. This is done by drawing a right-angled triangle to represent a situation.


447

WORKED Example 21 A 5.2 m building casts a 3.6 m shadow. Calculate the angle of elevation of the sun, correct to the nearest degree. THINK 1 2

Label the sides opp, adj and hyp. Choose the tan ratio because we are given the length of the opposite and adjacent sides.

3

Write the formula.

4

Substitute for opposite and adjacent.

5

Make θ the subject of the equation.

6 7

5.2 m

WRITE

Calculate. Give a written answer.

θ 3.6 m opp 5.2 m

hyp

θ 3.6 m adj

opp tan θ = --------adj 5.2 tan θ = ------3.6 5.2 θ = tan−1 ------3.6 = 55° The angle of elevation of the sun is approximately 55°.

remember 1. The angle of elevation is the angle at which you look up to see an object. 2. The angle of depression is the angle at which you look down to see an object. 3. Problems can be solved by using angles of elevation and depression with the aid of a diagram. 4. Worded problems should be given an answer written in words.

448


13E WORKED

Example

18

Angles of elevation and depression

1 From a point 100 m from the foot of a building, the angle of elevation to the top of the building is 15°. Calculate the height of the building, correct to 1 decimal place. 15° 100 m

2 The angle of elevation from a ship to an aeroplane is 60°. The aeroplane is 2300 m due north of the ship. Calculate the altitude of the aeroplane, correct to the nearest metre. 60° 2300 m

3 From a point out to sea, a ship sights the top of a lighthouse at an angle of elevation of 12°. It is known that the top of the lighthouse is 40 m above sea level. Calculate the distance of the ship from the lighthouse, correct to the nearest 10 m. WORKED

Example

19

4 From a point 50 m from the foot of a building, Rod sights the top of a building at an angle of elevation of 37°. Given that Rod’s eyes are at a height of 1.5 m, calculate the height of the building, correct to 1 decimal place.

40 m 12°

1.5 m

5 Richard is flying a kite and sights the kite at an angle of elevation of 65°. The altitude of the kite is 40 m and Richard’s eyes are at a height of 1.8 m. Calculate the length of string the kite is flying on, correct to 1 decimal place.

x

37° 50 m

x 40 m 65° 1.8 m

WORKED

Example

20

6 Bettina is standing on top of a cliff, 70 m above sea level. She looks directly out to sea and sights a ship at an angle of depression of 35°. Calculate the distance of the ship from shore, to the nearest metre.

35° 70 m

7 From an aeroplane flying at an altitude of 4000 m, the runway is sighted at an angle of depression of 15°. Calculate the distance of the aeroplane from the runway, correct to the nearest kilometre. 8 There is a fire on the fifth floor of a building. The closest a fire truck can get to the building is 10 m. The angle of elevation from this point to where people need to be rescued is 69°. If the fire truck has a 30 m ladder, can the ladder be used to make the rescue?

15° 4000 m

69° 10 m

449


9 From a navy vessel, a beacon which is 80 m D above sea level is sighted at an angle of elevation of 5°. The vessel sailed towards the beacon and 80 m thirty minutes later the beacon is at an angle of elevation of 60°. 5° 60° C A B Use the diagram on the right to complete the following. a Calculate the distance that the vessel was from the beacon when the angle of elevation to the beacon was 5° (the distance AC). b Calculate the distance that the vessel sailed in the 30 minutes between the two readings. 10 A 12 m high building casts a shadow 15 m long. Calculate the angle of elevation of the sun, to the nearest degree. 21

WORKED

Example

12 m

θ 15 m 4000 m

θ 1500 m

12 The angle of elevation to the top of a tower is 12° from a point 400 m from the foot of the tower. a Draw a diagram of this situation. b Calculate the height of the tower, correct to 1 decimal place. c Calculate the angle of elevation to the top of the tower from a point 100 m from the foot of the tower.

Calculation of heights To measure the heights of trees and buildings around your school, try the following. 1 Measure your height at eye level. 2 Take a clinometer and from a point measure the angle of elevation to the top of the tree or building. 3 Measure your distance from the foot of the tree or building. 4 Use trigonometry to calculate the height, remembering to add your height at eye level to the result of the calculation.

T SHEE

Work

11 An aeroplane that is at an altitude of 1500 m is 4000 m from a ship in a horizontal direction, as shown at right. Calculate the angle of depression from the aeroplane to the ship, to the nearest degree.

13.2

450


Proportional diagrams

4 km

In many cases, we need only an approximate measurement for a practical problem. This type of answer can be obtained by drawing a scale diagram. Consider the situation where a hiker walks 6 km due north, turns and walks 4 km due east. By 6 km drawing a diagram using a scale of 1 cm = 1 km, we can obtain an approximate measurement for the distance the hiker is from the starting point. By measurement, we can see that the distance the hiker is from the starting point is approximately 7.2 km. With a protractor, we can draw a scale diagram to solve problems involving angles. Suppose that the angle of elevation to the top of a tree is 40° from a point 12 m from the foot of the tree. h By measurement, the height of the tree is approximately 10 m. 40° 12 m In many situations a quick check of the accuracy of an answer is useful and can be made by using a scale drawing. In such cases the drawing would need to be only approximately to scale. Suppose that you were told that the angle of depression from the top of a 50 m cliff to a ship out to sea was 15°. You were then told that this ship is 1 km from shore. 15° 50 m

Using this diagram, we would estimate that the ship is only 190 m from shore. Such a diagram is a useful check to a calculation.

Checking with a proportional diagram Draw diagrams roughly to scale to check the results to the previous investigation. Such diagrams are used to develop car rally courses, cross-country running courses and orienteering events.

Using proportional diagrams Plan a track for a cross-country run or orienteering event around your school. 1 Measure the length of each leg and the angle involved in each turn. 2 On a scale diagram, draw the course. 3 By measuring your diagram, calculate the approximate length of the course.


451

summary Pythagoras’ theorem • When finding the hypotenuse of a right-angled triangle, use the formula: c2 = a2 + b2. • To find a shorter side of a right-angled triangle use: a2 = c2 − b2 or b2 = c2 − a2.

Trigonometric formulas for right-angled triangles opp • tan θ = --------adj opp • sin θ = --------hyp adj • cos θ = --------hyp • SOHCAHTOA — this acronym will help you remember trigonometric formulas.

Steps to find a side of a right-angled triangle • • • • •

Label the sides of the triangle opposite, adjacent and hypotenuse. Choose the correct ratio. Substitute given information. Make the unknown side the subject of the equation. Calculate.

Steps to find an angle in a right-angled triangle • • • • •

Label the sides of the triangle opposite, adjacent and hypotenuse. Choose the correct ratio. Substitute given information. Make the unknown angle the subject of the equation. Calculate by using the inverse trigonometric functions.

Angles of elevation and depression • The angle of elevation is the angle we look up from the horizontal to see an object. • The angle of depression is the angle we look down from the horizontal to see an object. • Problems are solved using angles of elevation and depression by the same methods as for all right-angled triangles.

Proportional diagrams • A scale diagram can be drawn to obtain a reasonable estimate of a distance or angle. • A diagram that is drawn roughly to scale can be used to check that an answer is reasonably accurate.

452


CHAPTER review 13A

1 Find the length of the side marked with a pronumeral, in each case writing your answer correct to 2 decimal places. a b c 32 cm

n

m

4.8 m

9.2 m

p

26 cm 9.2 m

d

3.2 m q

e

f

1.9 km

7.25 cm

t

1.3 km

r

17.25 cm

0.6 m

2.4 m

13A

2 To travel between the towns of Bolong and Molong, you need to travel west along a road for 45 km, then north along another road for another 87 km. Calculate the straight-line distance between the two towns.

13A

3 A rope is 80 m long and runs from a cliff top to the ground, 45 m from the base of the cliff. Calculate the height of the cliff, to the nearest metre.

13B

4 Calculate each of the following, correct to 4 decimal places. a sin 46° b tan 76°42′ c 4.9 cos 56° 5.69 2.5 e -----------------f ---------------------d 8.9 sin 67°3′ cos 75° tan 9°55′

13B

5 Calculate θ, correct to the nearest degree, given that: a cos θ = 0.5874 b tan θ = 1.23

c

sin θ = 0.8.

13B

6 Calculate θ, correct to the nearest minute, given that: a cos θ = 0.199 b tan θ = 0.5

c

sin θ = 0.257.

13C

7 Find the length of each side marked with a pronumeral, correct to 1 decimal place. a b c 6 cm m

3.9 m 78°

9° q

d

e

22° 12.6 cm

x

f

6.8 m 65°

12.6 cm 22°

32° q n

7.8 cm

t


g

h

g

453

i z 83°30'

138 mm

h 2.9 m

4.8 cm 26°42'

j

77°18'

k

4.32 m

l 38.5 m

k

63 km

16°8'

j

85°12' m

29°51'

8 A rope that is used to support a flagpole makes an angle of 70° with the ground. If the rope is tied down 3.1 m from the foot of the flagpole, find the height of the flagpole, correct to 1 decimal place.

13C

9 A dirt track runs off a road at an angle of 34° to the road. If I travel for 4.5 km along the dirt track, what is the shortest distance back to the road (correct to 1 decimal place)?

13C

10 A fire is burning in a building and people need to be rescued. The fire brigade’s ladder must reach a height of 60 m and must be angled at 70° to the horizontal. How long must the ladder be to complete the rescue?

13C

11 Find the size of the angle marked θ in each of the following, giving your answer correct to the nearest degree. a b c 2.3 m θ

13D

116 cm

43 cm

19 m

16 m

4.6 m

θ

θ

12 Find the size of the angle marked θ in each of the following, giving your answer correct to the nearest minute. a b c 10.8 m 2.9 m θ

θ

4.6 m 6.1 m

13D

11.9 cm

θ

13.8 cm

13 A kite on an 80 m string reaches a height of 50 m in a strong wind. Calculate the angle the string makes with the horizontal.

13D

14 There is 50 m of line on a fishing reel. When all the line is out, the bait sits on the bed of a lake and has drifted 20 m from the boat. Calculate the angle that the fishing line makes with the vertical.

13D

15 The top of a building is sighted at an angle of elevation of 40°, when an observer is 27 m back from the base. Calculate the height of the building, h correct to the nearest metre.

13E 40° 27 m

454 13E


16 Hakam stands 50 m back from the foot of an 80 m telephone tower. Hakam’s eyes are at a height of 1.57 m. Calculate the angle of elevation that Hakam must look to see the top of the tower.

80 m

θ 50 m

1.57 m

Practice examination questions 1 multiple choice Which of the triangles drawn below is not right angled? A B 13 cm 5 cm

5 cm

3 cm 12 cm

C

4 cm

D 7m

19 m

9m

17 cm

8 cm

15 cm

2 multiple choice Look at the triangle at right. 9m 9 9 Statement 1. cos θ = ----Statement 2. tan θ = ----41 40 Which of the above statements is true? A 1 only B 2 only C both 1 and 2

41 m

θ 40 m

D neither statement

3 multiple choice Which of the following statements is correct? A cos 30° = tan 60° B cos 30° = sin 60° C cos 30° = sin 30° D cos 60° = sin 60° 4 multiple choice 3 The exact value of cos θ = ------- . The angle θ = 2 A 30° B 45° C 60° 5 On a bushwalk starting at point A, Sally walks 4.2 km due west to point B then turns due south for a distance of 3.1 km to point C. a Calculate the distance, AC, that Sally must walk to return to her starting point. b Calculate the direction that Sally must walk, represented by the angle θ, correct to the nearest degree.

CHAPTER

test yourself

13

D 90° 4.2 km

B

A

3.1 km

θ C

6 A hot-air balloon takes off and after 30 minutes of flying reaches an altitude of 2000 m. At that time, the angle of depression to 15° its launch pad is 15°. 2000 m a Calculate the horizontal distance that the balloon has travelled in that half-hour (correct to the nearest 100 m). b Calculate the angle of depression to the launch pad after the balloon has travelled 15 km in one direction (assuming that it maintains its altitude of 2000 m).

The language of chance

14 syllabus reference Probability 1 • The language of chance

In this chapter 14A Informal description of chance 14B Sample space 14C Tree diagrams 14D Equally likely outcomes 14E Using the fundamental counting principle

areyou

READY?

Are you ready?


14.1

Understanding chance words

14.2

Understanding a deck of cards

14.3

Probability scale I

14.4

Probability scale II

14.5

Listing the sample space

1 For each of the following events, specify whether the chance of the event occurring is certain, likely, fifty-fifty, unlikely or impossible. a A coin is tossed and it lands on Heads. b A fair die is rolled and a number less than 5 is obtained. c Two dice are rolled and a total of 1 is obtained. d Your family wins the lottery. e The maximum temperature on a summer’s day in Sydney will be greater than 0oC.

2 For a standard deck of 52 playing cards, state the number of: a red cards b jacks c black queens d kings of diamonds e eights f number cards greater than 7.

3 For each of the probabilities listed below, state whether the event would be certain, likely, fiftyfifty, unlikely or impossible. a 0.2 b 0.5 c 0 d 1 e 0.9

4 List the events A, B and C below in order from least likely to occur to most likely to occur. A – tossing a coin and having it land Heads B – winning Lotto C – a baby being born on either Saturday or Sunday

5 List the sample space (possible outcomes) for each of the following. a rolling a die b tossing a coin c spinning a circular spinner numbered from 1 to 5

Chapter 14 The language of chance

457

Informal description of chance You have booked a ski holiday to Thredbo for the middle of July. What is the chance that there will be enough snow on the ground for you to ski? There is no exact answer to this question, but by looking at the amount of snow in Thredbo during July over past years, we know that there is a very good chance that there will be enough snow to ski again this year. We can say that it is very likely that we will be able to ski during July at Thredbo. Terms such as ‘very likely’, ‘almost certain’, ‘unlikely’ and ‘fifty-fifty’ are used in everyday language to describe the chance of an event occurring. For the purposes of probability, an event is the outcome of an experiment that we are interested in. We can describe an outcome as a possible result to the probability experiment. Imagine that you are playing a board game and it is your turn to roll the die. To win the game you need to roll a number less than 7. If you roll one die, you must get a number less than 7. We would describe the chance of this event occurring as certain. When an event is certain to occur, the probability of that event occurring is 1. Now let’s consider an impossible situation. certain In a board game you have one last throw of the die. To win you almost certain must roll a 7. We know that this cannot be done. We would say that probable this is impossible.

1

fifty-fifty

1 2

When an event is impossible, the probability of the event is 0.

unlikely The chance of any event occurring will often be somewhere between being certain and impossible, and we use a variety of very unlikely terms to describe where the chance lies in this range as shown in impossible 0 the figure at right. We use these terms based on our general knowledge of the world, the total possible outcomes and how often an event occurs.

WORKED Example 1 Describe the chance of each of the following events occurring. a Tossing a coin and it landing Heads b Rolling a 6 with one die c Winning the lottery d Selecting a spot (numbered) card from a standard deck THINK

WRITE

a There is an equal chance of the coin landing Heads and Tails. b There is only one chance in six of rolling a 6. c There is only a very small chance of winning the lottery. d There are more spot cards than picture cards in a deck.

a The chance of tossing a head is fifty-fifty. b It is unlikely that you will roll a 6. c It is very unlikely that you will win the lottery. d It is probable that you will select a spot card.

458


You will need to use these terms to describe events that are more likely to occur than others.

WORKED Example 2 Mrs Graham is expecting her baby to be born between July 20 and 26. Is it more likely that her baby will be born on a weekday or a weekend? THINK

WRITE

There are 5 chances that the baby will be born on a weekday and 2 chances that it will be born on a weekend.

It is more likely that Mrs Graham’s baby will be born on a weekday.

The term frequency refers to how often an event occurs. We use our knowledge about possible outcomes to order outcomes from the most frequent to the least frequent.

WORKED Example 3 A card is chosen from a standard deck. List the following outcomes in order from least likely to most likely. Selecting a picture card Selecting an Ace Selecting a diamond Selecting a black card THINK 1 2 3 4

WRITE

There are 12 picture cards in the deck. There are 4 aces in the deck. There are 13 diamonds in the deck. There are 26 black cards in the deck.

The order of events in ascending order of likelihood: Selecting an Ace Selecting a picture card Selecting a diamond Selecting a black card.

In the above examples, we have been able to calculate which event is more likely by counting the number of ways an event may occur. This is not always possible. In some cases we need to use general knowledge to describe the chance of an event occurring. Consider the following probability problems. ‘The letters of the alphabet are written on cards and one card is selected at random. Which letter has the greatest chance of being chosen, E or Q?’ Each letter has an equal chance of being chosen because there is one chance that E will be chosen and one chance that Q will be chosen. ‘Stacey sticks a pin into a page of a book and she writes down the letter nearest to the pin. Which letter has the greater chance of being chosen, E or Q?’


459

This question is more difficult to answer because each letter does not occur with equal frequency. However, we know from our experience with the English language that Q will occur much less often than most other letters. We can therefore say that E will occur more often than Q. This is an example of using your knowledge of the world to make predictions about which event is more likely to occur. In this way, we make predictions about everyday things such as the weather and which football team will win on the weekend.

WORKED Example 4 During the 2006 NRL season, the Brisbane Broncos won 9 of their first 12 games. In Round 13 they played South Sydney who had won 0 of their first 12 games. Who would be more likely to win? THINK

WRITE

Brisbane Broncos have won more games than South Sydney.

Brisbane Broncos would be more likely to win, based on their previous results. (Footy note: South Sydney won the game 34–14. Brisbane was more likely to win the game but nothing in football is certain.)

This is one example of past results being used to predict future happenings. There are many other such examples.

WORKED Example 5 Weather records show that it has rained on Christmas Day 12 times in the last 80 years. Describe the chance of it raining on Christmas Day this year. THINK

WRITE

It has rained only 12 times on the last 80 Christmas Days. This is much less than half of all Christmas Days.

It is unlikely that it will rain on Christmas Day this year.

remember 1. The chance of an event occurring ranges from being certain to impossible. 2. (a) An event that is certain has a probability of 1. (b) An event that is impossible has a probability of 0. 3. There are many terms that we use to describe the chance of an event occurring, such as improbable, unlikely, fifty-fifty, likely and probable. 4. Sometimes we can describe the chance of an event occurring by counting the possible outcomes, while other times we need to rely on our general knowledge to make such a description.

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14A 14.1

WORKED

Example

SkillS

HEET

1 Understanding chance words

14.2

SkillS

HEET

SkillS

14.3 Probability scale I

14.4

SkillS

HEET

1 Describe the chance of each of the following events occurring, using an appropriate probability term. a Selecting a ball with a double-digit number from a bag with balls numbered 1 to 40 b Selecting a female student from a class with 23 boys and 7 girls c Selecting a green marble from a barrel with 40 blue marbles and 30 red marbles d Choosing an odd number from the numbers 1 to 100 2 For each of the events below, describe the chance of it occurring as impossible, unlikely, even chance (fifty-fifty), probable or certain. a Rolling a die and getting a negative number b Rolling a die and getting a positive number c Rolling a die and getting an even number d Selecting a card from a standard deck and getting a red card e Selecting a card from a standard deck and getting a spot (numbered) card f Selecting a card from a standard deck and getting an ace g Reaching into a moneybox and selecting a 30c piece h Selecting a blue marble from a bag containing 3 red, 3 green and 6 blue marbles

Understanding a deck of cards

HEET

Informal description of chance

3 Give an example of an event which has a probability that could be described as: a certain b probable c even chance d unlikely e impossible.

Probability scale II

WORKED

Example

2

4 Is it more likely that a person’s birthday will occur during a school term or during the school holidays? 5 For each event on the left, state whether it is more likely, less likely or equally likely to occur than the event on the right. a Fine weather Christmas Day Wet weather Christmas Day b A coin landing Heads A coin landing Tails c Rolling a total of 3 with two dice Rolling a total of 7 with two dice d Winning a raffle made up of 50 tickets Winning a raffle made up of 200 tickets e Winning a prize in the Lotto draw Not winning a prize in the Lotto draw

WORKED

Example

3

6 A die is thrown and the number rolled is noted. List the following events in order from least likely to most likely. Rolling an even number Rolling a number less than 3 Rolling a 6 Rolling a number greater than 2 7 Write the following events in order from least to most likely. Winning a raffle with 5 tickets out of 30 Rolling a die and getting a number less than 3 Drawing a green marble from a bag containing 4 red, 5 green and 7 blue marbles Selecting a court card (ace, king, queen, jack) from a standard deck Tossing a coin and having it land Heads


WORKED

Example

4

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8 Before meeting in the cricket World Cup in 2007, Australia had beaten Bangladesh in 10 of the last 11 matches. Who would be more likely to win on this occasion? 9 Which of the following two runners would be expected to win the final of the 100 m at the Olympic Games? Carl Bailey — best time 9.92 s and won his semi-final Ben Christie — best time 10.06 s and 3rd in his semi-final Give an explanation for your answer. 10 multiple choice A stack of 26 cards has the letters of the alphabet written on them. Vesna draws a card from that stack. The probability of selecting a card that has a vowel written on it could best be described as: A unlikely B even chance C probable D almost certain 11 multiple choice Which of the following events is the most likely to occur? A Selecting the first number drawn from a barrel containing 20 numbered marbles B Selecting a diamond from a standard deck of cards C Winning the lottery with one ticket out of 150 000 D Drawing the inside lane in the Olympic 100-metre final with eight runners 12 multiple choice The ski season opens on the first weekend of June. At a particular ski resort there has been sufficient snow for skiing on that weekend on 32 of the last 40 years. Which of the following statements is true? A It is unlikely to snow at the opening of the ski season this year. B There is a fifty-fifty chance that it will snow at the opening of the ski season this year. C It is probable that it will snow at the opening of the ski season this year. D It is certain to snow at the opening of the ski season this year.

13 On a production line, light globes are tested to see how long they will last. After testing 1000 light globes, it is found that 960 will burn for more than 1500 hours. Wendy 5 purchases a light globe. Describe the chance of the light globe burning for more than 1500 hours.

WORKED

Example

14 Of 12 000 new cars sold last year, 1500 had a major mechanical problem during the first year. Edwin purchased a new car. Describe the chance of Edwin having a major mechanical problem in the first year. 15 During an election campaign, 2000 people were asked for their voting preferences. One thousand said that they would vote for the government. If one person is chosen at random, describe the chance that they would vote for the government.

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Common descriptions of chance The English language has many colourful expressions to describe the chance of an event occurring. Consider the following expressions and research them to answer the questions. 1 ‘That will happen once in a blue moon.’ a What is a blue moon? b How often does a blue moon occur? 2 ‘There is Buckley’s chance of that happening.’ a Who was Buckley? b How did this saying originate? Are there any similar expressions that you can think of? What are their origins?

Sample space At some time in our lives, most of us have tossed or will toss a coin. Many sports begin with the toss of a coin.

What is the chance that the coin will land on Heads? Most people would correctly say fifty-fifty. We need to develop a method of accurately describing the probability of an event. Before we can calculate probability, we need to be able to list all possible outcomes in a situation. This is called listing the sample space. When tossing a coin, the sample space has two elements: Heads and Tails. To calculate a probability, we need to know the elements of the sample space and how many elements are in the sample space.

WORKED Example 6 List the sample space for rolling a die. THINK

WRITE

The sample space is the numbers 1 to 6.

S = {1, 2, 3, 4, 5, 6}


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In many cases, several elements of the sample space may be the same. In such cases, we can distinguish between the number of elements in the sample space and the number of distinct (different) elements.

WORKED Example 7 In a barrel there are 4 red marbles, 5 green marbles and 3 yellow marbles. One marble is drawn from the barrel. a List the sample space. b How many elements are in the sample space? c How many distinct elements in the sample space? THINK

WRITE

a List each marble in the barrel.

a S = {red, red, red, red, green, green, green, S = green, green, yellow, yellow, yellow} b The sample space has 12 elements.

b Count the number of elements in the sample space. c Count the number of different elements in the sample space.

c The sample space has 3 distinct elements.

In some probability elements there may be more than one element in the sample space that gives us the desired outcome. Favourable outcomes are the elements from the sample space that will meet the requirements for an event to occur.

WORKED Example 8 Tegan is playing a board game. To win the game, Tegan must roll a number greater than 2 with one die. a List the sample space. b List the favourable outcomes. THINK

WRITE

a List all possible outcomes for one roll of a die. b List all elements of the sample space that are greater than 2.

a S = {1, 2, 3, 4, 5, 6} b E = {3, 4, 5, 6}

remember 1. The sample space is the list of all possible outcomes in a probability experiment. 2. The number of elements in a sample space is the total number of possible outcomes. 3. In the sample space, there are sometimes several elements that are the same. We may be asked to count the number of distinct (different) elements in the sample space. 4. Favourable outcomes are the elements from the sample space that meet the requirements for a certain event to occur.

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14B WORKED

14.5

SkillS

HEET

Example

6

Sample space

1 The numbers 1 to 10 are written on cards that are turned face down. The cards are shuffled and one is chosen. List the sample space. 2 For each of the following probability experiments, state the sample space. a Tossing a coin b Rolling a die c The total when rolling two dice d Choosing a letter of the alphabet e The day of the week on which a baby could be born f The month in which a person’s birthday falls

Listing the sample space

3 For each of the following probability experiments, state the number of elements in the sample space. a Choosing a card from a standard deck b Selecting the winner of a 15 horse race c Selecting the first ball drawn in a Lotto draw (The balls are numbered 1–44.) d Drawing a raffle ticket from tickets numbered 1 to 1500 e Selecting a number between 100 and 1000, inclusive f Drawing a ball from a bag containing 3 yellow, 4 red and 4 blue balls WORKED

Example

7

4 The letters of the word MISSISSIPPI are written on cards and turned face down. A card is then selected at random. a List the sample space. b How many elements are in the sample space? c How many distinct elements in the sample space? 5 A card is to be selected from a standard deck. a How many elements does the sample space have? b How many different elements are in the sample space if we are interested in: i the suit of the card? ii the colour of the card? iii the face value of the card?

WORKED

Example

8

6 Jane is playing a game of snakes and ladders. It is her turn to roll the die and to win she needs a number greater than 4. a List the sample space for this roll of the die. b List the favourable outcomes for this roll of the die. 7 A bag holds 60 black marbles and 40 white marbles. Tony is to choose one of these marbles from the bag. Tony wants to select a white marble. a How many elements are in the sample space? b How many favourable outcomes are contained in the sample space? 8 multiple choice A bag contains 5 blue discs, 9 red discs and 6 yellow discs. To win a game, Jenny needs to draw a yellow disc from the bag. How many elements are in the sample space? A 3 B 6 C 14 D 20


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9 multiple choice To win a game Jenny needs to draw a yellow disc from the bag in question 8. How many favourable outcomes are there? A 3 B 6 C 14 D 20 10 multiple choice A raffle has 100 tickets. Chris buys 5 tickets in the raffle. Which of the following statements is correct? A There are two elements in the sample space. B There are five favourable outcomes. C There are 100 elements in the sample space. D Both B and C 11 New South Wales are playing Queensland in a State of Origin match. a List the sample space for the possible outcomes of the match. b How many elements in the sample space? c Is each element of the sample space equally likely to occur? Explain your answer. 12 A bag contains five 20c pieces, four 10c pieces and one 5c piece. A coin is selected at random from the bag. Without replacing the first coin, a second coin is then selected. a List the sample space for the first coin selected. How many elements does the sample space have? b Assume that the first coin chosen was a 20c piece. List the sample space for the second coin chosen. c Assume that the first coin chosen was a 10c piece. List the sample space for the second coin chosen. d Assume that the first coin chosen was the 5c piece. List the sample space for the second coin chosen. 13 Write down an example of an event that has 4 elements in the sample space. 14 Write down an example of an event that has 10 elements in the sample space but only 4 distinct elements.

2 How many times in 100 trials would you expect each element of the sample space to occur?

E E

Dice

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Die rolling

sheet

3 Conduct 100 trials of the experiment and see how closely your results match the expected results. Experiment 1. Tossing a coin Experiment 2. Rolling a die Experiment 3. Selecting a card from a standard deck and noting its face value Experiment 4. Selecting a card from a standard pack and noting its suit

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1 List the sample space.

Coin toss lister

E

For each of the four probability experiments below, answer the following questions.

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Matching actual and expected results

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1 1 A die is rolled. Describe the chance that the uppermost face is 4. 2 A card is drawn from a standard pack. Describe the chance of selecting a black card. 3 A bag contains four $1 coins and seven $2 coins. Describe the chance that a coin drawn at random from the bag will be a $2 coin. 4 A barrel containing balls numbered 1 to 100 has one ball selected at random from it. How many elements are in the sample space? 5 Five history books, three reference books and ten sporting books are arranged on a shelf. A book is chosen at random from the shelf. How many elements are in the sample space? 6 For the example in question 5, how many distinct elements does the sample space have? 7 For the example in question 5, if you want a sporting book, how many favourable outcomes are there? 8 Copy and complete. If an event is certain, then the probability of it occurring is . 9 Copy and complete. If an event is impossible, then the probability of it occurring is . 10 If Jane needs to select an ace from a standard deck to win the game, how many favourable outcomes are there?


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Tree diagrams A multi-stage event is where there is more than one part to the probability experiment. Tree diagrams are used to find the elements in the sample space in a multi-stage probability experiment. Consider the case of tossing two coins. How many elements are there in the sample space? We draw a tree diagram to develop a system that will list the sample space for us. 1st coin

2nd coin Heads Tails

Heads

Heads Tails

Tails

The tree diagram branches out once for every stage of the probability experiment. At the end of each branch, one element of the sample space is found by following the branches that lead to that point. Therefore, when two coins are tossed, the sample space can be written: S = {Heads-Heads, Heads-Tails, Tails-Heads, Tails-Tails} There are four elements in the sample space; Heads-Tails and Tails-Heads are distinct elements of the sample space.

WORKED Example 9 A coin is tossed and a die is rolled. List all elements of the sample space. THINK 1 2

WRITE

Draw the branches for the coin toss. From each branch for the coin toss, draw the branches for the die roll.

Coin toss Head

Tail

3

List the sample space by following the path to the end of each branch.

