ROBERT ROWLAND
4TH EDITION
MATHS QUEST Preliminary Mathematics General
Fourth edition published 2013 by John Wiley & Sons Australia, Ltd 42 McDougall Street, Milton, Qld 4064 First edition published 2000 Second edition published 2008 Third edition published 2011 Typeset in 10/12pt Times LT Std © John Wiley & Sons Australia, Ltd 2000, 2008, 2011, 2013 The moral rights of the author have been asserted. National Library of Australia CataloguinginPublication data Author: Title: Edition: ISBN: Target Audience: Subjects: Dewey Number:
Rowland, Robert, 1963– Maths quest preliminary mathematics general / Robert Rowland. 4th ed. 978 1 118 51176 3 (pbk.) 978 1 118 51173 2 (eBook.) 978 1 118 51177 0 (flexisaver.) For secondary school age. Mathematics — Textbooks. Mathematics — Study and teaching (Secondary) 510
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Contents Introduction vi About eBookPLUS viii Acknowledgements ix
Chapter 3
Taxation
3A Calculating allowable deductions
Chapter 1
Earning money
3B
1
1A Calculating salary payments
Exercise 1A
2
3D
1C
3E
1E 1F 1G
1H ■ ■ ■ ■
3C
1
1B Calculating wages
1D
3 Exercise 1B 5 Commission and royalties 8 Exercise 1C 9 Payment by piece 12 Exercise 1D 13 Working overtime 14 Exercise 1E 16 Government allowances 19 Exercise 1F 20 Additions to and deductions from gross pay 22 Exercise 1G 23 Budgeting 27 Exercise 1H 29
3F ■ ■ ■ ■
Statistics and society, data collection and sampling 109 4A Statistical processes
4C
Chapter 2
Exercise 2A
2D
2E 2F 2G ■ ■ ■ ■
4D
43
functions 48 Exercise 2B 50 Calculation of compound interest 53 Exercise 2C 55 Calculating compound interest from a table of compounded values 57 Exercise 2D 60 Share dividends 62 Exercise 2E 62 Graphing share performance 64 Exercise 2F 66 Inflation and appreciation 68 Exercise 2G 69
Summary 72 Chapter review 73 ICT activities 76 Answers 77
4E
43
46
2B Graphing simple interest
2C
Summary 103 Chapter review 104 ICT activities 106 Answers 107
4B
2A Calculation of simple interest
81
Exercise 3A 83 Taxable income 85 Exercise 3B 87 Medicare levy 90 Exercise 3C 90 Calculating tax 91 Exercise 3D 94 Calculating GST and VAT 97 Exercise 3E 98 Graphing tax functions 101 Exercise 3F 101
Chapter 4
Summary 35 Chapter review 36 ICT activities 39 Answers 40
Investing money
81
■ ■ ■ ■
109 Exercise 4A 112 Target populations and sampling Exercise 4B 116 Population characteristics 118 Exercise 4C 120 Types of data 122 Exercise 4D 123 Bias 125 Exercise 4E 128
114
Summary 132 Chapter review 133 ICT activities 136 Answers 137
Chapter 5
Displaying single data sets
141
5A Types of graphs and stemandleaf
plots 141 Exercise 5A 144 5B Frequency tables and cumulative frequency 146 Exercise 5B 149 5C Range and interquartile range, deciles and percentiles 154 Exercise 5C 158 5D Fivenumber summaries 164 Exercise 5D 166
5E Appropriate graphs and misuse of graphs
Exercise 5E ■ ■ ■ ■
173
169
Chapter 9
Similar figures and trigonometry
Summary 177 Chapter review 178 ICT activities 182 Answers 183
9A Similar f igures and scale factors
285 Exercise 9A 287 9B Solving problems using similar f igures 291 Exercise 9B 293 9C Calculating trigonometric ratios 297 Exercise 9C 301 9D Finding an unknown side 302 Exercise 9D 305 9E Finding angles 308 Exercise 9E 309 9F Applications of rightangled triangles 312 Exercise 9F 315 Proportional diagrams 316
Chapter 6
Summary statistics
189
6A Calculating the mean
189 Exercise 6A 193 6B Median and mode 198 Exercise 6B 201 6C Standard deviation 205 Exercise 6C 208 6D Best summary statistics 212 Exercise 6D 214 ■ ■ ■ ■
■ ■ ■ ■
Summary 220 Chapter review 221 ICT activities 226 Answers 227
Probability
7C 7D 7E ■ ■ ■ ■
231
Exercise 7A 233 Units of measurement and scientific notation 234 Exercise 7B 237 Ratios 239 Exercise 7C 240 Rates 242 Exercise 7D 245 Percentage change 247 Exercise 7E 248
Summary 250 Chapter review 251 ICT activities 253 Answers 254
Perimeter, area and volume Exercise 8A
259
8B Area of plane shapes
262 Exercise 8B 264 8C Field diagrams 268 Exercise 8C 270 8D Applications of area 271 Exercise 8D 271 8E Volume of prisms 273 Exercise 8E 275 ■ ■ ■ ■
iv
Summary 279 Chapter review 280 ICT activities 282 Answers 283
Contents
10B 10C 10D 10E 10F 10G
10H
Chapter 8
8A Perimeter of plane shapes
325
10A Multistage events
231
7A Measurements as approximations 7B
Summary 317 Chapter review 318 ICT activities 321 Answers 322
Chapter 10
Chapter 7
Measurement
285
257
257
10I ■ ■ ■ ■
325 Exercise 10A 327 The fundamental counting principle 329 Exercise 10B 331 Probability statements 333 Exercise 10C 335 Relative frequency 336 Exercise 10D 338 Equally likely outcomes 341 Exercise 10E 342 The probability formula 343 Exercise 10F 345 Writing probabilities as decimals and percentages 348 Exercise 10G 349 Range of probabilities 351 Exercise 10H 353 Complementary events 356 Exercise 10I 357
Summary 360 Chapter review 361 ICT activities 364 Answers 365
Chapter 11
Algebraic manipulation
369
11A Operations with algebraic expressions
369 Exercise 11A 371 11B Further multiplication and division 372 Exercise 11B 373 11C Expanding and simplifying algebraic expressions 374 Exercise 11C 375
11D Substitution
376 Exercise 11D 377 11E Solving linear equations Exercise 11E 381
■ ■ ■ ■
13F Digital transfer rates
439 Exercise 13F 439 13G Random selection 440 Exercise 13G 441
379
Summary 384 Chapter review 385 ICT activities 387 Answers 388
■ ■ ■ ■
Chapter 12
ICT ACTIvITIEs — ProjECTsPlus
Modelling linear relationships 12A Graphing linear functions 12B 12C
12D 12E
■ ■ ■ ■
Summary 442 Chapter review 443 ICT activities 445 Answers 446
391
pro0135 The cost of piracy
391
Chapter 14
Exercise 12A 393 Gradient and yintercept 395 Exercise 12B 398 Drawing graphs using gradient and intercept 400 Exercise 12C 403 Simultaneous equations 404 Exercise 12D 407 Practical applications of linear functions 409 Exercise 12E 411
Focus study: Mathematics and driving 451 Section 1 — Costs of purchase and insurance 451 14A Depreciation of new cars 452 Exercise 14A 453 14B Insurance and stamp duty 454 Exercise 14B 456 14C Financing your vehicle 457 Exercise 14C 460 Section 2 — Running costs and depreciation 462 14D Fuel costs 462 Exercise 14D 463 14E Straight line depreciation 465 Exercise 14E 466 14F Declining balance method of depreciation 467 Exercise 14F 468 Section 3 — Road safety 470 14G Speed and stopping distances 470 Exercise 14G 473 14H Blood alcohol concentration (BAC) 474 Exercise 14H 476
Summary 414 Chapter review 415 ICT activities 418 Answers 419
Chapter 13
Focus study: Mathematics and communication 425 Section 1 — Mobile phone bills 425 13A Calculating costs 425 Exercise 13A 427 13B Reading a mobile phone bill 430 Exercise 13B 431 13C Phone usage 432 Exercise 13C 433 13D Choosing the best mobile phone plan for your needs 435 Exercise 13D 435 Section 2 — Digital download and file storage 13E Units of storage 436 Exercise 13E 438
448
■ ■ ■ ■
Summary 478 Chapter review 479 ICT activities 482 Answers 483
ICT ACTIvITIEs — ProjECTsPlus
436
pro0136 Causes of accidents
486
Glossary 489 Index 492
Contents
v
Introduction Maths Quest Preliminary Mathematics General is the fourth edition in a series specifically designed for the Mathematics General Stage 6 Syllabus 2012. There are five strands and two Focus studies: • Strand: Financial Mathematics • Strand: Data and Statistics • Strand: Measurement • Strand: Probability • Strand: Algebra and Modelling • Focus study: Mathematics and Communication • Focus study: Mathematics and Driving There is a suite of resources available: • a student textbook with accompanying eBookPLUS • a teacher edition named eGuidePLUS • flexisaver versions of all print products • a Solutions Manual containing fully worked solutions to every question.
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.
MATHS QUEST Preliminary Mathematics General
4TH E D ITION
ROWLAND
Exercises contain many carefully graded skills and application problems, including multiplechoice questions. Crossreferences to relevant worked examples appear beside the first ‘matching’ question throughout the exercises. Each exercise also contains new further development questions.
4T H E D I T I ON
Worked examples in a Think/Write format provide a clear explanation of key steps and suggest a process for solutions. Technology is incorporated into worked examples to demonstrate judicious use.
Preliminary Mathematics General
Clear, concise theory sections contain worked examples and highlighted important text.
ROBERT ROWLAND
Investigations, including spreadsheet investigations, provide further learning opportunities through discovery. 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 (multiplechoice, short answer and analysis) that help consolidate students’ learning of new concepts. Technology is fully integrated, in line with Board of Studies recommendations. As well as graphics calculators, Maths Quest features spreadsheets and interactivities.
student website — eBookPlus The accompanying eBookPLUS contains the entire student textbook in HTML plus additional exercises. Students may use the eBookPLUS on any device, and cut and paste material for revision or the creation of notes for exams, tablets. WorkSHEET icons link to editable Word documents that may be completed on screen or printed and completed by hand. Interactivity icons link to dynamic animations which help students to understand difficult concepts. Test yourself tests are also available and answers are provided for students to receive instant feedback.
vi
Introduction
Teacher edition eGuidePlus The eGuidePLUS contains everything in the eBookPLUS and more. Two tests per chapter, fully worked solutions to WorkSHEETs, the work program and other curriculum advice in editable Word format are provided. Maths Quest is a rich collection of teaching and learning resources within one package. Maths Quest Preliminary Mathematics General provides ample material, such as exercises, analysis questions, investigations, worksheets and technology files, from which teachers may set assessment tasks.
Maths Quest Preliminary Mathematics General solutions Manual
MATHS QUEST Preliminary Mathematics General SOLUTIONS MANUAL
ANITA CANN
4 T H EDIT IO N
The Solutions Manual contains fully worked solutions to every question in the student textbook. Students are provided with explanations as well as the solution process. The PDFs of the Solutions Manual are available on eBookPLUS.
ROBERT CAHN
Introduction
vii
About eBookPLUS Next generation teaching and learning This book features eBookPLUS: an electronic version of the entire textbook and supporting multimedia resources. It is available for you online at the JacarandaPLUS website (www.jacplus.com.au).
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About eBookPLUS
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Acknowledgements
ix
ChAptEr 1
Earning money ChAptEr ContEntS 1A 1B 1C 1D 1E 1F 1G 1h
Calculating salary payments Calculating wages Commission and royalties Payment by piece Working overtime Government allowances Additions to and deductions from gross pay Budgeting
1A Calculating salary Methods of payment
payments
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
There are 26 fortnights in a year, so we divide $46 800 by 26.
2
Evaluate.
WritE
Fortnightly pay = $46 800 ÷ 26 = $1800
ChAptEr 1 • Earning money
1
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
WritE
1
There are 12 months in a year, so multiply $3500 (monthly pay) by 12.
2
Evaluate.
Annual salary = $3500 × 12 = $42 000
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
WritE
1
Calculate the weekly pay by dividing the salary by 52.
Weekly pay = $41 560 ÷ 52 = $799.23
2
Calculate the hourly rate by dividing the weekly pay by 42.
Hourly rate = $799.23 ÷ 42 = $19.03
Exercise 1A
Calculating salary payments
1 WE1 Toni is paid a salary of $44 200 per annum. Calculate Toni’s fortnightly pay. 2 Roger is paid a salary of $49 920 per annum. Calculate Roger’s weekly pay. DiGitAl DoC Spreadsheet doc1439 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. 6 Copy and complete the table below for food production employees.
Annual salary $30 000 $39 500 $42 250 $54 350 $86 475
Weekly pay
Fortnightly pay
Monthly pay
7 WE 2 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. 2
Maths Quest Preliminary Mathematics General
10 MC 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 WE 3 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. 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. 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.
Further development 15 Calculate the monthly salary that is equivalent to earning $500 per week. 16 Dan earns $1045 per fortnight while Brian earns $2100 per month. Brian claims that he earns $10 per
month more than Dan. Explain why Brian is not correct. 17 Sandra is a lawyer who works an average 40 hours per week for her $78 500 per annum salary.
Stephanie also works as a lawyer and receives the same salary as Sandra. Stephanie asks for a salary increase on the basis that she must work 45 hours per week to complete her workload. a What would be an equivalent salary that Stephanie would ask for? b Is it necessarily correct to say that since Stephanie works more hours than Sandra that she works harder than Sandra? 18 The monthly salary that Alan earns in his job as a sports broadcaster is $4550. Calculate the equivalent
fortnightly salary. 19 Michelle is earning $74 000 per year at the time that she retires. In retirement she is paid a fortnightly
pension that is equivalent to 65.75% of her salary at retirement. Calculate the amount of Michelle’s fortnightly pension. 20 Garry’s superannuation package allows for a lump sum payment of 8.3 times his final salary. a At retirement Garry’s gross weekly salary was $765.70. Calculate his annual salary. b Calculate the amount of Garry’s lump sum payment.
1B
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.
ChAptEr 1 • Earning money
3
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
Multiply $13.65 (the hourly rate) by 38 (the number of hours worked).
WritE
Wage = $13.65 × 38 = $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
Divide $483.84 (the wage) by 42 (number of hours worked).
WritE
Hourly rate = $483.84 ÷ 42 = $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. 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 35hour 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
4
Calculate Ryan’s normal pay by multiplying $9.45 (hourly rate) by 35 (number of hours worked).
Maths Quest Preliminary Mathematics General
WritE
Normal pay = $9.45 × 35 = $330.75
Allowance = $0.86 × 12 = $10.32
2
Calculate the wet weather allowance by multiplying 0.86 (the wet weather allowance) by 12 (number of hours worked in the wet).
3
Add the normal pay to the wet weather allowance to calculate the total pay.
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.
Exercise 1B
Calculating wages
1 WE 4 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.
DiGitAl DoC Spreadsheet doc1439 payroll calculations
2 Copy and complete the table below by calculating the wage of each of the workers.
Name
Hourly Hours rate worked
A. Smith
$14.52
40
B. Brown
$16.45
38
N. Tran
$15.95
37.5
A. Milosevic $20.10
41
$18.04
36
L. McTavish
Wage
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 fulltime workers
to compensate for their lack of holidays and sick leave. a A fulltime 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 MC Which of the following workers earns the highest wage for the week? A B C D
Dylan, who works 35 hours at $13.50 per hour Lachlan, who works 37 hours at $12.93 per hour Connor, who works 38 hours at $12.67 per hour Cameron, who works 40 hours at $12.19 per hour
7 WE 5 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. ChAptEr 1 • Earning money
5
9 Copy and complete the table below.
Name A. White B. Black C. Green D. Brown E. Scarlet F. Grey
Wage $416.16 $538.80 $369.63 $813.96 $231.30 $776.72
Hours worked 36 40 37
Hourly rate
$19.38 $15.42 $20.44
10 Calculate the hourly rate of a casual worker who earns $250.80 for 20 hours work. 11 MC Which of the following workers is paid at the highest hourly rate? A B C D
Melissa, who works 35 hours for $366.45 Belinda, who works 36 hours for $376.20 April, who works 38 hours for $399.76 Nicole, who works 40 hours for $419.60
12 MC Which of the following people worked the greatest number of hours? A B C D
SuLi, who earned $439.66 at $11.57 per hour Denise, who earned $576.00 at $14.40 per hour Vera, who earned $333.20 at $9.52 per hour Camille, who earned $707.25 at $17.25 per hour
13 WE 6 Richard works as an electrical linesman and is paid $10.94 per hour for a 38hour week. When
he has to work at heights he is paid a 46c per hour ‘height allowance’. Calculate Richard’s pay in a week where 15 hours are spent working at heights. 14 Ingrid works as an industrial cleaner and is paid $14.60 per hour for a 35hour 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 40hour week. Calculate
Zoe’s hourly rate of pay. 17 Harlan earns $23.80 per hour in his job as a teacher’s aide. He works school hours which are 8:30 am
to 3:30 pm 5 days per week. a Calculate Harlan’s fortnightly wage. b Julie is a teacher on a salary of $81 000 per year. In addition to school hours Julie does 20 hours of preparation and marking per week during a school term. Calculate the difference in hourly rate of pay for Julie and Harlan.
Further development 18 Juanita has a casual job and is paid $10.80 per hour on weekdays and $14.85 per hour on weekends.
Calculate Juanita’s weekly pay when she works from 5:00 pm until 10:00 pm on Thursday and from 8:30 am until 1:00 pm on Saturday. 19 Naomi works the following hours in one week.
Monday 5:30 pm to 9:00 pm Tuesday 6:45 pm to 9:00 pm Wednesday 8:15 pm to 10:30 pm Naomi is paid $12.45 per hour up until 9:00 pm and $15.16 per hour after that. Calculate Naomi’s total pay for the week. 20 Katie earns an annual salary of $56 750 and works an average 42 hours per week. a Bill is paid the equivalent hourly rate as a wage. Calculate this amount. b Calculate the number of hours per week that Bill would need to work to earn $60 000 per year. 6
Maths Quest Preliminary Mathematics General
21 John works a 40 hour week at a rate of $12.76 per hour. John receives a 5% pay rise, so decides to
reduce his working hours by 5%. a How many hours will John now work per week? b John believes that his gross pay will remain the same. Is John correct? Use calculations to justify your answer. 22 Tracey works a 40 hour week at a rate of $18.49 per hour. a Show that Tracey’s gross weekly pay is $739.60 b Tracey receives a 5% pay rise. • Tracey’s employer says the new hourly rate is $19.41. • Tracey says that her total pay should now be $776.58. Explain how each figure is calculated. c What is the total pay under the employer’s plan? d What is the hourly rate under Tracey’s plan? Is it possible to pay Tracey at this hourly rate? e Who is correct? Justify your answer. 23 Frank is a butcher who earns $14.92 per hour for a 38 hour working week. a Calculate Frank’s total weekly pay. b Frank receives a 3.9% wage increase. Calculate Frank’s new hourly rate of pay correct to 4 decimal places. c Frank wants to earn $650 per week. Calculate the number of hours that he needs to work to achieve this goal.
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 Preliminary Mathematics General eBookPLUS.
DiGitAl DoC Spreadsheet doc1440 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). ChAptEr 1 • Earning money
7
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.
1C
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.
WorkED ExAMplE 7
Jack is an author who is paid a royalty of 12% of all sales. Calculate the royalty that Jack earns in a year where his book has sales to the value of $150 000. think
Calculate 12% of $150 000.
WritE
Commission = 12% of $150 000 = 12 ÷ 100 × $150 000 = $18 000
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.
8
Maths Quest Preliminary Mathematics General
think
WritE
1
Calculate 5% of $75 000.
5% of $75 000 = $3750
2
Calculate the balance of the sale.
Balance = $235 000 − $75 000 = $160 000
3
Calculate 2.5% of $160 000.
2.5% of $160 000 = $4000
4
Add up each portion to calculate the commission.
Commission = $3750 + $4000 = $7750
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.
Commission = 2% of $12 250 = 2 ÷ 100 × 12 250 = $245
2
Add the $250 to the commission to calculate her pay.
Pay = $250 + $245 = $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
WritE
1
Calculate the amount on which commission is to be paid.
$84 000 − $50 000 = $34 000
2
Find 2% of this amount.
Commission = 2% of $34 000 = 2 ÷ 100 × $34 000 = $680
3
Add the $300 to the commission to calculate Tony’s pay.
Pay = $300 + $680 = $980
Exercise 1C
Commission and royalties
1 WE 7 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.
3 Darren is a songwriter who is paid a royalty. If Darren has songs that have sales that total $400 000,
calculate his royalty if it is paid at a rate of: a 1% b 3%
c 3.4%. ChAptEr 1 • Earning money
9
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. 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. 6 MC 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 Lindsey is a salesperson who is paid 1.5% commission on all sales. a Calculate Lindsey’s earnings in a week where her sales total $95 000. b Calculate the sales that Lindsey must make in order to earn $1650. DiGitAl DoC Spreadsheet doc1443 Calculations with percentages
8 George is paid royalty on all sales. Given that George earns $450 on sales of $10 000: a calculate the percentage royalty that George is paid b calculate the royalty that George would be paid on sales of $15 400 c calculate the sales that George would need to make in order to earn $810. 9 MC 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% 10 WE 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. 11 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.
12 Using the sliding scale for commission shown in question 10, calculate the commission on a property
that sells for: a $90 000
b $140 000
c $600 000.
13 WE 9 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. 14 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. 15 MC A group of sales representatives each have $10 000 in sales for a week. Who earns the most money? A B C D
10
Averil, who is paid a commission of 8% Bernard, who is paid $250 plus 6% commission Cathy, who is paid $350 plus 4% commission Darrell, who is paid $540 plus 2.5% commission
Maths Quest Preliminary Mathematics General
16 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? 17 WE10 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. 18 Neville is a doortodoor 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.
Further development 19 MC 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 20 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? 21 Fiona is an auctioneer and has the choice of payment packages.
Package A — 4.4% commission on sales. Package B — An annual salary of $87 500 Package C — A wage where she is paid at a rate of $40 per hour. Fiona found that in the past year sales totalled $2 million and involved an average 44 hours work per week. Assume Fiona works 52 weeks per year. a Based on last year’s figures, what package should Fiona select? b What would be the minimum value of sales required to make Package A the best option? c What is the minimum number of hours that needs to be worked each week to make Package A the best option? d What reasons might Fiona’s employer have for wanting her to accept Package A? 22 J. L. Booker real estate agents pay its salespeople a commission of 0.25% of all sales. Ray Black real
estate agent pays a commission of 0.5% on all sales above $200 000. a Calculate the commission earned on selling a block of land for $250 000 for J. L. Booker compared to selling a block of land for Ray Black. b Calculate the commission earned on selling a house for $500 000 for J. L. Booker compared to selling a block of land for Ray Black. c Find the sale price for which the salesperson would earn the same commission under either employer. 23 Tom earns $1650 in a week. This includes a retainer of $350 and a commission component. a If Tom is paid a flat rate of commission and sales total $65 000, calculate the rate of commission
paid. b If Tom receives commission only for sales above $15 000 calculate the rate of commission that he
receives. 24 If Geoff is paid a flat 4% of all sales as commission and Linda is paid $250 plus 2.4% commission,
calculate the weekly sales for which both would receive the same pay. 25 Henrietta is paid a retainer and receives a 4% commission on all sales. Given that Henrietta
earned $830 on sales of $12 000 and $950 on sales of $15 000 calculate the amount of Henrietta’s retainer. ChAptEr 1 • Earning money
11
1D
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
Multiply the pay rate by the number of cars detailed.
WritE
Pay = $2.25 × 24 = $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
WritE
1
Divide 3500 by 1000 to calculate the number of thousand brochures delivered.
3500 ÷ 1000 = 3.5
2
Multiply 3.5 by $23.00 to calculate what Holly is paid.
Holly’s pay = 3.5 × $23.00 = $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. 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. 12
Maths Quest Preliminary Mathematics General
think
WritE
a Multiply 21 (the number of baskets) by $4.40 (the pay
per basket). b Divide $92.40 (total pay) by 8 (number of hours
worked).
Exercise 1D
a Pay = 21 × $4.40
= $92.40
b Hourly rate = $92.40 ÷ 8
= $11.55
payment by piece
1 WE11 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. 6 WE12 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. 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 WE13 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 of fruit. b If it takes Denise 8 hours to pick the fruit, calculate the equivalent hourly rate of pay. 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.
Further development 13 George is paid $1.20 for each toy that he assembles, up to the first 100. For the next 50 he receives
$1.60 per toy. For any further toys that he assembles he is paid $2. Calculate his gross pay for a day in which he assembles: a 98 toys b 136 toys c 167 toys. 14 Paul types manuscripts for a publishing company. Paul is paid $7 per page for the first 50 pages, $9 per page from 51–100 pages and $10 per page thereafter. Calculate his gross pay for a period in which he types: a 38 pages b 68 pages c 140 pages. ChAptEr 1 • Earning money