Die roll 1 2 3 4 5 6 1 2 3 4 5 6

S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, S = T5, T6}

Click on the PowerPoint icon on the Maths Quest General Mathematics Preliminary Course CD-ROM to see worked example 9 appear step by step. In many cases, the second branch of the tree diagram will be different from the first branch. This occurs in situations such as those outlined in the following worked examples, where the first event has an influence on the second event. The card chosen first can then not be chosen in the second event.

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WORKED Example 10 The numbers 2, 4, 7 and 8 are written on cards and are chosen to form a two-digit number. List the sample space. THINK 1

2

3

WRITE

Draw the first branch of the tree diagram to show each possible first digit. Draw the second branch of the tree diagram to show each possible second digit. When drawing the second branch, the digit from which the tree branches can’t be repeated. List the sample space by following the tree to the end of each branch.

1st digit 2nd digit 4 7 2 8 2 7 4 8 2 4 7 8 2 4 8 7

S = {24, 27, 28, 42, 47, 48, 72, 74, 78, 82, 84, 87}

By clicking on the PowerPoint icon on your Maths Quest General Mathematics Preliminary Course CD-ROM, you can see the tree diagram drawn step by step. Each question must be read carefully, to see if repetition is possible or not. In the above example, the numbers cannot be repeated because we are drawing two cards without replacing the first card. In examples such as tossing two coins, it is possible for the same outcome on both coins. 1st coin 2nd coin 3rd coin When drawing a tree diagram, the tree needs Heads to branch once for every stage of the experiment. Heads Tails Heads When we roll two dice, there are two levels to the Heads Tails tree diagram. If we were to toss three coins, there Tails would be three levels to the diagram, as shown Heads Heads at right. Tails Tails

Tails

Heads Tails


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WORKED Example 11 Four children go exploring. a Draw a tree diagram to list all possible combinations of boys and girls. b How many elements are in the sample space? c How many elements of the sample contain 3 boys and a girl? THINK

WRITE

a Draw the tree diagram.

a

1st child 2nd child 3rd child Boy Boy Girl Boy Boy Girl Girl Boy Boy Girl Girl Boy Girl Girl

4th child Boy Girl Boy Girl Boy Girl Boy Girl Boy Girl Boy Girl Boy Girl Boy Girl

b List the sample space by following the paths to the end of each branch.

b S = {BBBB, BBBG, BBGB, BBGG, BGBB, S = BGBG, BGGB, BGGG, GBBB, GBBG, S = GBGB, GBGG, GGBB, GGBG, GGGB, S = GGGG}

c Count the number of elements that contain 3 boys and 1 girl.

c There are four elements of the sample space which contain 3 boys and 1 girl.

By clicking on the PowerPoint icon on your Maths Quest General Mathematics Preliminary Course CD-ROM, you can see the tree diagram drawn step by step.

remember 1. A tree diagram is necessary in any example where there is more than one stage to the probability experiment. 2. The tree diagram must branch out once for every stage of the probability experiment. 3. Once the tree is drawn, the sample space is found by following the branches to each end.

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14C WORKED

Example

9

Tree diagrams

1 Two coins are tossed. Use a tree diagram to list the sample space. 2 On three red cards, the numbers 1, 2 and 3 are written. On three blue cards, the same numbers are written. A red card and a blue card are then chosen to form a two-digit number. Draw a tree diagram to list the sample space. 3 A family consists of 3 children. Use a tree diagram to list all possible combinations of boys and girls. 4 A coin is tossed and then a die is rolled. a How many elements are in the sample space? b Does it make any difference to the sample space if the die is rolled first and then the coin is tossed?

WORKED

Example

10

5 The digits 1, 3, 4 and 8 are written on cards. Two cards are then chosen to form a twodigit number. List the sample space. 6 Darren, Zeng, Melina, Kate and Susan are on a committee. From among themselves, they must select a chairman and a secretary. The same person cannot hold both positions. Use a tree diagram to list the sample space for the different ways the two positions can be filled. 7 A tennis team consists of six players, three males and three females. The three males are Andre, Pat and Yevgeny. The three females are Monica, Steffi and Lindsay. A male and a female must be chosen for a mixed doubles match. Use a tree diagram to list the sample space. 8 Chris, Aminta, Rohin, Levi and Kiri are on a Landcare group. Two of them are to represent the group on a field trip. Use a tree diagram to list all the different pairs that could be chosen. (Hint: In this example, a pairing of Chris and Aminta is the same as a pairing of Aminta and Chris.)

WORKED

Example

11

9 Four coins are tossed into the air. a Draw a tree diagram for this experiment. b Use your tree to list the sample space. c How many elements have an equal number of Heads and Tails? 10 multiple choice Three coins are tossed into the air. The number of elements in the sample space is: A 3 B 6 C 8 D 9 11 multiple choice A two-digit number is formed using the digits 4, 6 and 9. If the same number can be repeated, the number of elements in the sample space is: A 3 B 6 C 8 D 9 12 multiple choice A two-digit number is formed using the digits 4, 6 and 9. If the same number cannot be used twice, the number of elements in the sample space is: A 3 B 6 C 8 D 9


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13 When two coins are tossed there are three elements in the sample space, 2 Heads, 2 Tails or 1 Head and 1 Tail. Is this statement correct? Explain why or why not. 14 A two-digit number is to be formed using the digits 2, 5, 7 and 8. a If the same number can be used twice, list the sample space. b If the same number cannot be repeated, list the sample space. 15 The numbers 1, 2, 5 and 8 are written on cards and placed face down. a If two cards are chosen and used to form a two-digit number, how many elements are in the sample space? b If three cards are chosen and used to form a three-digit number, how many elements are in the sample space? c How many four-digit numbers can be formed using these digits? 16 A school captain and vice-captain need to be elected. There are five candidates. The three female candidates are Tracey, Jenny and Svetlana and the male candidates are Richard and Mushtaq. a Draw a tree diagram to find all possible combinations of captain and vice-captain. b How many elements are in the sample space? c If boys are filling both positions, how many elements are there? d If girls are filling both positions, how many elements are there? e If students of the opposite sex fill the positions, how many elements are there? 17 Two dice are rolled. a Use a tree diagram to calculate the number of elements in the sample space. b Steve is interested in the number of elements for each total. Copy and complete the table below. 3

4

5

6

7

8

9

10

11

12

No. of elements

T SHEE

Work

2

Total

14.1

c How many elements of the sample space have a double number?

Two-stage experiments Result

E

Coin toss lister

No. of times

2 Heads

E

2 Tails

Die rolling

Does this match your expected outcome? 2 Roll two dice 100 times and record the total of the two dice in a copy of the table below. 2

3

4

5

6

7

8

9

10

11

12

No. of times Percentage Compare your results with your answer to question 17 in the previous exercise.

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1 Head, 1 Tail

Total

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1 Toss two coins 100 times. Copy and complete the table below.

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Equally likely outcomes Below is the field for the 2006 Melbourne Cup. Melbourne Cup Odds Horse

Odds

Horse

Yeats

11-2

Zipping

9-1

Delta Blues

16-1

Dizelle

25-1

Railings

50-1

Ice Chariot

Tawqeet

5-1

Kerry O’Reilly

Odds

200-1 50-1

Geordieland

15-1

Zabeat

Headturner

66-1

Art Success

Short Pause

200-1

Demerger

100-1

20-1

Glistening

80-1

Activation

200-1 40-1

Land ‘n’ Stars

200-1

Mandela

20-1

Mahtoum

200-1

Dolphin Jo

80-1

On a Jeune

20-1

Pop Rock

5-1

Maybe Better Efficient

9-1 SCR

There were 23 horses in this race after Efficient was scratched. The sample space therefore has 23 elements. However, in this case, each outcome is not equally likely. This is because each horse in the race is not of equal ability. Some horses have a greater chance of winning than others. It is true in many practical situations that each outcome is not equally likely to occur. The weather on any day could be wet or fine. Each outcome is not equally likely as there are many factors to consider, such as the time of year and the current weather patterns.


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In each probability example, it is important to consider whether or not each outcome is equally likely to occur. In general, when the selection is made randomly then equally likely outcomes will result.

WORKED Example 12 In a rugby league match between Brisbane and Parramatta there are three possible outcomes: Brisbane win, Parramatta win and a draw. Is each outcome equally likely? Explain your answer. THINK

WRITE

Each team may not be of equal ability and draws occur less often than one of the teams winning.

Each outcome is not equally likely as the teams may not be of equal ability and draws are fairly uncommon in rugby league.

In some cases we need to use tree diagrams to calculate if each outcome is equally likely. A statement may seem logical, but unless further analysis is conducted, we can not be sure.

WORKED Example 13 When two coins are tossed there are three possible outcomes, 2 Heads, 2 Tails and one of each. Is each outcome equally likely? THINK 1

WRITE

There is more than one coin being tossed and so a tree diagram must be drawn.

1st coin Heads Tails

2

There are actually four outcomes, two of which involve 1 Head and 1 Tail. Therefore each of the outcomes mentioned is not equally likely to occur.

2nd coin Heads Tails Heads Tails

Each outcome is not equally likely. There are two chances of getting one Head and one Tail. There is only one chance of getting 2 Heads and one chance of getting 2 Tails.

remember 1. Each element of the sample space will not always be equally likely. 2. Outcomes will be equally likely if a selection is random. When other factors influence the selection, each outcome is not equally likely. 3. When there is more than one event involved, examine the tree diagram to determine if events described are equally likely.

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14D WORKED

Example

12

Equally likely outcomes

1 A tennis match is to be held between Lindsay and Anna. There are two possible outcomes, Lindsay to win and Anna to win. Is each outcome equally likely? Explain your answer. 2 There are 80 runners in the Olympic Games marathon. The sample space for the winner of the race therefore has 80 elements. Is each outcome equally likely? Explain your answer. 3 The numbers 1 to 40 are written on 40 marbles. The marbles are then placed in a bag and one is chosen from the bag. There are 40 elements to the sample space. Is each outcome equally likely? Explain your answer. 4 For each of the following, state whether each element of the sample space is equally likely to occur. a A card is chosen from a standard deck. b The result of a volleyball game between two teams. c It will either rain or be dry on a summer’s day. d A raffle with 100 tickets has one ticket drawn to win first prize. 5 For each of the following, state whether the statement made is true or false. Give a reason for your answer. a Twenty-six cards each have one letter of the alphabet written on them. One card is then chosen at random. Each letter of the alphabet has an equal chance of being selected. b A book is opened on any page and a pin is stuck in the page. The letter closest to the pin is then noted. Each letter of the alphabet has an equal chance of being selected. 6 multiple choice In which of the following is each member of the sample space equally likely to occur? A Kylie’s softball team is playing a match that they could win, lose or draw. B A bag contains 4 red counters and 2 blue counters. One counter is selected from the bag. C The maximum temperature on a January day will be between 20°C and 42°C. D A rose that may bloom to be red, yellow or white is planted in the garden.


WORKED

Example

13

475

7 A couple have two children. They could have two boys, two girls or one of each. The sample space therefore has three elements that are all equally likely. Is this statement correct? Explain your answer. 8 In a game two dice are rolled and the total of the two dice is the player’s score. a What is the sample space for the totals of two dice? b Is each element of the sample space equally likely to occur? 9 A restaurant offers a three-course meal. The menu is shown below. Entree

Main course

Dessert

Prawn cocktail Oysters Soup

Seafood platter Chicken Supreme Roast beef Vegetarian quiche

Pavlova Ice-cream

a A diner selects one plate from each course. Draw a tree diagram to determine the number of elements in the sample space. b Is each element of the sample space equally likely to occur? 10 There are 10 horses in a race. Ken hopes to select the winner of the race. a How many elements in the sample space? b Is each element of the sample space equally likely to occur? Explain your answer. c Loretta selects her horse by drawing the names out of a hat. In this case, is the sample space the same? Is each element of the sample space equally likely to occur? Explain your answer.

2 1 Describe the chance of selecting an ace from a standard deck of cards. For questions 2–5: A bag contains 3 black marbles, 4 white marbles and a red marble. 2 How many elements in the sample space? 3 How many distinct elements in the sample space? 4 If Julie needs to draw a red marble from the bag, how many favourable outcomes are there? 5 Is each element of the sample space equally likely to occur? 6 A pair of twins is born. Draw a tree diagram and then list the sample space for all possible combinations of boys and girls. 7 Amy and Katherine are in Year 11, while Luke, Matthew and John are in Year 12. One Year 11 student and one Year 12 student are to represent the school at a conference. List the sample space for all pairs that could be chosen. 8 A two-digit number is formed using the digits 1, 2 and 3. How many elements does the sample space have if the same digit can be used twice? 9 A two-digit number is formed using the digits 5, 6, and 7. How many elements are in the sample space if the same digit cannot be used twice? 10 A student has an exam in Mathematics that she could either pass or fail. Is each element of the sample space equally likely to occur?

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Using the fundamental counting principle A three-course meal is to be served at a 21st birthday party. Guests choose one plate from each course, as shown in the menu below. Entree

Main course

Dessert

Beef broth Calamari

Spaghetti Roast chicken Pasta salad Grilled fish

Ice-cream Banana split Strawberries

In how many different ways can the three courses for the meal be chosen? There are two possible choices of entree, four choices for main course and three Entree dessert choices. To find the sample space for all possible outcomes, we draw a tree diagram. By following the path to the end of each branch we can see that there are 24 Beef broth elements in the sample space. If we simply need to know the number of elements in the sample space, we multiply the number of possible choices at each level. Number of elements = 2 × 4 × 3 Number of elements = 24 There are 24 ways in which the three-course meal can be chosen. This multiplication principle is called the fundamental counting principle.

Main course Spaghetti Roast chicken


Pasta salad


Grilled fish


Spaghetti


Roast chicken


Pasta salad


Grilled fish


Calamari

The total number of ways that a succession of choices can be made is found by multiplying the number of ways each single choice could be made.

Dessert Ice-cream Banana split Strawberries

The fundamental counting principle is used when each choice is made independently of every other choice. That is, when one selection is made it has no bearing on the next selection. In the case above, the entree that is chosen has no bearing on what main course or dessert is chosen.


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WORKED Example 14 A poker machine has three wheels. There are 20 symbols on each wheel. In how many different ways can the wheels of the poker machine finish, once they have been spun? THINK 1

2

There are 20 possibilities for how the first wheel can finish, 20 for the second wheel and 20 for the third wheel. Multiply each of these possibilities together. Give a written answer.

WRITE Total outcomes = 20 × 20 × 20 Total outcomes = 8000

There are 8000 different ways in which the wheels of the poker machine can land.

WORKED Example 15 In Year 11 at Blackhurst High School, there are four classes with 20, 22, 18 and 25 students in them respectively. A committee of four people is to be chosen, one from each class to represent Year 11 on the SRC. In how many ways can this group of four people be chosen? THINK 1

2

There are 20 possible choices from the first class, 22 from the second, 18 from the third and 25 from the fourth class. Multiply these possibilities together. Give a written answer.

WRITE Total possible outcomes = 20 × 22 × 18 × 25 Total possible outcomes = 198 000

The committee of four people can be chosen in 198 000 different ways.

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Sometimes we need to reconsider examples that have some type of restriction placed on the possible selections.

WORKED Example 16 If number plates consist of 3 letters and 3 digits, how many different plates are possible if the first letter must be A, B or C, and the first digit cannot be 0 or 1?

THINK 1 2 3 4 5 6

WRITE

There are 3 possible first letters. There are 26 possible second and third letters. There are 8 possible first digits. There are 10 possible second and third digits. Multiply all these possibilities together. Give a written answer.

Total number plates = 3 × 26 × 26 × 8 × 10 × 10 Total number plates = 1 622 400 There are 1 622 400 possible number plates under this system.

remember 1. The fundamental counting technique allows us to calculate the number of different ways that separate events can occur. 2. This method can be used only when each selection is made independently of the others. To use this method, we multiply the number of ways that each selection can be made.


14E WORKED

Example

14

479

Using the fundamental counting principle

1 A poker machine has four reels, with 15 symbols on each wheel. If the wheels are spun, in how many ways can they finish? 2 Consider each of the following events. a A 10c coin and a 20c coin are tossed. In how many ways can they land? b A red die and blue die are cast. How many ways can the two dice land? c A coin is tossed and a die is rolled. How many possible outcomes are there? 3 A briefcase combination lock has a combination of three dials, each with 10 digits. How many possible combinations to the lock are there? 4 In the game of Yatzee, five dice are rolled. In how many different ways can they land? 5 Some number plates have two letters followed by 4 numbers. How many of this style of plate are possible? 6 multiple choice Personalised number plates have six symbols that can be any combination of letters or digits. How many of these are possible? A 1 000 000 B 17 576 000 C 308 915 776 D 2 176 782 336 7 multiple choice A restaurant menu offers a choice of four entrees, six main courses and three desserts. If one extra choice is offered in each of the three courses, how many more combinations of meal are possible? A 3 B 68 C 72 D 140

WORKED

Example

15

8 There are 86 students in Year 11 at Narratime High School. Of these, 47 are boys and 39 are girls. One boy and one girl are to be chosen as school captains. In how many different ways can the boy and girl school captain be chosen? 9 A travel agency offers Queensland holiday packages flying with QANTAS and Virgin Blue, travelling in First, Business and Economy class to Brisbane, the Gold Coast, The Great Barrier Reef and Cairns for periods of 7, 10 and 14 days. How many holiday packages does the traveller have to choose from?

480


10 A punter at the racetrack tries to pick the daily double. This requires her to pick the winner of race 6 and race 7. How many selections of two horses can she make if there are: a eight horses in each race? b 12 horses in each race? c 14 horses in race 6 and 12 in race 7? d 16 horses in race 6 and seven in race 7? e 24 horses in race 6 and 16 horses in race 7? 11 A poker machine has five wheels and 20 symbols on each wheel. a In how many ways can the wheels of the poker machine finish when spun? b There are 4 aces on the first wheel, 5 on the second wheel, 2 on the third wheel, 6 on the fourth wheel and 1 on the fifth. In how many ways can five aces be spun on this machine? 12 Radio stations on the AM band have a call sign of a digit from 2 to 9, followed by two letters. a How many radio stations could there be under this system? b In NSW all stations begin with a 2. How many stations are possible in NSW? 13 At a shoe store a certain pair of shoes can be bought in black, brown or grey; lace up or buckle up; and in six different sizes. How many different pairs of shoes are possible? 14 Home telephone numbers in Australia have eight digits. a How many possible home telephone numbers are there? b If a telephone number can’t begin with either a 0 or 1, how many are possible? c Freecall 1800 numbers begin with 1800 and then six more digits. How many of these are possible? d A certain mobile network has numbers beginning with 015 or 018 followed by six digits. How many numbers can this network have? 15 Madako can’t remember his PIN number for his bank account. He knows that it has four digits, does not begin with nine, is an odd number and that all digits are greater 16 than five. How many possible PIN numbers could he try?

WORKED

Example

16 multiple choice Postcodes in Australia begin with either 2, 3, 4, 5, 6, 7 or 8 followed by three more digits. How many of these postcodes can there be? A 70 B 1000 C 7000 D 10 000 17 Nadia goes to a restaurant that has a choice of 8 entrees, 15 main courses and 10 desserts. a How many combinations of entree, main course and dessert are possible? b Nadia is allergic to garlic. When she examines the menu she finds that three entrees and four main courses are seasoned with garlic. How many possible choices can she make without choosing a garlic dish?

Work

T SHEE

14.2

18 Bill is trying to remember Tom’s telephone number. It has eight digits and Bill can remember that it starts with 963 and finishes with either a 4 or a 6. How many possible telephone numbers are there for Tom? 19 A representative from each of six classes must be chosen to go on a committee. There are four classes of 28 students, a class of 25 students and a class of 20 students. How many committees are possible?


481

summary Informal description of chance • The chance of an event occurring can be described as being from certain (a probability of 1) to impossible (a probability of 0). • Terms used to describe the chance of an event occurring include improbable, unlikely, fifty-fifty, likely and probable. • The chance of an event occurring can be described by counting the possible outcomes and sometimes by relying on our general knowledge.

Sample space • A sample space is a list of all possible outcomes to a probability experiment. • It includes every possible outcome even if some outcomes are the same.

Tree diagrams • Tree diagrams are used to list the sample space when there is more than one stage to a probability experiment. • The tree must branch out once for each stage of the probability experiment.

Equally likely outcomes • Equally likely events occur when the selection method is random. • Events will not be equally likely when other factors influence selection. For example, in a race each person will not have an equal chance of winning, as each runner will be of different ability.

The fundamental counting principle • This principle can be used to count the number of elements in a sample space of a multi-stage experiment. • The total number of possible outcomes is calculated by multiplying the number of ways each stage of the experiment can occur.

482


CHAPTER review 14A

1 Graham and Marcia are playing a game. To see who starts they each take a card from a standard deck. The player with the higher card starts. Graham takes a five. Describe Marcia’s chance of taking a higher card.

14A

2 Describe each of the following events as being either certain, probable, even chance (fiftyfifty), unlikely or impossible. a Rolling a die and getting a number less than 6 b Choosing the eleven of diamonds from a standard deck of cards c Tossing a coin and it landing Tails d Rolling two dice and getting a total of 12 e Winning the lottery with one ticket

14A

3 Give an example of an event which is: a certain b impossible.

14A

4 The Chen family are going on holidays to Queensland during January. Are they more likely to experience hot weather or cold weather?

14A

5 List each of the events below in order from most likely to least likely. Winning a lottery with 1 ticket out of 100 000 tickets sold Rolling a die and getting a number greater than 1 Selecting a blue marble out of a bag containing 14 blue, 15 red and 21 green marbles Selecting a picture card from a standard deck

14A

6 Mark and Lleyton are tennis players who have played eight previous matches. Mark has won six of these matches. When they play their ninth match, who is more likely to win? Explain your answer.

14B

7 The letters of the word SAMPLE are written on cards and placed face down. One card is then selected at random. List the sample space. 8 List the sample space for each of the following probability experiments. a A coin is tossed. b A number is selected from the numbers 1 to 18. c The four aces from a deck of cards are selected. One of these cards is then chosen. d A bag contains 4 black marbles, 3 white marbles and 5 green marbles. One marble is then selected from the bag.

14B 14B

9 To win a game, Sarah must roll a number greater than 4 on the die. a List the sample space. b List the favourable outcomes. 10 For each of the following, state: i the number of elements in the sample space ii the number of favourable outcomes. a At the start of a cricket match, a coin is tossed and Steve calls Heads. b Anne selects a card from a standard deck and needs a number less than 9. (Aces count as 1.) c A bag contains 3 red, 8 blue and 4 black discs. Florian draws a disc from the bag and must not draw a black disc.


483

11 Two coins are tossed. Draw a tree diagram to find the sample space. 12 Two dice are rolled. How many elements are in the sample space? 13 A two-digit number is formed using 5, 6, 7 and 9, without repetition. a Use a tree diagram to list the sample space. b If Dan wants to make a number greater than 60, how many favourable outcomes are there?

14C 14C 14C

14 Mary, Neville, Paul, Rachel and Simon are candidates for an election. There are two positions, president and vice-president. One person cannot hold both positions. a List the sample space. b If Paul is to hold one of the positions, how many elements are in the event space?

14C

15 A school must elect one representative from each of three classes to sit on a committee. In 11A the candidates are Tran and Karen. In 11B the candidates are Cara, Daisy, Henry and Ian. In 11C the candidates are Bojan, Melina and Zelko. a List the sample space. b If there is to be at least one boy and at least one girl on the committee, how many elements are in the sample space?

14C

16 A greyhound race has eight runners. a How many elements are in the sample space? b Is each element of the sample space equally likely to occur? Explain your answer.

14D

17 For each of the following, explain if each element of the sample space is equally likely to occur. a There are 150 000 tickets in a lottery. One ticket is drawn to win first prize. b There are twelve teams contesting a hockey tournament. One team is to win the tournament. c A letter is chosen from the page of a book.

14D

18 A poker machine has five wheels. Each wheel has 15 symbols on it. In how many ways can the wheels land?

14E

19 There are four roads that lead from town A to town B, and five roads that lead from town B to town C. In how many different ways can I travel from town A to town C?

14E

20 The daily double requires a punter to select the winner of two races. How many selections are possible if there are 16 horses in the first leg and 17 in the second leg? 21 At a restaurant, a patron has the choice of five entrees, eight main courses and four desserts. In how many ways can they choose their meal?

14E

22 Jake has a bike chain that has a dial with four wheels, with 10 digits on each wheel. a How many different combinations are possible? b Jake has forgotten his combination. He can remember that the first digit is 5, and the last digit is odd. How many different combinations could there be to his chain?

14E

23 The dial to a safe consists of 100 numbers. To open the safe, you must turn the dial to each of four numbers that form the safe’s combination. a How many different combinations to the safe are possible? b How many different combinations are possible if no number can be used twice?

14E

484


Practice examination questions 1 multiple choice Jason and Kylie are playing a game of Monopoly. To move your piece, you roll two dice and move the same number of places as the total of the two dice. Kylie needs a total of 7 to land on Mayfair. The chance of Kylie rolling a 7 could best be described as: A impossible B unlikely C fifty-fifty D probable 2 multiple choice To win a game, Rhonda must roll a number greater than 3 with a single die. Which of the following statements is correct? A The sample space has 3 elements and there are 3 favourable outcomes. B The sample space has 3 elements and there are 6 favourable outcomes. C The sample space has 6 elements and there are 3 favourable outcomes. D The sample space has 6 elements and there are 6 favourable outcomes. 3 multiple choice A three-digit number is to be formed using the digits 3, 6, 7 and 8. The same number cannot be used more than once. How many three-digit numbers can be formed? A 4 B 12 C 24 D 64 4 multiple choice For which of the following events is each element of the sample space equally likely to occur? A The likelihood of selecting the winning Lotto combination B The likelihood of the weather being wet or fine C The likelihood of passing your next Maths exam D The likelihood of being successful in a job interview 5 multiple choice One person from each Year 11 class is to be elected to the Student Representative Council. If there are four classes in Year 11 with 23, 20, 19 and 25 people in these classes, the number of possible combinations of four representatives is: A 87 B 348 C 218 500 D 874 000 6 At a school athletics carnival, a RED YELLOW BLUE GREEN relay team must be selected. Below is the list of students who Richard Andrew Boris Milan qualified and the house for which Stan Frank Harry they compete. There must be one Ned Danny member of the relay team from Voula each house. a Draw a tree diagram to show all possible relay teams. b How many elements are in the sample space? c If Ned is to be in the relay team, how many favourable outcomes are there? d Describe the chance of Milan being in the relay team. e Is each element of the sample space equally likely to occur? Explain your answer.

CHAPTER

test yourself

14

7 Gino, Dennis, Kurt and Colin make up a tennis team. Two of them are to represent the club at a tournament. a Draw a tree diagram, to find the sample space for all possible teams. b List the favourable outcomes if Colin is to be in the team. c If the two selected players are to play two players selected from a group of six from another club, in how many ways can the final four players be chosen?

Relative frequency and probability

15 syllabus reference Probability 2 • Relative frequency and probability

In this chapter 15A Relative frequency 15B Single event probability 15C Writing probabilities as decimals and percentages 15D Range of probabilities 15E Complementary events

areyou

READY?

Are you ready?


15.1

Converting fractions to decimals

1 Convert the following fractions to decimals. a

15.2

15.3

b

3 --4

c

3 -----10

d

17 -----20

Converting decimals to percentages

2 Convert each of the following decimals to percentages. a 0.3 b 0.65 c 0.04

d 0.125

Simplifying fractions

3 Simplify each of the following fractions. a

15.4

1 --5

5 -----20

b

28 -----32

c

160 --------200

d

18 -----50

d

1 --3

Converting fractions to percentages

4 Write each of the following fractions as percentages. a

1 --4

15.5

Probability scale

15.6

Theoretical probability

b

3 --5

c

5 --8

5 For each of the following events, specify whether the chance of the event occurring is certain, fifty-fifty or impossible. a Two dice are rolled and a total of 14 is obtained. b A coin is tossed and it lands on Tails. c The month of July will follow June. d The maximum temperature on a summer’s day in Sydney will be less than 0°C. e A fair die is rolled and a number less than 4 is obtained.

6 In a box of chocolates, there are 6 soft-centred chocolates, 4 hard-centred chocolates and 5 nutcentred chocolates. A chocolate is selected at random. a What is the probability of selecting a hard-centred chocolate? b What is the probability of selecting a soft-centred chocolate? c What is the probability of selecting a nut-centred chocolate? d What is the sum of the probabilities calculated in parts a, b and c?

Chapter 15 Relative frequency and probability

487

Relative frequency You are planning to go skiing on the first weekend in July. The trip is costing you a lot of money and you don’t want your money wasted on a weekend without snow. So what is the chance of it snowing on that weekend? We can use past records to estimate that chance. If we know that it has snowed on the first weekend of July for 54 of the last 60 years, we could say that the chance of snow this year is very high. To measure that chance, we calculate the relative frequency of snow on that weekend. We do this by dividing the number of times it has snowed by the number of years we have examined. In this case, we can say the relative frequency of snow on the first weekend in July is 54 ÷ 60 = 0.9. The relative frequency is usually expressed as a decimal and is calculated using the formula: number of times an event has occurred Relative frequency = --------------------------------------------------------------------------------------------number of trials In this formula, a trial is the number of times the probability experiment has been conducted.

WORKED Example 1 The weather has been fine on Christmas Day in Sydney for 32 of the past 40 years. Calculate the relative frequency of fine weather on Christmas Day. THINK 1 Write the formula.

2 3

Substitute the number of fine Christmas Days (32) and the number of trials (40). Calculate the relative frequency as a decimal.

WRITE Relative frequency = number of times an event has occurred --------------------------------------------------------------------------------------------number of trials -----Relative frequency = 32 40

Relative frequency = 0.8

The relative frequency is used to assess the quality of products. This is done by finding the relative frequency of defective products.

WORKED Example 2 A tyre company tests its tyres and finds that 144 out of a batch of 150 tyres will withstand 20 000 km of normal wear. Find the relative frequency of tyres that will last 20 000 km. Give the answer as a percentage. THINK 1 Write the formula.

2 3 4

Substitute 144 (the number of times the event occurred) and 150 (number of trials). Calculate the relative frequency. Convert the relative frequency to a percentage.

WRITE Relative frequency = number of times an event has occurred --------------------------------------------------------------------------------------------number of trials --------Relative frequency = 144 150

Relative frequency = 0.96 Relative frequency = 96%

488


Relative frequencies can be used to solve many practical problems.

WORKED Example 3 A batch of 200 light globes was tested. The batch is considered unsatisfactory if more than 15% of globes burn for less than 1000 hours. The results of the test are in the table below. No. of hours

No. of globes

less than 500

4

500–750

12

750–1000

15

1000–1250

102

1250–1500

32

more than 1500

35

Determine if the batch is unsatisfactory. THINK 1 2

3

4 5 6

WRITE

Count the number of light globes that burn for less than 1000 hours. Write the formula.

Substitute 31 (number of times the event occurs) and 200 (number of trials). Calculate the relative frequency. Convert the relative frequency to a percentage. Make a conclusion about the quality of the batch of light globes.

31 light globes burn for less than 1000 hours. Relative frequency = number of times an event has occurred --------------------------------------------------------------------------------------------number of trials 31 Relative frequency = -------200

Relative frequency = 0.155 Relative frequency = 15.5% More than 15% of the light globes burn for less than 1000 hours and so the batch is unsatisfactory.

remember 1. The relative frequency is used to estimate the probability of an event. 2. The relative frequency, usually expressed as a decimal, is a figure that represents how often an event has occurred. 3. The relative frequency is calculated using the formula: number of times an event has occurred Relative frequency = --------------------------------------------------------------------------------------------- . number of trials 4. The relative frequency can also be written as a percentage and is used to solve practical problems.


15A WORKED

Example

Relative frequency

1 At the opening of the ski season, there has been sufficient snow for skiing for 37 out 15.1 of the past 50 years. Calculate the relative frequency of sufficient snow at the beginning of the ski season. Converting

SkillS

HEET

1

489

fractions

2 A biased coin has been tossed 100 times with the result of 79 Heads. Calculate the to decimals relative frequency of the coin landing Heads.

4 Farmer Jones has planted a wheat crop. For the wheat crop to be successful, farmer Jones needs 500 mm of rain to fall over the spring months. Past weather records show that this has occurred on 27 of the past 60 years. Find the relative frequency of: a sufficient rainfall b insufficient rainfall. WORKED

Example

2

decimals to percentages

5 Of 300 cars coming off an assembly line, 12 are found to have defective brakes. Calculate the relative frequency of a car having defective brakes. Give the answer as a percentage. 6 A survey of 25 000 new car buyers found that 750 had a major mechanical problem in the first year of operation. Calculate the relative frequency of: a having mechanical problems in the first year b not having mechanical problems in the first year. 7 On a production line, light globes are tested to see how long they will last. After testing 1000 light globes, it is found that 960 will burn for more than 1500 hours. Wendy purchases a light globe. What is the relative frequency that the light globe will: a burn for more than 1500 hours? b burn for less than 1500 hours? 8 multiple choice A study of cricket players found that of 150 players, 36 batted left handed. What is the relative frequency of left-handed batsmen? A 0.24 B 0.36 C 0.64 D 0.76 9 multiple choice Four surveys were conducted and the following results were obtained. Which result has the highest relative frequency? A Of 1500 P-plate drivers, 75 had been involved in an accident. B Of 1200 patients examined by a doctor, 48 had to be hospitalised. C Of 20 000 people at a football match, 950 were attending their first match. D Of 50 trucks inspected, 2 were found to be unroadworthy.

HEET

15.2 SkillS

3 Of eight Maths tests done by a class during a year, Peter has topped the class three Converting times. Calculate the relative frequency of Peter topping the class.

490


10 During an election campaign, 2000 people were asked for their voting preferences. One thousand and fifty said that they would vote for the government, 875 said they would vote for the opposition and the remainder were undecided. What is the relative frequency of: a government voters? b opposition voters? c undecided voters? 11 Research over the past 25 years shows that each November there is an average of two wet days on Sunnybank Island. Travelaround Tours offer one-day tours to Sunnybank Island at a cost of $150 each, with a money back guarantee against rain. a What is the relative frequency of wet November days as a percentage? b If Travelaround Tours take 1200 bookings for tours in November, how many refunds would they expect to give? 12 An average of 200 robberies takes place each year in the town of Amiak. There are 10 000 homes in this town. a What is the relative frequency of robberies in Amiak? b Each robbery results in an average insurance claim of $20 000. What would be the minimum premium that the insurance company would need to charge to cover these claims? 13 A car maker recorded the first time that its cars came in for mechanical repairs. The results are in the table below. 3

WORKED

Example

Time taken

No. of cars

0–3 months

5

3–6 months

12

6–12 months

37

1–2 years

49

2–3 years

62

more than 3 years

35

The assembly line will need to be upgraded if the relative frequency of cars needing mechanical repair in the first year is greater than 25%. Determine if this will be necessary. 14 For the table in question 13 determine, as a percentage, the relative frequency of: a a car needing mechanical repair in the first 3 months b a car needing mechanical repair in the first 2 years c a car not needing mechanical repair in the first 3 years.


491

15 A manufacturer of shock absorbers measures the distance that its shock absorbers can travel before they must be replaced. The results are in the table below. No. of kilometres

No. of shock absorbers

0–20 000

1

20 000–40 000

2

40 000–60 000

46

60 000–80 000

61

80 000–100 000

90

The relative frequency of the shock absorber lasting is 0.97 for a certain guaranteed distance. What is the maximum distance the manufacturer will guarantee so that the relative frequency of the shock absorbers lasting is greater than 0.97? 16 A soccer team plays 40 matches over a season and the results (wins, losses and draws) are shown below. W W W D L L L D W L W D L D W W L L L D W W D L L W W W L D L D D L W W W D D L a Put this information into a table showing the number of wins, losses and draws. b Calculate the relative frequency of each result over a season.

Researching relative frequencies Choose one of the topics below (or another of your choice) and calculate the relative frequency of the event. Most of the information needed can be found from books or the Internet. 1 Examine weather records and find out the relative frequency of rain on New Year’s Eve in Sydney. 2 Choose your favourite sporting team. Find the relative frequency of them winning over the past three seasons. 3 Find the relative frequency of the stock market rising for three consecutive days. 4 Check the NRL or AFL competitions and find the relative frequencies of win, loss and draw for each team.

492


Single event probability In chapter 14 we discussed the chances of certain events occurring. In doing so, we used informal terms such as probable and unlikely. While these terms give us an idea of whether something is likely to occur or not, they do not tell us how likely they are. To do this, we need an accurate way of stating the probability. We stated earlier that the chance of any event occurring was somewhere between impossible and certain. We also said that: • if an event is impossible the probability was 0 • if an event is certain the probability was 1. It therefore follows that the probability of any event must lie between 0 and 1 inclusive. A probability is a number that describes the chance of an event occurring. All probabilities are calculated as fractions but can also be written as decimals or percentages. Probability is calculated using the formula: number of favourable outcomes P ( event ) = ---------------------------------------------------------------------------total number of outcomes The total number of favourable outcomes is the number of different ways the event can occur, while the total number of outcomes is the number of elements in the sample space.

WORKED Example 4 Zoran is rolling a die. To win a game, he must roll a number greater than 2. List the sample space and state the number of favourable outcomes. THINK 1 2

WRITE

There are 6 possible outcomes. The favourable outcomes are to roll a 3, 4, 5 or 6.

S = {1, 2, 3, 4, 5, 6} There are 4 favourable outcomes.

Consider the case of tossing a coin. If we are calculating the probability that it will land Heads, there is 1 favourable outcome out of a total of 2 possible outcomes. Hence we can then write P(Heads) = 1--- . This method is used to calculate the probability of any 2 single event.

WORKED Example 5 Andrea selects a card from a standard deck. Find the probability that she selects an ace. THINK 1 2 3

There are 52 cards in the deck (total number of outcomes). There are 4 aces (number of favourable outcomes). Write the probability.

WRITE

P(ace) = P(ace) =

4 -----52 1 -----13


493

WORKED Example 6 In a barrel there are 6 red marbles, 2 green marbles and 4 yellow marbles. One marble is drawn at random from the barrel. Calculate the probability that the marble drawn is red. THINK 1 There are 12 marbles in the barrel (total number of outcomes). There are 6 red marbles in the barrel (number of 2 favourable outcomes). 3 Write the probability.

WRITE

P(red) = P(red) =

6 -----12 1 --2

Some questions have more than one favourable outcome. In these cases, we need to add together each of these outcomes to calculate the number of which are favourable.

WORKED Example 7 On a bookshelf there are 4 history books, 7 novels, 2 dictionaries and 5 sporting books. If I select one at random, what is the probability that the one chosen is not a novel? THINK 1 There are 18 books on the shelf (total number of outcomes). 2 Seven of these books are novels, meaning that 11 of them are not novels (number of favourable outcomes). Write the probability. 3

WRITE

P(not a novel) =

11 -----18

Some questions do not require us to calculate the entire sample space, only the sample space for a small part of the experiment.

WORKED Example 8 The digits 1, 3, 4, 5 are written on cards and these cards are then used to form a four-digit number. Calculate the probability that the number formed is: a even b greater than 3000. THINK a

1 2 3 4

If the number is even the last digit must be even. There are four cards that could go in the final place (total number of outcomes). Only one of these cards (the 4) is even (number of favourable outcomes). Write the probability.

WRITE a

P(even) =

1 --4

Continued over page

494


THINK

WRITE

b

b

If the number is greater than 3000, then the first digit must be a 3 or greater. There are four cards that could go in the first place. Three of these cards are a 3 or greater. Write the probability.

1 2 3 4

P(greater than 3000) =

3 --4

remember 1. The sample space is the list of all possible outcomes in a probability experiment. 2. The event space is a list of all favourable outcomes to a probability experiment. 3. The probability of an event is calculated using the formula: number of favourable outcomes P(event) = ---------------------------------------------------------------------------total number of outcomes

15B 15.3

SkillS

HEET

WORKED

Example

4 Simplifying fractions

Single event probability

1 A coin is tossed at the start of a cricket match. Manuel calls Heads. List the sample space and the number of favourable outcomes. 2 For each of the following probability experiments, state the number of favourable outcomes. a Rolling a die and needing a 6 b Rolling two dice and needing a total greater than 9 c Choosing a letter of the alphabet and it being a vowel d The chance a baby will be born on the weekend e The chance that a person’s birthday will fall in summer

495


3 For each of the following probability experiments, state the number of favourable outcomes and the total number of outcomes. a Choosing a red card from a standard deck b Selecting the winner of a 15 horse race c Selecting the first ball drawn in a Lotto draw (The balls are numbered 1 to 44.) d Winning a raffle with 5 tickets out of 1500 e Selecting a yellow ball from a bag containing 3 yellow, 4 red and 4 blue balls WORKED

Example

4 A coin is tossed. Find the probability that the coin will show Tails.

5

5 A regular die is cast. Calculate the probability that the uppermost face is: a 6 b 1 c an even number d a prime number e less than 5 f at least 5. 6 A barrel contains marbles with the numbers 1 to 45 written on them. One marble is drawn at random from the bag. Find the probability that the marble drawn is: a 23 b 7 c an even number d an odd number e a multiple of 5 f a multiple of 3 g a number less than 20 h a number greater than 35 i a square number. 7 Many probability questions are asked about decks of cards. You should know the cards making up a standard deck. 8 7 6 5 4 3 2 09 J1 Q

K

K

A

A

A

K

K

Q

Q

2

2

A

8 7 6 5 4 3 2 09 J1

Q

8 7 6 5 4 3 2 09 J1

8 7 6 5 4 3 2 09 J1

2

2

A card is chosen from a standard deck. Find the probability that the card chosen is: a the ace of diamonds b a king c a club d red e a picture card f a court card. WORKED

Example

6

8 A bag contains 12 counters: 7 are orange, 4 are red and 1 is yellow. One counter is selected at random from the bag. Find the probability that the counter chosen is: a yellow b red c orange. 9 The digits 2, 3, 5 and 9 are written on cards. One card is then chosen at random. Find the probability that the card chosen is: a the number 2 b the number 5 c even d odd e divisible by 3 f a prime number.

10 In a bag of fruit there are 4 apples, 6 oranges and 2 pears. Larry chooses a piece of fruit from the bag at random but he does not like pears. Find the probability that Larry 7 does not select a pear.

WORKED

Example

11 The digits 2, 3, 5 and 9 are written on cards. They are then used to form a four-digit number. Find the probability that the number formed is: 8 a even b odd c divisible by 5 d less than 3000 e greater than 5000.

WORKED

Example

12 multiple choice A die is cast. The probability that the number on the uppermost face is less than 4 is: A

1 --6

B

1 --3

C

1 --2

D

2 --3

496


13 multiple choice When a die is cast, which of the following outcomes does not have a probability equal to 1--- ? 2 A The number on the uppermost face is greater than 3. B The number on the uppermost face is even. C The number on the uppermost face is at least a 3. D The number on the uppermost face is a prime number. 14 multiple choice A card is chosen from a standard deck. The probability that the card chosen is a court card is: A

1 -----52

B

1 -----13

C

3 -----13

D

4 -----13

15 multiple choice When a card is chosen from a standard deck, which of the following events is most likely to occur? A choosing a seven B choosing a club C choosing a picture card D choosing a black card 16 One thousand tickets are sold in a raffle. Craig buys five tickets. a One ticket is drawn at random. The holder of that ticket wins first prize. Find the probability of Craig winning first prize. b After the first prize has been drawn, a second prize is drawn. If Craig won first prize, what is the probability that he now also wins second prize? 17 A lottery has 160 000 tickets. Janice buys one ticket. There are 3384 cash prizes in the lottery. a What is the probability of Janice winning a cash prize? b If there are 6768 consolation prizes of a free ticket for being one number off a cash prize, what is the probability that Janice wins a consolation prize? c What is the probability that Janice wins either a cash prize or a consolation prize? 18 A number is formed using all five of the digits 1, 3, 5, 7 and 8. What is the probability that the number formed: a begins with the digit 3? b is even? c is odd? d is divisible by 5? e is greater than 30 000? f is less than 20 000? 19 Write down an example of an event which has a probability of: a 1--b 1--c 2--- . 2

4

5

20 A three-digit number is formed using the digits 2, 4 and 7. a Explain why it is more likely that an even number will be formed than an odd number. b Which is more likely to be formed: a number less than 400 or a number greater than 400?

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Comparing probabilities with actual results

E


Tossing a coin

In this activity, we compare the probability of certain events to practical results. You may be able to do a simulation of these activities on a spreadsheet. 1 Tossing a coin a If we toss a coin P(Heads) = 1--- . Therefore, if you toss a coin, how many 2 Heads would you expect in: i 4 tosses? ii 10 tosses? iii 50 tosses? iv 100 tosses? b Now toss a coin 100 times and record the number of Heads after: i 4 tosses ii 10 tosses iii 50 tosses iv 100 tosses. Combine your results with the rest of the class. How close to 50% is the total number of Heads thrown by the class?

Number 1 2 3 4 5 6

Occurrences

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2 Rolling a die When you roll a die, what is the probability of rolling a 1? The probability for each number on the die is the same. Roll a die 120 times and record each result in the table below.

Rolling a die

Percentage of throws

How close are the results to the results that were expected?

Number 2 3 4 5 6 7 8 9 10 11 12

Occurrences

Percentage of throws

Do you notice anything different about the results of this activity, compared to the others?

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Rolling two dice

sheet

3 Rolling two dice Roll two dice and record the total on the faces of the two dice. Repeat this 100 times and complete the table below.

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1 1 A die is rolled. Find the probability that the uppermost face is 4. 2 A card is drawn from a standard pack. Find the probability of selecting a jack. 3 A bag contains four $1 coins and seven $2 coins. Find the probability that a coin drawn at random from the bag will be a $2 coin. 4 A barrel, containing balls numbered 1 to 100, has one ball selected at random from it. Find the probability that the ball selected is a multiple of 3. 5 Five history books, 3 reference books and 10 sporting books are arranged at random on a shelf. What is the probability of a sporting book being on the left-hand end of the bookshelf? 6 A coin is tossed 10 times with a result of 7 Heads and 3 Tails. The relative frequency of this coin landing Heads is 0.7; true or false? 7 A coin is tossed 10 times with a result of 7 Heads and 3 Tails. The probability of this 7 - ; true or false? coin landing Tails is ----10

8 In 60 rolls of a die, there have been 12 sixes. What is the relative frequency of rolling a six? 9 During a football season a team has won 15 matches and lost 5. Calculate the relative frequency of the team winning. 10 A car assembly line finds that five in every 1000 cars have faulty paintwork. If I purchase one of these cars, find, as a percentage, the relative frequency that the paintwork is faulty.


499

Writing probabilities as decimals and percentages In our exercises so far, we have been writing probabilities as fractions. This is the way that most mathematicians like to express chance. However, in day-to-day language, decimals and percentages are also used. Therefore, we need to be able to write probabilities as both decimals and percentages. When writing a probability as a decimal, we use the same formula and divide the numerator by the denominator to convert to a decimal.

WORKED Example 9 If I select a card from a standard deck, what is the probability of selecting a heart, expressed as a decimal? THINK 1 2 3 4

WRITE

There are a total of 52 cards in the deck (elements of the sample space). There are 13 hearts in the deck (elements of the event space). Write the probability. Convert to a decimal.

-----P(heart) = 13 52 P(heart) = 0.25

The chance of an event occurring is commonly expressed as a percentage. This is the percentage chance of an event occurring. When writing a probability as a percentage, we take the fractional answer and multiply by 100% to convert to a percentage.

WORKED Example 10 In a bag there are 20 counters: 7 are green, 4 are blue and the rest are yellow. If I select one at random, find the probability (as a percentage) that the counter is yellow. THINK 1 2 3 4

WRITE

There are 20 counters in the bag (elements of the sample space). There are 9 yellow counters in the bag (elements of the event space). Write the probability. Convert to a percentage.

9 - × 100% P(yellow counter) = ----20 P(yellow counter) = 45%

remember 1. Sometimes it is necessary to write a probability as a decimal or a percentage. 2. To write a probability as a decimal, we calculate the probability as a fraction, then divide the numerator by the denominator to convert to a decimal. 3. To write a probability as a percentage, we calculate the probability as a fraction, then multiply by 100% to convert to a percentage.

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15C SkillS

HEET

WORKED

Example

9 Converting fractions WORKED Example to 10 decimals

15.4

SkillS

HEET

15.1

Converting fractions to percentages

Writing probabilities as decimals and percentages

1 A die is rolled. What is the probability of rolling an even number, expressed as a decimal? 2 A barrel contains 40 marbles. There are 10 blue marbles, 15 red marbles and 15 white marbles. A marble is selected at random from the barrel. Calculate, as a percentage, the probability of selecting a red marble. 3 Write down the probability that a tossed coin will land Tails: a as a decimal b as a percentage. 4 A student is rolling a die. Write down each of the following probabilities as decimals, correct to 2 decimal places. a Getting a 1 b Getting an odd number c Getting a number greater than 4 5 For rolling a die, write down the following probabilities as percentages. Give your answers correct to 1 decimal place. a Getting a 3 b Getting an even number c Getting a number less than 6 6 From a standard deck of cards, one card is selected at random. Write down the probability of each of the following as a decimal (correct to 2 decimal places where necessary). a Selecting the king of hearts b Selecting a spade c Selecting any 5 d Selecting a red card e Selecting a court card (any king, queen or jack — the jack is also called a knave) 7 When selecting a card from a standard deck, what would be the probabilities of the following, written as percentages? Give your answers correct to 1 decimal place. a Selecting a jack of clubs b Selecting a diamond c Selecting any 2 d Selecting a black card e Selecting a court card

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8 multiple choice A raffle has 400 tickets. Sonya has bought 8 tickets. The probability that Sonya wins first prize in the raffle is: A 0.02 B 0.08 C 0.2 D 0.8 9 multiple choice In a class of 25 students, there are 15 boys and 10 girls. If a student is chosen at random from the class, the probability that the student is a boy is: A 10% B 15% C 40% D 60% 10 multiple choice Which of the following does not describe the chance of selecting a diamond from a standard deck of cards? -----A 13 B 0.13 C 0.25 D 25% 52

5

1

4

2 3

12 The board game in question 11 has the following rules. A player spinnning a 2 or a 5 is out of the game. A player spinning a 3 collects a treasure and automatically wins the game. Write down the probability, as a percentage, that with the next spin a player: a wins the game b is out of the game c neither wins nor is out of the game.

T SHEE

Work

11 The diagram on the right shows a spinner that can be used in a board game. When the player spins the spinner, what is the probability of getting the following results (expressed as a decimal)? a A5 b An even number c An odd number d A number greater than one

15.1

502


Range of probabilities Consider the following problem: A die is cast. Calculate the probability that the uppermost face is a number less than 7. We know this is certain to occur but we will look at the solution using the probability formula. There are 6 elements in the sample space and 6 elements in the event space. Therefore: P(no. less than 7) =

6 --6

=1 When the probability of an event is 1, the event is certain to occur. Now let’s consider an impossible situation: A die is cast. Calculate the probability that the uppermost face is a number greater than 7. There are 6 elements in the sample space and there are 0 elements in the event space. Therefore: P(no. greater than 7) =

0 --6

=0 When the probability of an event is 0, the event is impossible. All probabilities therefore lie in the range 0 to 1. An event with a probability of --1- has an even chance of occurring or not 2 occurring. The range of probabilities can be seen in the figure at right. This figure allows us to make a connection between the formal probabilities that we calculated in the previous exercise, and the informal terms we used in chapter 14. The closer a probability is to 0, the less likely it is to occur. The closer the probability is to 1, the more likely it is to occur. 0 ≤ P(E) ≤ 1

certain almost certain probable fifty-fifty

very unlikely impossible

For the following probabilities, describe whether the event would be certain, probable, fifty-fifty, unlikely or impossible. -----a 4--b 0 c 18 36

THINK is less than --1- and is therefore unlikely 2 to occur. b A probability of 0 means the event is impossible. 1 ------ = --- . Therefore, the event has an even c 18 36 2 chance of occurring. a

4 --9

WRITE a The event is unlikely as it has a probability of less than 1--- . 2 b The event is impossible as it has a probability of 0. c The event has an even chance of occurring as the probability = 1--- . 2

1 2

unlikely

WORKED Example 11 9

1

0


503

WORKED Example 12 In a batch of 400 televisions, 20 are defective. If one television is chosen, find the probability of it not being defective and describe this chance in words. THINK 1 2

3

WRITE

There are 400 televisions (elements of the sample space). There are 380 televisions that are not defective (number of favourable outcomes). Write the probability.

P(not defective) = P(not defective) =

4

Since the probability is much greater than 1--- and very close to 1, it is very 2 probable that it will not be defective.

380 --------400 19 -----20

It is very probable that the television chosen will not be defective.

There are many situations where this will occur. You need to be able to recognise when you can and cannot measure the probability. You cannot measure probability when each outcome is not equally likely.

WORKED Example 13 State whether the following statements are true or false, and give a reason for your answer. a The probability of correctly selecting a number drawn out of a barrel between 1 and 10 is

1 ------ . 10

b The weather tomorrow could be fine or rainy, therefore the probability of rain is 1--- . 2

THINK

WRITE

a Each outcome is equally likely.

a True, because each number is equally likely to be selected. b False, because there is not an equal chance of the weather being fine or rainy.

b Each outcome is not equally likely.

remember 1. Probabilities range from 0 to 1. A probability of 0 means that the event is impossible, while a probability of 1 means the event is certain. 2. By calculating the probability, we are able to make a connection with the more informal descriptions of chance. 3. The rules of probability can be applied only when each outcome is equally likely to occur.

504


15D 15.5

SkillS

HEET

Probability scale

WORKED

Example

11

Range of probabilities

1 For each of the probabilities given below, state whether the event would be impossible, unlikely, even chance, probable or certain. a

7 -----14

d 1 g

19 -----36

b

10 -----13

c

3 --8

e

37 -----40

f

25 -----52

i

12 -----25

h 0

2 For each of the events below, calculate the probability and hence state whether the event is impossible, unlikely, even chance, probable or certain. a Rolling a die and getting a negative number b Rolling a die and getting a positive number c Rolling a die and getting an even number d Selecting a card from a standard deck and getting a red card e Selecting a card from a standard deck and getting a spot card f Selecting a card from a standard deck and getting an ace g Reaching into a moneybox and selecting a 30c piece h Selecting a blue marble from a bag containing 3 red, 3 green and 6 blue marbles 3 Give an example of an event with a probability which is: a certain b probable c even chance d unlikely e impossible. 4 The probabilities of five events are given below. Write these in order from the most likely to the least likely event. 7 -----13

8 -----19

9 -----18

13 -----20

6 -----25

5 By calculating the probability of each, write the following events in order from least to most likely. A — Winning a raffle with 5 tickets out of 30 B — Rolling a die and getting a number less than 3 C — Drawing a green marble from a bag containing 4 red, 5 green and 7 blue marbles D — Selecting a court card from a standard deck E — Tossing a coin and having it land Heads 6 multiple choice The probabilities of several events are shown below. Which of these is the most likely to occur? A

1 --2

B

19 -----36

C

22 -----45

D

20 -----32

7 multiple choice Cards in a stack have the letters of the alphabet written on them (one letter per card). Vesna draws a card from the stack. The probability of selecting a card that has a vowel written on it could best be described as: A impossible B unlikely C even chance D probable


505

8 multiple choice For which of the following events can the probability not be calculated? A Selecting the first number drawn from a barrel containing 20 numbered marbles B Selecting a diamond from a standard deck of cards C Winning the lottery with one ticket out of 150 000 D Selecting the winner of the Olympic 100-metre final with 8 runners WORKED

Example

12

9 In a batch of 2000 cars that come off an assembly line, 50 have faulty paintwork. A car is chosen at random. a Find the probability that it has faulty paintwork. b Describe the chance of buying a car with faulty paintwork. 10 A box of matches has on the label ‘Minimum contents 50 matches’. The quality control department of the match manufacturer surveys boxes and finds that 2% of boxes have less than 50 matches. Find the probability that a box contains at least 50 matches and describe this chance. 11 A box of breakfast cereal contains a card on which there may be a prize. In every 100 000 boxes of cereal the prizes are: 1 new car 5 Disneyland holidays 50 computers 2000 prizes of $100 in cash 50 000 free boxes of cereal All other boxes have a card labelled ‘Second Chance Draw’. Describe the chance of getting a card labelled: a new car b free box of cereal c any prize d ‘Second Chance Draw’.

12 For each of the following determine whether the statement is true or false, giving a reason for your answer. 13 4 -. a The probability of selecting an ace from a standard deck of cards is ----52 1 -. b The probability of selecting the letter P randomly from a page of a book is ----26 c In a class of 30 students, the probability that Sam tops the class in a Maths test is 1 ------ . 30 d In a class of 30 students the probability that Sharon’s name is drawn from a hat is 1 ------ .

WORKED

Example

30

506


2 1 A coin is tossed. Find the probability of the coin landing Heads. 2 A card is drawn from a standard deck. Find the probability that the card selected is a diamond. 3 Three events have probabilities most likely.

1 10 --- , -----2 19

and

4 --- . 9

List these from the least likely to the

4 A tennis club has 40 members, of which 25 are female. If one member is chosen at random, find the probability (as a percentage) that the member is female. 5 For the tennis club in question 4, what is the probability (as a decimal) that the member chosen is male? 6 A card is drawn from a standard deck. Find the probability that the card selected is either a king or a queen (as a decimal to 3 decimal places). 7 A card is drawn from a standard deck. Find the probability that the card selected is a picture card (as a percentage to 1 decimal place). Copy and complete questions 8–10. 8 If an event is certain then the probability of it occurring is

.

9 If an event is impossible then the probability of it occurring is 10 An event has a probability of described as .

9 ------ . 10

.

The likelihood of the event occurring could be

Graphing results 1 Weather statistics Use the internet to find the number of wet days in Sydney during each month of the last five years. Copy and complete the table below for each month of the year. Year

No. of wet days

Relative frequency

Draw a radar chart to graph the month against the relative frequency of rain. 2 Sporting results Choose a sporting competition such as the AFL or NRL. Use the current or most recent season to calculate the relative frequency of each team winning. Choose an appropriate graph to display the results. (If you are using a spreadsheet, you can easily update your results each week.) 3 Topic of interest Choose a topic of interest. Research your area thoroughly and display your findings in graph form.


507

Complementary events When tossing a coin, we know there are two elements in the sample space. P(Heads) = 1--- and P(Tails) = 1--- . The total of the probabilities is 1. Now consider a 2 2 slightly more difficult problem.

WORKED Example 14 In a bag with 10 counters, there are 7 black, and 3 white counters. If one counter is selected at random from the bag, calculate: a the probability of selecting a white counter b the probability of selecting a black counter c the total of the probabilities. THINK

WRITE

a There are 10 counters of which 3 are white. b There are 10 counters of which 7 are black. 3 7 - and ------ together. c Add -----

a P(white) =

3 -----10

b P(black) =

7 -----10

10

10

3 - + c Total = ----10 Total = 1

7 -----10

In any probability experiment the total of all probabilities equals 1. We can use this rule to help us make calculations. In the above example, the chance of selecting a black counter and the chance of selecting a white counter are said to be complementary events. Complementary events are two events for which the probabilities have a total of 1. In other words, complementary events cover all possible outcomes to the probability experiment. When we are given one event and asked to state the complementary event, we need to describe what must happen for the first event not to occur.

WORKED Example 15 For each of the following events, write down the complementary event. a Tossing a coin and getting a Head b Rolling a die and getting a number less than 5 c Selecting a heart from a standard deck of cards THINK

WRITE

a There are two elements to the sample space, Heads and Tails. If the coin does not land Heads, it must land Tails.

a The complementary event is that the coin lands Tails. Continued over page

508


THINK

WRITE

b There are 6 elements to the sample space — 1, 2, 3, 4, 5, and 6. If we do not get a number less than 4 we must get either a 5 or a 6. c As we are concerned with only the suit of the card, there are four elements to the sample space: hearts, diamonds, clubs and spades. If we do not get a heart we can get any other suit.

b The complementary event is that we get a number greater than 4.

c The complementary event is that we do not get a heart.