13
15 Ravi delivers newspapers and is paid $28 per 1000 newspapers delivered. a What is Ravi paid for delivering 3500 newspapers? b How many must he deliver to earn $100? 1 c If it takes Ravi 3 2 hours to deliver 1000 newspapers, find his equivalent hourly rate of pay. 16 Garry delivers pizzas. He is paid $4 per delivery between 5:00 pm and 9:00 pm and $5.50 per pizza
after 9:00 pm. a Calculate his earnings on a night where he makes 12 deliveries between 6:00 pm and 9:00 pm and 4 deliveries between 9:00 pm and 10:30 pm. b Calculate his average hourly earnings between 6:00 pm and 10:30 pm. 17 Tina packs canned goods into boxes as they come off a production line. During a shift she is paid
20 cents per box up to 600 boxes and 25 cents per box thereafter. Tina is offered the chance to be paid a wage of $16.09 per hour. Given that Tina works an 8hour shift: a which package should Tina choose if she packs 500 boxes per shift? b which package should Tina choose if she packs 800 boxes per shift? c for both packages to be equivalent, how many boxes would Tina have to pack? 18 MC If a pieceworker earns x cents per article, up to 100 articles and y cents per article thereafter, then
his total earnings in dollars for producing 145 articles is: DiGitAl DoC WorkSHEET 1.1 doc10307
A 100x + 45y
1E
B 100y + 45x
C
100 x + 45 y 100
45 x + 100 y 100
D
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: 1
time and a half, which means that the person is paid 1 2 times the usual rate of pay, 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 1 overtime factor, which is 1 2 for time and a half and 2 for double time. 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
Multiply $9.78 (the normal hourly rate) by 1 1 2 (the overtime factor for time and a half).
14
Maths Quest Preliminary Mathematics General
WritE 1
Time and a half rate = $9.78 × 1 2 = $14.67
To calculate the pay for a period of time worked at time and a half or double time, we multiply the 1 normal pay rate by the overtime factor (either 1 2 or 2) and then by the 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 1 12
Multiply $12.84 (the normal pay rate) by (the overtime factor) and by 6 (hours worked at time and a half).
1
Pay = $12.84 × 1 2 × 6 = $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. WorkED ExAMplE 16
Natasha works as a waitress and is paid $11.80 per hour for a 38hour 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 1
Calculate Natasha’s normal pay.
2
Calculate Natasha’s pay for 5 hours at time and a half.
3
Add the normal pay and the time and a half pay together.
WritE
Normal pay = $11.80 × 38 = $448.40 1
Time and a half = $11.80 × 1 2 × 5 = $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
Gina is employed as a car assembly worker and is paid $10.40 per hour for a 36hour week. If Gina works overtime, the first 6 hours are paid at time and a half and the remainder at double time. Calculate Gina’s pay in a week where she works 45 hours.
ChAptEr 1 • Earning money
15
think
WritE
1
Calculate the number of hours overtime Gina worked.
Overtime = 45 − 36 = 9 hours
2
Of these nine hours, calculate how much was at time and a half and how much was at double time.
Time and a half = 6 hours Double time = 3 hours
3
Calculate Gina’s normal pay.
Normal pay = $10.40 × 36 = $374.40
4
Calculate what Gina is paid for 6 hours at time and a half.
Time and a half = $10.40 × 1 2 × 6 = $93.60
5
Calculate what Gina is paid for 3 hours at double time.
Double time = $10.40 × 2 × 3 = $62.40
6
Calculate Gina’s total pay by adding the time and a half and double time payments to his normal pay.
Total pay = $374.40 + $93.60 + $62.40 = $530.40
Exercise 1E
1
Working overtime
1 WE14 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. 2 Carmen works as a waitress and is paid $11.42 per hour. Calculate Carmen’s rate per hour on a Sunday
when she is paid double time. 3 Gareth works as a train driver and is normally paid $11.48 per hour. For working on public holidays he
is paid double time and a half (overtime factor = 2 1). Calculate Gareth’s hourly rate of pay on a public 2 holiday.
4 WE15 Ben works in a hotel and is paid $11.88 per hour. Calculate the total amount Ben will earn for
an 8hour shift on Saturday when he is paid at time and a half. 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. 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
Pay
7 MC 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 C $14.93
B $14.92 D $19.90
8 MC 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 16
Maths Quest Preliminary Mathematics General
9 MC 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 WE 16 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 4 hours overtime at time and a half. 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 35hour 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
Time and a half hours
Double time hours
W. Clark
$8.60
38
4
—
A. Hurst
$9.85
37
—
6.5
S. Gannon
$14.50
38
5
2.5
G. Dymock
$16.23
37.5
4
1.5
D. Colley
$24.90
36
6
8.5
Name
Total pay
14 MC 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 MC Patricia works a 35hour 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 16 WE 17 Steven works on a car assembly line and is paid $12.40 for a 36hour working week. The
first 4 hours overtime he works each week is paid at time and a half with the rest paid at double time. Calculate Steven’s earnings for a week in which he works 43 hours. 17 Kate works as a computer technician and is paid $18.56 per hour for a 38hour 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
Thursday — 7 hours
Friday — 10.5 hours.
Wednesday — 12 hours
Further development 19 Yvette is on a salary equivalent to $700 per week. She is offered the chance to accept a wage of
$14.65 per hour for a 38 hour week, and time and a half for any overtime. a Calculate the number of whole hours that Yvette needs to work to ensure that she does not receive less money if she accepts the offer. b Yvette estimates that she works an average 41 hours each week. If Yvette is to move from a salary to a wage, what is the minimum hourly rate that she should ask for? 20 Judy is paid $16.58 per hour for a 36 hour working week. Any overtime that Judy works is paid at a rate of
time and a half. Calculate the number of hours that Judy worked in a week where she earned $721.23. ChAptEr 1 • Earning money
17
21 Jack is a casual worker who is paid time and a quarter for all hours worked. The standard rate of pay is
$12.76 per hour. Jack works an average 36 hours per week. a Calculate Jack’s average weekly earnings. b Jack is paid the penalty rate as he is not paid for sick days and holidays. Jack has the opportunity to accept a permanent job, paid at normal rates; however, he would receive 10 sick days and 4 weeks holiday per year. Is Jack better off financially by taking the permanent job? Explain your answer. 22 Tim is a builder who works a 38 hour working week. Tim is paid time and a half for any overtime
worked. Calculate Tim’s hourly rate of pay given that he receives $746.36 for 44 hours work. 23 Carla is a nurse and works a 40 hour week. On a particular Monday, which is a public holiday, Carla is
paid double time and a half. Carla’s pay for the week will be equivalent to how many normal hours? 24 Georgia earns $18.45 for a 38 hour working week and is paid time and a half for any overtime worked.
Lily earns $16.76 for a 36 hour working week and is paid time and a half for the first four hours of overtime worked and double time thereafter. Calculate the minimum number of hours that Lily needs to work to earn more than Georgia who works a total of 42 hours.
Computer Application 2 Wages DiGitAl DoC Spreadsheet doc1448 Wages_2
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 Preliminary Mathematics General eBookPLUS.
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.) 18
Maths Quest Preliminary Mathematics General
1F
Government allowances
Many people rely on government allowances for their income, or at least to supplement their income. In this activity we look at some of these allowances and how they are calculated. To complete the investigation, go to the website www.centrelink.gov.au and answer the following questions. inVEStiGAtE: Youth allowance
1. What is the youth allowance and who is eligible to receive it? 2. How do you claim youth allowance? 3. Youth allowance is subject to an income test. What is an income test? 4. What is the income bank? 5. What is the assets test that applies to youth allowance?
The following table shows the amount of youth allowance paid under various circumstances. The table is correct at the time of publishing; however, you can go to the eBookPLUS and this table and all other material, including the worked examples and exercises, will be kept current. The maximum fortnightly payment is
If you are single with no children, under 18 years and living at home
$220.40
single with no children, under 18 years and required to live away from home
$402.70
single with no children, 18 years and over and living at home
$265.00
single with no children, 18 years and over and required to live away from home
$402.70
single with children
$527.50
partnered with no children
$402.70
partnered with children
$442.10
income limits The income test reduces the amount of youth allowance received, depending upon your weekly income. The rate at which the youth allowance decreases is described in the table below.
Job seekers Students and Australian apprentices
Fortnightly income range $62–250 Above $250 $236–316 Above $316
Reduction 50 cents in the dollar 60 cents in the dollar 50 cents in the dollar 60 cents in the dollar
WorkED ExAMplE 18
Angelo is 17 years old, single, has no children and lives at home while he works as an apprentice electrician. a Calculate the maximum amount of youth allowance that Angelo can receive. b If Angelo earns $300 per fortnight calculate the amount of his youth allowance payment. think
a Look up the table for a single 17 year old
who lives at home.
WritE
a Maximum youth allowance = $220.40
ChAptEr 1 • Earning money
19
b 1 Look up the table to find the income limit.
b Lower income limit = $236
2
Find the amount by which his income exceeds this limit.
$300 − $236 = $64
3
Calculate the reduction by multiplying the excess income by 0.5.
Reduction = $64 × 0.5 = $32
4
Subtract the reduction from the maximum allowance.
Angelo’s youth allowance = $220.40 − $32 = $188.40
WorkED ExAMplE 19
Riana is single, 17 years old with no children, lives at home and is a job seeker a Calculate the maximum amount of youth allowance that Riana can receive. b If Riana earns $280 per fortnight calculate the amount of her youth allowance payment. think
WritE
a Look up the table for a single under 18 year old
who lives at home and has no children. b 1 Look up the table to find the income limit.
a Maximum youth allowance = $220.40 b Lower income limit = $62,
upper income limit = $250
2
Find the amount by which her income exceeds the $250 threshold.
$280 − $250 = $30 above the upper limit $250 − $62 = $188 between $62 and $250
3
That part between $62 and $250 reduces her income by 50c in the dollar while the remainder reduces it by 60c in the dollar.
Reduction = $188 × 0.5 + $30 × 0.6 = $94 + $18 = $112
4
Subtract the reduction from the maximum allowance.
Riana’s youth allowance = $220.40 – $112 = $108.40
The above worked examples show the typical Centrelink calculations for the youth allowance. There are several other government payments such as Austudy, Abstudy, fares allowance and the aged pension. Similar calculations can be applied to each of these.
Exercise 1F
Government allowances
1 WE18 Katrina is a single 18 year old, with no children, and lives at home while she works as an
apprentice baker. a Calculate the maximum amount of youth allowance that Katrina can receive. b If Katrina earns $310 per fortnight calculate the amount of her youth allowance. 2 Calculate the amount of youth allowance payable to Greg, who is 17 years old, single with no children,
lives away from home and is employed as an apprentice carpenter earning $280 per fortnight. 3 Benjamin, 20 years old, partnered with a child, is earning $210 per fortnight. Calculate the amount of
youth allowance that he is entitled to. 4 What is the maximum fortnightly amount that can be earned by a single apprentice who is under
18 years old, living at home with no children: a if they are to receive the full youth allowance? b a part youth allowance? 5 A single job seeker with no children who is under 18 years of age and living at home earns $90 per
fortnight for delivering pamphlets to letterboxes. a Does this person receive the full youth allowance? b How much more can this person earn per fortnight before the youth allowance cuts out completely? 20
Maths Quest Preliminary Mathematics General
6 WE19 Jenny is 20 years old, has no children, lives away from home and is a job seeker. Calculate the
amount of youth allowance paid to Jenny given that she has income of $260 per fortnight. 7 Calculate the youth allowance payable in each of the following circumstances.
a b c d e f g h
Family situation Single, no children, under 18, at home, in an apprenticeship. Single, no children, under 18, at home, job seeker Single, no children, 18 and over, at home, and in an apprenticeship. Single, no children, 18 and over, at home, job seeker Single, no children, 18 and over, away from home, in an apprenticeship. Partnered, no children and a job seeker. Partnered, with dependants, and in an apprenticeship. Partnered, with dependants, job seekers
Income per fortnight $190.00 $112.50 $615.80 $212.90 $526.80 $275.00 $751.00 $394.75
8 Austudy is payable to people 25 years or older who are studying fulltime or are in an apprenticeship. It
is paid at the following rates and is subject to the same income and test as the youth allowance for a job seeker. If you are The maximum fortnightly payment is single $402.70 single, with children $527.50 partnered, no children $402.70 partnered, with children $442.10 Calculate the amount of Austudy payable to each of the following. a b c d
Status Single Single, with children Partnered, with children Partnered, no children
Fortnightly income Nil $153.50 $327.75 $279.80
9 Johann is a 30 year old who has gone back to university to fulltime study. Johann is partnered with no
child. Johann works casually for $230 per fortnight. a Does Johann receive the full amount of Austudy? b What amount of Austudy will he receive per fortnight? 10 MC Jade is a single 19 year old with no children who lives at home. The maximum amount of youth
allowance that Jade is entitled to is A $194.50 B $233.90
C $265.00
D $465.60
11 MC Which of the following people is entitled to the full youth allowance? A B C D
Andrea who is single, living at home and a job seeker earning $75 per fortnight. Bryce who is partnered with dependants is an Australian apprentice who earns $290 per fortnight. Cathy who is single, longterm unemployed, aged 24, who earns $240 per fortnight. David, who is single with no dependants, lives away from home, is a job seeker and earns $50 per fortnight.
Further development 12 Frank is a student who is over 18 years old and lives at home. a What is the maximum amount of youth allowance that Frank can receive? b What is the maximum amount that Frank can earn before the youth allowance begins to reduce? c What is the maximum amount that Frank can earn before his youth allowance cuts out completely? 13 Josie is single, living away from home, 28 years old, longterm unemployed but is returning to fulltime
study. a Calculate the amount of Austudy that Josie receives. b If Josie earns the most money possible from casual work without affecting her payment, what would be her total fortnightly income? ChAptEr 1 • Earning money
21
Additions to and deductions from gross pay 1G
Although we may calculate a person’s pay, this is not the amount that is actually received by that person. The amount that we calculate based on their wage or salary is called gross pay or gross wage. From gross pay several deductions may be made for items such as tax, union fees, private health insurance and superannuation. Many of you who are considering tertiary education may be aware that you may be left with what is called a HECS debt on completing a university course. HECS (higher education contribution scheme) is paying back the cost of your tertiary education to the government, although payment only needs to be made once your annual income passes a certain level. Repayment is through HELP (higher education loan program) and is a percentage of income that increases as annual income increases. The amount of money that you actually receive each week is called your net pay and is calculated by subtracting all deductions from your gross pay. WorkED ExAMplE 20
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
From $643.60 (gross pay) subtract $144.46 (tax), $57.92 (superannuation) and $11.40 (union fees).
WritE
Net pay = $643.60 − $144.46 − $57.92 − $11.40 = $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 21
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
Divide $324.60 (the annual union fee) by 52.
2
Round the answer off to the nearest cent.
WritE
Weekly deduction = $324.60 ÷ 52 = $6.24
WorkED ExAMplE 22
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
Calculate 9% of $1320 (gross pay).
WritE
Superannuation = 9% of $1320 = 9 ÷ 100 × $1320 = $118.80
When employees take annual leave,, they may receive an annual leave loading. When on holidays, such 1 employees are paid an additional 17 2 % of their gross pay for up to 4 weeks.
22
Maths Quest Preliminary Mathematics General
WorkED ExAMplE 23
Russell is a newspaper printer and is paid $14.75 per hour for a 36hour working week. a Calculate Russell’s pay for a normal working week. b Calculate Russell’s total pay for his 4 weeks 1 annual leave if he receives a 17 2 % annual leave loading on the 4 weeks pay.
think
WritE
a Normal pay = $14.75 × 36
a Multiply $14.75 (hourly rate) by 36 (hours
= $531.00
worked). b 1 Multiply $531.00 (weekly pay) by 4 to find
his normal pay for 4 weeks. 2
Calculate the annual leave loading by finding 1 17 2 % of $1692.
b Normal 4 weeks pay = $531.00 × 4
= $2124.00 1
Annual leave loading = 17 2 % of $2124.00 1
= 17 2 ÷ 100 × $2124.00 = $371.70
3
Add $371.10 (annual leave loading) to $2124 (normal 4 weeks pay).
Exercise 1G
gross pay
Total holiday pay = $2124.00 + $371.70 = $2495.70
Additions to and deductions from
1 WE 20 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. Gross pay $345.00 $563.68 $765.90 $1175.60 $2500.00
Deductions $89.45 $165.40 $231.85 $429.56 $765.40
Net pay
ChAptEr 1 • Earning money
23
3 David works in a mine and is paid a wage of $15.75 per hour
4
5
6
7
8
9 10 11
12
for a 36hour 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. 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. WE 21 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. 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. Dorothy is paid a wage of $13.45 per hour for a 38hour 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. 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. WE22 Sabrina earns a weekly wage of $623.50. She puts 9% of this wage into a superannuation fund. Calculate the amount that Sabrina pays in superannuation. 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. Rex is paid $11.12 per hour for a 38hour 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. The table below shows the rate at which HECS–HELP is to be paid. HELP repayment income (HRI) Below $49 096 $49 096–$54 688 $54 689–$60 279 $60 280–$63 448 $63 449–$68 202 $68 203–$73 864 $73 865–$77 751 $77 752–$85 564 $85 565–$91 177 $91 178 and above
Repayment rate Nil 4% of HRI 4.5% of HRI 5% of HRI 5.5% of HRI 6% of HRI 6.5% of HRI 7% of HRI 7.5% of HRI 8% of HRI
Calculate the amount of the annual HECS–HELP payment for a person whose HRI is: b $72 500 c $82 670 d $142 456.
a $32 000 24
Maths Quest Preliminary Mathematics General
13 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 Calculate the amount of HECS–HELP that Raylene must pay annually and hence the deduction
that is made each fortnight for HECS–HELP. d Raylene pays union fees of $486 per annum and has private health insurance of $1323.70
deducted from her pay fortnightly. Calculate the amount of the fortnightly deduction made for union fees and also for health insurance. e If Raylene pays $1009.22 in fortnightly tax, as well as the above deductions, calculate her weekly net pay. 14 WE23 LiangYi earns $13.60 per hour for a 38hour working week. a Calculate the amount LiangYi will earn in a normal working week. b Calculate the total amount LiangYi will receive for his 4 weeks annual leave if he receives a 1
17 2 % holiday loading. 15 Paula is paid an annual salary of $45 800. a Calculate Paula’s gross weekly salary. 1 b Calculate the total amount Paula will receive for her 4 weeks annual leave if she is paid a 17 2 %
holiday loading.
16 Leon is paid $12.95 per hour for a 36hour working week. a Calculate Leon’s weekly wage. 1 b Leon takes one week’s holiday for which he is given a 17 2 % loading. Calculate the holiday loading. c If Leon pays $83.24 in tax, calculate his net pay for his week’s holiday. 17 Scott is paid an annual salary of $68 500. a Calculate Scott’s salary for a 4week period. 1 b Calculate how much holiday loading Scott will receive for this 4week period if it is paid at 17 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 4week period. d If Scott pays $1779.92 in tax for this 4 weeks, calculate his net pay for the 4week holiday.
Further development 18 Tyrone is paid $15.65 per hour for a 40 hour working week. Tyrone pays 23% of his gross pay in tax
and 6% in superannuation. a Calculate Tyrone’s net pay. b Tyrone receives a 5% pay rise. This causes his tax to increase to 24% of his gross pay and his superannuation rises to 7% of his gross pay. Calculate the increase in Tyrone’s net pay as a percentage correct to 1 decimal place. 19 Ricky has a job that pays him an annual salary of $55 000. When negotiating a pay rise the company for
which Ricky works asks the employees to give up their 17.5% holiday loading in return for the pay rise. What is the minimum percentage pay increase that is required so that the employees will not be worse off? 20 Nancy is paid a fortnightly salary of $2397.68. Nancy’s employer also contributes an amount equal to
6% of her salary to her superannuation fund and provides a car under a lease arrangement which has a value of $724.50 per month. Calculate the total value of Nancy’s salary package. 21 Frank earns an annual salary of $90 000. He must pay 8% of his gross annual salary in superannuation
payments and 32% of his gross annual salary in taxation. Grace earns 85% of Frank’s salary but only pays 6% in superannuation and 30% in taxation. What percentage of Frank’s net annual salary is Grace’s net annual salary? 22 Richard earns an annual salary of $76 000. He pays 28% of his gross annual income in taxation and has
no other deductions. a What is Richard’s net annual salary? b Richard needs a new computer and chooses to salary package the computer. This means that his employer buys the computer for him and deducts the cost from his gross salary. As the computer is used for work purposes its value is not taxed. Given that the computer he chooses has a value of $3000 calculate the amount by which his net salary is reduced. ChAptEr 1 • Earning money
25
23 Indore has a gross annual salary of $62 750. Indore’s salary package includes a car that has a monthly
cost of $859.60. a Find the annual cost of packaging the car. b Calculate Indore’s net annual salary given that her only deduction is 27% in tax. c Calculate the amount that the car actually costs Indore from her net pay, per month, given that the car cost is taken from her gross salary and that this deduction causes her tax to fall to 25%.
Computer Application 3 Wages template 1. Load your spreadsheet Wages_2 and add the Deductions and Net Pay columns. Alternatively, access the spreadsheet Wages_3 from the Maths Quest Preliminary Mathematics General eBookPLUS.
DiGitAl DoC Spreadsheet doc1452 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. DiGitAl DoC Spreadsheet doc1453 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 Preliminary Mathematics General eBookPLUS. 26
Maths Quest Preliminary Mathematics General
1h
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 Wages
$700
Total
$700
Expenditure Rent Groceries Bills Car loan Car running costs Entertainment Credit card Savings Total
$150 $100 $100 $75 $50 $60 $50 $115 $700
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 24
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
WritE
1
Calculate the total of the previous years bills.
Annual total = $136 + $187 + $169 + $105 = $597
2
To calculate the weekly amount, divide $597 by 52.
Weekly amount = $597 ÷ 52 = $11.48
3
Increase $11.48 by 10%.
110% of $11.48 = $12.62
4
Make a practical approximation of the answer.
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 25
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. ChAptEr 1 • Earning money
27
think
WritE
1
Calculate the total annual amount for electricity.
Electricity = $110 × 6 = $660
2
Calculate the total annual amount for telephone.
Telephone = $95 × 4 = $380
3
Calculate the total annual amount for car insurance.
Car insurance = $254 × 2 = $508
4
Calculate the total annual amount for rates.
Rates = $1250
5
Find the annual total for all of these bills.
Total = $660 + $380 + $508 + $1250 = $2798
6
Increase $2798 by 10%.
110% of $2798 = 110 ÷ 100 × $2798 = $3077.80
7
Divide $3077.80 by 26.
Fortnightly allowance = $3077.80 ÷ 26 = $118.38
8
Round off and give a written answer.
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 26
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
a 1 2
Calculate the annual mortgage amount.
a Annual mortgage = $600 × 12
= $7200
Fortnightly amount = $7200 ÷ 26 = $276.92
Calculate the fortnightly amount by dividing by 26.
b 1 Calculate total expenses. 2
WritE
b Total expenses = $276.92 + $110 + $250 + $70 + $80
= $786.92
Savings = $950 − $786.92 = $163.08
Calculate savings by subtracting all expenses from $950.
c Draw up a budget by listing income and
expenses in two columns.
c
Income Wages $950
Total
$950
Expenditure Mortgage $276.92 Bills $110 Groceries $250 Car $70 Entertainment $80 Savings $163.08 Total $950