We can use our knowledge of complementary events to simplify the solution to many problems. The probability of an event and its complement will always add to give 1. We can use the result: P(an event does not occur) = 1 − P(the event does occur)

WORKED Example 16 Jessie has a collection of 50 CDs. Of these, 20 are by a rap artist, 10 are by heavy metal performers and 20 are dance music. If we select one CD at random, what is the probability that it is: a a heavy metal CD? b not a heavy metal CD? THINK

WRITE

a Of 50 CDs, 10 are by heavy metal performers.

a P(heavy metal CD) = =

b This is the complement of selecting a heavy metal CD. Subtract the probability of selecting a heavy metal CD from 1.

10 -----50 1 --5

b P(not heavy metal) = 1 − P(heavy metal) =1− =

1 --5

4 --5

remember 1. The complement of an event is the event that describes all other possible outcomes to the probability experiment. 2. The sum of the probability of an event and its complement equals 1. 3. To calculate the probability of an event, subtract the probability of its complementary event from 1.


15E WORKED

Example

Complementary events

1 A die is rolled. a List the sample space. b Write down the probability of each event in the sample space. c What is the total of the probabilities?

15.6 SkillS

HEET

14

509

Theoretical probability

2 A barrel contains 20 marbles. We know that 7 of them are blue, 8 are red and the rest are yellow. a One marble is selected from the barrel. Calculate the probability that it is: i blue ii red iii yellow. b Calculate the total of these probabilities. WORKED

Example

15

3 For each of the following, state the complementary event. a Winning a race b Passing a test c Your birthday falling on a Monday 4 Match each event in the left-hand column with the complementary event in the righthand column. A coin landing Heads A coin landing Tails An odd number on a die A spot card from a standard deck A picture card from a standard deck Not winning 1st prize in the raffle A red card from a standard deck A team not making the last 4 Winning 1st prize in a raffle with 100 tickets An even number on a die Making the last 4 teams in a 20 team tournament A black card from a standard deck 5 For each pair of events in question 4, calculate: a the probability of the event in the left-hand column b the probability of its complementary event c the total of the probabilities. 6 You are rolling a die. Write down the complementary event to each of the following. a Rolling an even number b Rolling a number greater than 3 c Rolling a number less than 3 d Rolling a 6 e Rolling a number greater than 1 7 In a barrel there are balls numbered 1 to 45. For each of the following, write down the complementary event. a Choosing an odd-numbered ball b Choosing a ball numbered less than 20 c Choosing a ball that has a number greater than 23 d Choosing a ball that is a multiple of 5 8 In a barrel there are 25 balls, 15 of which are coloured (10 pink and 5 orange). The rest are black. What is the complementary event to selecting: a a black ball? b a coloured ball? c a pink ball?

510


9 multiple choice Wilson rolls two dice. He needs to get a 6 on at least one of the dice. What is the complementary event? A Rolling no sixes B Rolling 2 sixes C Rolling 1 six D Rolling at least 1 10 multiple choice The probability of rolling at least one six is mentary event? A

9 -----36

B

11 -----36

11 ------ . 36

C

What is the probability of the comple-

25 -----36

D 1

11 In a barrel with 40 marbles, 20 are yellow, 15 are green and 5 are orange. If one marble is selected from the bag find the probability that it is: 16 a orange b not orange.

WORKED

Example

12 In a barrel there are 40 balls numbered 1 to 40. One ball is chosen at random from the barrel. a Find the probability that the number is a multiple of 5. b Use your knowledge of complementary events to find the probability that the number is not a multiple of 5. 13 There are 40 CDs in a collection. They can be classified as follows. 18 heavy metal 6 rock 10 techno 6 classical If one CD is chosen at random, calculate the probability that it is: a heavy metal b not heavy metal c classical d not classical e heavy metal or rock f techno or classical. 14 In a golf tournament there are 40 players. Of these, 16 are Australian and 12 are American. If they are all of the same skill level, find the probability that the tournament is: a won by an Australian b won by an American c not won by an Australian d not won by an American e not won by an Australian or an American.


511

15 After studying a set of traffic lights, Karen found that in every 100 seconds they were red for 60 seconds, amber for 5 seconds and green for 35 seconds. If you were to approach this set of lights calculate the probability that: a they will be green b you will need to stop. 16 In a game of Scrabble there are 100 lettered tiles. These tiles include 9 ‘A’s, 12 ‘E’s, 9 ‘I’s, 8 ‘O’s and 4 ‘U’s. One tile is chosen. Find the probability that it is: a an ‘E’ b a vowel c a consonant. 17 From past performances it is known that a golfer has a probability of 0.7 of sinking a putt. What is the probability that he misses the putt?

3 1 A card is drawn from a standard deck and its suit noted. List the sample space for this experiment. 2 Andrew needs to ring Sandra but he has forgotten the last digit. Find the probability that he can correctly guess the number. 3 If Andrew knows that the last digit of a telephone number is not a 7 or a 0, what is the probability of guessing the number? 4 What is the probability of correctly guessing the 4-digit PIN number to a bank account card? 5 A bead is selected from a bag containing 3 red beads, 4 yellow beads and 8 blue beads. Find the probability that the bead selected is blue. 6 What is the probability that the bead selected in question 5 is not blue? 7 A number is chosen between 1 and 20. Find the probability the number chosen is a multiple of 3. 8 A number is chosen between 1 and 20. Find the probability the number chosen is a multiple of 5. 9 Find the probability that the number chosen is not a multiple of 5. 10 Find the probability that the number chosen is not a square number.

T SHEE

Work

18 A basketballer is about to take a shot from the free-throw line. His past record shows that he has a 91% success rate from the free-throw line. What would be the relative frequency (as a percentage) of his: a being successful with the shot? b missing the shot?

15.2

512


summary Relative frequency • Relative frequency describes how often an event has occurred. • It is found by dividing the number of times an event has occurred by the total number of trials.

Single event probability • The probability of an event can be found using the formula: number of favourable outcomes P(event) = ---------------------------------------------------------------------------total number of outcomes • Probabilities are usually written as fractions but can also be expressed as decimals or percentages.

Range of probabilities • Probabilities range from 0 (impossible) to 1 (certain). The use of a fraction for a probability can help us describe, in words, the chance of an event occurring.

Complementary events • The complement of an event is the event that describes all other possible outcomes to the probability experiment. • The probability of an event and its complement add to give 1. • The probability of an event can often be calculated by subtracting the probability of its complementary event from 1.


513

CHAPTER review 1 From every 100 televisions on a production line, two are found to be defective. If you choose a television at random, find the relative frequency of defective televisions.

15A

2 It is found that 150 of every thousand 17-year-old drivers will be involved in an accident within one year of having their driver’s licence. a What is the relative frequency of a 17-year-old driver having an accident? b If the average cost to an insurance company of each accident is $5000, what would be the minimum premium that an insurance company should charge a 17-year-old driver?

15A

3 The numbers 1 to 5 are written on the back of 5 cards that are turned face down. Michelle then chooses one card at random. Michelle wants to choose a number greater than 2. List the sample space and all favourable outcomes.

15B

4 A barrel contains 25 balls numbered from 1 to 25. One ball is drawn from the barrel. Find the probability that the marble drawn is: a 13 b 7 c an odd number d a square number e a prime number f a double-digit number.

15B

5 A card is to be chosen from a standard deck. Find the probability that the card chosen is: a the 2 of clubs b any 2 c any club d a black card e a court card f a spot card.

15B

6 A video collection has 12 dramas, 14 comedies, 4 horror and 10 romance movies. If I choose a movie at random from the collection, find the probability that the movie chosen is: a a comedy b a horror c not romance.

15B

7 The digits 5, 7, 8 and 9 are written on cards. They are then arranged to form a four-digit number. Find the probability that the number formed is: a 7895 b odd c divisible by 5 d greater than 7000 e less than 8000.

15B

8 A raffle has 2000 tickets sold and has two prizes. Michelle buys five tickets. a Find the probability that Michelle wins 1st prize. b If Michelle wins 1st prize, what is the probability that she also wins 2nd prize?

15B

9 A barrel contains marbles with the numbers 1 to 40 on them. If one marble is chosen at random find, as a decimal, the probability that the number drawn is: a 26 b even c greater than 10.

15C

10 A carton of soft drinks contains 12 cola, 8 orange and 4 lemonade drinks. If a can is chosen at random, find the probability, as a percentage, that the can chosen is: a cola b orange c not orange.

15C

11 If an event has a probability of 5--- , would the event be unlikely, fifty-fifty or probable?

15D 15D

9

12 When 400 cars are checked for a defect, it is found that 350 have the defect. If one is chosen at random from the batch, find the probability that it has the defect and hence describe the chance of the car having the defect.

514


15E

13 State the event that is complementary to each of the following. a Tossing a coin that lands Tails b Rolling a die and getting a number less than 5 c Choosing a blue ball from a bag containing 4 blue balls, 5 red balls and 7 yellow balls

15E

14 A barrel contains 20 marbles of which 6 are black. One marble is selected at random. Find the probability that the marble selected is: a black b not black.

15E

15 The probability that a person must stop at a set of traffic lights is of not needing to stop at the lights?

15E

16 On a bookshelf there are 25 books. Of these, seven are fiction. If one book is chosen at random, what is the probability that the book chosen is non-fiction?

7 ------ . 12

What is the probability

Practice examination questions 1 multiple choice A number is chosen at random from the numbers 1 to 50. The probability that a multiple of 3 is chosen is: A

3 -----50

B

8 -----25

C

16 -----34

D

34 -----50

2 multiple choice The probability of a footballer kicking a goal is --5- . The chance of him kicking the goal could 8 best be described as: A unlikely B fifty-fifty C probable D certain 3 multiple choice An Olympic Games shooter hits a target with 46 out of 50 shots. The relative frequency of him hitting his target is: A 0.03 B 0.46 C 0.50 D 0.92 4 multiple choice The probability of a missile hitting its target is A

1 -----25

B

8 -----25

17 ------ . 25

C

The probability of missing the target is:

17 -----25

D 1

5 Theo, Marcus, Olivia, Ben and Kelly are the finalists in a contest run by a music store. Two names will be drawn. The first name will win two tickets to a ‘Silverchair’ concert. a What is the probability that Marcus wins the Silverchair tickets? b Ben and Kelly agree that if either win a prize, they will take the other to the concert. What is the probability that Ben and Kelly will attend the concert? c What is the probability that neither Ben nor Kelly win the tickets?

CHAPTER

test yourself

15

6 A navy ship fires 60 missiles at a target and hits the target 42 times. a Find the relative frequency of a missile hitting its target. b What is the relative frequency of a missile missing its target? c Describe in words the chance of a missile hitting its target.

Glossary

515

Glossary Adjacent — The side next to the angle used for reference in a right-angled triangle. Allowance — An extra payment made to a worker for working in unfavourable conditions. Angle of depression — The angle through which you must look down from the horizontal to sight an object. Angle of elevation — The angle through which you must look up from the horizontal to sight an object. Annual leave — A period of time that each permanent employee is allowed each year for holidays. Annual leave loading — An extra payment of 17 1--- % of the gross pay made to employees 2 when they take their annual leave. Appreciation — The amount by which an item grows in value over time. Bar graph — A graph where categorical data are displayed in horizontal bars, with the categories on a vertical axis and quantity on the horizontal axis. Bias — Bias occurs when the results of a survey are influenced by outside factors such as a poorly chosen sample. In such a case, one set of circumstances is more prevalent than in the wider population. Bimodal — A set of scores for which two scores occur most often. Box-and-whisker-plot — A method of graphically displaying a five-number summary. The plot is drawn to scale with the box representing the interquartile range and the whiskers representing the range. Within the box, the median is also shown. Budget — A list of a person’s income and expenses. A personal budget is made to try to avoid spending more than is earned. A balanced budget is where income equals expenditure. Casual rate — A higher rate of pay to compensate casual workers for the lack of holiday and sick pay. Categorical data — Data which are not numerical and are put into categories such as types of car.

Census — Data gathered from the entire population. Central tendency — A method for describing a typical score in a data set. There are three measures of central tendency — mean, median and mode. Column graph — Similar to a bar graph, but the data are displayed in vertical columns. Commission — Payment made to a salesperson. A commission is usually paid as a percentage of sales. Complementary events — Two events that cover all possible outcomes to a probability experiment. The sum of the probabilities of complementary events is 1. Compound interest — A form of interest payment. The interest paid at the end of one period is added to the principal before the next interest calculation is made. Compound interest can be calculated using the formula A = P(1 + r)n. Compound value interest factor — The amount to which $1 will amount under a compound interest investment. Compounded value — see Future value. Compounding period — The length of time between interest payments in a compound interest investment. Concentration — The amount of one substance that is contained in another. The concentration of a mixture is usually stated as a mass/mass rate or a mass/volume rate. Congruent figures — A special case of similar figures where the scale factor is equal to 1. Congruent figures are identical in shape and size. Continuous data — Data that can take any value within a given range. Cosine ratio — The ratio of the adjacent side and hypotenuse in a right-angled triangle. Cubic — A function where x is raised to the power of 3. Cumulative frequency — A progressive total of the frequencies.

516

General Mathematics

Cumulative frequency histogram — A histogram drawn of the cumulative frequencies. No space is left before the first column of a cumulative frequency histogram. Cumulative frequency polygon — A line graph drawn from the corner of the axes to the top right corner of each column on a cumulative frequency histogram. Data — Information before it is organised. Database — An organised set of data on a population. Debenture — A form of investment offered by private companies to raise money. Debentures are for a fixed period of time and simple interest is paid. Decile — A band of 10% of scores in a data set. Decreasing function — A function for which the dependent variable decreases as the value of the independent variable increases. The graph of a decreasing function decreases from left to right. Deductions — A sum of money that is deducted from an employee’s gross pay before receiving net pay. Dependent variable — In a function, the dependent variable is the variable for which the value is obtained by substitution of another variable, the independent variable. Direct linear variation — This occurs when one quantity varies directly with another. In other words, the value of one quantity can always be calculated by multiplying the other quantity by a constant amount. Discrete data — Discrete data are where the data can take only certain values, usually whole numbers. Dividend — A payment made to a shareholder in a company. The company distributes all or part of its profit by paying an amount per share called a dividend. Dividend yield — This is the company dividend expressed as a percentage of the value of each share. Dot plot — A graph where the data set is displayed as dots on a number line. Double time — A penalty rate that pays the employee twice the normal hourly rate. Elevation — A scale drawing of what a building will look like from one side.

Enlargement — A figure is drawn similar to, but larger than the original. The corresponding sides will be in equal ratio and all corresponding angles will be equal. Equally likely outcomes — These occur when each element of the sample space for a probability experiment is equally likely to occur. Event — An occurrence that is being examined in a probability experiment. External sources — Data sources that are used when data previously gathered, often by a different person, are used in the statistical investigation. Extrapolating — Extending a graph so as to make a prediction about future trends. Favourable outcomes — Elements from the sample space that meet the requirement for an event to occur. Five-number summary — A summary of a data set consisting of the lower extreme, lower quartile, median, upper quartile and upper extreme. Frequency — The number of times an event occurs. Frequency histogram — A graph suitable for statistical (quantitative) data. It is a column graph drawn with scores or class centres on the horizontal axis and frequency on the vertical axis. A half unit (half column width) space is drawn before the first column with no other gaps between columns. Frequency polygon — A line graph often drawn on the same axes as a frequency histogram. The line is drawn from the corner of the axes to the centre of each column. Frequency table — A table displaying statistical data. For ungrouped data the table will have columns for score, tally, frequency and possibly cumulative frequency. For grouped data the score column will be replaced with a class column and a class centre column. Function — A rule that connects an independent variable with a dependent variable such that there is at most one value of the dependent variable for every value of the independent variable. Fundamental counting principle — The number of elements of the sample space for a multi-stage probability experiment is found by multiplying the number of ways each stage can occur. This is the fundamental counting principle.

Glossary

Future value — The amount to which an investment will grow under compound interest. Goods and Services Tax — A tax that is levied on the price of all items other than fresh food. The GST is levied at a rate of 10%. Gradient — The rate of increase (or decrease) in the dependent variable per one unit increase in the independent variable. Gross pay — A person’s earnings before any deductions are taken out. Group certificate — A statement of gross income and the PAYE tax deducted from that income throughout the financial year. It is given to the employee by the employer at the end of each financial year. Grouped data — A data set tabulated in small groups rather than as individual scores. Histogram — A column graph that displays the frequency for a set of scores. Hypotenuse — The longest side of a rightangled triangle. The hypotenuse is opposite the right angle. Income — Money received by a person, usually in exchange for labour or the result of an investment. Income tax — Tax that is paid on all income received. Increasing function — A function for which the dependent variable increases as the value of the independent variable increases. The graph of an increasing function increases as we look at it from left to right. Independent variable — In a function, the independent variable is the variable for which any value can be substituted and which will produce the value of another variable, the dependent variable. Indirect tax — Any tax that is not paid directly to the government by the taxpayer. For example, the GST is an indirect tax because it is paid to the retailer who then passes it on to the government. Inflation — A percentage amount that describes the average rise in prices over one year. Information — When data are processed, presented and conclusions have been drawn they become information. Interest — A payment made for the use of money. It is paid to a depositor by a financial institution or by a borrower to a financial institution. Interest is usually expressed as a rate per annum.

517

Interest rate — The percentage amount of interest paid per annum. Internal sources — The person conducting the statistical investigation gathers the data. Interpolate — Drawing a graph using data found at the end points. Interquartile range — A number that represents the spread of a data set. The interquartile range is calculated by subtracting the lower quartile from the upper quartile. Investment bonds — A form of investment offered by the State or federal government to raise money. Money is invested for a fixed amount of time and simple interest is paid. Linear function — A function that is a straight line when drawn. Line graph — A graph used to show the way in which one quantity changes with respect to another. The graph is drawn by marking the data points on a set of axes and joining them with straight lines. Lower extreme — The lowest score in the data set. Lower quartile — The lowest 25% of scores in a data set. Mean — The average of a data set, found by totalling all the scores then dividing by the number of scores. Median — The middle score or the average of the two middle scores in a data set. Medicare levy — A payment made as part of our tax system that covers the cost of basic health care services. The basic levy is 1.5% of gross income; however, low income earners pay the levy at a reduced rate. Mode — The score in a data set with the highest frequency. Multi-stage event — This occurs when there is more than one part to a probability experiment. For example, tossing two coins can be considered as tossing one coin then tossing another, therefore there are two parts to this experiment. Net pay — The amount of money actually received by an employee after all deductions have been subtracted from the gross pay. Offset — In a traverse study, an offset is a line perpendicular to the transversal. It is drawn from the transversal to a vertex on the area being surveyed. Ogive — Another term for cumulative frequency polygon.

518

General Mathematics

Opposite — The side opposite to the angle used for reference in a right-angled triangle. Ordinary rate — The normal hourly rate for a wage earner. Outcome — A possible result to a probability experiment. Overtime — This is when a person earns more than the regular hours each week. PAYE — Pay As You Earn. The method usually applied to the collection of tax. Payment by piece (Piecework) — Payment for the amount of work completed. Penalty rate — A higher rate of pay made to a person who is working overtime. Per annum — per year Percentage chance — The probability of an event expressed as a percentage. Percentage error — The maximum error in a measurement as a percentage of the measurement given. Piecewise linear function — A linear function that follows different rules for different values of the independent variable. Piecework — see Payment by piece. Poll — A collection of information obtained by questionnaire. Polygon — A line graph displaying the frequency for a set of scores. Population — An entire group of people or objects to which a statistical inquiry is applied. Power — An index. The number to which a base is raised, indicating the number of times the base is multiplied by itself. Prefix — The first part of a word. In measurement, the prefix indicates the relative size of the units of measurement. Present value — The current value of an investment. Prism — A solid shape with a constant crosssection. Principal — The amount on which interest calculations are made. Probability — A number between 0 and 1 that describes the chance of an event occurring. Proportional to — Two quantities are proportional to each other (in proportion) when one quantity can be found by multiplying the other by a constant amount.

Pyramid — A solid shape with a plane shape as its base and triangular sides meeting at an apex. Quadratic — A function with a greatest power of 2. Quality control — A statistical process used by companies to ensure that their product meets the required standard. Quantitative data — Data that can be measured. A numerical value can be assigned to them. Quartile — 25% of the data set. The upper quartile is the top 25% of the data set and the lower quartile is the bottom 25% of the data set. Questionnaire — A set of questions completed for a statistical investigation. Radar chart — A type of line graph drawn around a central point. The categories are labelled in a circle and data points marked on each line emanating from the centre. The points are then joined. A radar chart is suitable to show a pattern that is likely to repeat. For example, sales made during each month of the year. Random sample — The members of the sample are chosen by a method in which luck is the only factor in deciding which members are to participate in the sample. Range — A number that represents the spread of a data set. The range is calculated by subtracting the smallest score from the largest score. Rate — A comparison of two quantities of a different type. Rate of change — The change in one quantity per one unit change in another. Ratio — A comparison of two quantities of the same type. Reduction — A similar figure, drawn smaller in size than the original. Relation — A rule connecting two variables. Relative frequency — A number between 0 and 1, usually a decimal, which describes how often an event has occurred. The relative frequency is found by dividing the number of times an event has occurred by the total number of trials. Retainer — A fixed payment usually paid to someone receiving commission. They receive the retainer regardless of the number of sales made.

Glossary

Royalty — A royalty is a payment made to the owner of a copyright such as a musician or author. The royalty is usually a percentage of sales. Salary — A form of payment where a person is paid a fixed amount to do their job. A salary is usually based on an annual amount divided into weekly or fortnightly instalments. Sample — When data are gathered from a portion of the population, that is taken to be representative of the whole population. Sample space — A list of all possible outcomes to a probability experiment. Scale factor — A number by which the side lengths on the first of two similar figures is multiplied by to obtain the measurements on the second of the figures. Score — Each piece of quantitative data is a score. Sector graph — A graph where a circle is cut into sectors. Each sector then represents a section of the data set. Each sector is the same proportion of the circle as the part of the data set it represents. Shares — A share is a part ownership of a company. Shares are traded on the stock exchange and fluctuate in value daily. The return from investing in shares comes from both the dividend and the share rising in value. Significant figures — The number of non-zero digits to which a number is approximated. Similar figures — Two or more figures with corresponding angles equal and corresponding sides in the same ratio. Simple interest — Interest that is paid without any interest payments being added to the principal before the next interest calculation. Simple interest is calculated using the formula I = Prn. Sine ratio — The ratio of the opposite side and the hypotenuse in a right-angled triangle. Standard deviation — A measure of the spread of a data set. The standard deviation is found on a calculator using either the population standard deviation or the sample standard deviation. Statistical inquiry — The process of gathering statistics. Statistics — Numerical facts compiled to describe a data set. Stem-and-leaf plot — A method of displaying a data set where the first part of a number is

519

written in the stem and the second part of the number is written in the leaves. Step function — A linear function for which the rule changes as the value of the independent variable changes. Strata — A group within a population that reflects the characteristics of the entire population. Stratified sample — The group to participate in the sample is chosen so that is has similar characteristics as the entire population, for example the same percentage of men and women. Summary statistic — A number such as the mean, median or mode which describes a data set. Systematic sample — The members of the sample are chosen according to some organised pattern. Tangent ratio — The ratio of the opposite side and the adjacent side in a right-angled triangle. Taxable income — The amount of income upon which the amount of tax due is calculated. Taxable income is calculated by subtracting any allowable tax deductions from the total gross income. Tax deduction — An amount that can be deducted from gross income before income tax is calculated. Tax deductions are allowed for work-related expenses and other items such as charity donations. Tax return — A form completed by every taxpayer at the end of the financial year, which states all income earned, any allowable tax deductions and all taxes already paid. The total amount of tax that should have been paid is then calculated. The taxpayer then either receives a tax refund or must pay a tax debt. Time and a half — A penalty rate where the employee is paid 1 1--- times the normal hourly 2 rate. Traverse survey — A survey done to calculate the area of an irregularly shaped block of land. A diagonal is constructed and offsets divide the shape into triangles and quadrilaterals from which the area can be calculated. Tree diagram — A method of listing the sample space for a multi-stage probability experiment. The diagram branches once for each stage of the experiment at each level showing all possible outcomes to each stage.

520

General Mathematics

Trial — The number of times a probability experiment has been conducted. Trigonometry — A branch of mathematics in which sides and angles of triangles are calculated. Ungrouped data — Data for which each score is individually tabulated. Unitary method — A method used in ratio and percentages. Upper extreme — The highest score in a data set.

Upper quartile — The highest 25% of scores in a data set. Value Added Tax — Similar to the GST, a VAT is levied on the cost of goods and services in many countries. The rate of VAT varies from country to country. Wage — A form of payment that is based on an hourly rate. y-intercept — The value of y when a function crosses the vertical axis.

Answers CHAPTER 1 Earning money

6 D 9

Are you ready?

10 13 15 16

Exercise 1A — Calculating salary payments

Weekly pay

Fortnightly pay

Monthly pay

$30 000

$576.92

$1153.85

$2500.00

$39 500

$759.62

$1519.23

$3291.67

$42 250

$812.50

$1625.00

$3520.83

$54 350

$1045.19

$2090.38

$4529.17

$86 475

$1662.98

$3325.96

$7206.25

$23 920 C a $634.62 a $688 45 hours

3 $4500 c $3228.33

8 $32 500 9 $69 600 11 $14.28 b $17.63 b Karina ($16.38 per hour)

Exercise 1B — Calculating wages 1 $518 2

Hours worked

Hourly rate

A. White

$416.16

36

$11.56

B. Black

$538.80

40

$13.47

C. Green

$369.63

37

$9.99

D. Brown

$813.96

42

$19.38

E. Scarlet

$231.30

15

$15.42

F. Grey

$776.72

38

$20.44

$12.54 $422.62 a $349.65 a $471.20

11 C 12 D 14 $548.80 b 43 hours b $11.78


1 $1700 4 a $745 5 $1653.84 6 Annual salary

7 10 12 13 14

8 39

Wage

Name

1 a 24 months b 156 weeks c 21 fortnights d 5 years 2 a $611 b $2619.39 c $27 493.44 d $876.92 3 a 0.34 b 0.79 c 0.04 d 0.672 e 0.0825 f 0.175 4 a $35 b $356 c $1620 d $571.25 5 a $12.84 b $32.25 6 12.25 hours or 12 hours and 15 minutes 7 a 14% b 65% c 20% 8 a $616 b $1176 c $2944

2 $960 b $1490

7 $9.68

1 4 7 10

$344.16 $95.04 $2692.31 $24.18

2 $518.70 5 $12.87 8 $4706.67

1 2 3 4 7 10 11 12 13 14 15 16 17 18

$2000 a $260 a $400 $1425 C a $4125 $950 a $390 B a $400 $625 a $300 D a $462.50

b $327 b $1200 5 $4140 8 $8125 b $5375

c $568.79 c $1360 6 B 9 $800 c $16 875

b $490

c $649

b $500

c $8000

c $406.80

b $372.50

c Bonito ($705)

Exercise 1D — Payment by piece $75.95 a $103.50 b $94.50 c $85.50 d $81 $175 4 $135 5 $325.50 6 $327 a $92 b $138 c $87.40 d $114.08 a $57.75 b $125.13 c $103.95 $22.80 a $11 b $27.80 c $19.32 a $134.40 b $16.80 a $174 b $21.75/h c $29.00/h

Name

Wage

A. Smith

$14.52

40.5

$580.80

B. Brown

$16.45

38.5

$625.10

N. Tran

$15.95

37.5

$598.13

A. Milosevic

$20.10

41.5

$824.10

1 2 3 7 8 9 10 11 12

L. McTavish

$18.04

36.5

$649.44


c $501.5 c $600.75

d $8125

b $345

Hours worked

1 4 7 10

$455.84 $3307.69 $570 $1840

2 $346.15 5 $32 406.40 8 $208.55

3 $740.38 6 $487.20 9 $60.30

1A ➔

b $566.40 b $499.50 b $225.42

3 $218.40 6 $903.85 9 $846.15

Exercise 1C — Commission and royalties

Hourly rate

3 a $424.80 4 a $432 5 a $17.34

answers

521

Answers

1D

answers

522

Answers

Exercise 1E — Working overtime 1 $16.95 4 $142.56 6

2 $22.84 5 $157.20

3 $28.70

Ordinary Overtime Hours rate rate Worked

Name

Pay

A. Nguyen

$8.90

Time and a half

4.5

$53.40

M. McDonnell

$9.35

Double time

6.5

$112.20

F. Milosevic

$11.56

Time and a half

7.5

$121.38

J. Carides

$13.86

Time and a half

6.5

$135.14

Y. Robinson

$22.60

Double time

5.5

$248.60

7 C 10 $543.52 13

8 A 11 $721.60

9 B 12 $596.25

Ord. rate

Normal hours

W. Clark

$8.60

38

4

A. Hurst

$9.85

37

.5

6

$482.65

S. Gannon

$14.50

38

5

2

$717.75

37.5

4

1.5

$754.70

36

6

8

$1518.90

$24.90

14 C 17 a $705.28 18 $455.78

15 A b $788.80

Double time hours

3 4 5 6 7 8 9 10 11

1 4 7 8

10

$378.40

11 12

16 $595.20 c $928

Deductions

Net pay

$345.00

$89.45

$255.55

$563.68

$165.40

$398.28

$765.90

$231.85

$534.05

$1175.60

$429.56

$746.04

$2500.00

$765.40

$1734.60

$360.45 a $2519.23 $7.58 $56.02 a $511.10 a $2153.85 $56.12 $167.26 a $422.56

1 4 7 10

Total pay

Exercise 1F — Additions to and deductions from gross pay 1 $360.90 2 Gross pay


9

Name

D. Colley

$2884.62 b $367.79 Union fees = $18.69; health insurance = $50.91 $1438.01 $516.80 b $2428.96 $880.77 b $4139.61 $466.20 b $81.59 c $464.55 $5269.23 b $922.12 c $96.15 $4315.28