To do work on budgeting you will need to be able to interpret the information on various household bills. 28
Maths Quest Preliminary Mathematics General
WorkED ExAMplE 27
Look at the extract from a sample telephone bill below.
a What is the to tal of thebill? c How much of the bill is for service and equipment? think
b For what period are the call charges? WritE
a Look in the box labelled ‘Total amount payable’.
a The total of the bill is $154.10.
b Look at the dates following ‘Local Calls’.
b The calls were for the period 5 Jan. to 4 Apr.
c Look at the amount next to ‘Service & Equipment’.
c The cost for service and equipment
was $51.45.
Exercise 1h
Budgeting
1 WE24 Vesna gets her telephone bill quarterly. Last year her four bills were $89.50, $103.40, $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. 2 Christopher pays $1360 each year in council rates. Calculate how much he should budget for each
fortnight for council rates. 3 Isabelle pays $34.65 per month in car insurance. Calculate the amount that she should budget each
week for car insurance. 4 Tristan’s mortgage repayments are $750 per month. Calculate the amount that Tristan should budget for
each fortnight to cover his mortgage bill. 5 WE25 Mr and Mrs Banks have the following bills.
Electricity Telephone Car insurance House insurance Council rates
$130 every quarter $108 per quarter $35 per month $29.50 per month $1100 per year
DiGitAl DoC Spreadsheet doc1455 Budgets
ChAptEr 1 • Earning money
29
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. 7 WE26 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. 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. 9 WE27 Look at the extract from a sample telephone bill below.
a b c d 30
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?
Maths Quest Preliminary Mathematics General
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 offpeak 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 f
meter? What was the previous reading of the offpeak meter?
ChAptEr 1 • Earning money
31
11 Look at the extracts from a sample gas bill below.
a b c d
32
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?
Maths Quest Preliminary Mathematics General
12 Look at the extract from the sample bill for council rates below.
a b c d
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?
ChAptEr 1 • Earning money
33
Further development 13 Petria earns $13.60 per hour in her casual job at McDonalds. Petria works an average 12 hours per
14
15
16
17
18 DiGitAl DoC WorkSHEET 1.2 doc10308
34
week. a Calculate Petria’s gross weekly pay. b Petria spends an average $30 per week on her mobile phone. Calculate the percentage of her gross pay that she spends on her phone. Olivia has a home telephone that costs her an average $65.00 per month. She also spends an average $30 per week on her prepaid mobile phone. Calculate the weekly saving if Olivia replaces both phones with a mobile plan that costs $119.95 per month. Damien has insurance costs of $562.50 for home and contents insurance with GOI Insurance, $1650 per year for health insurance with HBF insurance and $739.60 for car insurance with RMNA Insurance. GOI offers a 5% discount on its $1720 health insurance premium and a 10% discount on its $812.60 car insurance premium if Damien switches all policies to GOI. HBF will match its competitor’s premiums and discount the health insurance by 5% if all policies are with them. Which, if either, offer should Damien choose and what will be his total cost? Mr and Mrs Forrester have a combined net income of $1650 per week. Their weekly expenses include $600 per week for mortgage repayments and $450 per week in household bills. a What is the percentage of net pay that the Forresters allocate to their mortgage repayment and household bills? b Over the next year the Forresters’s pay increases by 3%. Their mortgage repayment remains unchanged; however, they allocate an extra 10% to cover their bills. Calculate the new percentage of net pay allocated to each of these items. Mr and Mrs. Marone have a $300 000 mortgage. They have a combined net income of $6000 per month and repay the loan at a rate of $2121 per month. a What percentage of their net income is allocated to repaying the home loan? b After a series of interest rate rises, their loan repayment increases to $2316 per month. Calculate the change in percentage of income allocated to the loan. Economists often talk of the cost of items in ‘real terms’. This is the percentage increase or decrease in purchasing power after wage rises and cost increases are factored. If the average income is $1500 per week and average expenses are $1350 per week, and wages rise by 5% and expenses increase by 10%, calculate the loss of income as a percentage in real terms.
Maths Quest Preliminary Mathematics General
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.
Working overtime
• Overtime is paid when the employee works more than the regular hours each week. Usually the employee will be paid at either: 1 time and a half — 1 2 times the normal hourly rate, or double time — twice the normal hourly rate.
Government allowances
• Government payments can be received as youth allowance, aged pension and for study purposes. • The Centrelink website can be used to look up the amount of these payments. • Most government payments are subject to an income test. In such cases the amount of the payment reduces once income reaches a certain level. At another, higher level of income the allowance cuts out altogether. • Payments are also subject to an assets test which works in a similar way. If you have more than a certain level of assets, payment is reduced.
Additions to and deductions from gross pay
• 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. 1
• Employees receive an extra 17 2 % when they take their annual leave. This is called 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.
ChAptEr 1 • Earning money
35
Chapter review M U lt ip l E C h oiCE
1 Which of the following is the highest salary? A B C D
$961.48 per week $1923.12 per fortnight $4165.00 per month $50 000 per annum
2 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 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 4 Janelle works a 38hour week at a rate of $14.50 per hour. When Janelle takes her 4 weeks annual leave 1
she is paid a loading of 17 2%. Janelle’s weekly wage, when she takes her leave, is: A $551 B $647.43 C $2204 D $2589.70 5 Which of the following employees is paid a wage? A B C D
Anthony, who receives a payment of 4% of the value of all sales each month Beth, who is paid fortnightly at a rate of $13.50 per hour Carmel, who is paid weekly based on an annual amount of $37 500 Damien, who is paid $1.2 million for every movie that he appears in
6 Cherry is single, for whom the full youth allowance is $402.70 per fortnight. The income test reduces
the amount of youth allowance by 50c in the dollar for fortnightly income between $62 and $250, and 60c in the dollar thereafter. Cherry has fortnightly income of $286.60, so the youth allowance she receives is: A $230.74 B $255.74 C $259.40 D $286.74 Sh ort AnS WEr
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. 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 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 fulltime bartender who works a 36hour week. Calculate his weekly wage. b Len is a casual bartender who works 16 hours a week. Calculate Len’s weekly wage. 36
Maths Quest Preliminary Mathematics General
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 9 10 11 12 13 14
15
16
17
18
19
20
21
he wanted to earn $500. 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. 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. 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. Hong has an afterschool job detailing cars. Hong is paid $11.75 for every car that he details. Calculate what Hong is paid for detailing 29 cars. 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. 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. 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. 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 Chaoping 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. Eddie works as a shop assistant and is paid an ordinary rate of $10.54 per hour for a 36hour 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. Use the table on page 19 to find the amount of youth allowance paid to: a Terry, who is 17 years old, single with no children, lives at home and is seeking a job. Terry is not subject to an income or assets test. b Kerry, who is 22 years old, an apprentice chef earning $328.60 per fortnight. Kerry lives away from home and is single with no children. 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. 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. 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. ChAptEr 1 • Earning money
37
22 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. 23 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. 24 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. E x t EnDED r E SponS E
1 Ken works as a pest inspector. Ken is paid a wage of $15.40 per hour. a If Ken works a normal 36hour 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
DiGitAl DoC Test Yourself doc10309 Chapter 1
38
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 1 annual leave loading of 17 2 %. 2 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.
Maths Quest Preliminary Mathematics General
ICT activities 1A
Calculating salary payments
DiGitAl DoC • Spreadsheet (doc1439): Payroll calculations (page 2)
1B
Calculating wages
DiGitAl DoCS • Spreadsheet (doc1439): Payroll calculations (page 5) • Spreadsheet (doc1440): Wages_1 (page 7)
1C
Commission and royalties
DiGitAl DoC • Spreadsheet (doc1443): Calculations with percentages (page 10)
1D
payment by piece
DiGitAl DoC • WorkSHEET 1.1 (doc10307): Perform calculations related to earning money. (page 14)
1E
1G Additions to and deductions from gross pay DiGitAl DoCS • Spreadsheet (doc1452): Wages_3 (page 26) • Spreadsheet (doc1453): Wages template (page 26)
1h
Budgeting
DiGitAl DoCS • Spreadsheet (doc1455): Budgets (page 29) • WorkSHEET 1.2 (doc10308): Perform calculations about earning money and budgeting. (page 34)
Chapter review • Test Yourself Chapter 1 (doc10309): Take the endofchapter test to check your progress. (page 38)
To access eBookPLUS activities, log on to www.jacplus.com.au
Working overtime
DiGitAl DoC • Spreadsheet (doc1448): Wages_2 (page 18)
ChAptEr 1 • Earning money
39
Answers CHAPTER 1 EArninG MonEY Exercise 1A
9
Calculating salary payments 2 $960
1 $1700 3 $4500 4 a $745 b $1490 c $3228.33 5 $1653.85 6
Annual salary
Weekly Fortnightly Monthly pay pay 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
7 9 10 12 13 14 15 16
17
18 19 20
$23 920 8 $32 500 $69 600 C 11 $14.28 a $634.62 b $17.63 a $688 b Karina ($16.38 per hour) 45 hours $2166.67 Compare the annual salaries. Dan’s equivalent monthly salary is found by multiplying by 26 and dividing by 12, giving $2264.17, $164.17 per month more than Brian. a $88 312.50 b Not necessarily correct. It is possible that Sandra works more efficiently than Stephanie, completing the same or more work in a shorter period of time. $2100 per fortnight $1871.35 a $39 816.40 b $330 476.12
Exercise 1B
1 $518 2
Calculating wages
Name
Hourly Hours rate worked
A. Smith
$14.52
40
$580.80
B. Brown
$16.45
38
$625.10
N. Tran
$15.95
37.5
$598.13
A. Milosevic
$20.10
41
$824.10
L. McTavish
$18.04
36
$649.44
3 a $424.80 c $501.50 4 a $432 c $600.75 5 a $17.34 6 D 8 39
40
Wage
Name A. White B. Black C. Green D. Brown E. Scarlet F. Grey
Wage $416.16 $538.80 $369.63 $813.96 $231.30 $776.72
Hours worked 36 40 37 42 15 38
Hourly rate $11.56 $13.47 $9.99 $19.38 $15.42 $20.44
$12.54 11 C 12 D $422.62 14 $548.80 a $349.65 b 43 hours a $471.20 b $11.78 a $1666 b $4.52 $120.83 19 $103.67 a $25.98 b 44.4 hours a 38 hours b John will earn $1.28 less each week. Current pay = $510.40, new pay = $509.12 22 a 40 × $18.49 = $739.60 b The employer figure is $18.49 × 1.05 = $19.4145 rounded down to $19.41. Tracey’s figure $739.60 × 1.05 = $776.58 c $776.40 d $19.4145 — it is possible to pay this amount by multiplying this hourly rate by the number of hours worked and rounding to the nearest cent after the calculation not before the calculation. e Tracey is correct as under the employer’s plan the pay rise is less than 5%. 23 a $566.96 b $15.5019 c 42 hours. 10 13 15 16 17 18 20 21
Exercise 1C
Commission and royalties
1 $2000 2 a $260 c $568.79 3 a $400 c $1360 4 $1425 6 B 7 a $1425 8 a 4.5% c $18 000 9 C 11 $800 12 a $4125 c $16 875 13 $950
b $566.40 b $499.50 b $225.42 7 $9.68
Maths Quest Preliminary Mathematics General
b $327 b $1200 5 $4140 b $110 000 b $693 10 $8125
Exercise 1D
payment by piece
1 $75.95 2 a $103.50 c $85.50 3 $175 4 $135 5 $325.50 6 $327 7 a $92 c $87.40 8 a $57.75 c $103.95 9 $22.80 10 a $11 c $19.32 11 a $134.40 12 a $174 c $29.00/h 13 a $117.60 c $234.00 14 a $266 c $1200 15 a $98 c $8.00/h 16 a $70 17 a The wage b The piecework c 635 boxes 18 C Exercise 1E
1 $16.95 3 $28.70 5 $157.20
b $5375
6
14 a $390 b $490 c $649 15 B 16 a $400 b $500 c $8000 d $8125 17 $625 18 a $300 b $345 c $406.80 19 D 20 a $462.50 b $372.50 c Bonito ($705) 21 a Package C b $2 080 000 c 43 hours d She has an incentive to sell more. 22 a J. L. Booker — $625. Roy Black — $250 b J. L. Booker — $1250, Roy Black — $1500 c $400 000 23 a 2% b 2.6% 24 $15 625 25 $350
b $94.50 d $81
b $138 d $114.08 b $125.13
b $27.80 b $16.80 b $21.75/h b $177.60 b $512 b 3572 b $15.56/h
Working overtime
2 $22.84 4 $142.56
Ordinary rate
Overtime rate
Hours worked
Pay
$8.90
Time and a half
4
$53.40
M. Donnell
$9.35
Double time
6
$112.20
F. Milosevic
$11.56
Time and a half
7
$121.38
J. Carides
$13.86
Time and a half
6.5
$135.14
Y. Robinson
$22.60
Double time
5.5
$248.60
Name A. Nguyen
7C 10 $543.52 13
8 A 11 $721.60
Name W. Clark A. Hurst S. Gannon
9 B 12 $596.25
Ord. rate
Normal hours
Time and a half hours
$8.60
38
4
$9.85
37
$14.50
38
13
$732.25 $717.75
$16.23
37.5
4
1.5
$754.70
36
6
8.5
$1543.80
Exercise 1F
9 10 12
6 2
$24.90
permanent job.
8
$492.50
G. Dymock
22 $15.88 24 43.2 hours
6 7
Total pay
D. Colley
14 C 15 A 16 $595.20 17 a $705.28 b $788.80 c $928 18 $455.78 19 a 45 hours b $16.47 20 41 hours 21 a $574.20 b Jack will earn less by taking the
1 2 3 4 5
5
Double time hours
a $265
23 52 hours
Government allowances b $228
$380.70 $442.10 a $236 a No b A further $367 $302.70 a $220.40 c $45.12 e $236.22 g $141.10 a $402.70 c $301.45 a No C a $265 c $691 a $402.70
5 $7.58 6 $56.02 7 a $511.10 c $407.26 8 a $2153.85 c $1448.41 9 $56.12 10 $167.26 11 a $422.56 c $310.01 12 a $0 c $5786.90 13 a $2884.62 c $4875
b $44.84
b $44.37 b $4350 d $11 396.48 b $367.79
$187.50
b $616.67
b d f h b d b 11 b
$195.15 $189.55 $293.70 $261.25 $481.75 $290.82 $318.70 D $236
14 15 16 17 18 19 20 21 22 23
b $638.70
health insurance = $50.91 e $625.26 a $516.80 b a $880.77 b a $466.20 b c $464.55 a $5269.23 b c $96.15 d a $444.46 b 1.35% $74 774.06 90.67% a $54 720 b a $10 315.20 b c $540.12/month
Exercise 1h
2
1 3 4 6 7
Additions to and deductions from gross pay 1 $360.90
Gross pay
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 b $1434.08
$2869.60.
16 a Mortgage — 36.4%, bills — 27.3% b Mortgage — 35.3%, Bills — 29.6% 17 a 35.35% b Increase of 3.25% 18 4.29% loss of income
ChAptEr rEViEW MUltiplE ChoiCE
d Union fees = $18.69;
Exercise 1G
3 $360.45 4 a $2519.23
b $5.11
10 a $198.17 b $60.70 c 55 d 1753 e 10 000 f 58 552 11 a $143.75 b 72 days c 1.24c/MJ d 161 12 a $1007.70 b $90 000 c $91.50 d 4, $251.90 13 a $163.20 b 18.38% 14 $17.32 15 The HBF offer will give a total cost of
$2428.96 $4139.61 $81.59 $922.12 $4315.28 2.0%
$2160 $45 807.50
Budgeting $8.42 2 $52.31 $8.00 $346.15 5 $59.78 $484.19 a $240 b Check with your teacher. 8 a $120.77 b $415.38 c $378.85 d Check with your teacher. 9 a $181.60 b 5 Oct. to 4 Jan. c $29.35 d $14
1 B 4 B
2 C 5 B
3 B 6 D
Short AnSWEr
1 $761.54 3 $15 912 5 a $522 c $588.24 6 a $442.80 7 $11.40/hr 9 $395.20 10 $798.50 12 $340.75 13 $98 14 a $21.30 15 a $98.10 16 a $368.60 c $426.80 17 $590.24 18 a $220.40 19 $311.64 20 $4.05 21 a $1855.77 b $176.30 22 a $631.55 b $13.26 c $72.63 d $405.36 23 $135 24 $57.12
2 $2592.31 4 $22.29/hr b $594.70 b $220.80 8 43 hours 11 $534
b b b d
$28.40 $109 $426.80 $485
b $355.14
ExtEnDED rESponSE
1 a b c d 2 a b c
$554.40 $685.30 $572.57 $2605.68 $1830.77 $164.77 $1119.22
ChAptEr 1 • Earning money
41
ChapTer 2
Investing money ChapTer ConTenTs 2a 2B 2C 2d 2e 2F 2G
Calculation of simple interest Graphing simple interest functions Calculation of compound interest Calculating compound interest from a table of compounded values Share dividends Graphing share performance Inflation and appreciation
2a
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
WriTe
1
Write down the simple interest formula.
I = Prn
2
Write down the values of P, r (converting the percentage to a decimal) and n.
P = $5000 r = 0.04 n=3
3
Substitute into the formula.
I = $5000 × 0.04 × 3
4
Calculate.
= $600
ChapTer 2 • Investing money
43
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 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
WriTe/display
Method 1 1
Write down the formula for simple interest.
I = Prn
2
Write down the value of P, r and n.
P = $12 000, r = 0.095, n = 5
3
Substitute the values into the given formula.
I = $12 000 × 0.095 × 5
4
Calculate the simple interest.
5
Write down the formula for the total amount.
6
Substitute the values into the given formula.
= $12 000 + $5700
7
Calculate.
= $17 700
Method 2
44
1
From the MENU select TVM.
2
Press 1 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 ! SET UP, highlight DATE MODE and press 1 for 365.
4
Press w 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. 1: SI is simple interest. 2: SFV is future value, in other words the principal plus interest. In this example, as we want the simple interest, we press 1 for SI.
Maths Quest Preliminary Mathematics General
= $5700 A=P+I
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
1
18 months = 12 = 1 2 years. Worked example 3
Calculate the simple interest earned on an investment of $7600 at 5.2% p.a. for 9 months. Think
WriTe
1
Write down the formula for simple interest.
I = Prn
2
Write down the value of P, r and n (converting 9 3 9 months to 12 = 4 year).
P = $7600, r = 0.052, n =
3
Substitute the values into the given formula.
I = $7600 × 0.052 ×
4
Calculate the simple interest.
3 4
3 4
= $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
WriTe
1
Write down the formula for simple interest.
I = Prn
2
Write down the value of I, P and r.
I = $650, P = $8375, r = 0.0625
3
Substitute the values into the given formula.
$650 = $8375 × 0.0625 × n
4
Simplify the RHS of the equation.
5
Make n the subject of the equation.
6
Calculate the value of n in years.
= 1.2418 years
7
Convert 0.2418 years to months by multiplying the decimal by 12.
≈ 15 months
8
Answer the question.
= $523.4375 × n n=
$650 $523.4375
It will take approximately 1 year and 3 months to earn $650 in simple interest.
inVesTiGaTe: Costs of banking
There are many different reasons why most people need to maintain at least one bank account. Different accounts are designed by the banks to meet people’s needs. The most frequent of these are: 1 Transactional accounts — these are accounts that people use to have their pay deposited into, and for regular withdrawals and shopping using EFTPOS. 2 Savings accounts — accounts where people make regular deposits and save towards a specific target. A Christmas club account is an example of this type of account. 3 Investment accounts — accounts where money is locked away, usually at a higher rate of interest, for a fixed period of time.
ChapTer 2 • Investing money
45
Bank accounts pay interest on the amount of money in these accounts, but may also charge fees which can be calculated in different ways. Find an example of each of the types of accounts listed above and answer the questions below. 1 What is the interest rate payable on this 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 an ATM card or a cheque book?) 4 What are the fees on the account? (For example, the account may have a monthly fee or a fee based on the number and type of transactions.) 5 Are the fees applied differently to deposits and withdrawals? Are they levied differently for overthecounter, ATM and EFTPOS transactions? 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 (semiannually) 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.
Calculation of simple interest
exercise 2a
1 We1 Veronica invests $4000 for 3 years at 5% p.a. Calculate the simple interest earned. diGiTal doC GC program — Casio doc1544 interest
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. 1
e $1000 for 1 2 years at 6% p.a. f diGiTal doC GC program — TI doc1545 interest
1
$1750 for 5 4 years at 7.45% p.a.
3 Julie has $40 000 to invest. She invests it for 5 years in a State Government bond that pays 5.6% p.a.
interest. Calculate the simple interest that Julie will earn: a each year b for the whole 5 years of the investment. 4 We2 Brian has a $10 000 inheritance that he wants to invest. He invests his money in government
diGiTal doC Spreadsheet doc1546 interest
bonds for 3 years at 8% p.a. Calculate: a the simple interest earned b the value of his investment on maturity. 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. 6 Frank is 7 years old and starts a savings account with the local bank. He has $140 with which to start
the account. 3 a If the interest rate is 3 4 %, calculate the amount of interest Frank will receive after 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. 1 c Find the total value of the investment after 4 2 years. 8 We3 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 mC The simple interest paid on an investment of $5750 at 4.6% p.a. for 2 years is: a $529.00 C $6291.17
46
Maths Quest Preliminary Mathematics General
B $541.17 d $6279.00
10 mC 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 C $3937.50
B $826.25 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 We4 How long, to the nearest month, will it take to earn $2400 simple interest, if $16 410 is invested
at 9.75% p.a.? 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 mC 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 C $20 000
B $12 200 d $21 200
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 semiannually (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 semiannually (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?
Further development 3
18 Silvia invested $15 000 that she won in Lotto into a government bond that pays 8 4 % simple interest
provided she keeps the bond for 5 years. What is the total value of Silvia’s bond at the end of 5 years?
19 Silvia’s bond allows her to withdraw part of the money invested during the term of the loan but will
only pay 5.5% on the funds that were withdrawn over the time it was invested. After 2 years Silvia withdraws $3000 to buy a big screen TV. Calculate the interest earned on the entire investment. ChapTer 2 • Investing money
47
20 Mandy invested $12 000 in a fund paying 9.5% p.a. simple interest over a 4 year period. a Calculate the interest that Mandy has earned. b Martin earned the same amount of interest as Mandy but did so at 9% p.a. in only three years.
Calculate the amount that Martin invested. 21 Carly has $3000 to invest. Her aim is to earn $630 in interest. Given that she earns 4.5% simple interest paid monthly, for what period of time will she need to invest the money? 22 Ted claims that if simple interest is paid, it makes no difference to the final financial outcome how often interest is paid. Is Ted correct? Explain your answer. 23 Chris has a sum of money invested at 5% p.a. simple interest. Chris believes if he can invest his money at 10% p.a. he will have twice as much money at the maturity of the investment. Is Chris correct? Explain your answer.
2B
Graphing simple interest functions
No. of years Interest
1 $500
2 $1000
3 $1500
4 $2000
5 $2500
The amount of interest earned can be graphed by the linear function at right. Note that the gradient of this graph is 500, which is the amount of one year’s interest, or 5% of the principal.
Interest ($)
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. 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 Interest
1
2
3
4
5
b Graph the interest earned against the number of years the money is invested. Think
WriTe/draW
Method 1 calculate the interest earned on $6000 at 4% p.a. for 1, 2, 3, 4 and 5 years. b Draw the graph with Years on the horizontal
axis and Interest on the vertical axis.
a No. of years
Interest b
Interest ($)
a Use the simple interest formula to
1 $240
1500 1000 500 0
0
1
2 3 4 Years
Method 2
48
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 = 240n
3
From the MENU select GRAPH.
Maths Quest Preliminary Mathematics General
2 $480
5
3 $720
4 5 $960 $1200
4
Delete any existing function and enter Y1 = 240X.
5
To draw up the axes press ! 3 for V‑Window and enter the setting shown at right.
6
Press w to return to the previous screen and then press 6 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
a 1 Use the simple interest formula
WriTe/draW
a
to calculate the interest earned on $12 000 at 4% p.a. for 1, 2, 3, 4 and 5 years. 2
3
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. 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.
No. of years
1
2
Interest (4%)
$480
$960
Interest (5%)
$600
$1200 $1800 $2400 $3000
Interest (6%)
$720
$1440 $2160 $2880 $3600
b Draw a line graph for each investment. b
3
4
5
$1440 $1920 $2400
Interest ($)
4000 Interest (6%) Interest (5%) Interest (4%)
3000 2000 1000 0
0
1
2 3 4 Years
5
ChapTer 2 • Investing money
49
exercise 2B
Graphing simple interest functions
1 We5 $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 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 can be 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. 6 We6 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. No. of years
1
2
3
4
5
Interest (5%) Interest (6%) Interest (8%) b Show this information in graph form. 7 Theo has $50 000 to invest. Theo investigates the website www.whichbank.com.au, which has an
Interest ($)
interactive component. Theo enters the figure $50 000 and the following graph is displayed. 40 000 35 000 30 000 25 000 20 000 15 000 10 000 5000 0
The Whichbank advantage
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. 50
Maths Quest Preliminary Mathematics General
8 The graph below shows the interest earned on an investment of $10 000 with two different banks.
3500
Interest ($)
3000 2500
Bank A
2000
Bank B
1500 1000 500 0
1
2 3 4 Number of years
5
Which bank pays the higher rate of interest? Explain how you know this. 9 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. 10 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. 11 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.
Further development 12 The graph below right shows the interest earned under a simple interest investment. a Find the gradient of this line. I (interest) b How does this relate to the investment? $500 (5, 480) c Given that the interest rate is 6% p.a. calculate $400
the amount of the initial investment.
$300 $200 $100 0
1
2
3
4
5 n (years)
13 The graph below right shows the growth of an investment under simple interest. a What was the amount of money invested? A (amount $) b What is the interest rate earned? 5000 c Find the value of the investment after 4000
8 years.
3000
(5, 300)
2000 1000 0
1
2
3
4
5 n (years)
ChapTer 2 • Investing money
51
14 ‘When graphing interest earned and the value of the investment on maturity on the same axes the lines
will be parallel.’ Is this statement correct? Explain your answer. 15 $5000 is to be invested at 5% p.a. simple interest. A graph of the investment at maturity (A) is to be
drawn against the number of years of the investment (n). a What will be the vertical intercept? b What will be the gradient? c Write the equation of the line. 16 Find the growth equation of an investment of $4000 at 7% p.a. simple interest. 17 Sandra says that when graphing interest, the graph is a direct variation. a Is Sandra correct? b Will a growth equation be a direct variation? Explain your answer.
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 Preliminary Mathematics General ebook open the spreadsheet ‘Interest’. diGiTal doC Spreadsheet doc1546 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 2B. 6. Save the spreadsheet as Simple Interest. 52
Maths Quest Preliminary Mathematics General
2C
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.
inTeraCTiViTy int0810 Compound interest
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
WriTe
1
Write the initial principal.
Initial principal = $10 000
2
Calculate the interest for the 1st year.
1st year’s interest = 10% of $10 000 = $1000
3
Calculate the 2nd year’s principal by adding the 1st year’s interest to the initial principal.
2nd year’s principal = $10 000 + $1000 = $11 000
4
Calculate the 2nd year’s interest.
2nd year’s interest = 10% of $11 000 = $1100
5
Calculate the 3rd year’s principal by adding the 2nd year’s interest to the 2nd year’s principal.
3rd year’s principal = $11 000 + $1100 = $12 100
6
Calculate the 3rd year’s interest.
3rd year’s interest = 10% of $12 100 = $1210
7
Calculate the future value of the investment by adding the 3rd year’s interest to the 3rd year’s principal.
Future value = $12 100 + $1210 = $13 310
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. Year
1
2
3
4
5
6
7
8
Simple interest
$1000
$2000
$3000
$4000
$5000
$6000
$7000
$8000
Compound interest
$1000
$2100
$3310
$4641
$6105
$7716
$9487
$11 436
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 After 2 years
A1 = 10 000 × 1.1 A2 = A1 × (1.1) = 10 000 × 1.1 × 1.1 = 10 000 × 1.12
(increasing $10 000 by 10%) (substituting the value of A1)
ChapTer 2 • Investing money
53
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. In the financial world, the terms future value (FV ) and present value (PV ) are sometimes used instead of amount and principal. After 3 years
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
WriTe
1
Write down the formula for the future value.
A = P(1 + r)n
2
Write down the value of P, r (as a decimal) and n.
P = $12 000, r = 0.07, n = 5
3
Substitute into the formula.
A = $12 000 × 1.075
4
Calculate.
= $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 sixmonthly, 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. Worked example 9
Calculate the future value of an investment of $6000 at 8% p.a. for 2 years with interest compounded quarterly. Think
54
WriTe
1
Write down the formula for the future value.
A = P(1 + r)n
2
Write down the value of P, r (as a decimal) and n.
P = $6000, r = 0.02, n = 8
3
Substitute into the formula.
A = $6000 × 1.028
4
Calculate.
Maths Quest Preliminary Mathematics General
= $7029.96
exercise 2C
Calculation of compound interest
1 We7 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. 2 Suzanne is to invest $15 000 for 2 years at 7% p.a. with interest paid annually. a Calculate the future value of the investment by calculating the simple interest for each year
separately. b Find the amount of interest earned.
diGiTal doC Spreadsheet doc1552 Compound interest
3 We8 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.
4 Use the compound interest formula to calculate the future value of each of the following investments
with interest compounded annually. a $4000 at 5% p.a. for 3 years b $8000 at 3% p.a. for 5 years c $18 000 at 8% p.a. for 4 years d $11 500 at 5.5% p.a. for 3 years e $8750 at 6.25% p.a. for 6 years 5 We9 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. 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
amount of interest earned. 8 Vicky invests $30 000 in a oneyear 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 sixmonthly. 1 b $7500 for 3 2 years at 5.6% p.a. with interest compounded quarterly. 1
c $152 000 for 2 2 years at 7.2% p.a. with interest compounded sixmonthly. 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 mC 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 mC 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
ChapTer 2 • Investing money
55
12 mC 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 sixmonthly d compound interest at 7.2% p.a. with interest compounded quarterly 13 mC An investment of a sum of money is made over a 6 year term at an interest rate of 8% p.a.
compounded sixmonthly. The future value of the investment is $15 049.70. The initial principal (the sum of money invested) is a $900 B $8500 C $9400 d $11 000 14 Brittany has $13 500 to invest. An investment over a 2year 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 oneyear 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. 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. 17 An investment has a future value of $25 000 after 3 years at 8% p.a. with interest compounded
annually. Find the initial principal.
Further development 18 mC The greatest return on a compound interest investment will be made if interest is compounded: a monthly C sixmonthly
B quarterly d annually
19 Find the amount at maturity for each of the following investments under compound interest. a $3000 at 8% p.a. for 2 years interest compounded quarterly b $2000 at 6% p.a. for 5 years interest compounded sixmonthly c $5000 invested at 12% p.a. for 4 years interest compounded monthly d $6800 invested at 9% p.a. for 6 years interest compounded quarterly 20 Andrea invests $2050 for 4 years. For the first two years of the investment the interest rate is 6% p.a.
and for the second two years the interest rate rises to 8% p.a. Given that interest is paid quarterly, calculate the value of Andrea’s investment at maturity. 21 Lily wishes to have $24 000 in her bank account after 6 years. If Lily can invest at 15.5% p.a. with interest compounded quarterly, calculate the amount of money that she needs to place in the investment. 22 Rita invests $10 000 at 8% p.a. simple interest. Shaun invests $9000 at 8% p.a. with interest compounded annually. a Which investment is worth more at the end of 5 years and by how much? b Find the number of whole years taken for the value of Shaun’s investment to exceed Rita’s. 23 Michelle and Jack each invest $15 000 at 7.2% p.a. Michelle earns simple interest, Jack earns diGiTal doC WorkSHEET 2.1 doc10310
56
compound interest. a Create a table to show the value of each investment at 5 year intervals for 30 years. b Find the amount of interest earned by each person in the 1st and 30th year. c Compare the growth in the value of each investment over the 30 year period.
Maths Quest Preliminary Mathematics General
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 Preliminary Mathematics General ebook open the spreadsheet ‘Interest’. diGiTal doC Spreadsheet doc1552 Compound 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). 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.
Calculating compound interest from a table of compounded values 2d
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. ChapTer 2 • Investing money
57
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. 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.
= $4560 × 1.170
5
Calculate.
= $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. 58
Maths Quest Preliminary Mathematics General
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 1
Calculate the interest rate per period and number of interest periods.
2
Look up the CVIF for 3% with 6 interest periods.
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
Give a written answer.
WriTe
Interest rate per period = 3% Interest periods = 6 CVIF = 1.194 CV = PV × CVIF $3000 = PV × 1.194 PV =
$3000 1.194
= $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.
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
1
Calculate the interest rate per period.
Interest rate per period = 4%
2
Write the value of PV and CV.
PV = $2500, CV = $3200
3
Write the formula.
4
Substitute the values of PV and CV.
$3200 = $2500 × CVIF
5
Make CVIF the subject of the formula.
CVIF =
6
Calculate the value of CVIF.
7
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.
Seven interest periods will be required.
8
Calculate the length of time for seven interest periods.
It will take 3 12 years for $2500 to grow to $3200.
CV = PV × CVIF
$3200 $2500
= 1.28
ChapTer 2 • Investing money
59
Calculating compound interest from a table of compounded values exercise 2d
1 We10 Toshika has $10 000 to invest for 4 years. The bank offers her 7% p.a. with interest inTeraCTiViTy int2400 simple and compound 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 a Use the CVIF table to calculate the compounded value of Greg’s investment. b Find the amount of interest earned. 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 interest earned on 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 sixmonthly c $250 at 6% p.a. for 5 years with interest compounded sixmonthly 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 d 4 interest periods
b 2 interest periods e 6 interest periods
c 3 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. $900 at 2.5% p.a. for 3 years with interest compounded annually $2340 at 5% p.a. for 2 years with interest compounded sixmonthly $7200 at 10% p.a. for 1 year with interest compounded quarterly $11 000 at 10% p.a. for 2 years with interest compounded quarterly $5750 at 10% p.a. for 1 12 years with interest compounded quarterly
a b c d e
8 mC 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 mC 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
10 We11 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 compounded annually, to have $10 000 in 3 years. Give your answer correct to the nearest dollar.
60
Maths Quest Preliminary Mathematics General
11 We12 How long will it take $2000 to grow to $2500 when invested at 8% p.a. with interest 12
13 14
15
compounded sixmonthly? 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 sixmonthly 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 sixmonthly. 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.) 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 sixmonthly 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 sixmonthly Bruce, Keith and Max each have $10 000 to invest over a 5year 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 sixmonthly. 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.
Further development 16 Use the compound interest formula to find the value of $1 invested at 12% for 1 to 10 interest periods. 17
18 19
20
21
Give each answer correct to 3 decimal places. Use your answers to question 16 to answer each of the following questions. Find the value of an investment of: a $3000 at 12% p.a. for 4 years with interest compounded annually b $7560 at 12% p.a. for 8 years with interest compounded annually c $12 500 at 24% p.a. for 4 years with interest compounded sixmonthly. Use your answers to question 16 to determine the number of interest periods required for an investment to double in value at an interest rate of 12% per interest period. Use the table on page 58 to determine which will give the greatest amount on maturity. An investment at 7% for 5 interest periods OR an investment at 5% for 7 interest periods. Explain your choice of answer. Consider an investment at 4% per interest period. The CVIF table on page 58 gives the values for up to 10 interest periods. Calculate the CVIF values for 11 to 20 periods giving each answer correct to 3 decimal places. Use your answer to question 20 to find: a the value of an investment of $4500 at 4% p.a. for 12 years b the interest earned on an investment of $6250 at 16% p.a. for 5 years with interest c the number of interest periods for an investment to double at 4% per interest period. ChapTer 2 • Investing money
61
2e
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 pershare basis. Worked example 13
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
WriTe
1
The dividend is calculated by dividing the profit by the number of shares.
Dividend = $34 200 000 ÷ 90 000 000 = $0.38
2
Give a written answer.
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 14
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%.
exercise 2e
Dividend yield =
0.25 5.42
× 100%
= 4.61%
share dividends
Unless stated otherwise, for the calculations in this exercise, assume that companies distribute all their profits as dividends. 1 We13 A company has issued 20 million shares and makes an aftertax 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. 62
Maths Quest Preliminary Mathematics General
3 A company makes an aftertax 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 aftertax 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. 5 A company makes a beforetax (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 aftertax 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 aftertax 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 with 42 million shares has a gross profit of $72.4 million. a Find the net profit given that the company pays tax at a rate of 36%. b The company decides to keep 25% of the net profit for future projects and distributes the rest to
the shareholders. Calculate the dividend per share in cents correct to 1 decimal place. 8 A company declares a dividend of 14 cents per share and there are 23.4 million shares in the company. a Find the net profit of the company. b Given that the company paid tax at the rate of 36% find the gross profit of the company. 9 A company declares a dividend of 78c. If there are 4.2 million shares in the company, calculate the
aftertax profit of the company. 10 We14 A company with a share price of $10.50 declares a dividend of 48c per share. Calculate the
dividend yield for this company. 11 Copy and complete, correct to 1 decimal place, the table below.
Dividend
Share price
$0.56
$8.40
$0.78
$7.40
$1.20
$23.40
$1.09
$15.76
$0.04
$0.76
Dividend yield (%)
12 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. 13 mC Which of the following companies paid the highest dividend yield? a B C d
Company A has a share value of $4.56 and pays a dividend of 35c/share. Company B has a share value of $6.30 and pays a dividend of 62c/share. Company C has a share value of $12.40 and pays a dividend of $1.10/share. Company D has a share value of 85c and pays a dividend of 7.65c/share.
14 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.
Further development 15 Rank each of the following share performances in order
from best to worst. a 23c per share dividend at a share price of $3.46 B 71c per share dividend at a share price of $8.29 C $1.23 per share dividend at a share price of $12.39 d $2.30 per share dividend at a share price of $19.49
ChapTer 2 • Investing money
63
16 Jerry buys 1000 shares in Cannington Ltd at $5.60 each. The company pays a dividend of 57 cents per
share. He also buys 500 shares in Warragul Ltd at $23.45 which pays a dividend of $1.98 per share. a Calculate the dividend yield for Cannington Ltd. b Calculate the dividend yield for Warragul Ltd. c Calculate the overall dividend yield for Jerry’s investment. 17 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. 18 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 aftertax profit of the company. 19 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). 20 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. 21 A company that has 18.6 million shares has an after tax profit of $25.7 million. a Calculate the dividend paid to the shareholders. b Calculate the dividend yield given that the share price is $10.60. c To raise funds the company issue another two million shares. Given that the company increases its
profit by 10% in the next year calculate the dividend yield. 22 Explain why the dividend yield is a better indicator of share performance than the actual amount of
the dividend.
2F
Graphing share performance
Because shares offer no guaranteed returns, we can only use the past performance of a share to try to predict its future performance. This is done by graphing the value of the share at regular intervals and then drawing a line of best fit to try to monitor the trend. By continuing the line of best fit you can make a prediction for future share prices. This is called extrapolating information from the graph. Interpolate is the opposite of extrapolate and occurs when drawing a graph using data found at the end points. Worked example 15
Share price ($)
The graph shows the share price of a company over a 3‑month period. 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. 4.40 4.20 4.00 3.80
64
Maths Quest Preliminary Mathematics General
ec D
ov
Month
1
N
ct
1
O 1
1
Se
pt
3.60
Think
WriTe/draW
a Draw a line on the graph, which best
a
4.80 Share price ($)
fits between the points marked.
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
b The predicted share price is $4.80.
months and read the predicted share price.
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. Worked example 16
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
$11.72
a On a set of axes plot the share price for each month and draw a line of best fit. b Predict the share price in June of the following year.
a 1 Draw up a set of axes and plot
WriTe/draW
a Share price ($)
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 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
Think
Month
ChapTer 2 • Investing money
65
Draw a straight line on the graph that best fits in with the marked points.
12.40 12.20 12.00 11.80 11.60 11.40 11.20 11.00 10.80 10.60 10.40 10.20 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 ($)
2
Month
b 1 Extend the line of best fit for six
b
months. Predict the share price by reading from the line of best fit.
exercise 2F
The predicted share price is $12.35.
Graphing share performance Share price ($)
1 We15 The graph at right shows the movement in a share price
over a 2month 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 November.
6.50 6.30 6.10 5.90
2 The graph at right shows the movement in a share price over
Month
4.50 4.00 3.50 3.00 2.50 1– M a 1– y Ju n 1– Ju 1– l Au 1– g Se p 1– t Oc 1– t No 1– v De c
Share price ($)
a 6month period. a Copy the graph into your book and on it draw a line of best fit. b Use your graph to predict the value of the share on 1 February.
1– Ju n
1– M ay
5.70 1– Ju l
2
Month
3 The graph at right shows the movement in a share
1.14 Share price ($)
price over a 9month 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.
1.12 1.10 1.08 1.06 1.04
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
1.02
Month
66
Maths Quest Preliminary Mathematics General
4 We16 The table below shows the share price of a large multinational company over a 12month
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.
Further development 6 Shares are considered by most people to be a riskier investment than putting money in the bank. a Suggest a reason why this may be the case. b How is this demonstrated by the share graphs? 7 Explain why predictions are made about future share performance by using a line of best fit. 8 When drawing a line of best fit: a What type of line is normally drawn? b Is this type of graph necessarily the best? Explain your answer. 9 A share price has lost value over a period of years. A line of best fit is drawn on the graph of the
declining price. a How will the vertical intercept relate to the share price? b What do you know of the gradient of the line of best fit? 10 The graph below shows the fluctuating share price over a five year period. a Find the equivalent simple interest rate to the growth in share price. b Find the equivalent compound interest rate to the growth in share price. $20
(5, $20)
$10 0
5 ChapTer 2 • Investing money
67
inVesTiGaTe: 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.
2G
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 17
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
Increase $35 000 by 5%.
WriTe
Future price = 105% of $35 000 = 105 ÷ 100 × $35 000 = $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 18
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
WriTe
1
Write the values of P, r and n.
P = $800, r = 0.04, n = 5
2
Write down the compound interest formula.
A = P(1 + r)n
3
Substitute the values of P, r and n.
= $800 × (1.04)5
4
Calculate.
= $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. 68
Maths Quest Preliminary Mathematics General
Worked example 19
Jeremy 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.
P = $250, r = 0.2, n = 10
2
Write down the compound interest formula.
A = P(1 + r)n
3
Substitute the values of P, r and n.
= $250 × (1.2)10
4
Calculate and round off to the nearest $10.
= $1550
exercise 2G
inflation and appreciation
1 We17 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% 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? 5 We18 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). ChapTer 2 • Investing money
69
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 mC 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 We19 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 5 years, calculate the value of the shirt (to the nearest $10). 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). 12 The 1968 Australian 2c piece is very rare. If a coin collector purchased one in 2012 for $400 and the
value of the coin increases by 15% per year, calculate its value in 2025 (to the nearest $10). 13 Inflation figures are generally released every quarter. If the average inflation rate is 0.9% per quarter,
find the cost of each of the following items after 3 years. a A newspaper that now costs $1.10. b A loaf of bread that now costs $3.20 c A pair of jeans that now costs $86 d A television that now costs $1650 e A house that now costs $350 000 14 The inflation rate is predicted to average 2.3% p.a. for 2 years and then 3.5% for 3 years. Given that the
price of an iPod is $250 today estimate the cost of the iPod at the end of the 5 year period.
Further development 15 During a severe recession the economy goes into a state of deflation. This is where average prices fall.
If, over a 2 year period the deflation average is 0.2% per quarter, find the price of a new car that was priced at $35 000 at the beginning of the period.
16 At the beginning of 2011 the average wage was $745 per week and the average basket of groceries cost
$143.50. a What percentage of the average wage was the cost of the average basket of groceries? b Over the next three years inflation is expected to be 4.1% p.a. Estimate the cost of the average basket of groceries at the end of the three years. c Over the same period of time wages are expected to rise by only 3% p.a. Estimate the average wage at the end of the three year period. d Economists say that wages over the three years have dropped ‘in real terms’. By comparing the cost of an average basket of groceries and wages at the beginning and end of the three year period explain what the economists mean. 70
Maths Quest Preliminary Mathematics General
17 Eddie has $15 000 to invest. In three years he wants to buy a car that currently costs $18 000. Eddie
invests his money at 4.8% p.a. with interest compounded quarterly. Over the same period inflation is expected to average 0.85% per quarter. a How much more money does Eddie need to purchase the car at the beginning of the three year period? b Calculate how far short Eddie is of the money needed to purchase the car at the end of the three years. 18 Consider an investment of $1000 at 5% p.a. for one year in a period where inflation is running at 3.5% p.a. a What will the $1000 investment be worth after one year? b What will the cost of $1000 worth of goods be after one year? c Write the value of the investment after one year as a percentage of the cost of $1000 worth of goods after one year. d By what percentage has the investment grown in real terms?
diGiTal doC WorkSHEET 2.2 doc10311
ChapTer 2 • Investing money
71
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.
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 and appreciation
• 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 collectables and of memorabilia, which tend to rise at a rate greater than inflation.
72
Maths Quest Preliminary Mathematics General
Chapter review 1 The simple interest paid on $5600 at 5.6% for 3 years is: a $940.80 C $6540.80
m U lTip l e C ho iC e
B $994.46 d $6594.47
2 The compound interest paid on $5600 at 5.6% for 3 years with interest compounded annually is: a $940.80 C $6540.80
B $994.46 d $6594.47
3 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 In 2013, 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 s ho rT a n s W er
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 1 e $45 820 at 4.75% p.a. for 3 2 years 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 paid Dion in quarterly instalments, calculate the value of each interest payment. 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. 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. 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.