$477.75 $235.35 $101.40 $2408.28

2 $1663.08 5 $75.90 8 $436.10

3 $410.40 6 $19.02 9 $353.52

Exercise 1G — Budgeting

Time and a half hours

G. Dymock $16.23

12 a c d 13 a 14 a 15 a 16 a d

b $1434.08 b $5.11 b $44.84

c $407.26 c $1448.41

b $44.37

c $310.07

$8.42 2 $52.31 3 $8.00 $346.15 5 $59.78 6 $484.19 a $240 b Check with your teacher. a $120.77 b $415.38 c $378.85 d Check with your teacher. a $181.60 b 5 Oct. to 4 Jan. c $29.35 d $14 a $198.17 b $60.70 c 55 d 1753 e 10 000 f 58 552 a $143.75 b 72 c 1.24c d 161 a $1007.70 b $90 000 c $91.90 d 4, $251.90

Chapter review 1 5 6 7 10 13 14 15 16 17 20 21 22

$761.54 2 $2592.31 3 $15 912 4 $22.29 a $522 b $594.70 c $588.24 a $442.80 b $220.80 $11.40 8 43 hours 9 $395.20 $798.50 11 $534 12 $340.75 $98 a $21.30 b $28.40 a $98.10 b $109 a $368.60 b $426.80 c $426.80 d $485 $590.24 18 $311.64 19 $4.05 a $1855.77 b $176.30 a $631.55 b $13.26 c $72.63 d $405.36 $135 23 $57.12

Practice examination questions 1 B 2 C 3 B 5 a $554.40 b $685.30 d $2605.68 6 a $1830.77 b $164.77

4 B c $572.57 c $1119.22

CHAPTER 2 Units of measurement Are you ready? 1 a d 2 a 3 a 4 a 5 a 6 a

500 cm b 6200 m c 2 kg e 6250 kg f 2 days b 208 weeks c 0.5% b 3.3% c 2.1865 b 0.002 c 48.81 $840 b $2375 c 3.225 12 : 5 b 10 : 3 c 4 : 13

8.5 m 0.75 L 480 min 0.9% d 118.3 d 1.093 75 L d 3:2

Exercise 2A — Units of measurement 1 a e i 2 a e 3 a d g 4 a d g j

7 cm b 6m c 5 km d 90 mm 1200 cm f 9000 m g 8.6 cm h 9200 m 2.4 km j 64 mm k 1125 cm l 22 mm 8 kg b 3t c 7000 kg d 5000 g 9.5 t f 2.35 kg g 5500 kg h 4840 g 2L b 11 kL c 4000 L 15 000 mL e 4.8 kL f 8.65 L 7900 L h 12 800 mL 4 min b 6h c 3 days 120 h e 420 min f 180 min 730 or 731 days h 36 months i 208 weeks 1 --12- days k 168 h l 7 --12- h

5200 mm long, 2400 mm wide and 2500 mm high 150 kg 7 1800 L 26 hours and 40 minutes A 10 C a Seconds b Litres c Centimetres d Kilometres e Tonnes 12 a Kilolitres b 410 litres per day 5 6 8 9 11

Exercise 2B — Relative error 1 a 4.5 cm to 5.5 cm b 11.5 cm to 12.5 cm c 33.5 cm to 34.5 cm d 58.5 cm to 59.5 cm e 89.5 cm to 90.5 cm f 199.5 cm to 200.5 cm 2 a 35 m to 45 m b 85 m to 95 m c 245 m to 255 m d 295 m to 305 m e 995 m to 1005 m f 1995 m to 2005 m 3 a 5.25 cm to 5.35 cm b 9.75 m to 9.85 m c 7.15 km to 7.25 km d 4.95 mm to 5.05 mm e 9.85 km to 9.95 km f 0.05 m to 0.15 m 4 a 44.5 km to 45.5 km b 0.5 km c 1.11% 5 a 0.75% b 7.14% c 2.78% d 0.13% e 5% 6 a i 4 cm ii 3 cm iii 5 cm b i 12.5% ii 16.67% iii 10% 7 a i 40 mm ii 27 mm iii 52 mm b i 1.25% ii 1.85% iii 0.96% 8 a 1 h 15 min to 1 h 25 min b 5 minutes c 6.25% 9 B 10 C 11 C 12 D 13 a 29.5 m to 30.5 m b 19.5 m to 20.5 m c 98 m d 102 m e 2m 14 a 575.25 m2 b 625.25 m2 c 25.25 m2 15 750.4 mL 16 79.25 km 4.5 m 2 6800 m 9450 mL 5 240 s 560 cm 8 4000 cm 65.65 kg to 65.75 kg

3 6.8 kg 6 72 h 10 1.4%

Exercise 2C — Significant figures

Exercise 2D — Rates 1 a Distance/time b Money/mass c Money/time d Mass/capacity e Goals/games f Temperature/time g Distance/time h Mass/area i Money/capacity j Time/mass k Capacity/distance l Distance/time 2 a 90 km/h b $4/kg c 5 runs/over d 5°/h e $16.25/h f 22.5 m2/kg g 7.5 m/s h $44.60/day i 7.7 km/L j 2.875 goals/game k 12.5 kL/day l 77.6 c/L 3 C 4 $27.40/h 5 130 beats/min 6 87 km/h 7 a 196.9 s b 16.3 m/s 8 7.2 km/L 9 12 L/100 km 10 C 11 $471.66 12 900 mL 13 800 g of rice 400 g of pink salmon 1 egg (nearest whole number) 80 g of butter 40 g of plain flour 600 mL of milk 120 g of shredded cheese 100 g of breadcrumbs 14 200 g of brown sugar 60 g of oil 20 g of cocoa powder 80 g of self-raising flour 80 g of plain flour 60 g of choc bits 40 g of white chocolate 15 8 16 1.9 kg 17 a $14.40 b $72 18 54 km/h 19 a 25 m/s b 9 km/h c 8 L/km d 12.5 L/100 km 20 C 21 No, she has enough lawn food for only 40 m2. 22 a 9.6 km/L b 33.75 L

25 000 000 b 200 000 000 8 780 000 d 5000 700 000 f 287 0.033 b 0.004 c 0.001 50 6.257 e 68.3 f 0.0002 100 000 000 km b 150 000 000 km 147 000 000 km d 146 570 000 km

1 2 3 4 6 8

$742.50 a 24.9375 km b 547.4 kg c 3 h 21 min 36 s a 1 h 54 min b 2 h 16 min 48 s $181.50 5 225.625 kg Yes, each equals 66 kg. 7 $555.75 a $250 b 20%

10 Quick Questions 2 1 3 5 7 9

546 cm 0.069% 1.79 × 108 15 kg/h $292.60

2 4 6 8 10

724.5 kg to 725.5 kg 0.028 cm 1.0 × 10−6 15.6 m/s $12.96

1E

➔

1 a c e 2 a d 3 a c

4 D 5 D 6 C 7 a 4.7 × 108 km b 7.4 × 1012 mm c 1.3 × 104 t d 2.4 × 10−3 g e 2.2 × 10−1 L f 3 × 10−3 s 8 a Check with your teacher. b 4 c 230 000 000 km 9 a 2.4 × 103 m b 9.1 × 105 km c 4.6 × 107 mm d 4.9 × 102 t 10 e 3.45 × 10 kg f 8.11 × 109 g g 8.9 × 104 kL h 1.09 × 1013 L i 7.1 × 106 mL 10 2.05 × 10−7 mm 11 9.43 × 1012 km

Exercise 2E — Percentage change

10 Quick Questions 1 1 4 7 9

523

answers

Answers

2E

answers

524

Answers

Exercise 2F — Using ratios 1 a 5:1 b 10:9 c 3:8 d 7:3 e 5:3 f 3:4 g 3:16 h 1:13 i 16:11 j 25:6 k 4:5 l 5:3 m 4:1 n 2:5 o 2:3 p 20:1 q 3:8 r 3:10 s 4:1 t 3:50 u 5:2 v 20:13 w 9:2 x 20:3 2 D 3 B 4 12 5 1050 6 1260 kg 7 a 7.5 L b 8.5 L 8 320 9 84 10 C 11 Tom gets $700 and Rachael gets $300. 12 Natalie works 21 hours and Kathy works 14 hours. 13 10 14 25 000 15 B 16 a 2:5 b 600 17 a $9.30 b 5:3 c Sandra — $400 000, Kevin — $240 000 18 Sand 75 kg, cement 50 kg and gravel 25 kg 19 a 4:5:2 b 12 20 a 4:5 b Vicky pays $80 and Monica pays $100.

Chapter review 1 a 9 cm b 600 cm c 6700 m d 4.8 km e 69 mm f 1125 cm g 9 kg h 9.5 t i 4840 g j 11 kL k 4.55 L l 12 800 mL m 5 min n 4h o 4 days p 96 h q 1461 days r 60 months 2 112 kg 3 a 33.5 cm to 34.5 cm b 8.85 kg to 8.95 kg c 450 km to 550 km d 2.245 L to 2.255 L e 795 km to 805 km 4 a 1.5% b 0.6% c 10% d 0.2% e 0.6% 5 a 6 × 107 km b 4 × 105 mm 8 c 1.47 × 10 m d 6.43 × 105 t e 8.739 × 10−1 t f 5.74 × 10−4 g −3 g 2.874 × 10 mL h 5.874 × 10−3 g 6 a 5.2 × 106 mm b 9.1 × 104 kg 10 c 3.45 × 10 kg 7 a 25 c/L b 40 km/h c $8.40/h d 1.5 h/kg 8 630 km 9 33 h 10 31.5 L 11 $891 12 a 3:4 b 4:3 c 3:5 d 8:5 e 25:4 f 1:6 g 2:9 h 3:10 13 $360 14 $600:$1400 15 a 7:3 b Yasmin $175 000, Carrie $75 000 Practice examination questions 1 A 2 A 3 B 4 B 5 C 6 a 20.5 m b 50 mm c 0.24% d 1.64 × 108 7 a 55 800 000 km b 0.09% c 3200 km/h

CHAPTER 3 Applications of area and volume

4 a 2000 mm3 5 a 729 cm3

b 3 m3 b 1824 cm3

c 5 cm3

Exercise 3A — Review of area 1 a d 2 a d 3 a d 4 a d 5 a d 6 a d 7 a 8 a d 9 B 12 a 13 a

64 cm2 8.41 m2 27 m2 109.34 km2 54 m2 20.37 m2 72 m2 216.32 km2 67.5 cm2 179.91 km2 20 m2 60 m2 140 m2 144 m2 80 m2

b e b e b e b e b e b e b b

841 mm2 13.69 cm2 1026 mm2 130.9 m2 29.14 cm2 26.46 km2 612 cm2 3306 mm2 728.5 mm2 106.59 m2 4074 mm2 5.84 m2 36 m2 68.5 cm2

c f c f c f c f c f c f c c

10 A

11 B b $960.77 b 180 m2

34.56 m2 1.5 m 12 m 15 m

d 90 m2

c 18 m long and 15 m wide

Exercise 3B — Calculating irregular areas from a field diagram 1 a B

b 1125 m2

C 10 25 25 20 15

D

A

2 a Check with your teacher. b 3727.5 m2 c 234 m 3 a 1925 m2 b 2667.5 m2

c 3650 m2


22.09 m2 2 140.65 cm2 12 m 5 1.645 m2 2 20.025 cm 8 336 mm2 1 cm2 = 100 mm2

Exercise 3C — Solid shapes 1 Triangular prism 2 Triangular pyramid 3 a Rectangular prism b Rectangular pyramid c Hexagonal prism 4 Check with your teacher. 5

Are you ready? 1 a c 2 a 3 a

42.25 m2 9.12 m2 200 mm2 486 cm2

b 8.12 m2 d 34.2 m2 b 30 m2 c 5 ha b 952 cm2

12.96 km2 156.25 cm2 2914 cm2 40.32 m2 3116 mm2 6.845 m2 72.54 m2 6.4 m2 33.88 m2 0.585 m2 22.4 m2 26.82 cm2 104 m2 10 m2

6 a iii

b i

c ii

3 7 cm 6 16.83 m2 9 100 mm2

5 a 17.5 m2 d 17.5 m2 6 360 cm2 9 36 cm2

7 a

c 42.5 m2

b 70 e 30 L 7 28 cm2 10 95.6 m2

8 B

10 Quick Questions 2 0.44 m2 4224 m2 Square pyramid 486 cm2 568 cm2

29.52 m2 Triangular prism Triangular pyramid 236 cm2 160 cm2

b

1 3 5 7 9

c

Exercise 3E — Volume of a prism

8 a Square pyramid b Triangular prism

c Cone

9

a

b

c

d

1 a 120 cm3 d 256.68 m3 2 230.95 m3 3 a 125 cm3 d 24 389 mm3 f 551.368 m3 4 a 72 cm3 d 56.448 m3 5 a 1357.2 cm3 d 3817.0 cm3 f 13.3 cm3 6 a i 12 cm2 b i 24 cm2 c i 4.59 m2 d i 12.48 m2 7 a 187.5 m3 d 6.324 cm3 8 a i 200 cm2 b i 99 m2 c i 204 cm2 d i 153 m2 9 B 12 a 504 000 cm3 13 101.25 m3 14 a 150.796 m3 15 a 175 m2 16 a 2.8 m3 d 27.5 cm

2 4 6 8 10

b 228 cm3 e 286.74 mm3

answers

525

Answers

c 228 cm3 f 598.29 cm3

b 13.824 m3 c 2197 m3 3 e 179.406 144 m b e b e

4914 mm3 4050 mm3 339.3 m3 26 880.3 cm3

c 104.895 m3 f 4228.125 m3 c 29 772.9 cm3

ii ii ii ii

60 cm3 288 cm3 6.885 m3 156 m3 c 2280 m3

ii ii ii ii

2000 cm3 792 m3 1224 cm3 1836 m3 11 D

b 875 m3

10 D b 504 L

b 150 800 L b 17.5 m3 b 2800 L

c $796.25 c 24 429 L

Exercise 3F — Volume of other solids

Exercise 3D — Surface area b e b e

486 cm2 c 6144 cm2 23 064 mm2 f 47.04 m2 107.5 m2 c 3276 cm2 1444 cm2 f 1274 mm2 4 28 125 cm2

b 9400 cm3

c 186 cm3

b 560 cm3

c 120 m3

b 92 cm3

c 700 cm3

b 1810 cm3

c 2212 mm3

b 2144.7 cm3

c 8181.2 m3

9 C

10 D

b 2144.7 cm3

c 1436.76 cm3

b 84.8 cm3

c 480.7 cm3

b 65.5 cm3

c 85.15 cm3

2F

➔

1 a 150 cm2 d 43.74 cm2 2 a 122 m2 d 95.98 cm2 3 11.92 m2

1 a 6666.7 cm3 d 11.25 m3 2 a 96 cm3 d 100 cm3 3 a 100 cm3 d 208 mm3 4 a 262 cm3 d 77 585 cm3 5 254.5 cm3 6 a 904.8 cm3 d 137.3 m3 7 6.37 cm3 8 A 11 335.1 cm3 12 a 7 cm d 707.9 cm3 13 a 565.5 cm3 14 14 cm3 15 a 52.4 cm3

3F

answers

526

Answers

Chapter review 1 a d 2 a d 3 a

27.04 cm2 559 cm2 180 cm2 700 m2

b 112.77 cm2 e 787.5 cm2 b 141.05 cm2

b 2897.5 m2

C 25 25

c 2368 mm2 f 2.4 m2 c 78.125 cm2

D

40 B

36 30 A

4 3417.5 m2 5 a Rectangular prism c Square pyramid

b Triangular prism

6 a

b

c

7 Triangular prism 8 a 105.84 cm2 d 18.18 m2 9 a 54 m2 10 a 75 m2 d 1316.8 m2 11 a 37.5 m2 12 a 274.625 cm3 d 984 cm3 13 585 cm3 14 a 48 cm3 15 a 10 800 m3 d 2339 mm3

b 25.98 m2

c 19.44 m2

b 352 cm2 b 1650 m3

c 340 m2 c 1 650 000 L

b 2344 b 24 389 mm3 e 368 cm3

c 202.768 m3 f 57 906 mm3

b 798 mm3 b 10.95 m3 e 50 965 mm3

c 5.29 m3 c 19 658 mm3 f 179 594 cm3

Practice examination questions 1 D 2 D 4 a 100 cm2 b 400 cm3 d

3 C c 360 cm2

CHAPTER 4 Basic algebraic skills Are you ready? 1 a 46, 54, 62 c 48, 96, 192 2 a c 3 a d 4 a e 5 a 6 a 7 a

y=x+3 y = 4x + 3 7g x + 11 18 b 25 f 12x b 3x b x = 39 b

b –8, –13, –18 d 1, 1--2- , 1--4b y = 3x d y = 8 – 2x b 6y c 3gy e 15g + 17 f 4h + 4t –15 c –3 d 9 80 g 60 h 2 12ab c –5kp d 6m2n 15 c 4 d –7xy x = 17 c x=9 d x = 12

Exercise 4A — General number patterns 1 a e i 2 a d g j 3 a d g j

3 b 4 c 5 11 f 9 g 13 2.5 j 1--415, 18, 21 b 18, 22, 26 37, 44, 51 e 44, 55, 66 58, 71, 84 h −1, 3, 7 1 1--4- , 1 1--2- , 1 3--41 b 2 6 e 5 9 h 7

4 a d g i 5 a d g j 6 a c e g i j 7 a d g j 8 a

6, 5, 4 b 22, 20, 18 c 24, 20, 16 16, 10, 4 e −1, −6, −11 f −6, −13, −20 29, 20, 11 h −25, −32, −39 2.2, 1.5, 0.8 j −1, −1 3--4- , −2 1--22 b 3 c 4 5 e 7 f 3 −2 h 2 i 13 1.5 32, 64, 128 b 81, 243, 729 1280, 5120, 20 480 d 1250, 6250, 31 250 4802, 33 614, 235 298 f −162, −486, −1458 −48, 96, −192 h 1.6, 3.2, 6.4 85 683, 1 113 879, 14 480 427 20.25, 30.375, 45.5625 2 b 10 c 4 5 e 3 f 20 8 h 2 i −10 10 1 1 --b ----c 1--42 10

b 78.5 mL d 61

c 24, 29, 34 f 32, 41, 50 i 12, 14.5, 17 c 4 f 7 i 0.7

3 --4

d

1 --5

e

1 --3

f

g

1 --8

h

1 --2

1 i − ----10

j

1 -----10

9 a 100, 50, 25 c 6.25, 1.5625, 0.390 625 e 9, 3, 1 1 1 - , --------g 1--8- , ----64 512 5 a 78.5 cm3 c 65.4 cm3

d 7 h 4

1 -----20

b 1, 0.1, 0.01 d 1.2, 0.24, 0.048 f 2.5, 0.125, 0.006 25 1 1 1 - , ------ , -----h ----16 32 64

i −0.01, 0.001, −0.0001 j −0.5, −0.05, −0.005 10 a 1st term = 2, 5th term = 14, 10th term = 29 b 1st term = 45, 5th term = 29, 10th term = 9

c d e f g h i j 11 a b c d e f g h i j 12 a d g j 13 a c e g i

1st term = 4, 5th term = 324, 10th term = 78 732 1st term = 192, 5th term = 12, 10th term = 0.375 1st term = −25, 5th term = 7, 10th term = 47 1st term = −6, 5th term = −18, 10th term = −33 1st term = −2, 5th term = −512, 10th term = −524 288 1st term = −48, 5th term = −3, 10th term = 0.093 75 1st term = 1, 5th term = 15, 10th term = 55 1st term = 1, 5th term = 25, 10th term = 100 Add 5 to the previous term. Subtract 3 from the previous term. Multiply the previous term by 2. Divide the previous term by 2. Add 7 to the previous term. Subtract 13 from the previous term. Multiply the previous term by 2.5. Divide the previous term by 7. Add 3 to the previous term. Multiply the previous term by −2. 19 b 6 c 32 80 e −11 f −4 −12 h 9 i 27 36 2, 8, 14 b 2, 10, 50 45, 37, 29 d 2000, 400, 80 −12, −7, −2 f 15, 6, −3 20, −4, 0.8 h 6, −30, 150 80, 40, 20 j 6, −1 --12- , --38-

14 a 10 b 15 d 19 e 11 g 9 h 9 j 11 15 Check with your teacher.

c 8 f 9 i 12

d

e

f

g

h

i

j

4 a

b

c

Exercise 4B — Number pattern notation 1 a c e g i k 2 a c e g i k 3 a

b

c

1, 2, 3, 4, . . . 9, 18, 27, 36, . . . 1, 3, 5, 7, . . . 36, 59, 82, 105, . . . 9, 8, 7, 6, . . . 93, 86, 79, 72, . . . 2, 4, 8, 16, . . . −5, 25, −125, 625, . . . 1, 13, 61, 253, . . . 6, 12, 24, 48, . . . −1, 1, −1, 1, . . . −4, 16, −64, 256, . . .

b d f h j l b d f h j l

4, 8, 12, 16, . . . −6, −12, −18, −24, . . . 7, 11, 15, 19, . . . 1, 1 1--2- , 2, 2 1--2- , . . . 17, 14, 11, 8, . . . −3, −8, −13, −18, . . . 3, 9, 27, 81, . . . 10, 100, 1000, 10 000 7, 9, 13, 21, . . . 6, 18, 54, 162, . . . −3, 9, −27, 81, . . . 1, 7, −5, 19, . . .

d

e

f

g

n

1

2

3

4

5

Tn

9

16

23

30

37

n

1

2

3

4

5

Tn

3

9

27

81

243

n

1

2

3

4

5

Tn

4

6

10

18

34

n

−1

2

−3

4

− 5

Tn

−2

4

−8

16

−32

n

−1

−62

−623 −624

−3125

Tn

−5

−25

−125 −625

−3125

n

1

2

3

4

5

Tn

1

4

9

16

25

n

−1

−2

3

4

5

Tn

−2

−2

0

4

10

n

1

4

9

10

20

Tn

9

36

81

90

180

n

1

5

9

10

25

Tn

5

17

29

32

77

n

1

3

6

12

50

Tn

5

17

35

71

299

n

1

10

20

50

100

Tn

13

85

165

405

805

n

1

2

5

9

12

Tn

2

4

32

512

4096

n

1

3

6

8

59 010

Tn

2

26

728

6560

59 048

n

1

2

3

4

5

n

−1

4

−2187 59 010

−14 348 915

Tn

7

14

21

28

35

Tn

−3

81

−2187 59 049

−14 348 907

n

1

2

3

4

5

n

−1

2

15 626

−1 953 129

9 765 610

Tn

1

5

9

13

17

Tn

−5

25

15 625

−1 953 125

9 765 625

n

1

2

3

4

5

n

1

3

5

Tn

9

14

19

24

29

Tn

1

27

125

h

i

10

answers

527

Answers

20

1000 8000

➔

4A 4B

answers

528 j

5 6 7 8 9

a a a a a d 10 B 13 a d

Answers

n

1

2

−96

−10

−920

Tn

1

0

−24

−80

−360

11 b 26 b 8 b −4 b 50 b 0.097 656 25

38 −18 128 −60 25

c c c c c

122 −154 1024 −100 3.125

11 C b Tn = 3n − 1 e Tn = 2n

Tn = 4n Tn = 36 − 3n

d d d d

158 −370 4096 224

1 9, −11, 13 3 --12-

2 Add 8 to the previous term. 4 2, −6, −14, −22, −30

5 15 7 731 8

6 2, 6, 12, 20, 30

n

−1

3

7

10

15

Tn

−1

7

23

35

55

9 Yes, it is the 19th term. 10 Tn = 40 − 2n

Exercise 4C — Adding and subtracting like terms 6w 8n

c 12 mL

39e 4k −4m 22m −z 13k 13t 16b − 26 4r − 17 13d − 8 2w

95°F 29.2 973.14

c 22.46°F c 6.93 f 4.31

−8 236.6 80

c 176 f 75 i −85.4

$16.50

iii $34.50

18 229.61 0.1

c 214.94 c 2.5

2 Tn = 6n − 3 5 −m 8 7693.12

27, 33, 39 4p + 2q 35.34 1040.6

1 4 7 10

3 5r 6 −3v − 11w 9 3156.6

Exercise 4E — Multiplication and division of algebraic expressions 1 a 24 b 76 7 d q e p5 2 a 3×3×3×3×3×3 b 4×4×4×4×4 c 8×8×8×8×8×8×8 d m×m×m e y×y f j×j×j×j×j×j×j×j×j 3 a q7 b x11 d 47 e a6 6 g 6b h 10d 8 j 27j 6 k 8k4 7 9 m 20x y n 20m5n9 p 48j 4k10 q 50p5q3 4 4 a a b b4 d e4 e 6g2 3 2 g 8j k h 4m3n j 9r3

Exercise 4D — Substitution 1 a 68°F b 2 a 30 b d 136 e 3 1.335 4 a 600 b d 32.83 e g 100 h j 25.5 5 a C = 4.5 + 0.6d b i $7.50 ii 6 C 7 a 134.04 b 8 a 14 b

c 1.88 c 6.4 mL

10 Quick Questions 2 12 C c Tn = 50 − n f Tn = 3n − 1


1 a 2t b 3r c d 4x e 7q f 2 a m+m b n+n+n+n+n+n c s+s+s+s+s+s+s+s+s d w+w+w+w+w+w+w+w e y+y+y f r+r+r+r 3 a 12y b 19x c d 47j e 25k f g 8r h w i j 2w k −7s l mj n 6p o 4 a 10x + 7 b 4h − 4 c d 3p + 7q e 5 + 4w f g 4j + 8k h 3x − 15 i j 3a − 8 k 17b − 6 l m 10y − 6 n 3c − b o p 19e − 4p q 20t − 23s r

a 2--3b 1.81 a 3 mL b 3.6 mL 9 mL 10 mL a 19.2 mL b 12 mL a 24.6 — ideal weight b 31.6 — overweight c 14.9 — severely underweight

9 10 11 12 13 14

5 a d g 6 a

12

a 8k12 25p6q8 m4

k 12s2 b e h b

d 12q4

e

g 1024w26

h

j 6m6

k

20

c 92 f w3

c f i l o r c f i l

g c 49m12 f 7776r15s5 i n19 c 3 -------f 5 4r 8v i 1 ----------l 21y 4 2f 2 − 8f c r7 + r2s f 21y9 + 7y5z2 i 5gh − 20h5 l 4a2b2 + 28ab 25m5n5 − 15m2n7

7 a m2 + m b d w4 − 4w e g 20q4 − 36q h j 12x6 + 6x7 k m 6x2y + 6xy2 n o 8p3q − 8pq2 p q 18r7s8 − 18r4s6 r 14p5q5r − 49pq4r6

y15 s3 35g10 24m2 63a5b9 54r4s3t10 c2 6h4 7p8q3 t3 ---3 h4 512n6 25x2y10 p6 4s5 128x20 2 --z 5m2 + 6mn p10 − 5p5q 9a5 − 18a4b4 18t2s3 + 54t 6

Exercise 4F — Solving linear equations 1 a d g j m p

z = 43 y = 266 x = 18 k = 22 u = −180 c = 3 2--5-

b e h k n q

w = 88 r = 48 p = 19 573 b = −16 f = −40 v = 30.4

c f i l o r

q = 18 t = 331 e = −34 a = −12 d = 5.2 3 t = 6 ----10

2 B 3 a x = 4 3--5d x = −6 2--34 a d g j

a=6 s=4 e = −2 r = 11.6

mt = 6 p s = −1.55

b x = 6 3--7-

c x = 6 1--6-

6 e x = 3 ----13

f x=

b e h k

q = 17 w = 17 d = −2 f = −2 --67-

c f i l

n g = −3 -----q y = 13 32

2 --9

z=7 x = 24 c = −17 v = 1 --23-

o b = 12 r h = 9 3--7-

5 a s = 15 d f = 52 g k = 105 j j = 20 6 a y = 10 d s = −20 7 a Correct d Correct g Correct 8 a a=2 d d=3 g g=5 j j = 34 9 D 10 a b = 5 d p = 10 g t=5 j g=5 mz = 9 1--3-

b e h k b e b e h b e h k

v = 119 p = 88 v = 12 p = −273 p=6 w = 18 Incorrect Correct Correct b=9 e=5 h=2 k=3

c f i l c f c f

r = 48 s = 48 g=0 m = 36 q = 12 m = −2.5 Correct Incorrect

c f i l

c=4 f=3 i = −9 l=9

b e h k n

n=2 k = 26 n=4 g = −2 1 y = 2 ----18

c f i l o

h = 13 m = 31 r=5 v = −4 u = 3.95

11 a s = 36

b t=8

c u=

3 --8

dv=4

e w = 2 --29-

f x=4

g x = 10 2--3-

h m = −6

i p=2

30x + 20 25x + 4 12 a --------------------- = -----------------5 4

b $12

e f g h i j 2 a b c d e f g h i j 3 a c 4 a c e g h 5 a

b

c

Exercise 4G — Equations arising from substitution b=7 a l = 19 b b = 5.75 a l = 14 b l = 54 a 10 b 26.7 d −23.3 e −9.4 a a=2 b a = −23 d n = 86 a R = 10 b d=6 d h = 6.7 e b = 8.5 g F = 59 h u=4 j a=8 a 10 km b 18 km a C = 1.5 + 2.2m b i $3.70 ii $12.50 c i 3 min ii 7 min