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. 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. 8 Barry has an investment with a present value of $4500. The investment is made at 6% p.a. with interest
compounded sixmonthly. Calculate the value of the investment in 4 years. 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 sixmonthly 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 sixmonthly 1 e $5000 at 2.6% p.a. for 2 2 years with interest compounded quarterly 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. ChapTer 2 • Investing money
73
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. 12 Use the table of CVIF values on page 58 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 sixmonthly d $12 400 at 10% p.a. for 5 years with interest compounded sixmonthly e $2400 at 4% p.a. for 2 years with interest compounded quarterly 13 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. 14 A company that has an aftertax profit of $2.3 billion distributes this among its 156 million shares.
Calculate the dividend that this company will declare. 15 A company has a share price of $8.62. It declares a dividend of 45c per share. Calculate the dividend
yield on this share. 16 A company with a share price of 45c declares a dividend of 0.7c per share. Calculate the dividend yield
on this investment. 17 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. 18 The table below shows the fluctuations in a share price over a period of 1 year.
Month
Share price
January
$15.76
February
$16.04
March
$16.27
April
$16.12
May
$16.49
June
$16.39
July
$16.60
August
$16.77
September
$16.51
October
$16.71
November
$16.69
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. 19 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. 20 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. 21 In 1985, 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 2015 (to the nearest $100). 74
Maths Quest Preliminary Mathematics General
1 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.
e x Ten d ed res p o n s e
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. 2 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?
diGiTal doC Test Yourself doc10312 Chapter 2
ChapTer 2 • Investing money
75
ICT activities 2a
Calculation of simple interest
diGiTal doCs • GC program — Casio (doc1544): Interest (page 46) • GC program — TI (doc1545): Interest (page 46) • Spreadsheet (doc1546): Interest (page 46)
2B
Graphing simple interest functions
diGiTal doC • Spreadsheet (doc1546): Interest (page 52)
2C
Calculation of compound interest
diGiTal doCs • Spreadsheet (doc1552): Compound interest (pages 55, 57) • WorkSHEET 2.1 (doc10310): Apply knowledge of interest calculations to questions. (page 56) inTeraCTiViTy • int0810: Compound interest (page 53)
76
Maths Quest Preliminary Mathematics General
2d Calculating compound interest from a table of compounded values inTeraCTiViTy • int2400: Simple and compound interest (page 60)
2G
inflation and appreciation
diGiTal doC • WorkSHEET 2.2 (doc10311): Apply knowledge of interest and inflation rates to problems. (page 71)
Chapter review diGiTal doC • Test Yourself Chapter 2 (doc10312): Take the endofchapter test to check your progress. (page 75)
To access eBookPLUS activities, log on to www.jacplus.com.au
Answers CHAPTER 2 exercise 2a
Calculation of simple interest 1 $600 2 a $120 b $615 c $21 420 d $6656.85 e $90 f $684.47 3 a $2240 b $11 200 4 a $2400 b $12 400 5 $8648.80 6 a $5.25 b $145.25 7 a $448.63 b $224.32 c $9558.84 8 $850.50 9A 10 C 11 $131.86 12 18 months 13 $6250 14 C 15 a $627.13 b $12 542.50 16 a $2250 b $13 500 1 c 7 years 2
2
3
4
Interest ($)
5
2
3 4 Years
5
Interest ($) Interest ($)
5 a
0
1
2
3 Years
4
5
2 a
1
2
3
4
Interest ($)
16 000 14 000 12 000 10 000 8000 6000 4000 2000 0
c 1600 d $16 000
0 1 2 3 4 5 6 7 8 9 10 Years
Interest (3.75%)
1
$165.00
$192.50
$206.25
2
$330.00
$385.00
$412.50
3
$495.00
$577.50
$618.75
0
1
2 3 Years
4
4
$660.00
$770.00
$825.00
5
$825.00
$962.50
$1031.25
5
b
10 000
3.75% 3.5% 3%
1500 1000
5000 1
0
2 3 4 Years
500 0
5
0
1
2 3 4 Years
400
10 a $4500, $5000, $5200 b $875
300
11
200 100 1
0
2 3 4 Years
5
5
5000 6.2% 5.8% 5.2%
4500 4000 0
4000
5
Interest $1600 $3200 $4800 $6400 $8000
Interest (3.5%)
15 000
Investment ($)
0
Interest (3%)
1000
b
500
No. of years
2000
0
1000
b
1
3000
1500
No. of years
0
20 000
2000 Interest ($)
9 a
0
$400 $800 $1200 $1600 $2000
5
200
Interest ($)
1
2 3 4 Years
400
600
0
Interest ($)
No. of years
1
0
Eastpac = $35 000, NZA bank = $30 000 b Whichbank = 7.5%, Eastpac = 7%, NZA bank = 6% 8 Bank B as the investment grows quicker
800
4 288
1 a
b
5
4000
d
5000
7 a Whichbank = $37 500,
Graphing simple interest
functions
2 3 4 Years
8% 6% 5%
10 000
0 1
0
1000
c
$18 750 18 $21 562.50 $5580 a $4560 b $16 888.89 4 years and 8 months Ted is correct because interest is calculated only upon the principal and interest paid earns no interest. 23 Chris is incorrect. If the interest rate doubles the amount of interest paid will double, but the principal will remain the same.
Interest
2000
0
17 19 20 21 22
exercise 2B
4000
0
b
b
6000
Interest ($)
Interest ($)
3 a
Interest ($)
inVesTinG money
0
1
2
3
4
5
3000
Years
2000
12 a 96 b The interest earned each year c $1600 13 a $2000 b 10% p.a. c $3600 14 This statement is correct as the amount
1000 1
2
3 Years
4
5
6 a
No. of years
Interest (5%)
Interest (6%)
Interest (8%)
1
$1250
$1500
$2000
2
$2500
$3000
$4000
3
$3750
$4500
$6000
4
$5000
$6000
$8000
5
$6250
$7500
$10 000
being added in both cases is the annual interest. 15 a 5000 b 250 c A = 250n + 5000 16 A = 280n + 4000 17 a This is correct as the number of years is multiplied by the amount of interest per year. b This is not a direct variation as the graph does not pass through (0, 0).
ChapTer 2 • Investing money
77
$15 000
$15 000
5
$20 400
$21 235.63
10
$25 800
$30 063.47
15
$31 200
$42 561.12
20
$36 600
$60 254.15
25
$42 000
$85 302.33
30
$47 400
$120 763.26
b $1080, $1080; $1080, $8110.97 c $32 400; $ 105 763.26.
78
exercise 2e
1 2 3 4
share dividends 25c/share $1.50/share 6c/share 29.27c/share
Maths Quest Preliminary Mathematics General
21 22
Dividend yield
$0.56
$8.40
6.7%
$0.78
$7.40
10.5%
$1.20
$23.40
5.1%
$1.09
$15.76
6.9%
$0.04
$0.76
5.3%
2.91% B $364 D–C–B–A a 10.2% b 8.4% c 9% a 6.6% b $1.06/share a $1.44 b $7.4906 million 8.5% a 0.59% b $10.64 c 6.4c/share d 0.61% a $1.38/share b 13% c 12.9% Because the dividend yield expresses the share market dividend as a percentage of the share price hence making it comparable to other shares and other forms of investment.
exercise 2F Graphing share performance
1 a Share price ($)
0
17 18 19 20
Share price
7.30 7.10 6.90 6.70 6.50 6.30 6.10 5.90 5.70
1– M ay 1– Ju n 1– Ju l 1– Au g 1– Se pt 1– Oc t
Jack (CI)
12 13 14 15 16
Dividend
Month
b Approximately $7.60 2 a 4.50 4.00 3.50 3.00 2.50 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
Years Michelle (SI)
5 a $1.224 million b $2.176 million c 43.52c/share 6 a $9.28 million b $6.08 million c $0.76/share 7 a $46.336 million b 82.7c 8 a $3.276 million b $5.118 75 m illion 9 $3.276 million 10 4.57% 11
Share price ($)
23 a
exercise 2d Calculating compound interest from a table of compounded values 1 $13 110 2 a $12 580 b $4080 3 $46 800 4 $2599.20 5 a $3385 b $3195.90 c $86 d $1964.61 e $1669.50 f $7130 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 $6110, Max $6290 e Bruce 50%, Keith 61.1%, Max 62.9% 16 1.120, 1.254, 1.405, 1.574, 1.762, 1.974, 2.211, 2.476, 2.773, 3.106 17 a $4722 b $18 718.56 c $30 950 18 7 interest periods 19 5% for 7 interest periods has a greater CVIF value. 20 1.539, 1.601, 1.665, 1.732, 1.801, 1.873, 1.948, 2.026, 2.107, 2.191 21 a $7204.50 b $7443.75 c 18 interest periods
1–
exercise 2C Calculation of compound interest 1 $6655 2 a $17 173.50 b $2173.50 3 $2938.66 4 a $4630.50 b $9274.19 c $24 488.80 d $13 503.78 e $12 588.72 5 $70 555.25 6 $502 7 $2059.91 8 a 0.5% b $31 850.33 9 a $1003.69 b $9111.56 c $181 402.12 d $20 039.67 e $1 264 568.95 10 B 11 B 12 B 13 C 14 a $15 746.40 b $15 793.09 c $15 817.40 d $15 833.99 15 a 0.0219% b $108 320.72 c $8320.72 d $320.72 16 a $4720 b $4726.24 c Compounding interest 17 $19 845.81 18 A 19 a $3514.98 b $2687.83 c $8061.13 d $11 599.22 20 $2705.72 21 $9637 22 a Rita’s by $776.05 b 8 years
b Approximately $2.00
3 a
17 a b 18 a c
Share price ($)
2.04 2.02 2.00 1.18 1.16 1.14 1.12 1.10 1.08 1.06 1.04 1.02 1.00
mUlTiple ChoiCe
1 A 3 C
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 No 1– v 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
14.00
1 $1000 2 a $1296 d $4.05 3 a $7280 4 6.5% 5 15 months 6 a
No. of years
12.00 10.00 1– Ja 1– n Fe 1– b M 1– ar A 1– pr M 1– ay Ju 1– n J 1– ul Au 1– g Se 1– p O 1– ct No 1– v De 1– c Ja 1– n Fe 1– b M 1– ar A 1– pr M 1– ay Ju n
Share price ($)
16.00
Month
Interest b Interest ($)
b Approximately $15.00 5 a 14.50 14.00 13.50 13.00
b $14.50 6 a Because share prices go up and down. b The graph of the share price is irregular reflecting the rises and
falls.
7 The line of best fit is the average trend over a period of time. 8 a Straight line b Not always as the growth may be exponential (the same shape as
a compound interest graph) The purchase price It will be negative. 20% 14.87%
inflation and appreciation
1 $20 800 2 a $618 b $48.15 c $1.91 d $579.60 e $932.40 3 a $878.05 b $901.76 4 $117.90 5 $619 6 $2.52 7 $1.20 8 $122.80 9 D 10 $500 11 $2350 12 $2460 13 a $1.22 b $3.56 c $95.76 d $1837.29 e $389 728.38 14 $290.07 15 $34 443.90 16 a 19.26% b $161.88 c $814.08 d The average basket of groceries now takes up 19.88% of the
average wage, a greater proportion, hence the wage at the end of the period has less purchasing power than at the beginning of the period.
c $42 c $455
1
2
3
4
5
$450
$900
$1350
$1800
$2250
0 1 2 3 4 5 6 7 8 9 10 Years
c 450 d $4500
7 $2778.30 8 $5700.47 9 a $3932.39 b d $23 851.00 e 10 $756.94 11 a $12 024.02 b $12 052.04 c Compounding interest 12 a $7866 b d $20 199.60 e 13 $1.93/share 14 $14.74/share 15 5.22% 16 1.6% 17 0.82c/share 18 a Share price ($)
Month
4500 4000 3500 3000 2500 2000 1500 1000 500 0
b $2820 e $7617.58 b $39 780
$12 596.90 $5334.67
c $14 457.96
$13 216.44 $2599.20
c $4411.83
17.00 16.50
16.00 15.50 1– Ja 1– n Fe 1– b M 1– ar A 1– pr M 1– ay Ju 1– n J 1– ul Au 1– g Se 1– pt O 1– ct No 1– v De c
12.50 12.00 1– Ja 1– n F 1– eb M 1– ar A 1– pr M 1– ay Ju 1– n J 1– ul Au 1– g Se 1– pt O 1– ct No 1– v De c
Share price ($)
2 B 4 B
shorT ansWer
b Approximately $1.20 4 a
exercise 2G
b $1035 d 1.45%
ChapTer reVieW
Month
9 a b 10 a b
$3000 $2615.89 $1050 101.45%
Month
b Approximately $18.00
19 $83.44 20 $149.90 21 $7900
exTended response
1 a b c d e 2 a
$8130 No. The investment will be worth only $8121.43. No. Jaclyn could lose the money she has saved for her holiday. $9345 24.6% $32 136.75 b $32 600 c $436.17
ChapTer 2 • Investing money
79
ChapTer 3
Taxation ChapTer ConTenTS 3a 3B 3C 3d 3e 3F
Calculating allowable deductions Taxable income Medicare levy Calculating tax Calculating GST and VAT Graphing tax functions
3a
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 Go tax (PAYG 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 PAYG 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 drycleaning 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 drycleaning the overalls and $1.50 per week for workrelated telephone calls. Calculate Ken’s total deductions. Think
WriTe
1
Calculate Ken’s total drycleaning and telephone deductions.
Drycleaning = $5 × 52 = $260 Telephone = $1.50 × 52 = $78
2
Add up all of Ken’s deductions.
Deductions = $1400 + $25 + $200 + $260 + $78 = $1963
ChapTer 3 • Taxation
81
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 longterm costs of running a car such as registration, insurance, depreciation and maintenance. 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 74 c/km. Calculate the size of the tax deduction in a year where she travels 2547 km on workrelated matters. Think
Multiply the number of kilometres (2547) by the rate per kilometre (0.519). Be sure to convert the rate in cents to dollars.
WriTe
Travel deduction = 2547 × 0.74 = $1884.78
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 2010 for $3200. Each year he is allowed a 33% deduction for the depreciation of the computer. Calculate the tax deduction allowed in: a the 2010–11 financial year b the 2011–12 financial year. Think
WriTe
a The depreciation was 33% of the purchase price.
a Tax deduction = 33% of $3200
b 1 Calculate the value of the computer at the
b Computer value = $3200 − $1056
beginning of 2008–09, by subtracting the depreciation from the purchase price. 2
The depreciation was 33% of its value at the end of the last financial year.
= 0.33 × $3200 = $1056 = $2144
Tax deduction = 33% of $2144 = 0.33 × $2144 = $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 deductible. 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. 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
82
Calculate the amount of rent that Wendy pays for a year.
Maths Quest Preliminary Mathematics General
WriTe
Rent = $400 × 52 = $20 800
2
Calculate Wendy’s annual electricity bill.
Electricity = $326 × 4 = $1304
3
Calculate Wendy’s annual telephone bill.
Telephone = $276 × 12 = $3312
4
Add these expenses to calculate the tax deduction allowed.
Total tax deduction = $20 800 + $1304 + $3312 = $25 416
exercise 3a
Calculating allowable deductions
1 KiYeong has a gross annual income of $39 650 and allowable tax deductions of $934. Calculate
KiYeong’s taxable income. 2 Trevor has a gross annual income of $55 000. Trevor also earned $435 from other investments and has
allowable deductions of $1326. Calculate Trevor’s taxable income. 3 We1 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. 4 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 workrelated telephone calls per week. Calculate Jasmine’s total tax deductions. 5 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 drycleaned 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. 6 Matt works as a sports journalist. He claims $60 per week for telephone calls, $600 for a pair of
binoculars, $25 per week for admittance to sporting events and $1250 for travel. Calculate Matt’s total tax deductions. ChapTer 3 • Taxation
83
7 We 2 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 63 c/km. Calculate the tax deduction Rajid is allowed in a year where he claims 3176 km in workrelated travel. 8 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 workrelated travel). Engine capacity Up to 1.6 L More than 1.6 L and up to 2.6 L More than 2.6 L
Allowable deduction 63 c/km 74 c/km 75 c/km
Calculate the total tax deduction allowed for a person who claims: 2000 km in a Mazda2 with a 1.3 L engine 2645 km in a Toyota Corolla with a 1.8 L engine 1564 km in a Ford Focus with a 2.5 L engine 2900 km in a Holden Commodore with a 3.6 L engine. 9 Briony uses her car for work related matters. During the year she changed cars. Briony did 2943 km in her 1.4 litre Mazda 3 before upgrading to a 2.2 litre. She then did 1854 km in the new car. Find the total amount of Briony’s tax deduction for travel. a b c d
10 Calculate the difference in tax deductions allowed for 2700 km of travel between a 1.6 L vehicle and a
2.6 L vehicle. 11 We 3 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. 12 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. 13 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 2013–14 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 2013–14 financial year ii 2014–15 financial year iii 2015–16 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? 14 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 computer for only 12 of the financial year. A deduction of 40% p.a. is allowed for depreciation of the computer, but Catherine can claim only 8 of this. Calculate Catherine’s allowable tax deduction for the 12 computer. 15 We 4 Gabrielle owns a small boutique in a shopping mall. Her operational expenses are: • $325 per week rent • $280 per quarter in electricity • $185 per quarter in telephone bills. Calculate the total allowed in tax deductions for running this business.
84
Maths Quest Preliminary Mathematics General
Further development 16 Greg is a graphic designer who works from home. He has set up one room in the house as his office. In
17
18
19
20
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. Henry is a motor mechanic who runs his own garage. Henry has the following workrelated expenses: • $350 per week for rent on the garage • $590 per quarter for the electricity bill • $260 per quarter for the telephone bill • $75 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 workrelated trips in his van, which has a 3.0 L engine. Calculate Henry’s total tax deductions. John has a 2.6 litre car which he uses for work purposes. John does 4850 km travel in his car during the financial year. His total car expenses for the year are $10 325. He has the choice of using the cents per kilometre method of claiming his tax deduction, or onethird of his total expenses. (Refer to the table in question 8 on page 84.) a Which method should John choose? b What tax deduction will this give John? When travelling more than 5000 km each year, the onethird of all expenses method must be used to calculate the tax deduction. Len has a new Toyota Aurion that cost $40 000 and is allowed 15% depreciation in his expenses. a Calculate the depreciation that Len is allowed to claim. b If Len’s other expenses amount to $5850 calculate the total amount of Len’s tax deduction. Explain what is meant by the term tax deduction.
3B
Taxable income
A person’s taxable income is the income on which their tax is calculated. Most people have PAYG tax deducted from their wage or salary throughout the year. The amount of PAYG 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 PAYG 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.
ChapTer 3 • Taxation
85
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 2010–11 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.
Total income = $34 500 + $278.50 + $148.63 = $34 927.13
2
Calculate the taxable income by subtracting tax deductions from gross income.
Taxable income = $34 927.63 − $1846.30 = $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 fulltime jackeroo and plays two evenings per week in a band. Murray earns $471.52 per week from his fulltime 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 deductible. 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. Think
a 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.
b 1 Calculate the total number of trips made
between jobs.
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 = $30 770.12 b Number of trips = 2 × 52
= 104
2
Calculate the total kilometres travelled.
Total km = 104 × 15 = 1560 km
3
Calculate the deduction by multiplying kilometres travelled by the allowable rate.
Travel deduction = 1560 × 0.457 = $712.92
c 1 Calculate the total tax deduction. 2
86
WriTe
Calculate the taxable income by subtracting the tax deductions from the total income.
Maths Quest Preliminary Mathematics General
c Total deduction = $948.50 + $712.92
= $1661.42
Taxable income = $30 770.12 − $1661.42 = $29 108.70
exercise 3B
Taxable income
1 We5 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.
4
5
6
7
8
9
Janelle also earned $238.79 in interest from her bank accounts. She also had tax deductions totalling $2340.45. Calculate Janelle’s taxable income. 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. Janine has two parttime 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. 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. We6 Stefan worked as a hairdresser and earned a gross wage of $537.90 per week. He also worked parttime 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 74 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. mC Wayne’s gross fortnightly pay is $1156.60. He has tax deductions of $5 per week for drycleaning his work uniform, $50 per month in workrelated 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 Garry has a net annual income of $45 670 after tax deductions of $7450. Find Garry’s gross income.
Further development 10 Nicole has a gross income of $48 730 and a net income of $34 970. Calculate the amount that Nicole
has in tax deductions. 11 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 63 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. 12 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. ChapTer 3 • Taxation
87
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. 13 Georgia earns $13.40 per hour in her
job as a waitress. Georgia works a 38 hour week. a Calculate Georgia’s gross weekly wage. b Calculate Georgia’s gross annual wage given that she took 4 weeks holiday and was paid a holiday loading of 17.5% during that period. c Georgia had tax deductions of $12 per week for having her uniform cleaned and $175 for union fees. Calculate Georgia’s taxable income. 14 Ian is a telephone salesperson who is paid 15% commission on all sales in addition to a $250 per week
retainer. a Calculate Ian’s gross income given that his annual sales are $185 750. b Ian has the following tax deductions, $120 per week for the telephone, 10% of his household bills of $8750 for his home office and 780 km in travel at 74 cents per kilometre. Calculate Ian’s taxable income. 15 Explain what is meant by the terms: a gross income b taxable income.
Computer Application 1 Calculating taxable income We are going to use a prepared spreadsheet to calculate a person’s taxable income. diGiTal doC Spreadsheet doc1608 Tax calculator
88
Maths Quest Preliminary Mathematics General
1. From your Maths Quest Preliminary Mathematics General eBookPLUS, open 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. 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 170 House area (m2) 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. ChapTer 3 • Taxation
89
3C
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, xrays 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. Worked example 7
Calculate the Medicare levy for a person with an annual taxable income of $44 300. Think
WriTe
Medicare levy = 1.5% of $44 300 = 0.015 × $44 300 = $664.50
Calculate 1.5% of $44 300.
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 0–1 2 3 4 More than 4 dependent children
Surcharge income threshold $100 000 $101 500 $103 000 $104 500 $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
WriTe
1
Calculate the regular Medicare levy.
Medicare levy = 1.5% of $164 000 = 0.015 × $164 000 = $2460
2
Decide if the person must pay the Medicare levy surcharge.
This person’s income is over the $101 500 threshold for a person with two children.
3
Calculate the Medicare levy surcharge.
Surcharge = 1% of $164 000 = $1640
4
Calculate the total Medicare levy by adding the surcharge to the regular levy.
Total Medicare levy = $2460 + $1640 = $4100
exercise 3C
medicare levy
1 We7 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 90
Maths Quest Preliminary Mathematics General
c $94 000.
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. 4 mC 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 We8 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. 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. 8 James has an annual income of $250 000 and is single with no children. The cost of private health
diGiTal doC WorkSHEET 3.1 doc10313
insurance for James would be $950 per year. Calculate how much James would save each year by joining a private health fund as opposed to having to pay the Medicare levy surcharge.
Further development 9 James pays $597 in Medicare levy. Calculate his taxable income. 10 Tim has a gross income of $51 000 per year. He does not have private health insurance. a Calculate the Medicare levy that Tim must pay including the 1% surcharge. b Tim is able to find $1500 in tax deductions. This brings his income below the threshold for the
surcharge. Calculate the amount by which the Medicate levy is reduced. 11 What is the difference in the Medicare levy between a single person who earns $49 900 and someone
who earns $50 100? 12 Explain what the Medicare levy is. inVeSTiGaTe: 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?
3d
Calculating tax
The amount of PAYG tax payable is based on a table that shows the annual tax payable in 2011–12. 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 on this income
0–$6000
Nil
$6001–$37 000
15c for each $1 over $6000
$37 001–$80 000
$4650 plus 30c for each $1 over $37 000
$80 001–$180 000
$17 550 plus 37c for each $1 over $80 000
$180 001 and over
$54 550 plus 45c for each $1 over $180 000
ChapTer 3 • Taxation
91
From the 2011–12 taxation rate table above 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 $37 000 pay 15c for every dollar over $6000. Bracket 3: People who earn between $37 001 and $80 000 pay $4650 plus 30c for every dollar over $37 000. Bracket 4: People who earn between $80 001 and $180 000 pay $17 550 plus 37c for every dollar over $80 000. Bracket 5: People who earn over $180 000 pay $54 550 plus 45c for every dollar over $180 000. To calculate the amount of PAYG 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 $39 600. Think
WriTe
1
$39 600 is in the $37 001 to $80 000 tax bracket.
2
Tax payable is $4650 plus 30c (0.3) for each $1 over $37 000.
3
Calculate the amount over $37 000 by subtracting $37 000 from $39 600.
$39 600 − $37 000 = $2600
4
Apply the rule $4650 plus 30c for each $1 over $37 000.
Tax payable = $4650 + 0.3 × $2600
5
Calculate.
= $5430
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
a 1 $85 000 is in the $80 001 to $180 000 tax bracket.
WriTe
a
2
Subtract $80 000 from $85 000 to calculate the amount over $80 000.
$85 000 − $80 000 = $5000
3
Apply the rule $17 550 plus 37c for every dollar over $80 000.
Income tax = $17 550 + 0.37 × $5000 = $19 400
b Calculate the Medicare levy as 1.5% of $85 000.
b Medicare levy = 1.5% of $85 000
c Add the Medicare levy to the income tax to find the total
c Total tax = $19 400 + $1275
tax payable.
= 0.015 × $85 000 = $1275
= $20 675
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 PAYG tax. The yearly amount of tax is then calculated using this amount, then divided into a weekly payment. 92
Maths Quest Preliminary Mathematics General
Worked example 11
Trevor earns a gross wage of $772.70 per week. Calculate the amount of PAYG tax that is deducted by his employer. Think
WriTe
$772.70 per week = $40 180 per year
1
Calculate $772.70 per week as an annual amount by multiplying by 52 (ignore any cents).
2
$40 180 is in the $37 001 to $80 000 tax bracket.
3
Calculate the amount over $37 000 by subtracting $37 000 from $40 180.
$40 180 − $37 000 = $3180
4
Apply the appropriate rule to calculate the tax payable.
Annual tax = $4650 + 0.3 × $3180 = $5604
5
Calculate the Medicare levy as 1.5% of $40 180.
Medicare levy = 1.5% of $40 180 = 0.015 × $40 180 = $602.70
6
Calculate the total tax payable for the year.
Total yearly tax = $5604 + $602.70 = $6206.70
7
Divide the yearly tax into weekly instalments by dividing by 52.
Weekly tax = $6206.70 ÷ 52 = $119.36
PAYG 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 payment summary from each of their employers. A payment summary is a statement of gross earnings and the amount of PAYG tax that has been deducted from those earnings. In a tax return, all payment summaries are collected to find the total gross income and total PAYG 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. Worked example 12
Catherine’s gross annual salary as a veterinarian’s assistant is $44 500. She has paid $7567.70 in PAYG 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
a 1 Calculate taxable income by adding all
incomes and subtracting any tax deductions. 2
WriTe
a Taxable income = $44 500 + $560.40 − $2345
= $42 715.40
Taxable income = $42 715
Ignore cents in stating the taxable income.
b 1 $42 715 is in the $37 001 to $80 000 tax
b
bracket. 2
Subtract $37 000 from $42 715 to calculate the amount earned over $37 000.
$42 715 − $37 000 = $5715
3
Apply the appropriate rule to find the tax payable.
Income tax = $4650 + 0.3 × $5715 = $6364.50 ChapTer 3 • Taxation
93
4
Calculate the Medicare levy.
Medicare levy = 1.5% of $42 715 = 0.015 × $42 715 = $640.73
5
Calculate the total tax payable by adding the income tax and the Medicare levy.
Total tax = $6364.50 + $640.73 = $7005.23
c 1 Catherine has paid more tax than she needed to
c
so she gets a refund. 2
Calculate the size of the refund by subtracting the amount she should pay ($7005.23) from the amount paid ($7567.70).
3
Give a written answer.
$7567.70 − $7005.23 = $562.47
Catherine receives a refund of $562.47.
The Federal Labor Government, led by Prime Minister Julia Gillard, introduced a carbon tax which took effect from 1 July 2012. While this tax was to be paid by big businesses, households were compensated from any resulting price increases through the application of new taxation rates. The following rates for 2012–13 apply from 1 July 2012. Taxable income
Tax on this income
0–$18 200
Nil
$18 201–$37 000
19c for each $1 over $18 200
$37 001–$80 000
$3572 plus 32.5c for each $1 over $37 000
$80 001–$180 000
$17 547 plus 37c for each $1 over $80 000
$180 001 and over
$54 547 plus 45c for each $1 over $180 000
The above rates do not include the Medicare levy of 1.5%.
exercise 3d
Calculating tax
1 We 9 Use the income tax table on page 91 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 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). 94
Maths Quest Preliminary Mathematics General
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. 5 Brett earns $860.75 per week. a What is the gross annual amount? b Calculate the tax payable on that amount. 6 We10 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. 7 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 8 Kerry is a television executive who earns $349 000 per year. a What is Kerry’s gross weekly pay? b What is the annual amount of PAYG tax that Kerry must pay? c Calculate Kerry’s Medicare levy. d Calculate the amount of Kerry’s net weekly pay. 9 We11 Sandy earns a gross weekly pay of $478.60. Calculate the amount of PAYG tax deducted each
week by her employer (including Medicare levy). 10 Ashley earns a gross fortnightly pay of $2174.35. Calculate the amount of PAYG tax that Ashley’s
employer should deduct each fortnight. 11 mC 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 $4650 B $7500 C $8250 d $9300 12 mC Henry earns a gross pay of $1295.60 per fortnight. The amount of PAYG tax including the
Medicare levy that will be deducted from Henry’s pay over a full year will be: a $4152.84 B $4658.12 C $10 105.68 d $33 685.60 13 mC Ian receives a gross pay of $822.50 per week. The only deduction that Ian has taken from his
gross pay is tax including Medicare levy. Ian’s net weekly pay will be: a $122.72 B $135.05 C $687.45 d $699.78 14 Natasha has a gross monthly salary of $6780. Calculate the amount of tax that Natalie has deducted
from her salary each month including the Medicare levy. 15 We12 At right is a payment summary for Wendell Hancock.
Wendell has also earned $372.10 in interest from an investment and has tax deductions totalling $1298. a Calculate Wendell’s taxable income. 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.
PAYMENT SUMMARY
Wendell Hancock Gross income: $39 600.00 PAYG tax deducted: $6024.00
16 Raymond earns a gross weekly pay of $1748.90. a Calculate Raymond’s gross annual pay. b Calculate the amount of PAYG tax including the Medicare levy that would be deducted from
Raymond’s pay each week. ChapTer 3 • Taxation
95
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, including the Medicare levy based on his annual taxable income. d Calculate his refund or tax debt.
Further development 17 Vonda Flockhart is employed by day as a journalist and by night as a radio announcer. Her payment
summaries are shown below. Job 1
Job 2
PAYMENT SUMMARY
PAYMENT SUMMARY
Vonda Flockhart Journalist Gross income: $35 000.00 PAYG tax deducted: $4875.00
Vonda Flockhart Radio announcer Gross income: PAYG tax deducted:
$9605.00 $4322.25
a Calculate Vonda’s gross annual pay from both jobs and the total amount of PAYG 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. d Calculate the tax refund that Vonda is owed. 18 Jelena receives a gross weekly pay of $1350.52. a Calculate the amount of PAYG 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. 19 Lois has a gross annual income from her job as a journalist of $78 600. a Calculate the PAYG tax including Medicare levy that Lois should have deducted from her pay
over the year. b Lois has income of $400 per week from a property that she rents, but her expenses in running
that property total $23 500 per annum. Calculate Lois’ taxable income. Note: When expenses exceed the income this is called negative gearing. c Calculate the amount of tax including Medicare levy that Lois should pay. d Calculate the amount of tax refund that Lois should receive. 20 Clark is also a journalist and receives $68 600 per annum. a Calculate the PAYG tax including Medicare levy that Clark should have deducted from his pay
over the year. b Clark also has a part time job as a super hero. For this he is paid $175 per week but has not paid
any tax on this amount. Clark has a deduction of 4590 km in travel between jobs at a rate of 75 cents per kilometre. Calculate Clark’s taxable income. c Calculate the tax including Medicare levy that Clark should pay. d Calculate the amount of Clark’s tax refund or tax debt. 21 Jimmy is a photographer and has a gross salary of $x and has had the correct amount of tax deducted
from his fortnightly pay. At the end of the financial year Jimmy has deductions of $1340 but has earned $976 in interest from some investments. Explain whether Jimmy will receive a tax refund or have to pay a tax debt. 22 Explain what is meant by each of the following terms: a PAYG tax b tax return diGiTal doC Spreadsheet doc1608 Tax calculator
96
c tax refund.
Computer Application 2 Tax calculation 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.
Maths Quest Preliminary Mathematics General
1. From your Maths Quest Preliminary Mathematics General ebookPLUS, 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.
3e
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. Some items 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 preGST price of $127.50. Calculate the GST payable on the purchase of the bat. Think
Calculate 10% of $127.50.
WriTe
GST payable = 10% of $127.50 = 0.1 × $127.50 = $12.75
When calculating the amount required to purchase an item, you will need to add the GST to the pretax price. The quickest way to do this will be to calculate 110% of the pretax 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. ChapTer 3 • Taxation
97
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.
Total cost = 110% of $19.80 = 1.1 × $19.80 = $21.78
2
Give a written answer, rounding your answer to the nearest 5c.
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 pretax price of the item. This means that we need to divide the total cost by 110%, written as a decimal. Worked example 15
Calculate the pretax price of a car that costs $31 350, including GST. Think
WriTe
1
Total cost is 110% of the price.
2
Price is total cost divided by 1.1.
Price = $31 350 ÷ 1.1 = $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
WriTe
1
Calculate the cost of the car by multiplying the daily rate by the number of days.
Cost = $56.50 × 5 = $282.50
2
Add the VAT by calculating 112.5% of the cost.
Total cost = 112.5% of $282.50 = $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.
exercise 3e
Calculating GST and VaT
1 We13 Calculate the GST payable on a book that has a pretax price of $35.60. 2 Calculate the GST payable on each of the following items (prices given are pretax): a a bottle of dishwashing liquid at $2.30 b a basketball at $68.90 c a pair of cargo pants at $98.50 d a bus fare at $1.30 e a restaurant meal for which the bill totals $89.90. 98
Maths Quest Preliminary Mathematics General
3 Calculate the GST payable on each of the following items (correct to the nearest cent): a a barbecued chicken with a pretax price of $7.99 b a tin of shoe polish with a pretax price of $4.81 c a tin of dog food with a pretax price of 93c d a pack of toilet rolls with a pretax price of $6.25 e a pack of frozen pies with a pretax price of $3.36. 4 We14 A pair of sports shoes that cost $112.50 has 10% GST added to the cost. Calculate the 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 pretax): 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 b c d e
a camera priced at $240 a bus fare for $7.50 a whitewater rafting tour costing $376 a ski lift ticket costing $23.50 a new suitcase priced at $78.90.
ChapTer 3 • Taxation
99
7 We15 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. 9 We16 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 pretax 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.
Further development 12 When overseas tourists leave Australia they are entitled to a refund of GST charged on their purchases.
Calculate the amount of GST refund due for purchases totalling $5674. 13 Under a similar arrangement when Australians travel to the UK they are entitled to a refund of the VAT
which is levied at 15%. Judy has receipts that total £1856. a Calculate the amount of Judy’s refund in £. b Given that £1 = $A0.45 calculate Judy’s refund in $A. 14 Jason has been to the USA where the VAT is levied at different rates in different states. On leaving the
USA he has receipts for goods US$750 where the VAT was levied at 10%, US$450 where the VAT was levied at 11% and US$1677 from where the VAT was 7%. a Calculate the amount of Jason’s VAT refund in US$. b Given that A$1 = US$0.80 calculate the refund in A$. 15 An alcohol product has a wholesale price of $23.67. It is then subject to a 35% tax. After this tax a
retail mark up of 15% is added before the 10% GST is added. Find the final retail price of the alcohol. 16 An item is priced at $220 on the shelf. Keith says that the 10% GST on the item is $22, while Maxine
says that it is $20. Who is correct? Explain the reasoning. 17 GST stands for Goods and Services Tax. Explain what the difference between a ‘Good’ and a
‘Service’ is. 100
Maths Quest Preliminary Mathematics General
Graphing tax functions
3F
We can draw linear graphs to display the tax payable. This is possible for both the GST and income tax.
exercise 3F
Graphing tax functions
inTeraCTiViTY int2404 paYG tax graph
1 Draw a set of axes with the price on the horizontal axis and GST on the vertical axis, as shown. 50 GST ($)
40 30 20 10 0 0
50 100 150 200 250 300 350 400 450 500 Price ($)
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. 2 Draw a set of axes with ‘Income’ on the horizontal axis and ‘Income tax’ on the vertical axis, as shown
Income tax (× $1000)
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 (× $1000)
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 $34 000.
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 $37 500 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. 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 $180 500 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. ChapTer 3 • Taxation
101
Further development 3 A nation has a twotiered GST system. For the first $1000 of any item GST is levied at 5% and any
amount over $1000 is levied at 15%. a Draw a graph of this GST function. b Find the GST levied on an item costing: i $450 ii $1000 c Find the cost of an item for which the GST is: i $20 ii $100 4 A nation has the following tax system. Income
diGiTal doC WorkSHEET 3.2 doc10314
102
iii $3250. iii $355.
Tax
0–$10 000
Nil
$10 000–$50 000
Nil plus 25 cents for every $1 over $10 000
Over $50 000
$10 000 plus 40 cents for every $1 over $50 000
Draw a graph of this taxation system.
Maths Quest Preliminary Mathematics General
Summary Calculating allowable deductions
• Allowable tax deductions are amounts that are deducted from gross income, as they are not taxable. • Deductions are allowed for workrelated 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.
Calculating tax
• Pay As You Go (PAYG) 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.
Calculating GST and VaT
• 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.
ChapTer 3 • Taxation
103
Chapter review m U lT ip l e C h oiCe
1 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 Allan’s taxable income is $48 000. Allan’s tax for the year is (use the PAYG tax table on page 91
(including Medicare levy)): a $720 B $4200
C $8400
d $9120
3 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 Sh orT anS Wer
B $445.00
C $489.50
d $538.45
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 drycleaning of these overalls and $2.50 per week for workrelated telephone calls. Calculate Tony’s total tax deductions. 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 workrelated trips during the year. Calculate the tax deduction that Catherine can claim if she is entitled to claim at the rate of 63c per kilometre. 3 Brian runs a small bakery and has equipment in his bakery to the value of $45 000 at the beginning of
the 2012–13 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 2013–14 b 2014–15 c 2015–16. 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? 5 Verity is employed as a vet. Verity has the following tax deductions.
• $1500 for appropriate clothing • $5 per week for drycleaning • $2590 for new equipment • 28% depreciation on major equipment currently valued at $65 000 • 3287 km in travelling expenses (at a rate of 74 c/km) • $127.45 per month in telephone calls • $318 per quarter for electricity Calculate Verity’s total tax deductions. 6 Eddie has a gross annual salary of $46 000 and has tax deductions that total $2117. Calculate Eddie’s
taxable income. 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. 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. 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 63 c/km. Raymond had other tax deductions totalling $950. Calculate Raymond’s taxable income for the year. 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. 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. 104
Maths Quest Preliminary Mathematics General
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. 12 Use the table on page 90 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. 13 Use the tax table on page 91 to calculate the tax payable on a taxable income of $44 500. 14 Use the tax table on page 91 to calculate the tax payable on an income of: a $5000 b $19 357 c $35 670 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 PAYG tax that would be deducted from Brett’s wages. 16 Fiona has a gross fortnightly salary of $3367.90. Calculate the amount of PAYG tax that would be 17
18 19
20 21
deducted from Fiona’s pay each fortnight. Neville has a gross annual salary of $43 750. He has tax deductions totalling $3495. During the year Neville has had $7331.25 deducted in PAYG 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. An electric guitar has a pretax price of $990. Calculate the amount of GST payable on the guitar. 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. A round of golf costs $20 before tax. Calculate the total cost of the game after the 10% GST is added. The total cost of a restaurant meal was $123.75, including GST. Calculate the actual cost of the meal without the tax.
1 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
ex Ten d ed r eS p o n S e
Lleyton’s taxable income. c Calculate the Medicare levy for Lleyton. d Use the table on page 91 to calculate the income tax due for Lleyton. e If Lleyton’s employer has deducted $9438.60 in PAYG tax through the year, calculate Lleyton’s
tax refund or tax debt. 2 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?
diGiTal doC Test Yourself doc10315 Chapter 3
ChapTer 3 • Taxation
105
ICT activities 3B
Taxable income
diGiTal doC • Spreadsheet (doc1608): Tax calculator (page 88)
3C
medicare levy
diGiTal doC • WorkSHEET 3.1 (doc10313): Answer questions involving tax calculations. (page 91)
3d
Calculating tax
diGiTal doC • Spreadsheet (doc1608): Tax calculator (page 96)
106
Maths Quest Preliminary Mathematics General
3F
Graphing tax functions
inTeraCTiViTY • int2404: PAYG tax graph (page 101) diGiTal doC • WorkSHEET 3.2 (doc10314): Answer questions about taxation. (page 102)
Chapter review • Test Yourself Chapter 3 (doc10315): Take the endofchapter test to check your progress. (page 105)
To access eBookPLUS activities, log on to www.jacplus.com.au
Answers CHAPTER 3 exercise 3a Calculating allowable deductions 1 $38 716 2 $54 109 3 $1153.90 4 $1624.55 5 $1070.10 6 $6270 7 $2000.88 8 a $1260 b $1957.30 c $1157.36 d $2175 9 $3226.05 10 $324 11 a $1000 b $600 c $360 12 $3087.50 13 a i $30 000 ii $18 000 iii $10 800 b 2013–14 14 $960 15 $18 760 16 $712.27 17 $47 937.50 18 a Cents per kilometre b $3589 19 a $6000 b $3950 20 A tax deduction is subtracted from the gross income before any tax calculations are made. exercise 3B
Taxable income
1 $43 754.25 2 a $16 879.20 b $15 736.50 3 $43 568.34 4 a $24 219.40 b $22 262.60 5 a $20 503.60 b $20 075.20 6 a $42 500 b $2600 c $39 900 7 a $34 262.30 b $710.40 c $31 992.00 8A 9 $53 120 10 $13 760 11 a $74 280 b $8958.90 c $65 321.10 12 a $25 649 b $1470 c $1540 d $1349.40 e $21 289.60 13 a $509.20 b $26 834.84 c $26 083.84
14 a $40 862.50 b $33 170.30 15 a The amount of money earned without
any deductions
b That remaining after tax deductions
have been subtracted from gross income
exercise 3C
medicare levy
1 $598.05 2 a $648.75 b $851.18 c $1410 3 a $23 491 b $352.37 4 D 5 $106 000 6 $4600 7 $3500 8 $1550 9 $39 800 10 a $1275 b $532.50 11 $504 12 A tax to pay for Australia’s public medical
system
exercise 3d
Calculating tax
1 $4417.50 2 a $0 b $1901.25 c $3787.50 d $6846 e $22 304.50 f $67 150 3 a $32 639 b $3995.90 4 $8867.40 5 a $44 759 b $6977.70 6 a $4440 b $534 c $4974 7 a $2977.50 b $16 230 c $28 760 8 a $6711.54 b $130 600 c $5235 d $4099.33 9 $61.66 10 $404.23 11 D 12 B 13 C 14 $1606.13 15 a $38 674 b $5732.31 c $291.69 16 a $90 942 b $441.60 c $22 481.04 d Refund $481.94 17 a $44 605, $9197.25 b $41 513 c $6626.60 d $2570.65
18 a $15 671.51 b $2401.56 debt 19 a $18 309 b $75 900 c $17 458.50 d $850.50 20 a $15 159 b $74 257 c $16 940.96 d $1781.96 21 Jimmy will get a tax refund as his
deductions are greater than his additional income. This means that his taxable income will be less than the gross income. 22 a PAYG (Pay As You Go) is the tax that is paid at each pay period. b Tax return is where all income and PAYG tax is calculated at the end of the final year. c Tax refund is the balance paid to the taxpayer when more PAYG tax has been paid through the year than the amount calculated in the tax return. exercise 3e
Calculating GST and VaT
1 $3.56 2 a 23c b $6.89 c $9.85 d 13c e $8.99 3 a 80c b 48c c 9c d 63c e 34c 4 $123.75 5 a $126.39 b $32.89 c $16.17 d $5.45 e $6.47 6 a $30.00 b $0.94 c $47.00 d $2.94 e $9.86 7 $98.50 8 a $1.90 b 19c 9 $348.10 10 a $1.08 b $80.63 c $193.29 d $49.21 e $6.95 11 a $33 550 b $36 630 c $34 705 d $38 885 12 $515.82 13 a £242.09 b $537.98 14 a $US222.48 b $278.10 15 $40.42 16 Maxine is correct as the GST is added on
the wholesale price meaning that the retail price needs to be divided by 11. 17 A ‘Good’ is an item where a ‘Service’ is something that people do for the customer. exercise 3F
1 a
i $10
Graphing tax functions ii $20 iii $50
50
b
40 GST ($)
TaxaTion
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 ($)
ChapTer 3 • Taxation
107
Income tax ($' 000)
2
90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0 200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
Income ($' 000) a Nil b i $600 c i $4800 d i $26 600 e i $58 000 3 a
ii ii ii ii
ShorT anSWer
$4200 $9000 $58 000 $62 500
iii $18 000 iii $67 000
250
GST ($)
200 150 100 50 0
0
b i $22.50 c i $400 4
200 400 600 800 1000 1200 1400 1600 1800 2000 Amount ($)
ii $50 ii $1333.33
iii $387.50 iii $3033.33
35 000 30 000
Tax ($)
25 000 20 000 15 000 10 000 5000
0
10 00 0 20 00 0 30 00 0 40 00 0 50 00 0 60 00 0 70 00 0 80 00 0 90 00 0 10 00 00
0
Income ($)
ChapTer reVieW mUlTiple ChoiCe
1 B
108
2 D
3 B
Maths Quest Preliminary Mathematics General
1 $2415 2 $2849.49 3 a $12 600 c $6531.84 4 6 years 5 $27 783.78 6 $43 883 7 $70 420 8 a $28 483.60 9 a $34 409.30 10 $585 11 a $232.50 c $712.50 e $1485 12 a $1314 c $3900 13 $6900 14 a Nil c $4450.50 e $98 650 15 a $33 987.20 c $509.81 e $90.54 16 $833.18 17 a $40 225 c $1550.92 18 $99 19 a $1.13 c 12c e $1.25 20 $22
b $9072
b $26 465.60 b $31 952.34 b $435 d $1252.61 b $1515
b $2003.55 d $21 087.94 b $4198.08 d $4707.89
b $6220.88
b $36.90 d $3.75 21 $112.50
exTended reSponSe
1 a c e 2 a c
$50 440 b $49 321.80 $739.82 d $8346.30 $352.48 refund $330 b $360 NZ$9.60 cheaper to pay on arrival in New Zealand
Chapter 4
Statistics and society, data collection and sampling Chapter Contents 4a 4B 4C 4d 4e
Statistical processes Target populations and sampling Population characteristics Types of data Bias
4a
statistical processes
There are many cases in society where data needs to be analysed. Governments and businesses have data analysed regularly to try to 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 sets of data are analysed and decisions made about what areas need to have road works and what places need greater police supervision. Data are also analysed in areas such as business and sport. Shops will look at sales figures to determine stock and staffing requirements, while in sport player performances are measured statistically by coaches and the media. The purpose of completing a statistical enquiry is to turn raw data into meaningful information. Data are sets of facts that are collected, but limited data alone can have very little meaning. When lots of data are collected and presented and conclusions are drawn, the data becomes more useful information. There are six stages to completing a statistical investigation. Stage 1: Posing questions This first stage of the statistical process is to determine the final information required, then writing questions that will give us the answer and also allow for easy collation of findings and presentation of results. Consider the case where you represent the local council. Your job is to investigate what sporting facilities are needed in your local area. You will need to pose questions that accurately determine the needs of an area. Questions need to target specific needs and not be too vague. ‘What sporting facilities do you think are needed in this area?’ is Please rank the following open ended and is based only on opinion. The question invites a sporting needs in this area. range of responses that may be difficult to tabulate. The question does not analyse if the respondent would even use the facilities that Cricket nets they believe are needed. Tennis courts More relevant questions would be Golf course 1. Do you currently use the sporting facilities in the district? Netball courts 2. How many hours of sport do you play each week? Soccer field 3. What sports do you currently play? Other 4. For the sports that you currently play, are the existing facilities adequate? 5. Rank the following sporting needs in this area. A survey form Chapter 4 • Statistics and society, data collection and sampling
109
Stage 2: Collecting data Data can be collected using either internal or external sources. To collect data from an external source means that the data is obtained by doing research. For example, if you are researching data on share prices you might seek this information from the Australian Stock Exchange. If the data is 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 data that does not require a response from people. This may, for example, be observing the number of students who attend the school canteen at lunchtime. 2. Questioning — this is where the data is obtained by getting a response from people; for example, investigating the sporting facilities needed in the local area. Worked exaMple 1
Brendon is planning a skiing trip and needs to investigate the best places and best times to travel. Would he use internal or external sources to obtain this information? thInk
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Brendon would obtain his information from the Bureau of Meteorology.
Brendon would use an external source.
Stage 3: 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 usually made easier if the questionnaire is designed with ease of tabulation in mind. Worked exaMple 2
A Year 11 class was surveyed on their weekly income. The responses are shown below. $75 $56 $43
$115 $45 $79
$60 $83 $58
$54 $71 $89
$88 $40 $70
$0 $37 $105
$98 $87 $99
$102 $117 $55
Complete the table below. Income $0–$20
Tally
Frequency
$21–$40 $41–$60 $61–$80 $81–$100 $101–$120 thInk
Count the number of responses within each category and put a tally mark in the column.
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Income
Maths Quest Preliminary Mathematics General
Frequency
$0–$20