1 2 3 4 5 6

7 8

Chapter review 20, 24, 28 96, 192, 384 −16, −23, −30 3.125, 0.781 25, 0.193 125

c c f c

l = 2.8 37.8 −20.7 n = 5050

c w = 54 f D = 1080 i n = 16 c 8.5 km iii $19.10 iii 15 min

n

1

3

4

7

10

Tn

9

27

36

63

90

n

1

4

7

9

12

Tn

11

17

23

27

33

n

1

2

5

8

10

Tn

2

4

32

256

1024

n

−1

−23

6

−19 689 59 010

Tn

−3

−27

729

−19 683 59 049

6 a 4y b 17w d 6t e q 7 a 5m + 4n b 3a − 5 d 16m − 6 e 8x + 14 8 40 9 127 10 36.75 12 a w8 b a7 d 36q3 e 5p4 g 48x6 h 12r8 j 20a4b5 k 28g4h3 mb4 n f3 9r p 9y4 q ----s s 4a2

t 4m2

v a12 y k6

w 125m12 z 16q5

c f c f c f i l o

14r 14x 4k − 2l 10m − 16 11 120 24x8 30y2 30x12y10 48m5n6 9r3

r 4s2t4 7 u --------3 p3 x 27p9q15

4C ➔

1 a b c d

d

25, 36, 49 768, 3072, 12 288 0.1, 0.01, 0.001 15, 21, 28 −18, −9, 0 5, 8, 13 Begin with 3 and to get the next term add 4 to the previous. Begin with 6 and to get the next term multiply the previous by 2. Begin with 12 and to get the next term subtract 7 from the previous. Begin with 800 and to get the next term divide the previous by 4. The square numbers. Begin with 3 and to get the next term multiply the previous by 4. Begin with 1000 and to get the next term divide the previous by 10. The triangular numbers. Begin with −54 and to get the next term add 9 to the previous. Begin with 1 and to get the next term add the two previous terms. 45, 37, 29, 21, 13 b 6, 18, 54, 162, 486 −34, −22, −10, 2, 14 d 243, 162, 108, 72, 48, 32 7, 14, 21, 28, 35 b 5, 25, 125, 625, 3125 5, 13, 21, 29, 37 d −2, −4, −8, −16, −32 14, 8, 2, −4, −10 f 3, 9, 27, 81, 243 9, 99, 999, 9999, 99 999 −19, 23, −229, 1283, −7789

answers

529

Answers

4G

Answers

13 a m2 + 3m c 4x8 − 2x3 e 12p3q − 8pq5 14 a a = 67 d d = 966 g g = −33 j j = 16 mm = −9 p p = 65 15 a s = 5 dv=5 g y = −5 j b=8 16 a l = 8 d r = --12-

b e h k n q b e h k b

b 10p2 − 30pq d 6w5 − 6w3 f 21a14 − 63a6b2 b = 160 c c = 89 e = 12 f f = −76 h = 2 7--9i i = −11 k=9 l l = −9 n = 6.5 o t = 40 q = 12 r r = 35 t = 17.5 c u = 42 w=3 f x=9 z = −21.5 i a = 20 c=8 l d=8 w = 31 c F = 77

Practice examination questions 1 B 2 B 3 B 4 D 5 a 441.6 b 21.8 8x 4 x 6 a −x + y b -------c -------2 7 y 2y d x = −49

CHAPTER 5 Statistics and society Are you ready? 1

Score

Tally

Frequency

1

III

3

2

III|

5

3

II|| ||

7

4

IIII

4

5

II

2

6

III

3

2 a 7 d 3 3 a Type of meat

b 26 c Television e Reading and other Amount sold (kg)

Fraction

Angle size (°)

Lamb

10

1 --9

Beef

45

1 --2

180

Pork

5

1 -----18

20

Chicken

15

1 --6

60

Turkey

10

1 --9

40

Rabbit

5

1 -----18

20

90

1

360

Total b

40

Type of meat sold Rabbit (5) Lamb (10) Turkey (10) Chicken (15) Pork (5)

4 a 3.6%

Beef (45)

b 0.16%

c 49%

d 0.3%

Exercise 5A — Collecting data 1 a External b External d External e Internal g External h External 2 a Observation b Questioning d Observation e Questioning g Questioning h Observation 3 Check with your teacher. 4 Check with your teacher.

c Internal f Internal c Questioning f Observation

Exercise 5B — Organising data 1 Make Holden Ford Nissan Mazda Toyota Mitsubishi

Tally |||| ||| |||| ||| || ||| |||| || ||


2 Mark 4 5 6 7 8 9 10

Tally || |||| |||| | |||| |||| |||| ||| |

Frequency 2 4 6 9 5 3 1

3 Mark 40–49 50–59 60–69 70–79 80–89 90–99

Tally | || |||| |||| |||| ||| ||| ||


4 Check with your teacher.

Exercise 5C — Displaying data 1 Frequency

answers

530

9 8 7 6 5 4 3 2 1 0

i a ta n n d de For issa azd yo bish o su M N T H it Make of car M ol

2

Make of car Holden Ford Nissan Mazda Toyota Mitsubishi

Number of students

3

9 8 7 6 5 4 3 2 1 0

4

5 6 7 8 9 Spelling test results

10

4

4 Score 1 2 3 4 5 6 5 Number of CDs 0–4 5–9 10–14 15–19 20–24 25–29 30–34 35–39

40

–4 50 9 –5 60 9 –6 70 9 –7 80 9 –8 90 9 –9 9

Number of students

10 9 8 7 6 5 4 3 2 1

Maths exam mark

6

5 9 8 Frequency

Frequency 3 6 6 6 3 3 Number of students 3 9 9 3 4 1 0 1

Tally ||| |||| |||| |||| |||| ||| |||| | |

7 6 5 4 3 2 1 0 1 2 3 4 5 6 Number of people in a household

Frequency

10

Tally ||| |||| | |||| | |||| | ||| |||

531

answers

Answers

7 6 5 4

Number of people in a household

3 2

1

1

2

9

9 20

18

0– 21

9

4

0– 19

9 16

0– 17

9

0– 15

0– 13

14

12

10

0– 11 9

3 5

Number of customers Number of students

6 Marks on maths exam 40–49 50–59 60–69 70–79 80–89

10 9 8 7 6 5 4 3 2 1 0

0– 4 5 10 –9 – 15 14 – 20 19 – 25 24 – 30 29 – 35 34 –3 9

90–99

7

Number of students

6

Exercise 5D — Quality control 1 Accepted 2 Accepted 3 Unsatisfactory 4 Unsatisfactory 5 Unsatisfactory 6 Satisfactory

Number of CDs purchased Number of CDs purchased 0–4 5–9

Chapter review

10–14

c External

15–19 20–24

c Observation

25–29 30–34 35–39

5A ➔

1 a External b Internal d Internal 2 a Observation b Questioning d Questioning 3 Check with your teacher.

5D

answers

532

Answers

CHAPTER 6 Data collection and sampling Are you ready? 1 a 6% 2 a 4600 3 a 135

b 21.6% b 840 b 69

c 8.54% c 1200 c 109.4775

Exercise 6A — Target populations and sampling 1 Census — every member of the population participates. 2 Sample 3 a Sample b Sample c Census d Census e Sample 4 a Sample b Census c Census d Sample 5 Sample 6 163, 176, 381, 495, 9 7 827, 211, 417, 554, 207, 26, 810, 781, 192, 371 8 Check with your teacher. 9 a Systematic b Stratified c Systematic d Random e Stratified 10 A 11 C 12 34 males and 16 females 13 Year 7 — 8, Year 8 — 8, Year 9 — 8, Year 10 — 6, Year 11 — 5, Year 12 — 4

Exercise 6B — Population characteristics 1 Year 7 — 9, Year 8 — 11, Year 9 — 9, Year 10 — 8, Year 11 — 7, Year 12 — 6 2 36 men and 24 women 3 25 females and 15 males 4 2 senior management, 12 middle management and 36 clerks. 5 3 teachers and 47 students 6 Age Male Female 20–29 30–39 40–49 50–59

10 7 12 1

7 8 3 2

7 Year 7 — 16, Year 8 — 16, Year 9 — 14, Year 10 — 13, Year 11 — 12, Year 12 — 9 8 Major department store 60 votes, small department store 30 votes and small stores 2 votes 9 Year Boys Girls 7

9

9

8

10

11

9

10

10

10

9

8

11

7

7

12

6

6

10 Quick Questions 1 Census Sample Census Random sample Systematic sample Stratified sample 18 The members of the committee are not in the same proportion as the members of the parliament. 9 4 Liberal and 2 Labor 10 A stratified sample is chosen to fully represent the characteristics of the whole population, whereas a random sample may not do this, especially if the sample is small. 1 2 3 4 5 6 7 8

Exercise 6C — Bias 1 Check with your teacher. 2 Check with your teacher. 3 a Sample does not represent characteristics of population b No control over responses c Unrepresentative sample d Abnormal conditions e Only extreme groups in sample 4 The decrease in the value of the Australian dollar compared with the American dollar is accentuated by the large scale on the y-axis. The decrease is actually only 2 cents. The scale on the x-axis is not uniform (9 May, 11 May, 12 May). 5 What type of university tests? What do the terms ‘consistently’, ‘majority’, ‘more effective’, ‘most other’ mean? No hard evidence has been provided to support the claim. 6 a There would be many more student drivers in Year 12 than in Year 11 — perhaps also some in Year 10. b Students with part-time jobs are in lower year levels as well. c Residents not at the neighbourhood watch meeting have been ignored. d Other music students who play instruments and don’t belong to the choir have been excluded. e The composition of cars in a shopping centre car park is not representative of the cars on the road. f Females have been excluded. g Users of the local library would not reflect the views of teenagers.

Exercise 6D — Types of data 1 a d 2 a d 3 a d 4 a b c d e f g h

Quantitative b Categorical Categorical e Quantitative Ordinal b Ordinal Nominal e Nominal Continuous b Discrete Continuous e Continuous Quantitative, discrete Categorical, nominal Quantitative, continuous Quantitative, continuous Categorical, nominal Quantitative, continuous Quantitative, discrete Quantitative, continuous

c Quantitative f Quantitative c Nominal c Continuous

5

i Quantitative, discrete j Categorical, nominal k Quantitative, continuous l Quantitative, discrete m Quantitative, discrete n Categorical, nominal o Categorical, ordinal p Quantitative, continuous q Quantitative, discrete r Categorical, ordinal s Categorical, nominal t Categorical, ordinal a Quantitative and discrete b Categorical c Categorical d Quantitative and continuous e Quantitative and continuous f Quantitative and discrete Categorical and nominal Categorical and ordinal Quantitative and discrete C Categorical and ordinal Quantitative and continuous

6 7 8 9 10 11

10 Quick Questions 2 1 The question leads the responder to an expected answer of yes by using emotional words and ideas. 2 Should the tax rates for upper incomes be raised? 3 One side of the debate may be more motivated to call in, certain groups of the population may be misrepresented among viewers of this program, and low income groups may not feel they want to waste money on the phone call. 4 Random sample 5 Categorical and nominal 6 Quantitative and discrete 7 Quantitative and discrete 8 Quantitative and continuous 9 Check with your teacher. 10 Check with your teacher.

533

6 Check with your teacher. 7 a Categorical b Quantitative c Quantitative d Quantitative e Categorical 8 a Discrete b Continuous c Continuous d Discrete e Continuous 9 2000 10 750 11 a Barry — 2667 Viet — 1667 Mustafa — 1571 b 1968

answers

Answers

Practice examination questions 1 D 2 B 3 A 4 C 5 a Categorical b 1984 6 a Quantitative and continuous b Systematic c The data are influenced by factors that don’t make them representative of the whole population. d Take cans of paint from each of the five machines.

CHAPTER 7 Modelling linear relationships Are you ready? 1 a Not linear c Linear 2 a 3

b

3 a –7 4 a

y

b Linear d Not linear c –3

1 --3

b –5

c 2

y=x+1 1

x

−1

Exercise 6E — Estimating populations 4000 2 400 a 10 000 b 25 000 c 3663 No — estimated population 20 000 a 625 b 500 c 625 a 833 b 1000 c 882 d 905

1 3 4 5 6

b

y

Chapter review y = 2x − 4

2

x

−4

6A ➔

1 a Sample b Census c Census d Sample 2 Random sample — where the participants are chosen by luck Stratified sample — where the participants are chosen in proportion to the entire population Systematic sample — where a system is used to select the participants 3 a Systematic b Random c Stratified 4 Check with your teacher. 5 Year 7 — 12, Year 8 — 12, Year 9 — 11, Year 10 — 10, Year 11 — 8, Year 12 — 7

6E

Answers

c

8

y

Height (m)

6

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

x

9

(10, 2)

0 1 2 3 4 5 6 7 8 910 Years

Money earned ($)

3

600 500 400 300 200 100 0

y = 6 − 2x

(60, 600)

0 10 20 30 40 50 60 Hours worked

10 b x=3

c x=

2 --5

Exercise 7A — Graphing linear functions M 500 400 300 200 100 0

(5000, 250)

0 00 00 00 00 00 P 20 40 60 80 10 0

2 a $400.00 3 Y 90 000 80 000 70 000 60 000 50 000 40 000 30 000 20 000 10 000 0

4

C 100 90 80 70 60 50 40 30 20 10 0

160 140 120 100 80 60 40 20 0

7

C 100 90 80 70 60 50 40 30 20 10 0

b $475.00

1 a 2 2 D 3 a

0 0 0 0 0 20 40 60 80 100 Number of rolls of film

Y = 80A

400 350 300 250 200 150 100 50 0

d 0.5

(1000, 360)

0 00 00 00 00 0000 2 4 6 8 1100 No. of newspapers delivered

0 00 00 00 00 00 A 2 4 6 8 10

b 0.3 4 a 350

C = 1.5t (60, 90)

0 10 20 30 40 50 60 t

b $67.50 d = 15t

0 1 2 3 4 5 6 7 8 910 t C = 4 + 1.5d (60, 94)

300 250 200 150 100 50 0 –50 –100

c 60 (200, 300)

50 100 150 200 No. of patrons in a cinema

b 2: $2 increase in profit per person attending c −100: $100 cost is incurred before anybody attends the movie. 5 a

(10, 150)

0 10 20 30 40 50 60 d

c −3

b 16

(500, 40 000)

5 a $25.50 6 d

(1000, 6150)

Exercise 7B — Gradient and intercept

Payment ($)

1

7000 6000 5000 4000 3000 2000 1000 0 –1000 –2000

Profit ($)

5 a x=4

Profit ($)

answers

534

b

x

−0

1

2

3

y

−4

1

6

11

y 6 5 4 3 2 1

(2, 6) y = 5x – 4

0 –5–4–3–2–1–1 1 2 3 4 5 –2 –3 –4 –5

x

c Gradient = 5, y-intercept = −4

Exercise 7C — Drawing graphs using gradient and intercept

6

y 4

1 a Gradient = 2, y-intercept = 2 b Gradient = 3, y-intercept = −8 c Gradient = −4, y-intercept = 2

(1, –1)

y = 4 – 3x

7 a

y

e Gradient = --12- , y-intercept = 1

6 (1, 3)

f Gradient = − 3--2- , y-intercept = 3 y

y = 6 – 3x

b

1

1 –2 (1, –1)

y

x

–1 1–2

–3 y

x –3

y = 2x + 1 (1, 3)

b

y

x

y

2 (1, –3)

–6

c

y (1, 5) x

0 y = 5x

4

y

y = 1–2 x + 2

2

(2, 3) x

–4

5 a

y

y = –34 x – 1 (4, 2)

–1

y (3, 1) 0

c

y

2

2 –3 (2, –1) –4


2 −3 y = 3x + 6 y = 1--2- x − 3

9 Check with your teacher.

x

2 4 6 8

−3 5 y = 4 − 2x Check with your teacher.

10 Line must pass through one origin.

Exercise 7D — Graphing variations D 900 800 700 600 500 400 300 200 100 0 0 1 2 3 4 5 6 7 8 910 t

2 a

y = 3–2 x – 4

x

8 C 9 C 10 Some possible answers are: a y = 2x + 1 b y = –2x + 1 c y = –3x + 5 d y = 6x e y = –2x – 1 f y=6 g y = 2x – 1 Check other answers with your teacher. 11 Some possible answers are: a y = –x + 1 b y = 2x c y = – --12- x + 1 Check other answers with your teacher.

x

y = 1–3x

1

y = − –2 x + 4 8

1 x

4– 3

b

4

y = 3x – 6

x

(1, –5) y = –2x – 3

c

1 – 1–2

x

2

y = 2x – 3

3 a

x

4– 3

d Gradient = --34- , y-intercept = 3

2

535

answers

Answers

n 16 14 12 10 8 6 4 2 0

0 1 2 3 4 5 6 7 8 910 p

c n = 1.5p

7A ➔

b 1.5

7D

Answers

3 Cost ($)

y 90 80 70 60 50 40 30 20 10 0

4

Distance (km)

c y = 8x

350 300 250 200 150 100 50 0

30

70 60 50 40 30 20 10 0 0

5

0 20 40 60 80 100 s

b D = 3s W 500 400 300 200 100 0

6 Pay ($)

L (L) 70 60 50 40 30 20 10 0

900 800 700 600 500 400 300 200 100 0 0 10 20 30 40 50 60 No. of hours worked

4

8 12 16 20 g (m)

c 1.6 m

Income ($000)

0 00 00 00 00 00 A$ 1 2 3 4 5

Exercise 7E — Step and piecewise functions Fare ($)

50 45 40 35 30 25 20 15 10 5 0

0 10 20 30 40 50 60 70 80 90 10 110 0 12 0 13 140 0 15 0

$US 300 250 200 150 100 50 0

Tax payable ($000)

0

b 16.8 m

Exercise 7F — Simultaneous equations 1 4 kg of apples and 5 kg of bananas y 2 a y = 2x +1 10 8 6 4 2

0 –10–8–6–4–2–2 2 4 6 8 10 x –4 –6 y=7–x –8 –10

2.80 2.40 2.00 1.60 1.20 0.80 1

2

5

7

120 100 80 60 40 20 0

1

2 3 4 Time (h)

180 160 140 120 100 80 60 40 20 0

0 00 00 00 00 00 d (km) 1 2 3 4 5

b 30 L 7 a h (m)

8

1

0 50 100150200 No. of business cards

0 10 20 30 40 50 h

6 a

35

4 8 12 16 20 Number of calls

0 1 2 3 4 5 6 7 8 910 x

b 8 4 a D

5

40

25

Cost of printing ($)

3 a

Posting charge ($)

answers

536

2 3 4 Sections

5

6.00 5.50 5.00 4.50 4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50

b (2, 5) 3 a b 2x + 2y = 22

y

10 8 6 4 2

x 0 –10–8–6–4–2–2 2 4 6 8 10 –4 –6 –8 y = x – 5 –10

0.5 1 1.5 2 2.5 3 Mass of parcel (kg)

c Length = 8 cm, width = 3 cm

4 ab

s 800 700 600 500 400 300 200 100

7 a

s = 2t

y y = 2x – 1 1– 2

0

x

0 –1

t + s = 750 t

0 0 0 0 0 0 0 0 10 20 30 40 50 60 70 80

b

y

c Steve = $500, Theo = $250 5 ab M M = E – 21

6

(1, 3)

0

200

537

answers

Answers

x

2

150

y = 6 – 3x

100 50

c

E + M = 135

0

y

E 50 100 150 200

(2, 4)

3

c English = 78, Maths = 57

y = 1–2 x + 3

x

0

–6

Chapter review c 300 250 200 150 100 50 0

8 Cost ($)

1 a

01 2 3 45 67 8 h

45 40 35 30 25 20 15 10 5 0 0 10 20 30 40 50 Amount of petrol (L)

b $230

3 a

9 a

q

60 50 40 30 20 10 0 0

0 00 00 00 00 00 A 2 4 6 8 10

C ($) 90 80 70 60 50 40 30 20 10 0

10

0

4

8 12 16 20 k (km)

b $56 4 a Gradient = 2, y-intercept = 1 b Gradient = − 1--2- , y-intercept = −2

Profit ($)

c Gradient = −1, y-intercept = 3 5 a 500 400 300 200 100 0 –100

4

8 12 16 20 p

c q = 3p

b 3

11

Cost per minute (cents)

E 800 700 600 500 400 300 200 100 0

90 80 70 60 50 40 30 20 10 0

Height (m)

2

6 5 4 3 2 1 0

0 00 00 00 00 00 2 4 6 8 10 Distance of call (km)

012 3456789 Time (years)

40 80 120 160 200 No. of cakes sold

w 20 w = l – 10

15 10 5

l 0

5 10 15 20 2l + 2w = 40

c l = 15, w = 5

7E

➔

b 2.5, means that an extra $2.50 profit is made for every cake sold. c −80, means that there is an initial cost of $80 to run the stall which will be lost if no cakes are sold. 6 a Gradient = 3, y-intercept = −2 b Gradient = --34- , y-intercept = 7 c Gradient = −1, y-intercept = 5

12 a b

7F

Answers

Practice examination questions 1 C 2 B 3 D y 5 ae y = 2x – 3

4 B

Exercise 8B — Graphing simple interest functions 1 a

5 4 3 2 1

x

b

Are you ready? c 0.045 c 5460

d 0.0625

(5, 24883)

16 000 14 000 12 000 10 000 8000 6000 4000 2000 0

Interest ($)

5000

b 1.25 b $199.75 b 37.5%

5

b Interest ($)

4 a 5 5 a $72 6 a 7%

2 3 4 Time (years)

c 1.5 c $75 c 0.25%

Exercise 8A — Calculation of simple interest $615 $90 $11 200 $12 400

b $145.25 b $224.32

c $21 420 f $684.47

c

1

C $6250 $12 542.50 $13 500

2 $1260 6 $138 10 $173.58

2

3 4 Years

5

2

3 4 Years

5

800 600 400

0

1

4000 3000 2000 1000

c $9558.84

0

11 $131.86 14 C c 7 --12- years

d

3 $2700 7 $215.44

1

2

3 4 Years

2

3 4 Years

5

20 000

10 Quick Questions 1 1 $800 5 $7000 9 53c

0

15 000 10 000 5000

0

4 $1.90 8 $1998.75

0

4 288

4

5

$1600 $3200 $4300 $6400 $8000

0 1 2 3 4 5 6 7 8 9 10 Years

0

10 13 b b

3

2000

0

Interest ($)

b e b b

2

200

Interest ($)

1 $600 2 a $120 d $6656.85 3 a $2240 4 a $2400 5 $8648.80 6 a $5.25 7 a $448.63 8 $850.50 9 A 12 18 months 15 a $627.13 16 a $2250 17 $18 750

1

5

4000

0

1

3 4 Years

c 1600 d $16 000 3 a 6000

(4, 20736) (3, 17280) (2, 14400) (1, 12000)

0

2

Interest ($)

CHAPTER 8 Investing money

10000

1

No. of years

b

15000

0

Interest

c n = 0.6m

20000

$1200 $1600 $2000

Interest ($)

012 3456 m

25000

$800

5

2000

0

b 0.08 b 97

$400

4

500

2 a

1 a 0.12 2 a 61.88 3

3

1000

5 4 3 2 1 0

b 0.6

2

1500

c −3 f (2, 1)

b 2 d y = 2x − 3 6 a n

1

No. of years Interest

0 –5–4–3–2–1–1 1 2 3 4 5 –2 –3 –4 –5 y = 5 – 2x

Population

answers

538

1

5

9 a d 10 B 14 a c 15 a c 16 a c

400 300 200 100 0

b

Investment ($)

0

1

5

3500


3000 0

1

2 3 Years

4

5

No. of years 1 2 3 4 5 Interest (5%) $1250 $2500 $3750 $5000 $6 250 Interest (6%) $1500 $3000 $4500 $6000 $7 500 Interest (8%) $2000 $4000 $6000 $8000 $10 000 8% 6% 5%

10 000

Interest ($)

b

3 4 Years

4000

0

6 a

2

5000 0 0

1

2

3 4 Years

5

7 a Whichbank = $37 500, Eastpac = $35 000, NZA bank = $30 000 b Whichbank = 7.5%, Eastpac = 7%, NZA bank = 6% 8 a 1

No. of years

2

3

4

5

Interest (3%)

$165.00 $330.00 $495.00 $660.00

$825.00

Interest (3.5%)

$192.50 $385.00 $577.50 $770.00

$962.50

Interest (3.75%) $206.25 $412.50 $618.75 $825.00 $1031.25 3.75% 3.5% 3%

1500

Interest ($)

b

1000

$2051.28 $429.69 $6792.58 $14 700.69

1 4 7 10

2 $17 253 5 $2315.25 8 $6805.66

500

Exercise 8D — Calculating compound interest from a table of compounded values 1 $13 110 2 $12 580 3 $46 800 4 $2599.20 5 a $8385 b $9895.90 c $336 d $25 634.61 e $14 919.50 f $122 130 6 a 1.025 b 1.051 c 1.077 d 1.104 e 1.160 f 1.218 7 a $969.30 b $2583.36 c $7948.80 d $13 398 e $6670 8 D 9 C 10 $8636 11 3 years 12 a 1 year b 6 years c 5 years d 18 months e 4 years 13 16% 14 a 7% b 8% c 28% d 10% e 12% 15 a $15 000 b $16 110 c $16 290 d Bruce $5000, Keith $6100, Max $6290 e Bruce 50%, Keith 61.1%, Max 62.9%

1 a No. of years 0

1

2

3 4 Years

1 Future value $8400

5

b

9 a $4500, $5000, $5200 b $875 10 5000

6.2% 5.8% 5.2%

4500

3 $3437.50 6 $315.25 9 $6812.41

Exercise 8E — Graphing compound interest functions

0

Interest ($)

$1003.69 b $9111.56 c $181 402.12 $20 039.67 e $1 264 568.95 11 B 12 B 13 C $15 746.40 b $15 793.09 $15 817.40 d $15 833.99 0.0219% b $108 320.72 $8320.72 d $320.72 $4720 b $4726.24 Compounding interest

Future value ($)

Interest ($)

5 a

539

answers

Answers

3 $9261

4 5 $9724 $10 210

10 000 9500 9000 8500 8000 0 0

4000

2 $8820

1

2 3 4 Years

5

2 a No. of years

1 2 3 4 5 Future value $12 960 $13 997 $15 117 $16 326 $17 632

0 1

2

3

4

5

b

Years

Exercise 8C — Calculation of compound interest 2 $17 173.50 b $9274.19 e $12 588.72 6 $502 b $31 850.33

3 $2938.66 c $24 488.80

25 000 20 000 15 000 10 000 0 0 1 2 3 4 5 6 7 8 9 10 Years

7 $14 059.91 c $25 900

8A ➔

1 $6655 4 a $4630.50 d $13 503.78 5 $70 555.25 8 a 0.5%

30 000 Future value ($)

0

8E

b

20 000 15 000

Future value ($)

Answers

25 000

Future value ($)

3 a

Annually Six-monthly

5500 5000 4500 4000 0 0

0 0

b

1

2

3 4 Years

Future value ($)

2000

1

2

3 4 Years

5

Share price $8.40

Dividend yield 6.7%

$0.78 $1.20 $1.09 $0.04

$7.40 $23.40 $15.76 $0.76

10.5% 5.1% 6.9% 5.3%

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

$2060 $2122 $2186 $2252 $2318 $2388 $2460 $2534 $2610 $2688

FV

b Future value ($)

2700 2600 2500 2400 2300 2200 2100 2000 0

Share price ($)

1 a Future value ($)

1800 1600 1400

Month

Share price ($)

0 1 2 3 4 5 6 7 8 9 10 Years

3 $999 $1528 $2078

4 $1359 $2100 $2884

5 $1733 $2706 $3755

8000

1

7 a No. of years

2

3 4 Years

1 Annually $4240 Six-monthly $4244

5

2 $4494 $4502

3.00

3 $4764 $4776

b Approximately $2.00 3 a 2.04

Share price ($)

Interest (8%) Interest (6%) Interest (4%)

10 000

3.50 2.50

2 $653 $988 $1331

12 000

4.00

1– Ju 1– n J 1– ul A 1– ug Se 1– pt O 1– ct N 1– ov D e 1– c Ja 1– n Fe 1– b M 1– ar A 1– pr M ay

1 $320 $480 $640

Interest (4%) Interest (6%) Interest (8%)

0

7.30 7.10 6.90 6.70 6.50 6.30 6.10 5.90 5.70


600 500 400 300 200 100 0

0

c 6.4c/share d 0.61%

4 $5050 $5067

5 $5353 $5376

2.02 2.00 1.18 1.16 1.14 1.12 1.10 1.08 1.06 1.04 1.02 1.00 1– J 1– an F 1– eb M 1– ar A 1– pr M 1– ay J 1– un 1– Jul A 1– ug Se 1– pt O 1– ct N 1– ov D 1– ec J 1– an F 1– eb M 1– ar A 1– pr M 1– ay J 1– un 1– Jul A 1– ug Se 1– pt Oc t

Compound interest earned ($)

0 1 2 3 4 5 6 7 8 9 10 Years

6 a No. of years

b

b $10.64

12 $364 b $1.06/share b $7.4906 million

1– M ay 1– Ju n 1– Ju l 1– Au g 1– Se pt 1– Oc t

1200 0

11 B

Exercise 8G — Graphing share performance

Years

b

2.91% a 6.6% a $1.44 8.5% a 0.59%

10 13 14 15 16 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

5 a

5

Dividend $0.56

4 a Years

4

25c/share 2 $1.50/share 6c/share 4 29.27c/share a $1.224 million b $2.176 million c 43.52c/share a $9.28 million b $6.08 million c $0.76/share $3.276 million 8 4.57%

1 3 5 6 7 9 0

2 3 Years

Exercise 8F — Share dividends

3000

0

1

5

4000

Interest ($)

answers

540

Month

b Approximately $1.20

14.00 12.00 10.00

28 000 27 000 26 000 25 000 24 000 23 000 22 000 21 000 20 000

0 0

Month

Share price ($)

13.50 13.00

b $48.15 e $932.40 b $901.76 5 $619 8 $122.80 11 $2350

c $1.91

6 $2.52 9 D 12 $2460

b $2820 c $42 e $7617.58 b $39 780 c $455 5 15 months 1

2

$450

$900

16.50 16.00 15.50

3

4

22 $7900

Are you ready? 1 a 145 cm c 20 cm 2

5

b 5 cm d 2 years

$1350 $1800 $2250

Interest ($)

3

4500 4000 3500 3000 2500 2000 1500 1000 500 0 0 1 2 3 4 5 6 7 8 9 10 Years

13

9 10

3 4 5 6 7 8 9 10 11 12

b

12

8

CHAPTER 9 Displaying single data sets

Chapter review

10 11

5 6 7 Years

Practice examination questions 1 A 2 B 3 C 4 B 5 a $8130 b No. The investment will be worth only $8121.43. c No. Jaclyn could lose the money she has saved for her holiday. d $9345 e 24.6% 6 a $32 136.75 b $32 600 c $436.17

Exercise 8H — Inflation and appreciation

7 9

4

b Approximately $18.00 20 $83.44 21 $149.90

b $14.50

Interest

3

Month

Month

No. of years

2

1– Ja 1– n Fe 1– b M 1– ar A 1– pr M 1– ay Ju 1– n J 1– ul Au 1– g Se 1– pt O 1– ct No 1– v De c

12.50 12.00

1– Ja 1– n Fe 1– b M 1– ar A 1– pr M 1– ay Ju 1– n J 1– ul Au 1– g Se 1– pt O 1– ct No 1– v De c

Share price ($)

14.00

1 $1000 2 a $1296 d $4.05 3 a $7280 4 6.5% 6 a

1

c Approximately $29 600 14 $1.93/share 15 $14.74/share 16 5.22% 17 1.6% 18 81.8 c/share 19 a 17.00


1 $20 800 2 a $618 d $579.60 3 a $878.05 4 $117.90 7 $1.20 10 $500

541

answers

b

16.00

1– Ja 1– n Fe 1– b M 1– ar A 1– pr M 1– ay Ju 1– n J 1– ul Au 1– g Se 1– p O 1– ct No 1– v De 1– c Ja 1– n Fe 1– b M 1– ar A 1– pr M 1– ay Ju n

Share price ($)

4 a

Future value ($)

Answers

c 450 d $4500 $2778.30 8 $5700.47 a $3932.39 b $12 596.90 c $14 457.96 d $23 851.00 e $5334.67 $756.94 a $12 024.02 b $12 052.04 c Compounding interest a $7866 b $13 216.44 c $4411.83 d $20 199.60 e $2599.20 a No. of years

1

2

3

4

5

Future value $20 800 $21 632 $22 497 $23 397 $24 333

Score

Frequency

20

5

21

8

22

7

23

2

24

4 c 61.5

d 9.4

8F

➔

4 a 6 b 12.5 5 Key: 2 | 3 = 23 Stem Leaf 1 09 2 7 3 3348 4 56 5 2

8H

Answers

Exercise 9A — Frequency tables Score

Tally

Frequency

2

|||

3

3

||||

4

4 6

|||

3

7

|

1

Score

Tally

Frequency

4

|||

3

5

||||

5

6

||||

5

||||

8

|

1

|

1

Frequency

10.5–10.9

10.7

2

11.0–11.4

11.2

5

11.5–11.9

11.7

8

12.0–12.4

12.2

8

12.5–12.9

12.7

5

13.0–13.4

13.2

2

1 0 1 2 3 4 5 6 7 8 9 10 Scores

3

9

Class centre

Exercise 9B — Types of graphs

7

10

2

Score

Tally

Frequency

66

|

1

67

|||

3

25 26 27 28 29 30 31 32 33 34 35 Maximum temperature

3

6

Wins (180°)

69

||

2

Draws (72°)

70

|||

3

Losses (108°)

71

||||

5

||||

72

5

|| |||

68

||||

|

||||

||||

14

73

||||

5

74

||||

4

Rent (120°)

75

||

2

Bills (90°)

76

|

1

77

|||

3

Groceries (72°)

78

|

1 Tally

4

Car running costs (42°)

Class

Class centre

30–39

34.5

|

1

40–49

44.5

||||

5

50–59

54.5

||||

5

60–69

64.5

||||

|||

8

70–79

74.5

||||

|

6

80–89

84.5

|||

84.5

||

2 Frequency

Class

Class centre

Tally

0.6–0.8

0.7

||

2

0.8–1.0

0.9

|||

3

1.0–1.2

1.1

|||

3

1.2–1.4

1.3

1.4–1.6

1.5

1.6–1.8

1.7

1.8–2.0

1.9

||||

4

2.0–2.2

2.1

||||

4

|||| ||||

5

16 000 14 000 12 000 Year

3

90–99

||||

Savings (36°)

Frequency

20 0 20 0 0 20 1 0 20 2 0 20 3 0 20 4 05 20 0 20 6 07

4

6 3

7

3

|

|||

Population

2

||||

5

Class

|

6

||||

9

|

6

6

Account balance ($)

1

6

700 600 500 400 300 200 J FMAM J J A S O N D Month

7

Categories of employment

answers

542

Managers Professionals Para-professionals Tradespersons Clerks Salespersons Plant operators Labourers 100 200 300 400 500 600 700 800 900 Weekly average earnings ($)

No. of gold medals

8

4 a

18 16

14 12 10 8 6 4 2 0

l l a a y s i e e o y s h ink urn om ky Cit nic trea cow ele eou lon ant ne en els elbo 0 R 4 To ico Mu on Mos Ang 8 S arce Atl Syd Ath H 8 4 6 0 6 M x 6 s 9 B 2 e 52 M 19 19 M 97 976 980 Lo 1 92 199 200 200 1 1 1 84 19 956 68 19 1 9 9 1 1

September

5

May

August

20 15 10 5 0

June July

10

Frequency

Average monthly temperature (°C) January 30 February December 25 20 March 15 November 10 5 0 October April

b

Frequency

9

No. of matches in a box 47 48 49 50 51 52 53 54

6 a

40%

4.5 4.5 4.5 4.5 4.5 4.5 30 31 32 33 34 35 Length of fish (mm)

4.00 pm

7

1

9

4

10–12

11

15

12–14

13

18

14–16

15

12

16–18

17

8

18–20

19

2

No. of cars

Frequency


0

2

2

6.00 am 8.00 am

2.00 pm

10.00 am 12.00 pm

Frequency

b 1

Number of drivers (frequency)

Exercise 9C — Statistical graphs 14 12 10 8 6 4 2 0

7 a

16 14 12 10 8 6 4 2 0

30 25 20 15 10 5 0

Frequency

1

8

10

2

11

21

3

6

27

4

2

29

5

1

30

0 1 2 3 4 5 No. of cars

9A ➔

18 19 20 21 22 23 24 25 Age


1 2 3 4 5 6 No. of children in family

b No. of members

3

9 8 7 6 5 4 3 2 1 0

18 16 14 12 10 8 6 4 2 0 7 9 11 13 15 17 19 Time taken (s)

0 1 2 3 4 5 Number of mistakes (score)

2

Frequency

6–8

4.00 am

0%

Class centre

8–10

20% 6.00 pm

Frequency 3 5 10 15 7 5 4 1

47 48 49 50 51 52 53 54 No. of matches in a box

Time taken (seconds)

Percentage of televisions being watched 12.00 am 80% 2.00 am 10.00 pm 60% 8.00 pm

16 14 12 10 8 6 4 2 0

543

answers

Answers

9C

b

2

49

10

12

50

32

44

51

9

53

52

5

58

53

2

60

40 35 30 25 20 15 10 5 0

100 90 80 70 60 50 40 30 20 10 0

5. 15 5 25.5 35.5 45.5 .5

5. 15 5 2 5 .5 3 5 .5 4 5 .5 .5

35 30 25 20 15 10 5 0

5 Grouped Frequency

2

1 5.5, 15.5, 25.5, 35.5, 45.5 2 5, 20, 49, 86, 97 3 48 4 86 6 40 7


48


Frequency


5 .5 15 .5 25 .5 35 .5 45 .5

48 49 50 51 52 53 No. of jelly beans

Score

9

100 90 80 70 60 50 40 30 20 10 0

5.5 15 .5 25 .5 35 .5 45 .5


Score

8

Score

Score

Cumulative Length (cm) Class centre Frequency frequency

10 a

4–5

4.5

6

6

5–6

5.5

10

16

6–7

6.5

60

76

7–8

7.5

58

134

8–9

8.5

8

142

9–10

9.5

4

146

140 120 100 80 60 40 20 0

4.5 5.5 6.5 7.5 8.5 9.5 Length of oysters (cm)

Lifetime (hours)

Class centre Frequency


10 Ogive

Exercise 9D — Range and interquartile range 1 a d 2 a 3 a 4 a b 5 a c 6 a d 7 a d 8 a d 9 a

5 b 9 c 1 6.94 e 89 4 b 5 c 6 49 b 30 c 23 Sydney — 120 Collingwood — 40 Collingwood, because the range is lower. 9 b 8 The range for machine A is large, only because of one extreme score. 5 b 25 c 53 15 e 74 5 b 9 c 2 4 e 32 2 b 1 c 2 1 e 4 Score

Frequency


20–25

22.5

6

6

0

26

26

25–30

27.5

25

31

1

31

57

30–35

32.5

70

101

2

22

79

35–40

37.5

61

162 3

8

87

40–45

42.5

30

192

45–50

47.5

8

200

4

3

90

b Cumulative frequency

b

Frequency

200 180 160 140 120 100 80 60 40 20 0 Lifetime of battery (hours)


9 a

No. of jelly beans

60 50 40 30 20 10 0


b


8 a

Answers

22 .5 27 .5 32 .5 37 .5 42 .5 47 .5

answers

544

c 1 10 A

90 80 70 60 50 40 30 20 10 0

0

1

2 3 Score

d 2 11 B

4

12 D

13 B

14 a

b


IQ score 75–85 85–95 95–105 105–115 115–125 140 120 100 80 60 40 20 0

Class centre Frequency 80 12 90 25 100 50 110 24 120 13

Cumulative frequency 12 37 87 111 124

3 Key 10 | 1 = 101 Stem 8* 9* 9* 10* 10* 11* 11* 12* 12* 13*

c 50 d 10 15 a Maximum

b


temperature (°C) 0–5 5–10 10–15 15–20 20–25 25–30 30–35 35–40 400 350 300 250 200 150 100 50 0

No. of days 4 22 95 124 94 19 5 2

Cumulative frequency 4 26 121 245 339 358 363 365

2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 Maximum temperature (°C)

c Lower quartile = 13, upper quartile = 22, interquartile range = 9 d Median = 17.5 e 0–11 16 Check with your teacher. Answers depend on class size.

Exercise 9E — Stem-and-leaf plots 1 Key 0 | 6 = 6 Stem Leaf 0 6 1 3 5 7 8 2 0 0 5 6 6 7 8 9 3 1 2 2 8 4 3 6 5 2 2 Key 3 | 6 = 36

0 1 1 4 0

Leaf 6 7 8 8 9 9 0 0 1 2 2 2 2 2 3 5 6 7 7 8 8 8 0 2 2 2 3 4 5 7 8 9 6 8 8 1 2 5 2

7 11

14* | 8 = 14.8

Stem Leaf 13* 8 9 14* 0 2 3 3 14* 5 6 6 7 7 8 8 15* 1 2 2 15* 5 6 7 9 1.95 a 8.4 s b Lower quartile = 7.8 s, upper quartile = 8.85 s c 1.05 D 8 A 9 D 10 B a Key 20 | 6 = 20.6

b c d 12 a

Stem Leaf 20 2 7 21 4 6 7 8 22 2 8 8 8 23 0 1 2 6 6 24 6 6 7 8 25 1 22.9 Lower quartile = 21.75, upper quartile = 24.1 2.35 Key 8 | 2 = 82

Stem Leaf 8 2 4 5 5 8 9 0 1 2 3 4 10 1 1 3 3 7 11 0 0 1 4 5 12 6 6 13 2 b 100 c Lower quartile = 92, d 18

6 6 7 8 9 8 7

Upper quartile = 110

Exercise 9F — Five-number summaries 1 8, 15, 16.5, 18, 25 2 a 23, 44, 81.5, 83.5, 92 b 1, 2, 4, 6, 7 c 8, 29, 45, 72, 93 3 1, 3, 4, 5, 7 4 40, 65, 72, 78, 100 5 a 13 b 5 6 a 122 b 6 7 35

40

45 50

55 60

c 26 c 27

9D ➔

Stem 3 4 5 6 7 8

5 6

10* | 6 = 106

Leaf 6 8 2 5 5 5 6 6 6 8 9 2 2 2 3 7 7 7 8 8 0 1 2 5

4 Key 14 | 3 = 14.3

80 90 100 110 120 IQ Score

545

answers

Answers

9F

8 a d 9 a d 10 B 13 a b

Answers

148 92 58 27

b 56 e 28 b 31 e 8 11 C 22, 28, 35, 43, 48 20

25

30 35

40 45

Chapter review

c 90

1

Score 0 1 2 3 4 5 6

Frequency 6 8 5 3 2 0 1

2

Class 3.5–4.0 4.0–4.5 4.5–5.0 5.0–5.5 5.5–6.0 6.0–6.5 6.5–7.0

Class centre 3.75 4.25 4.75 5.25 5.75 6.25 6.75

c 43 12 D

50

14 a 10, 13.5, 22, 33.5, 45 b 10

15

20 25

15 a Class

30 35

40

45

Cumulative Class centre Frequency frequency

120–129

124.5

4

4

130–139

134.5

3

7

140–149

144.5

4

11

150–159

154.5

6

17

160–169

164.5

2

19

170–179

174.5

2

21

180–189

184.5

7

28

ABC (54°) SBS (10°) Channel 7 (96°) Channel 9 (144°) Channel 10 (56°)

4

Monday 70 60 50 40 30 20 10 0

No. of hamburgers sold

c i 155 d

ii 140

iii 182 Friday

120 130 140 150 160 170 180 190

Class


15–19

17

7

7

20–24

22

15

22

25–29

27

4

26

30–34

32

3

29

35–39

37

0

29

40–44

42

0

29

45–49

47

1

30

Thursday

5

10 8 6 4 2 0

25

6 a

15

Wednesday

12

30 20

Tuesday


Frequency

16 a

b

Frequency 1 4 5 2 4 3 1

3

30 25 20 15 10 5 0

12 4. 13 5 4. 14 5 4. 15 5 4. 16 5 4. 17 5 4. 18 5 4.5


b


answers

546

0 1 2 3 4 5 6 Number of sales

Class

Class centre

Frequency

10

5000–9999

7 500

1

5

10 000–14 999

12 500

5

15 000–19 999

17 500

9

20 000–24 999

22 500

3

25 000–29 999

27 500

2

30 000–34 999

32 500

2

0

17 22 27 32 37 42 47 Age

c 15

20

25 30

35 40

45

d Check with your teacher.

50

10 8 6 4 2 0

b Frequency

Frequency

b

00 00 00 00 00 0 0 75 12 5 17 5 22 5 27 5 32 5

No. of people at a football match

7 a Class centre Frequency


30–39

34.5

40–49

44.5

6

9

50–59

54.5

12

21

60–69

64.5

15

36

70–79

74.5

18

54

80–89

84.5

10

64

70 60 50 40 30 20 10 0

3

c 30 25 20 15 10 5 0

12 17 22 27 32 37 42 No. of admissions to hospital in a day

34.5 44.5 54.5 64.5 74.5 84.5 English exam mark

8 a 7 9 a i 25 b i 62.5 c i 1.1 10 a 2 11 a 61

ii ii ii b b

b 159 c 1.4 24 iii 27.5 iv 3.5 43 iii 84 iv 41 0.7 iii 1.5 iv 0.8 Lower = 1, upper = 3 c 2 Lower = 54, upper = 70 c 16

12 Key 2 | 1 = 21

13 14 15 16 17

7 6 5 4 3 2 1 0

12 17 22 27 32 37 42 No. of admissions to hospital in a day

3


b


Class

547

answers

Answers

Stem Leaf 2 1 1 3 4 8 8 8 9 9 9 3 0 3 4 5 5 5 6 8 8 8 8 9 4 0 0 0 1 1 2 2 4 5 5 5 6 8 9 a 28 b 38 c 12.5 6, 24, 41, 69, 91 0, 21, 26, 32, 50 a 43 b 43 c 14

d e 7 a d

Class

Are you ready? 1 a 4.4 2 3

3 a

3 A


ii 13 b 15 c 35 e Check with your teacher.

CHAPTER 10 Summary statistics

0 2 4 6 8 1012 14 16 18 20

Practice examination questions 1 C 2 B 4 C 5 A 6 a

34 i 23 45 7


b 1.4125

4

5

6

7

c 75.4

8

9

Score

Frequency

12

2

13

3

14

3

15

3

16

2

17

1

18

4

19

2

10

12

7

7

15–19

17

4

11

Class

Frequency

20–24

22

7

18

0–9

1

25–29

27

5

23

10–19

4

30–34

32

4

27

20–29

6

35–39

37

1

28

30–39

5

40–44

42

2

30

40–49

4

b

9F

➔

10–14

9F

answers

548

Answers

4 Key 2 | 1 = 21 Stem Leaf 2 0 0 0 0 4 4 5 7 8 9 9 3 1 1 1 2 6 9 4 0 1 1 1 1 4 6 6 6 6 8 8 9

9 a

Score (x)

Frequency (f)

f×x

8

2

16

Exercise 10A — Calculating the mean 1 a 5 b 26.5 d 7.72 e 376 2 72.6 3 125.7 c/L 5 Yes, mean mass is 45.035 g. 6 a Score Frequency

c 74.25

4

2

8

5

4

20

70

11

132

14

6

84

16

2

32

b 12 --1310 D 13 a

2

36

Σ f = 30

Σ f × x = 370

11 B

12 C

Class

Class centre (x)

Frequency (f)

31–40

35.5

1

35.5

f×x

6

5

30

7

9

63

41–50

45.5

3

136.5

8

3

24

51–60

55.5

4

222.0

9

5

45

61–70

65.5

7

458.5

71–80

75.5

11

830.5

10

2

20

Σ f = 30

Σ f × x = 210

b 7 7 a

No. of televisions sold

No. of weeks

f×x

16

4

64

81–90

85.5

2

171.0

91–100

95.5

2

191.0

Σ f = 30

Σ f × x = 2045

b 68.17 14 a Time

Class centre

No. of swimmers

f×x

50.01–51.00

50.5

4

202.0

51.01–52.00

51.5

12

618.0

17

4

68

18

3

54

52.01–53.00

52.5

23

1207.5

19

6

114

53.01–54.00

53.5

38

2033.0

140

54.01–55.00

54.5

15

817.5

55.01–56.00

55.5

3

166.5

Σ f = 95

Σ f × x = 5044.5

20

7

21

12

252

22

8

176

23

2

46

24

4

96

25

2

50

Σ f = 52

Σ f × x = 1060

1–10

5.5

12

Frequency (f)

f×x

11–20

15.5

6

4

0

21–30

25.5

5

35.5

7

b 20.4 8 a Score (x) 0

15 16 17 18 19 20

b 53.1 Check with your teacher. a 14.1 b 3.4 a 6.6 b 30.67 166.25 a 12.5, 13.5, 14.5, 15.5, 16.5 a Class Class centre

c 44.4 b 14.4 Frequency

1

9

9

31–40

2

18

36

41–50

45.5

9

51–60

55.5

9

61–70

65.5

5

71–80

75.5

5

3

b 2.3

7

12

18

4 1.81 m f×x

10

10

30

4

5

20

5

4

20

81–90

85.5

5

Σ f = 50

Σ f × x = 115

91–100

95.5

7

b 46.4

2.29 b 2.19 c 20.17 3.07 e 42.44 26.94 b 2.14 c 57.51 0.26 e 96.04 Sample b Population c Population Sample e Population 616.6 b Sample c 270.97 1.44 b Population c 0.48 x– = 4.9, σn = 1.0 b x– = 48.2, σn = 1.2 x– = 78.3, σn = 2.3 x– = 17.45, σn = 3.69 b x– = 14.95, σn = 7.49 x– = 56.02, σn = 14.26 Brianna: x– = 75, σn = 3.69 Katie: x– = 74, σn = 18.28 b Brianna is more consistent because she has a lower standard deviation. B 10 C x– = $1825, σn − 1 = 797 a 500 b x– = 455.3, σn − 1 = 88.9 Crinkle, because the standard deviation in the weight of each pack is lower and therefore you are more likely to get the correct amount.

1 a d 2 a d 3 a d 4 a 5 a 6 a c 7 a c 8 a

9 11 12 13

b


Exercise 10B — Standard deviation

Frequency

frequency

17

4

4

18

9

13

19

6

19

20

12

31

21

8

39

22

5

44

23

4

48

24

2

50

7 1

8 C

9 C

Days sickness

Frequency

0–4

10

10

5–9

12

22

10–14

7

29

15–19

6

35

20–24

5

40

25–29

3

43

30–34

2

45

b e b e b

8 110 148, 151 2.6 8

c 11 c No mode c 42, 44

b 22–28 Class centre

Frequency


0–50

25

8

8

75

9

17

100–150

125

12

29

150–200

175

15

44

200–250

225

6

50

250–300

275

4

54

300–350

325

2

56

350–400

375

2

58

60 50 40 30 20 10 0

25 75 125 175 225 275 325 375 Depth of snow (cm)

c 152 19 a Class

Class centre

Frequency


$200–$249

224.5

8

8

$250–$299

274.5

4

12

$300–$349

324.5

6

18

$350–$399

374.5

6

24

$400–$449

424.5

4

28

$450–$499

474.5

2

30

$500–$549

524.5

6

36

$550–$599

574.5

4

40

b $350–$399 c 40 35 30 25 20 15 10 5 0

5 5 5 5 5 5 5 5 22 27 32 37 42 47 52 57 Weekly wage ($)

$360

10 Quick Questions 1 1 6.2 6 77.5

2 6 7 78

3 5 8 78

4 2.25 9 1.12

5 2.39 10 1.12

10A ➔

b 10–14 12 a 2, 7, 12, 17, 22, 27, 32

b

10 A Cumulative frequency

7 12 17 22 27 32 No. of sick days

50–100


b 20 6 3 11 a

Score

2

(cm)


6 81 a 5 b 5.4 c 62 d 102 a 4 b 5.6 c The median is a better measure because one large score makes the mean larger than what is typical. 5 a Cumulative

c 10

13 a 5 d 0.4 14 a 17 d 72 15 a 4 16 17 17 a 17–20 18 a Depth

Exercise 10C — Median and mode 1 2 3 4

45 40 35 30 25 20 15 10 5 0

answers

549

Answers

10C

Answers

Exercise 10D — Best summary statistics

Chapter review


1 a $425 000 b $370 000 c $350 000 d The median, as the mean is inflated by one large score and the mode is the lowest price. 2 a 7.1 b 7 c 7 d The mode, as it is the size that sells the most. 3 a 23 550 b 20 000–30 000 c 10 000–20 000 d 200

b 64.875 b 9.46

c 7.7

d 35.8

Class

Class centre

Frequency

f×x

21–24

22.5

3

67.5

25–28

26.5

9

238.5

29–32

30.5

17

518.5

33–36

34.5

31

1069.5

100

37–40

38.5

29

1116.5

50

41–44

42.5

25

1062.5

45–48

46.5

19

883.5

49–52

50.5

10

505.0

Σ f = 143

Σ f × x = 5461.5

0

0 0 0 0 0 0 0 00 00 00 00 00 00 00 15 25 35 45 55 65 75 Crowd number

e 21 000 4 B 5 a

f Median

Class 0–4 5–9 10–14 15–19 20–24 25–29

Class centre 2 7 12 17 22 27


Cumulative frequency 16 22 26 28 29 30

b 6.8 c 0–4 d 0–4 6 Check with your teacher; answers depend on groupings used. 7 a Class Cumulative Class

centre

Frequency

frequency

1–15

8

1

1

Mean = 38.2 a 31.1 b 23.2 c 0.445 a 29.9 b 26.4 c 18.6 a 71.8 b Population c 17.3 a 1.95 b Sample c 0.89 a x– = 0.81, σn = 0.42 b x– = 67.25, σn = 75.3 c x– = 28.1, σn = 1.2 d x– = 27.5, σn = 7.03 9 a 27 b 6 c 3.2 d 5.5 e 128 10 a 2 b 56 c 68.5 11 a Class Cumulative 4 5 6 7 8

Class

centre

Frequency

frequency

30–39

34.5

18

18

40–49

44.5

34

52

50–59

54.5

39

91

16–30

23

13

14

31–45

38

2

16

60–69

64.5

45

136

46–60

53

0

16

70–79

74.5

29

165

61–75

68

5

21

80–89

84.5

10

175

76–90

83

4

25

90–99

94.5

5

180

20 15 10 5 0

ii 16–30

iii 16–30

b 50–59 c 180 Cumulative frequency

b i 42.2 c 25

8 23 38 53 68 83 Age of patients

27 e No Check with your teacher. Player A: 34.3 Player B: 41.8 Player B Player A: 32.5 Player B: 0 Player A Check with your teacher. 65 out of 80 earn less than the mean and another 10 only slightly more than the mean. Hence it is misleading. b The median and the mode would be useful in your submission as they are not distorted by the larger salaries as the mean is.

d f 8 a b c d e 9 a

1 a 5.2 2 a 7.025 3

150


answers

550

160 140 120 100 80 60 40 20 0

.5 .5 .5 .5 .5 .5 .5 34 44 54 64 74 84 94 Score

d 58 a 2 b 23, 27 a 2 b 15, 18 46–49 a 27.8 b 24.5 d Median 16 Check with your teacher. 12 13 14 15

Practice examination questions 1 A 2 A 3 D 4 B

c No mode

c 28

5 B

6 a Class centre

Frequency


$50 000– $75 000

$62 500

12

12

$75 000– $100 000

$87 500

18

30

$100 000– $125 000

$112 500

26

56

$125 000– $150 000

$137 500

24

80

$150 000– $175 000

$162 500

12

92

$175 000– $200 000

$187 500

8

100

Income

b d f 7 a b c d e

Exercise 11B — Solving problems using similar figures 1 6.25 m 2 a

1m 25 cm

9.5 m b 38 m 3 a

h

$120 000 c 35 622 $100 000–$125 000 e $100 000–$125 000 mean Text A: x– = 58.6, σn = 25.1 Text B: x– = 62.25, σn = 11.8 Population because the whole classes’ results have been used. Text B Text B — lower standard deviation Check with your teacher.

Are you ready? a 5:4 b 8:7 c 16 : 25 d 1:5 a XY b ∠CAB 4 ∠ABX = ∠ACX (given) ∠AXB = ∠AXC (given) ∠XAB = ∠XAC (angle sum of a triangle) ∴ LABX ||| LACX (equiangular) 5 a 1:3 b 10 : 9 1 2 3 4

Exercise 11A — Similar figures and scale factors Corresponding angles equal Corresponding angles equal Corresponding sides in equal ratio Corresponding sides in equal ratio Corresponding angles equal Corresponding angles equal They are similar. They are not similar. a 2:3 b 1 1--2-

10 a 2:5

11 2:3 12 200 13 2 000 000

h

CHAPTER 11 Similarity of two-dimensional figures

1 2 3 4 5 6 7 8 9

551

answers

Answers

4 7 8 9 10 11 12

3m b 1:6 10 m a 20 km d 51.2 km a 4 cm d 68 cm D B 20 m 4 cm

50 cm c 10.2 m 5 B b 36 km e 3.6 km b 9 cm e 1.6 cm

c f c f

6 60 km 26 km 1.6 km 4.4 cm 7.5 cm

Exercise 11C — House plans

1 a 20 m × 25 m b 10 m × 15 m 2 a 18 m × 12 m b 6.75 m × 4.5 m c Bed 1 — 4.5 m × 4.5 m 3 a 8.5 m b 3.5 m c 40° 4 Check with your teacher. 5 Check with your teacher.