1
$21–$40

2
$41–$60
 
7
$61–$80

4
$81–$100
 
6

4
$101–$120
110
Tally
Stage 4: 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 chapter 5. For now we will look specifically at column graphs and sector graphs. Worked exaMple 3
The table below shows the results of a survey on favourite sports. Sport AFL Basketball Cricket Netball Rugby League Rugby Union Soccer Tennis Show this information in: gr aph
a a olumn c
Frequency 6 2 7 2 3 1 2 1
b a sector graph.
thInk
a 1 Draw the horizontal axis showing each
3
a
sport. Draw a vertical axis to show frequencies up to 7. Draw the columns.
Frequency
2
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8 7 6 5 4 3 2 1 0
l e l t n r s FL tbal cke tbal agu nio cce nni i e e e e r U So T sk C N y L by b g Ba g u Ru R Sport
A
b 1 Calculate each angle as a fraction of 360°.
b
AFL =
6 24
× 360°
Basketball =
= 90° 7 Cricket = 24 × 360° = 105° Rugby League =
3 24
2 24
× 360°
= 30° 2 Netball = 24 × 360° = 30°
× 360°
= 45° 1 Rugby Union = 24 × 360° Soccer =
2 24
= 15° × 360°
= 30°
Tennis =
1 24
× 360°
= 15°
Chapter 4 • Statistics and society, data collection and sampling
111
2
Draw the graph.
Sport AFL Basketball Cricket Netball Rugby League Rugby Union Soccer Tennis
Stage 5: Analysing data and drawing conclusions Once the data have been organised and displayed, they need to be studied and conclusions drawn. It is at this stage of the statistical inquiry that the results can be reflected on and conclusions made. Stage 6: Writing the report The final stage is to collate all the earlier stages into a written report. The written report should: 1. Pose the questions that the statistical analysis is examining. 2. Explain how the data was collected, what type of data was collected and from what sources. 3. Include all tables used. 4. Use appropriate graphs to display the data. 5. Contain conclusions and recommendations, and reasons why these conclusions were reached.
exercise 4a
statistical processes
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
dIgItal doC Spreadsheet doc1505 Frequency tables
2 For each of the following, state whether an internal or external source has been used. a A football coaching assistant records the number of tackles made by each player in a match. b To analyse immigration trends, a researcher obtains records of the nations from which immigrants
have come. c Attendance records at the Royal Easter Show are gathered to plan the required number of trains
for each day of the next Show. d The Deputy Principal of a school collects records on the number of siblings each student at the
school has. 3 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 112
Maths Quest Preliminary Mathematics General
c d e f g h
The most popular football team in Year 11 The football team that attracts the largest crowds The number of students in your class with a learner’s permit The number of trees in your school grounds The average weekly income of Year 11 students The number of people who speed through an intersection
4 We 1 Rewrite the following openended questions so that the responses will be easier to tabulate. 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? 5 We 2 A class of students was asked to identify the make of car their family owned. Their responses are
shown below. Holden Ford Nissan Mazda Mitsubishi
Ford Holden Holden Toyota Toyota
Nissan Ford Holden Ford Holden
Mazda Mitsubishi Ford Holden Ford
Toyota Toyota Toyota Holden Ford
Holden Toyota Mazda Ford Toyota
Put these results into a table. 6 Display the following golf scores in an appropriate table.
70 71 72 69
70 66 70 72
67 73 70 72
72 70 69 71
67 69 72 70
71 70 70 73
73 66 71 69
66 66 70 73
72 71 71 70
69 71 72 68
7 We 3 The marks scored on a Maths exam, out of 100, by 25 Year 11 students are shown below.
87 54 71
44 60 83
95 66 74
66 69 81
78 66 69
69 77 70
66 79 57
92 66
78 71
8 The data below show the number of customers that entered a shop each day in a certain month.
114 178 169 141
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. 9 Draw a column graph to display the information from question 5. 10 Draw a sector graph to display the information from question 7. 11 The data below shows the changing temperature over a period of 12 hours.
Time Temperature (°C)
7 am 8 am 9 am 10 am 11 am 12 pm 1 pm 2 pm 3 pm 4 pm 5 pm 6 pm 7 pm 11
13
14
16
19
20
22
22
20
17
16
15
14
Plot these points on a graph and join the points. This type of graph is a line graph.
Further development 12 Explain the advantages of using external data sources. 13 Fiona and Suzi collect information about road transport in their county town: • Fiona has found information from the RTA about vehicle registration in NSW. • Suzi records the types of vehicles that pass through the main intersection during the day. a Which person has used an internal data source? b What is the advantage of Fiona’s data collection method? c State a benefit of Suzi’s data collection method. Chapter 4 • Statistics and society, data collection and sampling
113
14 The data below shows the marks, out of 100, from an exam taken by 40 students.
93 52 92 77
88 66 67 77
43 70 94 44
59 62 79 64
67 93 55 99
57 56 73 56
79 51 91 95
60 86 97 63
76 49 51 44
55 53 61 91
a What is the lowest score in the data set? b What is the highest score in the data set? c Mieka records the scores in a table which has been started below:
Score
Frequency
40
0
41
0
42
0
43
1
44
2
Explain why this is not a practical way to display the data. d Display the data from question 6 in a table using a class size of 10 beginning with 40–49. e Display the data from question 6 in a table using a class size of 5 beginning with 40–44. f Discuss the advantages and disadvantages of using a smaller class size. 15 For each of the following, state whether the data would best be displayed by a column, sector or line
graph. a The number of goals kicked by each player in a soccer team. b The breakdown of reasons for which the NRMA is called to assist broken down drivers. c The height of a plant as it grows over one year. d The numbers of various animals living in a national park.
4B
target populations and sampling
The first step in gathering the relevant data for a statistical investigation is to target the population that is 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. In this case we would survey doctors and other medical personnel, as well as a selection of patients who use these facilities. When starting an investigation, we must determine the quantity of information needed for the database. Consider the case of a company hired to calculate TV ratings. Does the company need to know what every household is watching? Obviously not; they ask a selection of homes to record their television viewing. Conversely, consider the case of selecting a commemorative Year 12 jersey at your school. In this case it would be reasonable to ask the opinion of every person in Year 12. Data can be collected in two ways: 1. Census — This is where an entire population is counted. Australians complete a census every five years. This is a survey of every household in the nation. In most statistical investigations a census involves surveying the entire target population. 2. Sample — A sample is a more practical way of obtaining data. Only a selection of the target population is surveyed; however, it is important that those selected are representative of the whole population. Before deciding whether to do a census or sample we need to consider whether it is feasible to obtain census data. Consider the case of someone who is analysing the NRL results for a season. The target population is the result of each match. This is recorded in a number of places and the data easily obtained. 114
Maths Quest Preliminary Mathematics General
Worked exaMple 4
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
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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 any sample is representative of the whole population the method of sampling is important. We will look at three methods of sampling. Method 1 — Random sample. In a random sample those to be surveyed are selected by chance. In such a sample every member of the target population should have an equal chance of being selected. If this method is used you should get a mix of the population that is representative of the whole. Calculators and spreadsheets generate random numbers in different ways. In many cases the random number will be a decimal between 0 and 1. To get a whole number we could multiply this decimal by the number of people in the target population and then poll the people that correspond to the random number generated. Worked exaMple 5
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
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1
Multiply the results of the random number generator by the size of the population.
0.983 × 750 = 737.25 0.911 × 750 = 683.25 0.421 × 750 = 315.75
2
Round up to whole numbers.
The 738th, 684th and the 316th people on the roll would be surveyed.
If the selection is not random there is a possibility that the sample may have bias. For example, if doing a survey at school you may • Have a tendency to survey people you know. • Choose an area where more students of a particular year tend to sit. • Choose more of one sex than the other. Method 2 — Stratified sample. In this sampling method the numbers in the survey from each subgroup are chosen in proportion to the whole population. Suppose that you are surveying 60 people from your school. Should you survey 10 people from each year? It is better to select the numbers from each year in proportion to the whole population. If, for example, 20% of students are in Year 7, then 20% of 60 (12) students from this year should be chosen. Worked exaMple 6
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? Chapter 4 • Statistics and society, data collection and sampling
115
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thInk 1
The population is 47% male and so 47% of the sample should be male. Find 47% of 60.
47% of 60 = 0.47 × 60 = 28.2
2
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.
53% of 60 = 0.53 × 60 = 31.8
3
There should be 28 males and 32 females in Adrian’s survey.
Method 3 — Systematic sample. When selecting a systematic sample the participants are chosen in an organised way. For example, tyres coming off a production line need to be tested for wear. Every 100th tyre produced may be selected.
exercise 4B
target populations and sampling
1 For each of the following describe what the target population for a survey would be. a The school endofyear committee wants to find out the preferred venue for the Year 12 farewell. b The local council wants to find out what sporting facilities are needed in the area. c A newspaper wants to predict the winner of the next federal election. d A group planning to build a preschool would like to know what facilities would attract
enrolments. e A recording label wants to predict the likely success of a grunge band. 2 We 4 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. 3 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? 4 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 halfyearly exam, every student’s mark is recorded. e To test the quality of tyres on a production line, every 100th tyre is road tested. 5 Below are a number of statistical investigations. For each state whether data would be available for the
entire target population a Arthur wants to research the average temperature in Sydney during each month. b The manufacturer wants to test the life span of every battery produced. c Carla wants to compare the average birth weight of baby boys and girls. d David wants to compare the batting average of all test cricketers. 6 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 7 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? 8 We5 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? 116
Maths Quest Preliminary Mathematics General
9 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? 10 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. 11 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. 12 MC 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. 13 MC Which of the following statistical investigations would be practical to complete by census? a B C d
A newspaper wants to know public opinion on a political issue. A local council wants to know if a skateboard ramp would be popular with young people in the area. An author wants a cricket player’s statistics for a book being written. An advertising agency wants to know the most watched program on television.
14 We6 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 should she choose, if she chooses to use a stratified sample? 15 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.
Further development 16 When completing a census, explain why it is more practical in most cases to use external sources. 17 Greg is trying to generate a random sample of people in the street. He does this by selecting every
fifth person that walks past him. a Explain why this sample is not random. b Will Greg’s method provide him with a mix of participants that is representative of the entire population? 18 In a quality control operation there are four production lines labelled A, B, C and D. Every 50th item
off production line B is selected for testing. a What method of sample selection is being used? b Is this the right method of sample selection? c Explain the fault in the sample selection method and suggest a better method. 19 To generate a random number between 1 and 100, Ricky’s calculator generates a decimal number
between 0 and 1. He multiplies this number by 100, adds 1 and takes the integer part of the result. a Demonstrate this process and find the resulting integer if Ricky starts with 0.739. b Explain the purpose of multiplying by 100. c Explain the purpose of adding 1. Chapter 4 • Statistics and society, data collection and sampling
117
20 To complete a survey, John collects a sample of 50 people all with surname ‘Smith’ from the telephone
book. John claims that since Smith is a very common surname his sample should be representative of the whole population. Give three reasons why this sample will not be representative of the whole population. dIgItal doC WorkSHEET 4.1 doc10316
21 Rod claims that a systematic sample is the best way to ensure a sample is representative of the entire
population. Give one reason for and one reason against Rod’s claim.
4C
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 crosssection of people according to socioeconomic 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. InvestIgate: 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 7
The table below 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. Year
Boys
Girls
7
96
102
8
85
87
9
92
88
10
80
74
11
71
75
12
69
65
493
491
Total Grand total
118
Maths Quest Preliminary Mathematics General
984
thInk 1
Write each strata as a fraction of the total and then multiply by the 60 to be chosen in the sample.
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Year 7 boys = 984 × 60 ≈ 5.85 102
Year 7 girls = 984 × 60 ≈ 6.22 85
Year 8 boys = 984 × 60 ≈ 5.18 87
Year 8 girls = 984 × 60 ≈ 5.30 92
Year 9 boys = 984 × 60 ≈ 5.61 88
Year 9 girls = 984 × 60 ≈ 5.37 80
Year 10 boys = 984 × 60 ≈ 4.88 74
Year 10 girls = 984 × 60 ≈ 4.51 71
Year 11 boys = 984 × 60 ≈ 4.33 75
Year 11 girls = 984 × 60 ≈ 4.57 69
Year 12 boys = 984 × 60 ≈ 4.21 65
Year 12 girls = 984 × 60 ≈ 3.96 2
To complete the table, round off each of these answers to the nearest whole number.
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. Chapter 4 • Statistics and society, data collection and sampling
119
exercise 4C
population characteristics
1 We7 The table below shows
the number of students in each year at a school. Year
No. of students
7
90
8
110
9
90
10
80
11
70
12
60
Total
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 3
4
5
6
completed, how many people of each sex participated? 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? 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? 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? The table below shows the age and sex of the staff of a corporation. Age
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. 7 The table below shows the number of students who are in each year level at a school. Year Number of students
7
8
9
10
11
12
187
192
168
157
137
108
If 80 students are to be selected to participate in a survey, how many should be chosen from each year level? 120
Maths Quest Preliminary Mathematics General
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
7
8
9
10
11
12
Boys
104
112
107
97
75
68
Girls
98
119
110
88
82
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
Boys
Girls
7 8 9 10 11 12
Further development 10 Use your calculator to generate five random numbers between 0 and 10. a Find the average of the five numbers. Compare your answer to others in your class by looking for
the highest and smallest results. b Now generate 20 random numbers between 0 and 10 and find the average. Compare your answers
again to others in the class. c What do you notice about the results to part (a) and (b)? 11 What does question 10 tell you about sample size? 12 The following two tables show information about the population of a school.
Year
No. of students
7
180
8
204
9
191
10
172
11
139
12
114
Male 400
Female 600
How can the information in both tables be incorporated into a stratified sample? 13 Jack wants to complete a survey on the same school population, however, as well as year and gender he
wants to include a third stratum of ethnic origin. a Explain why a stratified sample becomes more difficult when extra strata are added. b What would be the best method for Jack to obtain his sample? Chapter 4 • Statistics and society, data collection and sampling
121
14 The table below shows the number of people in each of four strata and the number selected to
participate in a survey from each group. In which of the four strata has the wrong number of participants been selected? Strata
No. in population
No. in survey
A
243
16
B
347
23
C
198
14
D
376
27
15 MC When constructing a stratified random sample Bettina completes the following steps which are
written in the incorrect order. a Uses a random number generator to select the participants from each stratum. B Multiplies each fraction by the number required for the sample. C Allocates a number to each person in the population. d Writes each strata as a fraction of the population. Write the steps A, B, C and D in the correct order. InvestIgate: 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 7 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.
4d
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. Worked exaMple 8
State whether the following pieces of data are categorical or quantitative. a The value of sales recorded at each branch of a fastfood outlet b The breeds of dog that appear at a dog show thInk
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a The value of sales at each branch can be measured.
a The value of sales are quantitative data.
b The breeds of dog at a show cannot be measured.
b The breeds of dog are categorical data.
Maths Quest Preliminary Mathematics General
There are two types of categorical data and two types of quantitative data. Data Categorical Data which are placed in categories; that is, nonnumerical form, such as hair colour, type of vehicle, and so on. Nominal Need subgroups to complete the description, such as hair colour: blond, brown and so on.
Ordinal Need a ranking to order the description, such as achievement levels: very high, high, satisfactory and so on.
Quantitative Data which are in numerical form; such as height, number of children in the family, and so on. Discrete 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.
InteraCtIvItY int0219 Classifying data
Continuous Measured in a continuous decimal scale, such as mass, temperature, length and so on.
Worked exaMple 9
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
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a 1 Determine whether the data are categorical or 2
represented by a number.
Determine whether the data are discrete or continuous.
The data are discrete as the number of absences can be counted and is an exact value.
b 1 Determine whether the data are categorical or 2
need to be placed in nonnumerical categories.
Determine whether the data are nominal or ordinal.
The data are nominal as there is no ranking or order involved. c The data are categorical as the pizza sizes need
quantitative.
to be ranked in order ranging from small to family.
Determine whether the data are nominal or ordinal.
The data are ordinal as pizzas are ranked in order of size.
d 1 Determine whether the data are categorical or 2
b The data are categorical as the types of vehicle
quantitative.
c 1 Determine whether the data are categorical or
2
a The data are quantitative as absences are
quantitative.
d The data are quantitative as room temperature
quantitative.
is represented by a number.
Determine whether the data are discrete or continuous.
The data are continuous as temperature can assume any value and measurement is involved.
exercise 4d
types of data
1 We8 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. Chapter 4 • Statistics and society, data collection and sampling
123
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. 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 sixyearolds 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 4 We9 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 lineup 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. 124
Maths Quest Preliminary Mathematics General
9 MC 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 categorical and nominal data C quantitative and discrete data d quantitative and continuous data 10 The following graph shows the number
Number of days in January
of days of each weather type for the Gold Coast in January.
14 12 10 8 6 4 2 0
ot
H
m ild ar M W Weather
ol
Co
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. Height (cm)
180 160 140 120 100
5 6 7 8 9 10 11 12 13 14 15 Age
Further development 12 Carl says that categorical data is words and quantitative data is numbers. Explain what Carl means 13 14 15 16 17
by this statement. Is this always correct? Danielle says that if quantitative data can be fractions then it must be continuous. Is Danielle correct? Explain your answer. Garry says that for quantitative data, ‘if you count it, it’s discrete and if you measure it, it’s continuous’. Explain what Garry means by this statement. a Explain why it would be inappropriate to use a line graph for categorical data. b What type of graph would be most appropriate? For what type of data is a line graph most appropriate? Give an example of each of the data types shown below. • Categorical and ordinal • Categorical and nominal • Quantitative and discrete • Quantitative and continuous
4e
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. Chapter 4 • Statistics and society, data collection and sampling
125
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 selfadministered 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 nonrandom method, as discussed earlier, generally tends to introduce an element of bias. 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 passersby. 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 phonein number, while the ‘No’ respondents are asked to ring a different phonein 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 nonresponse 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. 126
Maths Quest Preliminary Mathematics General
InvestIgate: 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. InvestIgate: 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 10:00 and 11:00 am on a Tuesday. 3 A newspaper conducting a Gallup poll on an election took a sample of 1000 people from Sydney. InvestIgate: 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 6 25 1 20
1995 9 50 1.5 50
2000 13 44 2.5 66
We shall use a spreadsheet to produce misleading graphs based on these data. Graph 1
2005 20 54 5 100 Graph 2
Graph 3 1 Enter the data as indicated in the spreadsheet above. 2 Graph the data using the Chart Wizard. You should obtain a graph similar to Graph 1. Chapter 4 • Statistics and society, data collection and sampling
127
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 10
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
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a 1 Consider phone book selection. 2 3
Consider those with no phone contact. Consider the hours of contact.
b 1 Consider the newspaper 2
circulation. Consider the urge to respond.
exercise 4e
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. 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.
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? 128
Maths Quest Preliminary Mathematics General
2 Rewrite parts a to d in question 1 so that the expected response is reversed. 3 We10 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 questionnaire
regarding sporting facilities at their school b Sampling using ‘phonein’ responses to an issue viewed on a television program c Promoting the results of a mailresponse 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 5 Comment on the following statement:
‘University tests have demonstrated that DoubleWhite 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 a Year 11 students b Year 12 students c Residents attending a neighbourhood d e f g
watch meeting Students in the school choir Cars in a shopping centre car park Males at a football match Users of the local library
Australian currency
4 Why does this graph produce a biased visual impression? 71c
Value of A$ compared with US$1
70c 69c
9 May 11 May 12 May Date
Population Student drivers Students with parttime jobs Residents of a suburb Music students in the school Models of Holden cars on the road Popular TV programs Popular teenage magazines
Further development
105
7 The graph at right shows the money spent by government
on health care in 2000, 2004 and 2008. a What impression does the graph give about health care funding? b Draw a graph that gives a more realistic impression of the decline in funding.
Health care cost ($ m)
104 103 102 101 100
8 The graph below shows employment growth.
2000
Growth of total employment, 1947–81
2004 Years
2008
Total employment (millions)
6 5 4 3 2 1 0 1947
1954
1961
1966
1971
1976
1981
a What impression has been given by the graph? b How has this impression been achieved? Chapter 4 • Statistics and society, data collection and sampling
129
9 The graph at right shows road 650
fatalities in Queensland over a 20year period. a What is the first impression that the graph gives? b How has this effect been achieved? c How has the change in the number of fatalities been exaggerated? 10 You are a manufacturer
Road fatalities, Queensland
600 550 500 450 400
1987 1984 1981 1978 1975 1972 1969 and your plant is Source: Qld Year Book, 1989, p. 205 and the Australian Bureau of Statistics. discharging heavy metals into a waterway. Your chemists do tests on the water every three months. The table below shows the results over a 2year period. Draw a graph that will show your company in the best light.
2009 Date Concentration (parts per million)
Jan. 7
Apr. 9
July 18
2010 Oct. 25
Jan. 30
Apr. 40
July 49
Oct. 57
11 The sector graph shows the breakup of workers compensation costs incurred by employers other than
government. a What fraction of the total costs are weekly compensation payouts and statutory lump sum claims? b What angle should be at the centre of this sector? c What angle is at the centre of this sector? d Why has this distortion of angle occurred? Discuss how this might be used to mislead the reader?
Breakup of nongovernment workers compensation costs Common law claims $143.5m
Total $202.8m
Common law fees and outlays $19m
Weekly compensation payouts and statutory lump sum claims $40.3m
Source: CourierMail, 21 September 1991.
12 The graph below shows how the $27 that a buyer pays for a CD is distributed among departments
involved in production and marketing. dIgItal doC WorkSHEET 4.2 doc10317
Where your $27 goes Other recording costs 65c Record company Distribution 56c sales process $1.27
Record company administration costs $1.54 Mechanical royalties $1.57
Record shop $7.40 Record company profit $1.54 Advertising $1.94
Sales tax $3.27
Production $3.40
Royalties and costs to artist $3.86
You are required to find out whether or not the graph is misleading, and fully explain and support any statements you make. 130
Maths Quest Preliminary Mathematics General
InvestIgate: 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. 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.
Chapter 4 • Statistics and society, data collection and sampling
131
Summary 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.
target populations and sampling
• 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.
types of data
• Data can be classified as being categorical or quantitative. • Categorical data are data that are nonnumerical. For example, a survey of car types is not numerical. • Categorical data can be nominal (unordered) or ordinal (ordered). • 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.
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.
132
Maths Quest Preliminary Mathematics General
Chapter review 1 Which of the following is an example of a census? a B C d
A newspaper conducts an opinion poll of 2000 people. A product survey of 1000 homes to determine what brand of washing powder is used Every 200th jar of Vegemite is tested to see if it is the correct mass. A federal election
M U ltIp l e C ho IC e
2 Which of the following is an example of a random sample? a B C d
The first 50 students to arrive at school take a survey. Fifty students’ names are drawn from a hat and those drawn take the survey. Ten students from each year of the school are asked to complete a survey. One class in the school is asked to complete the survey.
3 Which of the following is an example of categorical data? a B C d
The type of car that is in each home The number of cars in each home The distance travelled by a person’s car in a one year period The amount of money spent on petrol in a one year period
4 Which of the following is an example of continuous data? a B C d
The type of car that is in each home The number of cars in each home The distance travelled by a person’s car in a one year period The amount of money spent on petrol in a one year period
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
s ho rt a n s W er
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 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? 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. 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. 6 Draw a column and a sector graph to represent the results to question 4. Chapter 4 • Statistics and society, data collection and sampling
133
7 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. 8 Name and describe three different methods for selecting a sample. 9 Which method of sampling has been used for each of the following? a The qualitycontrol 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. 10 Use a random number generator to select 10 numbers between 1 and 1000. 11 The table below shows the number of students in each year of school.
Year
No. of students
7
212
8
200
9
189
10
175
11
133
12
124
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? 12 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 13 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 14 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. 134
Maths Quest Preliminary Mathematics General
1 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. What type of sample has been taken? 2 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.
e x ten d ed res p o n s e
dIgItal doC Test Yourself doc10318 Chapter 4
Chapter 4 • Statistics and society, data collection and sampling
135
ICT activities 4a
statistical processes
dIgItal doC • Spreadsheet (doc1505): Frequency tables (page 112)
4B
target populations and sampling
dIgItal doC • WorkSHEET 4.1 (doc10316): Answer questions on statistics and data. (page 118)
4d
types of data
InteraCtIvItY • Classifying data (int0219): Interact with data classification. (page 123)
136
Maths Quest Preliminary Mathematics General
4e
Bias
dIgItal doC • WorkSHEET 4.2 (doc10317): Answer questions on statistics and data. (page 130)
Chapter review • Test Yourself Chapter 4 (doc10318): Take the endofchapter test to check your progress. (page 135)
To access eBookPLUS activities, log on to www.jacplus.com.au
Answers CHAPTER 4 statIstICs and soCIetY, data ColleCtIon and saMplIng
10
Number of students
Marks on maths exam 40–49 50–59
exercise 4a
Tally 
8
Ford