Chapter review 1 2 3 4 5 6 7 8 9 10

Check with your teacher. Check with your teacher. Check with your teacher. a Check with your teacher. b 2:5 20 cm 20 m 22.5 m a 400 m b 2.4 km c 3.4 km 80 m × 128 m a 30 m × 34 m b 10.8 m × 14 m c 151.2 m2

Practice examination questions 1 C 2 D 4 a 7.68 m b 12.5 m 5 a Check with your teacher. b 12.4 m long and 11.8 m wide

3 B

10D ➔

b 2 1--2-

170 cm

11C

Answers

CHAPTER 12 Taxation

4 D 7 $3500

Are you ready? 1 2 3 4 5

a a a a a

$2058 $495 25% $75

b b b b

$24 960 $862.50 12.5% $1400

c c c c b

y

$1012.50 $528 d $708.75 150% d 6.4% $250 y

y = 2x + 1 –1 2

y = 4x – 2

x –2

1 2

x

y

c

5 5

x y=5–x

2 $1624.55 4 $1451.43 b $1208.77 c $811.72 b $600

c $360

ii $18 000

iii $10 800

10 $18 760 12 $40 074.50

Exercise 12B — Taxable income $43 754.25 a $16 879.20 $43 568.34 a $24 219.40 a $20 503.60 a $42 500 a $34 262.30 a $74 280 a $25 649 d $1349.40 10 A 1 2 3 4 5 6 7 8 9

b $15 736.50 b b b b b b e

$22 262.60 $20 075.20 $2600 $498.24 $8737.46 $1470 $21 289.60

c c c c

$39 900 $32 204.06 $65 542.54 $1540

$2260 $1105.50 $41 771 $24 570

2 $1581.22 5 6 years 8 $22 660

3 $1650 6 $36 520 9 $37 525.80

Exercise 12C — Medicare levy 1 $598.05 2 a $648.75 3 a $23 491

1 $5235 2 a Nil b $1901.25 c $3975.00 d $7896.00 e $23 740 f $71 200 3 a $32 639 b $4391.70 4 $9917.40 5 a $5280.00 b $534.00 c $5814.00 6 a $2977.50 b $17 280.00 c $30 590 7 $61.66 8 $477.23 9 C 10 B 11 C 12 a $38 674 b $6782.31 c $291.69 13 a $90 942.80 b $468.09 c $23 821.78 d $519.16 14 a Gross annual pay = $44 605 Total tax = $10 091.33 b $41 513.40 c $7676.60 d $2414.73 15 a $16 721.51 b Debt of $2401.56 1 4 7 10

$600 $385.05 $2034 $59 500

2 $360 5 $502.62 8 $4350

3 $442.01 6 Nil 9 $24 870

Exercise 12E — Calculating GST and VAT 1 $3.56 2 a 23c d 13c 3 a 80c d 63c 4 $123.75 5 a $126.39 d $5.45 6 a $30.00 d $2.94 7 $98.50 8 a $1.90 9 $348.10 10 a $1.08 d $49.21 11 a $33 550 d $38 885

b e b e

$6.89 $8.99 48c 34c

c $9.85

b e b e

$32.89 $6.47 $0.94 $9.86

c $16.17

c 9c

c $47.00

b 19c b $80.63 e $6.95 b $36 630

c $193.29 c $34 705

Exercise 12F — Graphing tax functions 1 a i $10 b 50

ii $20

iii $50

40


6 $4600


Exercise 12A — Calculating allowable deductions 1 $1153.90 3 $1070.10 5 a $914 d $1560.20 6 a $1000 7 $3087.50 8 a i $30 000 b 2012–2013 9 $960 11 $712.27

5 $106 000 8 $1550

Exercise 12D — Calculating tax

GST ($)

answers

552

b $851.18 b $352.37

c $1410

30 20 10 0

0 50 00 50 00 50 00 50 00 50 00 1 1 2 2 3 3 4 4 5 Price ($)

2 a Nil b i $600 ii $3600 c i $3600 ii $9600 d i $26 600 ii $58 600 e i $58 600 ii $63 100 See graph at top of page 553

iii $18 600 iii $67 600

Income tax ($' 000)

80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

answers

553

Answers

200

190

180

170

160

150

140

130

120

110

100

90

80

70

60

50

40

30

20

10

0

Income ($' 000)

Chapter review 1 3 4 6 8 9 10 11 12 13 14 15 16 17 18 19 20

$2865 2 $2148.43 a $12 600 b $9072 c $6531.84 6 years 5 $27 057.35 $43 883 7 $70 420 a $28 483.60 b $26 456.60 a $34 409.30 b $32 366.15 $585 a $232.50 b $435 c $712.50 d $1252.61 e $1485 a $1314 b $1515 c $3900 $7950.00 a Nil b $2003.55 c $5301.00 d $22 424.80 e $102 700 a $33 987.20 b $4796.00 c $509.81 d $5305.91 e $102.04 $882.28 a $40 255 b $7280.33 c $1100.92 $99 a $1.13 b $36.90 c 12c d $3.75 e $1.25 $22 21 $112.50

Practice examination questions 1 B 2 D 3 B 4 a $33 280 b $32 161.80 c $482.43 d $4248.30 e $834.90 refund 5 a $330 b $360 c NZ$9.60 cheaper to pay on arrival in New Zealand

CHAPTER 13 Right-angled triangles Are you ready? Opposite AB, adjacent BC, hypotenuse AC Opposite PR, adjacent PQ, hypotenuse QR Opposite YZ, adjacent XY, hypotenuse XZ 6.9 cm b 25.1 cm 4.74 b 18.5 c 0.378 507.182 e 10.80 f 0.7643 x = 12 b x = 13.05 c x = 3.444 16 x = 24 b x = 0.36 c x = 540 24° b 47° c 27° d 87°

Samos Island Egypt and Babylonia c2 = a2 + b2 Patterns in music Plimpton 322 A set of numbers that obeys Pythagoras’ theorem

1 2 3 4 5 6

Exercise 13A — Pythagoras’ theorem 1 2 3 4 5 6 7 11 15

a PR b YZ a 13 cm b 170 mm a 10.82 cm b 6.93 m a 5.7 cm b 1.9 m a 8.9 cm b 22.1 cm d 37.3 m a Yes b Yes B 8 C 9 13 m 3.73 m 12 2.2 m 13 7.5 m a Check with your teacher.

c c c c c

AB 61 m 14.20 km 3.9 m 47.4 mm

c No 10 3.23 m 14 7.07 cm b 368 m

Exercise 13B — Calculating trigonometric ratios 1 a 1.540 d 0.657 2 a 0.602 d 51.893 3 a 0.707 d 5.896 4 a 0.5 d 548.6 g 5.306 5 a 0.42 d 5.10 g 7.77 6 10° 7 a 44° 8 86°40′ 9 a 42°57′

b 17.663

c 40.460

b 2.092

c 15.246

b 0.247

c 6.568

b e h b e h

0.9659 64 1.374 1.56 2.87 73.30

c f i c f i

1 1.301 15.77 0.09 0.38 0.87

b 80°

c 57°

b 31°21′

c 16°5′


13 cm 6.409 m 133.55 63°26′

2 17.0 cm 5 0.8290 8 30°

3 22.4 cm 6 11.0 9 73°

12A ➔

1 a b c 2 a 3 a d 4 a 5 a 6 a

History of mathematics

13B

answers

554

Answers

Exercise 13C — Finding an unknown side 1 a opp

c 40°

α

θ

hyp

adj

adj

opp

c adj γ

opp hyp

148.1 mm 3 5.08 m 4 30.0 cm a 12.1 cm b 55.2 m c 9.43 km a 12.5 m b 89.3 mm c 10.1 m a 5.42 m b 1.35 km c 2.06 km d 18.4 mm e 3.20 cm f 66.5 m g 5.40 m h 5.39 km i 0.240 m j 41.6 km k 84.4 m l 13.2 cm 8 D 9 A 10 A 11 C 12 6 m 13 4.2 m 14 20 km 15 a b 30.3 cm 2 5 6 7

b 1.6 m

16 a 1.4 m

17 9.65 m 18 a 60 m

c 15.5 m

10 Quick Questions 2 2 22.4 m 5 32.9 cm 8 177.8 mm

3 26.5 m 6 25.3 m 9 97.5 m

Exercise 13D — Finding angles a 30° b a 32°48′ b a 53°8′ b a 50° b d 21° e 5 a 39°48′ b d 79°6′ e 6 A 7 B 10 75°31′ 11 8°38′

75° 45°3′ 55°35′ 32° 81° 80°59′ 63°1′ 8 C 12 13°

c c c c f c f

81° 35°16′ 45°27′ 33° 34° 13°30′ 19°28′ 9 37° 13 4°35′

Exercise 13E — Angles of elevation and depression 1 5 8 9 10

c 3.58 m f 2.47 m c 2.7400 f 14.2998 c 53° c 14°54′ c 13.6 cm f 14.6 m i 15.6 mm l 5.3 km 10 63.9 m c 68° c 59°35′ 16 57° 4 A

Are you ready?

b 58 m 15°

1 2 3 4

1 a 13.01 m b 18.65 cm d 15.65 cm e 2.30 km 2 98 km 3 66 m 4 a 0.7193 b 4.2303 d 8.1955 e 21.9845 5 a 54° b 51° 6 a 78°31′ b 26°34′ 7 a 37.9 cm b 3.8 m d 11.7 cm e 14.7 cm g 1.5 m h 4.9 cm j 7.5 m k 10.7 m 8 8.5 m 9 2.5 km 11 a 57° b 27° 12 a 23°4′ b 61°37′ 13 39° 14 24° 15 23 m

CHAPTER 14 The language of chance

60°

17 cm 21.5 m 12.5 mm 17 m

Chapter review

Practice examination questions 1 C 2 B 3 B 5 a 5.22 km b 54° 6 a 7500 m b 7°36′

24° 13.5 cm

1 4 7 10

12° 400 m

b

hyp

b 85 m

12 a

26.8 m 2 3984 m 3 190 m 4 39.2 m 42.1 m 6 100 m 7 15 km Yes, the ladder needs to be only 28 m long. a 914 m b 868 m 39° 11 21°

1 a Fifty-fifty b Likely c Impossible d Unlikely e Certain 2 a 26 b 4 c 2 d 1 e 4 f 12 3 a Unlikely b Fifty-fifty c Impossible d Certain e Likely 4 B–C–A 5 a S = {1, 2, 3, 4, 5, 6} b S = {H, T} c S = {1, 2, 3, 4, 5}

Exercise 14A — Informal description of chance 1 a Probable b Unlikely c Impossible d Fifty-fifty 2 a Impossible b Certain c Even chance d Even chance e Probable f Unlikely g Impossible h Even chance 3 Check with your teacher. 4 More likely during school term 5 a More likely b Equally likely c Less likely d More likely e Less likely 6 Rolling a 6 Rolling a number less than 3 Rolling an even number Rolling a number greater than 2 7 Winning a raffle with 5 tickets out of 30 Selecting a court card from a standard deck Drawing a green marble from a bag containing 4 red, 5 green and 7 blue marbles Rolling a die and getting a number less than 3 Tossing a coin and having it land Heads

8 9 10 13

Australia Carl Bailey because he has better past performances. A 11 B 12 C Probable 14 Unlikely 15 Fifty-fifty

Exercise 14B — Sample space

1 S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 2 a S = {Heads, Tails} b S = {1, 2, 3, 4, 5, 6} c S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} d S = {a, b, c, d, . . . ., y, z} e S = {Sun, Mon, Tue, Wed, Thu, Fri, Sat} f S = {Jan, Feb, Mar, . . . ., Dec} 3 a 52 b 15 c 44 d 1500 e 901 f 11 4 a S = {M, I, S, S, I, S, S, I, P, P, I} b 11 c 4 5 a 52 b i 4 ii 2 iii 13 6 a S = {1, 2, 3, 4, 5, 6} b E = {5, 6} 7 a 100 b 40 8 D 9 B 10 D 11 a S = {NSW win, Qld win, draw} b 3 c No, because the teams may not be of equal ability. 12 a S = {20, 20, 20, 20, 20, 10, 10, 10, 10, 5}, 10 b S = {20, 20, 20, 20, 10, 10, 10, 10, 5} c S = {20, 20, 20, 20, 20, 10, 10, 10, 5} d S = {20, 20, 20, 20, 20, 10, 10, 10, 10} 13 Check with your teacher. 14 Check with your teacher.


Unlikely 100 10 4

2 Fifty-fifty 5 18 8 1

3 Probable 6 3 9 0

Exercise 14C — Tree diagrams

2

3

4

5

6

7

8

9 10 11 12

No. of elements 1

2

3

4

5

6

5

4

Total

3

2

1

c 6

Exercise 14D — Equally likely outcomes 1 2 3 4 5 6 7 8 9 10

No. The players are not of equal ability. No. The runners are not of equal ability. Yes. The number is chosen randomly. a Yes b No c No d Yes a True. The letter is chosen randomly. b False. On a page of writing, each letter of the alphabet does not occur equally often. B No, there are two chances of a boy and a girl, as they could be born in either order. a S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} b No a 24 b No. The chance of each combination depends on people’s taste. a 10 b No. Each horse has a different rider and ability. c Yes. Yes. The selection of horse is made randomly.

10 Quick Questions 2 1 4 6 7 8

Unlikely 2 8 1 5 Yes S = {BB, BG, GB, GG} S = {AL, AM, AJ, KL, KM, KJ} 9 9 6

3 3

10 No

Exercise 14E — Using the fundamental counting principle 1 2 3 6 9 10 11 12 13 14 15 17 18

50 625 a 4 1000 D 72 a 64 d 112 a 3 200 000 a 5408 36 a 100 000 000 c 1 000 000 96 a 1200 20 000

b 36 4 7776 7 B b e b b

144 384 240 676

c 12 5 6 760 000 8 1833 c 168

b 80 000 000 d 2 000 000 16 C b 550 19 307 328 000

Chapter review 1 Marcia will probably get a higher card. 2 a Probable b Impossible c Even chance d Unlikely e Unlikely

13C ➔

1 S = {HH, HT, TH, TT} 2 S = {11, 12, 13, 21, 22, 23, 31, 32, 33} 3 S ={BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG} 4 a 12 b No 5 S = {13, 14, 18, 31, 34, 38, 41, 43, 48, 81, 83, 84} 6 S = {DZ, DM, DK, DS, ZD, ZM, ZK, ZS, MD, MZ, S = MK, MS, KD, KZ, KM, KS, SD, SZ, SM, SK} 7 S = {AM, AS, AL, PM, PS, PL, YM, YS, YL} 8 S = {CA, CR, CL, CK, AR, AL, AK, RL, RK, LK} 9 a Check with your teacher. b S = {HHHH, HHHT, HHTH, HHTT, HTHH, S = HTHT, HTTH, HTTT, THHH, THHT, THTH, S = THTT, TTHH, TTHT, TTTH, TTTT} c 6 10 C 11 D 12 B 13 The statement is not correct because there are four elements to the sample space. The one Head and one Tail can occur in either order. 14 a S = {22, 25, 27, 28, 52, 55, 57, 58, 72, 75, 77, 78, S = 82, 85, 87, 88} b S = {25, 27, 28, 52, 57, 58, 72, 75, 78, 82, 85, 87} 15 a 12 b 24 c 24

16 a S = {TJ, TS, TR, TM, JT, JS, JR, JM, ST, SJ, SR, S = SM, RT, RJ, RS, RM, MT, MJ, MS, MR} b 20 c 2 d 6 e 12 17 a 36 b

answers

555

Answers

14E

answers

556

Answers

3 Check with your teacher. 4 Hot weather 5 Rolling a die and getting a number greater than 1 Selecting a picture card from a standard deck Selecting a blue marble from a bag containing 14 blue, 15 red and 21 green marbles Winning the lottery with 1 ticket out of 100 000 tickets sold 6 Mark is most likely to win based on past performances. 7 S = {S, A, M, P, L, E} 8 a S = {Heads, Tails} b S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, S = 16, 17, 18} c S = {club, spade, diamond, heart} d S = {black, black, black, black, white, white, S = white, green, green, green, green, green} 9 a S = {1, 2, 3, 4, 5, 6} b E = {5, 6} 10 a i 2 ii 1 b i 52 ii 32 c i 15 ii 11 11 S = {HH, HT, TH, TT} 12 36 13 a S = {56, 57, 59, 65, 67, 69, 75, 76, 79, 95, 96, 97} b 9 14 a S = {MN, MP, MR, MS, NM, NP, NR, NS, PM, S = PN, PR, PS, RM, RN, RP, RS, SM, SN, SP, S = SR} b 8 15 a S = {TCB, TCM, TCZ, TDB, TDM, TDZ, THB, S = THM, THZ, TIB, TIM, TIZ, KCB, KCM, S = KCZ, KDB, KDM, KDZ, KHB, KHM, KHZ, S = KIB, KIM, KIZ} b 15 16 a 8 b No. Each greyhound is not of equal ability. 17 a Each is equally likely, as the winning ticket is selected randomly. b Each outcome is not equally likely, as the teams have different abilities. c Each letter is not equally likely, as each letter is not used equally often. 18 759 375 19 20 20 272 21 160 22 a 10 000 b 500 23 a 100 000 000 b 94 109 400 Practice examination questions 1 B 2 C 3 C 4 A 5 C 6 a S = {RABM, RAHM, RADM, RFBM, RFHM, RFDM, RNBM, RNHM, RNDM, RVBM, RVHM, RVDM, SABM, SAHM, SADM, SFBM, SFHM, SFDM, SNBM, SNHM, SNDM, SVBM, SVHM, SVDM} b 24 c 6 d Certain e No. Each runner is of different ability. 7 a S = {GD, GK, GC, DK, DC, KC} b E = {GC, DC, KC} c 90

CHAPTER 15 Relative frequency and probability Are you ready? 1 a 0.2 2 a 30% 3 a --14-

b 0.75 b 65% b --78-

c 0.3 c 4% c --45-

d 0.85 d 12.5% 9 d ----25

4 a 25%

b 60%

c 62.5%

d 33 1--3- %

5 a Impossible b Fifty-fifty d Impossible e Fifty-fifty 4 6 a ----b 2--5c 1--315

c Certain d 1

Exercise 15A — Relative frequency 1 4 5 6 7 8 10 11 12 13 14 15 16

0.74 2 0.79 3 0.375 a 0.45 b 0.55 4% a 0.03 b 0.97 a 0.96 b 0.04 A 9 A a 0.525 b 0.4375 c 0.0375 a 6.67% b 80 a 0.02 b $400 Yes, the relative requency is 27%. a 2.5% b 51.5% c 17.5% 40 000 km a Result Number Win Loss Draw

15 14 11

b Win = 0.375, Loss = 0.35, Draw = 0.275

Exercise 15B — Single event probability

1 S = {Heads, Tails}, 1 2 a S = {1, 2, 3, 4, 5, 6}, 1 b S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, 3 c S = {a, b, c, d, e, . . ., y, z}, 5 d S = {Sun, Mon, Tue, Wed, Thu, Fri, Sat}, 2 e S = {Jan, Feb, Mar, . . ., Nov, Dec}, 3 3 a 26, 52 b 1, 15 c 1, 44 d 5, 1500 e 3, 11 4 1--25 a

1 --6

b

1 --6

c

1 --2

d

1 --2

e

2 --3

f

1 --3

6 a

1 -----45

b

1 -----45

c

22 -----45

d

23 -----45

e

1 --5

f

1 --3

g

19 -----45

g

2 --9

i

2 -----15

7 a

1 -----52

b

1 -----13

c

1 --4

1 --2

e

12 -----52

f

3 -----13

1 -----12

b

4 -----12

c

7 -----12

d 8 a

9 a

1 --4

b

1 --4

c

1 --4

d

3 --4

e

1 --2

f

3 --4

11 a

1 --4

b

3 --4

c

1 --4

d

1 --4

e

2 --4

c 12 a b c d

10 -----12

10

12 C 16 a

10 Quick Questions 2 1

13 C

14 C

5 -----------1000

b

4 --------999

17 a

3384 ------------------160 000

b

6768 ------------------160 000

18 a

1 --5

d

1 --5

15 D c

10 152 ------------------160 000

b

1 --5

c

4 --5

e

4 --5

f

1 --5

19 Check with your teacher. 20 a Because there are two numbers which could go in the last place. b A number greater than 400

10 Quick Questions 1 1 --6

1

2

6 True

1 -----13

3

7 False

7 -----11

4

8 0.2

33 --------100

5

9 0.75

5 --9

10 0.5%

Exercise 15C — Writing probabilities as decimals and percentages 1 2 3 4 5 6 7 8 11 12

0.5 37.5% a 0.5 a 0.17 a 16.7% a 0.02 d 0.5 a 1.9% d 50% A a 0.2 d 0.8 a 20%

b 50% b 0.5 b 50% b 0.25 e 0.23 b 25% e 23.1% 9 D b 0.4

c 0.33 c 83.3% c 0.08 c 7.7% 10 B c 0.6

b 40%

c 40%

Exercise 15D — Range of probabilities 1 a d g 2 a c e

Even chance b Probable c Unlikely Certain e Probable f Unlikely Probable h Impossible i Unlikely 0, impossible b 1, certain 3 26 --- , even chance - , even chance d ----6 52 40 -----52

, probable

g 0, impossible

f

4 -----52

, unlikely

h

6 -----12

, even chance

3 Check with your teacher. 7 9 8 6 ------ , ------ , ------ , ------ , -----4 13 20 13 18 19 25 5 A, D, C, B, E 6 D 50 9 a ----------2000 10

98 --------100

7 B

8 D b Unlikely

1 --2

1 --4

3

6 0.154 10 Probable

4 --9

, 1--2- ,

10 -----19

4 62.5%

7 23.1%

8 1

Exercise 15E — Complementary events 1 a S = {1, 2, 3, 4, 5, 6} b 1--67 -----20

2 a i

ii

8 -----20

c 1 5 iii ----20

b1 3 a Losing a race b Failing a test c Your birthday not falling on a Monday 4 A coin landing Heads A coin landing Tails An odd number on a die An even number on a die A picture card from a A spot card from a standard deck standard deck A red card from a A black card from a standard deck standard deck Winning 1st prize in a Not winning 1st prize in raffle with 100 tickets the raffle Making the last 4 teams A team not making the in a 20 team last four tournament 3 1 1 1 - , --- , --------- , --5 a 1--2- , 1--2- , ----13 2 100 5 b

1 --2

, --12- ,

10 -----13

, --12- ,

99 --------100

,

4 --5

c 1 6 a Rolling an odd number b Rolling a number less than 4 c Rolling a number greater than 2 d Not rolling a 6 e Rolling a 1 7 a Choosing an even-numbered ball b Choosing a ball numbered greater than 19 c Choosing a ball that has a number less than 24 d Choosing a ball that is not a multiple of 5 8 a Selecting a coloured ball b Selecting a black ball c Not selecting a pink ball 9 A 10 C 11 a 1--8b 7--812 a

1 --5

b

4 --5

13 a

9 -----20

b

11 -----20

c

3 -----20

d

17 -----20

e

3 --5

f

2 --5

2 --5

b

3 -----10

c

3 --5

7 -----10

e

3 -----10

d

15A ➔

b Even chance

2

5 0.375 9 0

14 a

, very probable

11 a Very unlikely

Probable d Unlikely True, as there are 4 aces from 52 cards in the deck. False, as each letter does not occur equally often. False, as each student is not of equal ability. True, as the name is chosen randomly.

answers

557

Answers

15E

answers

558

Answers

15 a

7 -----20

b

13 -----20

16 a

3 -----25

b

21 -----50

17 0.3 18 a 91%

29 -----50

c

5

8 -----15

6

7 -----15

8

1 --5

9

4 --5

0.02 a 0.15 b $750 S = {1, 2, 3, 4, 5}, E = {3, 4, 5} 1 1 a ----b ----25 25

5 a d

1 -----10

c

3 --4

7 a

1 -----24

b

3 --4

c

1 --4

3 --4

e

1 --2

1 --------400

b

4 -----------1999

8 a 4

1 ---------------10 000

7

3 -----10 4 --5

10

Chapter review

1 --5

b

d

1 S = {heart, club, diamond, spade} 1 2 ----3 --1810

d

7 -----20

b 9%


1 2 3 4

6 a

9 a 0.025 10 a 50%

b 0.5 b 33 1--3- %

c 0.75 c 66 2--3- %

11 Probable 12 7--8- . It is probable that the car will have a defect. 13 a Tossing a coin that lands Heads b Rolling a die and getting a number greater than 4 c Not choosing a blue ball 3 7 14 a ----b ----10 10 15

5 -----12

16

18 -----25

c

13 -----25

f

16 -----25

Practice examination questions 1 B 2 C 3 D 5 a --15b --25-

6 a 0.7 b 0.3 c The missile will probably hit its target.

e

9 -----25

1 -----52

b

1 -----13

c

1 --4

1 --2

e

4 -----13

f

10 -----13

4 B c

3 --5

559

Index

Index abnormal conditions (sampling) 181 addition, algebraic expressions 125–6 adjacent 423 algebraic expressions adding and subtracting like terms 125–6 multiplication and division 131–3 algebraic formulas, substitution into 127 allowances 6 analysing data 148 and drawing conclusions 159 angle of depression 445, 446 angle of elevation 445–6, 447 annual leave 26 annual leave loading 26 appreciation 264–5 area 79–80 calculating irregular areas from a field diagram 85–6 area formulas 79 bar graphs 279 bias 180, 183–4 in questionnaire design 180 sampling 180–1, 183–4 statistical interpretation 181 bimodal scores 344 box-and-whisker plots 309–11 budgets 31–4 capacity, units of 48 casual rate 7 categorical data 186, 187 census 169 chance common descriptions of 462 informal description of 457–9 see also probability experiments classes 274 collecting data 151, 169–72 column graphs 156–7, 279 commission 11–13 complementary events 507–8 compound interest 241–4 calculating from a table of compounded values 248–50 compound interest formula 243, 264 compound interest functions, graphing 253–5 compound value interest factor (CVIF) 248–50 compounded value of an investment (CV) 241–4 calculating compound interest from a table of compounded values 248–50 compounding period 243 concentration (of a substance) 63–4 cones, volume 103 congruent figures 366 constant of variation 216 Consumer Price Index (CPI) 264 continuous data 187

cosine ratio 426–7 calculating 427 cube surface area 92 volume 97 cumulative frequency 285 calculating median from 341–2 cumulative frequency histogram 285 cumulative frequency polygon 285 cylinders, volume 98–9 data 150 analysing 149, 159 collecting 151, 169–72 displaying 156–8 organising 153–4 types of 186–8 database 169 debentures 234 decile 296 decreasing function 206 deductions 25 see also tax deductions dependent variable 201, 205, 206 direct linear variation 216 direct proportion 216 discrete data 187 displaying data 156–8 dividend yield 257 dividends 257–8 division algebraic expressions 132–3 index law 132 dot plots 277 double time 19 drawing conclusions 159 drawing graphs using gradient and intercept

211–14

elements (in a sample space) 462–3 enlargement factor 369 equally likely outcomes 472–3 equations arising from substitution 139–41 estimating populations 191–2 events (probability) 457–9 external sources (of data) 151 favourable outcomes 463 field diagrams 85–6 finding an unknown side (trigonometry) finding angles (trigonometry) 438–42 five-number summaries 308–10 frequency (of event) 458 frequency histograms 283–4 frequency polygon 284–5 frequency tables 273–5 calculating mean from 324–6 functions 201 fundamental counting principle 476–8

431–4

560

Index

future value of an investment

241–4

gradient and intercept 205–9 graphing 201–3 piecewise 219 point of intersection of two linear functions step functions 218–19 lower extreme 308 lower quartile 293, 304, 308

general term of the sequence 119–22 gradient 205–7 and constant of variation 216 and intercept to draw graphs 211–14 graphing compound interest functions 253–5 linear functions 201–3 share performance 260–2 simple interest functions 236–9 tax functions 409 variations 216–17 graphs bar 279 column 156–7, 279 dot plots 277 drawing using gradient and intercept 211–14 line 278 radar charts 279–80 sector 158, 278 statistical 283–7 types of 277–80 gross pay 385 additions and deductions from 25–6 grouped data 273 calculating mean from 326–7 groups 274 GST (Goods and Services Tax) 405–6 histograms 283–4 history of mathematics house plans 374–6 hypotenuse 418

mass, units of 48 maximum error 52–3 mean 323–7, 350 calculating from a frequency table 324–6 calculating from grouped data 326–7 measurement maximum error 52–3 percentage error 53 significant figures 57–60 units of 47–50 measures of central tendency 323 median 293, 295–6, 304, 308, 341–3, 350 calculating from cumulative frequency 341–2 Medicare levy 395–7, 404 misleading graphs 181, 182–3 mode 343–4, 350 multi-stage events 467 multiplication algebraic expressions 131–3 index law 132 negative gradient 206 net pay 25, 385 nets, solid shapes 89–90 nominal data 187 non-random sampling methods 180 non-response rate 181 number pattern notation 119–22 number patterns 115–16

417

income 3 income tax 385, 404 see also tax payable (PAYE) increasing function 206 independent variable 201, 205, 206 index laws 131–3 indirect tax 405 inflation 264 inflation rate 264 information 150 intercept 207–8 and gradient to draw graphs 211–14 interest 231 interest rate 231 internal sources (of data) 151 International System of Units (SI system) 47 interquartile range 293–4, 295–6, 305 investment bonds 234 irregular areas, calculating from a field diagram leading questions 180 length, units of 47 like terms, adding and subtracting 125–6 line graphs 278 linear equations, solving 134–7 linear functions drawing using gradient and intercept 211–14

221–2

observation 151 offset 85 ogive 285 opposite 423 ordinal data 187 ordinary rate (wages) 6 organising data 153–4 outcome 457 overtime 19–21

85–6

parallelogram, area 80 particular responses (sampling) 181 PAYE tax 385, 390 calculating 398–402 payment by piece 16–17 payment methods 3 penalty rates 19 per annum 3 percentage change 67 percentage error 53 pie charts 158, 278 piecewise linear functions 219 piecework 16–17 polls 169

Index

polygon, frequency 284–5 population standard deviation 333–4 populations 169 characteristics 174–6 estimating 191–2 posing questions 150 positive gradient 206 present value (PV) 248 principal 231 prisms 89 surface area 92–3 volume 97–8 privacy and ethical issues 163 probability 457, 492 range of 502–3 writing as decimals and percentages 499 probability experiments complementary events 507–8 equally likely outcomes 472–3 multi-stage events 467–9 relative frequency 487–8 sample space 462–3 single-event probability 492–4 using the fundamental counting principle 476–8 proportional diagrams 450 pyramids 89 surface area 93 volume 103 Pythagoras of Samos 417 Pythagoras’ theorem 417, 418–20 quality control 160–1 quantitative data 186–7, 273 quartiles 293 questioning 151 privacy and ethical issues 163 questionnaire design, bias in 180 questionnaires 169 radar charts 279–80 random sample 170–1 range 292–3, 295 rate of change 205 rates 61–4 ratios 69–71 rectangle, area 79 rectangular prism surface area 92–3 volume 98 relations 201 relative error 52–3 relative frequency 487–8 report writing 159 retainer 13 rhombus, area 80 right-angled triangles angles of elevation and depression 445–7 calculating trigonometric ratios 423–8 finding an unknown side 431–4 finding angles 438–42 proportional diagrams 450 Pythagoras’ theorem 418–20

royalties

11

salary payments 3–4 sample 169 sample space 462–3 favourable outcomes 463 tree diagrams 467–9 sample standard deviation 334–5 sampling bias 180–1, 183–4 sampling methods 170–2 scale factors 366, 369, 374–6 scores 273 sector graphs 158, 278 sequences 115–16 general term 119–22 share dividends 257–8 share performance, graphing 260–2 shares 257 significant figures 57–60 similar figures 365–7 to solve problems 371–2 similar triangles 365, 370 simple interest 231–4 simple interest formula 231 simple interest functions, graphing 236–9 simultaneous equations 221–2 sine ratio 425–6 calculating 426 single event probability 492–4 SOHCAHTOA 433 solid shapes 89–90 spheres, volume 104 square, area 70 standard deviation 333–6 interpreting 335–6 population 333–4 sample 334–5 statistical graphs 283–7 statistical interpretation bias 181 statistical organisations 164 statistical processes 150 collecting data 151 organising data 153–4 posing questions 150 statistics 149 stem-and-leaf plots 302–5 step functions 218–19 strata 171 stratified sample 171 substitution equations arising from 139–41 into algebraic formulas 127 subtraction, in algebraic expressions 125–6 summary statistics 350–1 surface area 92–4 surveys 169–70 privacy and ethical issues 163 sampling methods 170–2 systematic sampling 171 tangent ratio 423–5 calculating 424–5

561

562

Index

target populations 169 tax deductions 385–7 tax functions, graphing 409 tax payable (PAYE) 385, 390, 398–402 tax return 385 taxable income 385, 390–1, 393–4 and Medicare levy 395–7, 404 tax payable 398–402, 404 telephone polls 181 time, units of 49 time and a half 19 trapezium, area 80 traverse survey 85 tree diagrams 467–9 equally likely outcomes 473 trial (probability experiments) 487 triangle, area 79–80 trigonometric formulas 424, 426, 427, 433 trigonometric ratios calculating 423–8 finding an unknown side 431–4 finding angles 438–42

units of measurement 47–50 capacity 48 length 47 mass 48 time 49 upper extreme 308 upper quartile 293, 304, 308 value added tax (VAT) 406 vanishing point 900 variations, graphing 216–17 volume cones 103 cylinders 98–9 prisms 97–8 pyramids 103 spheres 104 wages 6–7 working overtime 19–21 written reports 159 y-intercept

ungrouped data

273

207

Preliminary General Maths Text Book

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