8
Nissan

2
Mazda

3

7
Mitsubishi

2
Score
Tally
Frequency
66

67 68
6
10 5
Time
12 Answers will vary but the major advantage
is access to large amounts of data. Suzi Large amounts of accurate data Data relates only to her town. 43 b 99 As most scores will have a frequency of 1 the table will not be any more informative than the list of scores.
13 a b c 14 a c
d
4
40–49
4

2
50–59
10

1
60–69
8
5
70–79
7
10
80–89
2
71

7
90–99
9
72

7
73

4
Mark
Tally
40–49
e
Class
Frequency
40–44
3
Frequency
45–49
1

1
50–54
4
50–59

2
55–59
6
60–69

9
60–64
5
8
65–69
3
70–74
2
75–79
5
80–84
0
85–89
2
90–94
6
95–99
3
70–79

80–89

3
90–99

2
8 Check with your teacher. 9 Frequency
15
Frequency
70
9 8 7 6 5 4 3 2 1 0
f The smaller class sizes provide more
i a ta n d n de For issa azd yo bish N M To itsu Make of car M
ol
H
20
Class
69
7
25
0
Frequency
Holden
Toyota
11
9 a 10 m a 11 m a 12 m pm 1 pm 2 pm 3 pm 4 pm 5 pm 6 pm 7 pm
Make
90–99
am am
4 5
80–89
8
3
70–79
Temperature (°C)
2
60–69
7
1
statistical processes a External b External c Internal d External e Internal f Internal g External h External a Internal b External c External d Internal a Observation b Questioning c Questioning d Observation e Questioning f Observation g Questioning h Observation Check with your teacher.
detailed data, however the larger class size provides a more compact summary of the data. 15 a Column b Sector c Line d Column
exercise 4B target populations and sampling 1 a All Year 12 students b Sporting players and clubs in the local district c All Australian citizens 18 years of age and older d Parents of preschool children e People in the age group to whom grunge music appeals 2 Census — every member of the population participates. 3 Sample 4 a Sample b Sample c Census d Census e Sample 5 a Data would be available from the Bureau of Meteorology b No data would be available as you cannot test every battery. c Not all data would be available. A sampling method would be needed. d Data would be available from records of matches played. 6 a Sample b Census c Census d Sample 7 Sample 8 163, 176, 381, 495, 9 9 827, 210, 417, 553, 207, 26, 809, 780, 192, 371 10 Check with your teacher. 11 a Systematic b Stratified c Systematic d Random e Stratified 12 A 13 C 14 34 males and 16 females 15 Year 7 — 8, Year 8 — 8, Year 9 — 8, Year 10 — 6, Year 11 — 5, Year 12 — 4 Choose the 40th student at random. 16 As a census generally involves gathering large quantities of information and an external source is more practical if the information is already there. It may be impractical to survey an entire population. 17 a There is a method to Greg’s selection and so his sample is systematic. b As there should be a good mix of people passing by his sample should be representative of the population. 18 a Systematic b No c Only one production line is being tested and faults on other lines may be overlooked. Selecting every 200th item off each line would be a better method. 19 a 74 b Multiplying by 100 gives 100 possible answers. c Adding one moves the possible results from 0–99 to 1–100 as required. 20 Answers will vary but reasons may include: many people from the same family, not a mix of the various ethnic groups that make up the population, not a mix of all areas of the country. 21 The systematic sample will have one characteristic in the correct proportion but other characteristics of the population may not be in that proportion.
Chapter 4 • Statistics and society, data collection and sampling
137
population characteristics
1 Year 7 — 9, Year 8 — 11, Year 9 — 9, 2 3 4 5 6
Year 10 — 8, Year 11 — 7, Year 12 — 6 36 men and 24 women 25 females and 15 males 2 senior management, 12 middle management and 36 clerks 3 teachers and 47 students Age
Male
Female
20–29
10
7
30–39
7
8
40–49
12
3
50–59
1
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 Answers will vary but for part (c) you
11 12
13
14 15
should notice the result for (b) is closer to the expected value of 5. The larger the sample the more likely results will reflect the whole population. The correct number of students are chosen from each year and then within each stratum 40% should be boys and 60% girls. a Having multiple strata you need to stratify within each group, making sample selection very tedious. b Jack should select a random sample as this should include all relevant strata. B and D C–D–B–A
4 a Quantitative, discrete b Categorical, nominal c Quantitative, continuous d Quantitative, continuous e Categorical, nominal f Quantitative, continuous g Quantitative, discrete h Quantitative, continuous 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 5 a Quantitative and discrete b Categorical c Categorical d Quantitative and continuous e Quantitative and continuous f Quantitative and discrete 6 Categorical and ordinal 7 Categorical and ordinal 8 Quantitative and discrete 9 C 10 Categorical and ordinal 11 Quantitative and continuous 12 This statement means that words are
13
14
15
exercise 4d
1 a b c d e f 2 a b c d e 3 a b c d e
138
types of data Quantitative Categorical Quantitative Categorical Quantitative Quantitative Ordinal Ordinal Nominal Nominal Nominal Continuous Discrete Continuous Continuous Continuous
16 17
generally used to describe categorical data and numbers which have a value generally describe quantitative data. The statement is generally true but not always correct. Danielle is not correct. This may be true in most cases but clothing that has half sizes is an example of discrete fractional data. To be continuous data must be able to take any value. Garry means that anything that is counted can only take certain values, usually whole numbers but anything that is measured can take any value within a reasonable range. a Categories have no ‘in between’ values so a line graph would have no meaning. There should not be links between the categories. The order of the categories is also generally not important. b Column graph Quantitative continuous data Answers will vary.
exercise 4e
Bias
1 Check with your teacher. 2 Check with your teacher. 3 a Sample does not represent b c d e
characteristics of population. No control over responses Unrepresentative sample Abnormal conditions Only extreme groups in sample
Maths Quest Preliminary Mathematics General
4 The decrease in the value of the
Australian dollar compared with the American dollar is accentuated by the large scale on the yaxis. The decrease is actually only 2 cents. The scale on the xaxis 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 parttime 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. 7 a That health care funding has been halved over the eight year period. b
110 100 90 Health care cost ($m)
exercise 4C
80 70 60 50 40 30 20 10 0
2000
2004 Years
2008
8 a That employment growth was linear in
that period.
b The scale on the horizontal axis is not
even.
9 a That fatalities are increasing. b By reversing the horizontal scale c By beginning the vertical scale at 400 10 Check with your teacher. 11 a 20% b 71.5° c 80° d The perspective magnifies some sectors
and diminishes others.
12 The graph is misleading as the perspective
accentuates sectors at the front because the graph is elliptical rather than circular.
Chapter revIeW
6
MUltIple ChoICe
2 B 4 C
Frequency
1D 3A short ansWer
1 a External c External 2 a Observation c Observation 3 Check with your teacher. 4 Score Tally
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
      Tally       
b d b d
Internal Internal Questioning Questioning Frequency 3 6 6 6 3 3 Number of students 3 9 9 3 4 1 0 1
10 Check with your teacher. 11 Year 7 — 12, Year 8 — 12, Year 9 — 11,
7 6 5 4 3 2 1 0
Year 10 — 10, Year 11 — 8, Year 12 — 7
1 2 3 4 5 6 Number of people in a household Number of people in a household 1 2 3 4 5 6
7 a Sample b Census c Census d Sample 8 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 9 a Systematic b Random c Stratified
12 a Categorical b Quantitative c Quantitative d Quantitative e Categorical 13 a Discrete b Continuous c Continuous d Discrete e Continuous 14 Check with your teacher. extended response
Categorical Random Quantitative and continuous Systematic 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.
1 a b 2 a b c
Chapter 4 • Statistics and society, data collection and sampling
139
Chapter 5
Displaying single data sets Chapter ContentS 5a Types of graphs and stemandleaf plots 5B Frequency tables and cumulative frequency 5C Range and interquartile range, deciles and percentiles 5d Fivenumber summaries 5e Appropriate graphs and misuse of graphs
types of graphs and stemandleaf plots 5a
Data can be displayed in many different ways. Most commonly data are displayed on some type of graph. Graphs such as column graphs, sector graphs and line graphs were looked at in chapter 4 and more can be found on these types of graphs using the skillsheets if you need them. In this section we are going to look at some displays that are of particular use to statisticians.
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. Worked example 1
The information below shows the sales in a department store over a year. Month
Sales ($m)
Month
Sales ($m)
January
2.8
July
1.8
February
1.7
August
1.1
March
1.1
September
1.6
April
1.2
October
1.9
May
1.3
November
2.5
June
1.6
December
3.4
Show these data in a radar chart.
Chapter 5 • Displaying single data sets
141
think
draW
1
Draw equal sectors of 30°.
2
Draw the sales from the centre using 1 cm = $0.5 million.
3
Plot the points.
4
Join each point with a straight line.
Sales ($m) January 3.5 3 2.5 2 1.5 1 0.5 0
December November
October
February March
April
September
May
August
June July
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 2
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
Draw a number line showing all numbers from 0 to 10.
2
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.
draW
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Stemandleaf plots A stemandleaf plot is a method of tabulating data that generally consists of two parts. 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 stemandleaf 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 stemandleaf 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.
142
Maths Quest Preliminary Mathematics General
It is convention to assemble the data in order of size, so this stemandleaf 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 Leaf 0 7 9 1 1 4 5 6 7 8 8 9 2 1 2 2 3 3 5 7 7 3 0 2 When preparing a stemandleaf 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 3
The information below shows the mass, in kilograms, of twenty 16yearold boys. 65 61
45 58
56 49
57 52
58 64
54 71
61 66
72 65
70 66
69 60
Show this information in a stemandleaf plot. think
Write
1
Make the ‘tens’ the stem and the ‘units’ the leaves.
2
Write a key.
Key: 5  6 = 56 kg
3
Complete the plot.
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
It is also useful to be able to represent data with a class size of 5. This could be done for the stemandleaf plot below 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 stemandleaf plot for the ‘possum’ data would appear as follows. Key: 1  1 = 1.1 kg Stem 0* 1 1* 2 2* 3
1*  5 = 1.5
Leaf 7 9 1 4 5 6 7 8 8 9 1 2 2 3 3 5 7 7 0 2
A stemandleaf 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 (see page 146) some generalisations are made when these values are calculated.
Chapter 5 • Displaying single data sets
143
Worked example 4
The following data give the length of gestation in days for 24 mothers. Prepare a stemandleaf plot of the data using a class size of 5. 280 287 285 276 266 292 288 273 295 279 284 271 292 288 279 281 270 278 281 292 268 282 275 281
think
Write
1
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.
2
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.
3
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 stemandleaf plot and are conveniently arranged in order of size, the plot can be used to locate the upper and lower quartiles and the median.
exercise 5a
types of graphs and stemandleaf plots
1 We1 The table below shows the average monthly temperature in Sydney.
144
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
Maths Quest Preliminary Mathematics General
Month
Temp (°C)
Show this information in a radar chart.
2 The table below shows the percentage of televisions that are being watched over a 24hour 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. 3 We 2 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
30 30
31 29
4 Below is the maximum temperature in Sydney each day during February.
28 35 33 34 30 32 31 30 31 29 Show this information on a dot plot.
27 29
28 28
29 25
30 26
26 30
28 31
33 29
5 We3 The data below give the number of errors made each week by 20 machine operators. Prepare a
stemandleaf diagram of the data using stems of 0, 1, 2 etc. 6 17
15 26
20 38
25 31
28 26
18 29
32 32
43 46
52 13
27 20
6 The data below give the time taken for each of 40 runners on a 10 km fun run.
Prepare a stemandleaf diagram for the data using a class size of 10 minutes. 36 66 42 71
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
7 We4 The typing speed of 30 word processors is recorded below. Prepare a stemandleaf diagram of
the data using a class size of 5. 96 88 124
102 86 95
92 107 98
96 111 102
95 107 108
102 108 112
95 103 120
115 121 99
110 107 121
108 96 130 Chapter 5 • Displaying single data sets
145
8 Twenty transistors are tested by applying increasing voltage until they are destroyed. The maximum
voltage that each could withstand is recorded below. Prepare a stemandleaf 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 Questions 9 and 10 refer to the stemandleaf plot below. Key: 12  1 = 1210 12*  5 = 1250 Stem Leaf 12 1 2 4 12* 5 7 7 9 9 13 0 1 1 2 3 4 4 13* 5 6 6 7 9 9 14 0 2 3 4 14* 0 1 9 mC The class size used in the stemandleaf plot is: a 1 B 10 C 33 10 mC The number of scores that have been recorded is: a 27 B 33 C 1210
d 50 d 1410
Further development 11 a Describe the advantages of displaying data in a dot plot. b The data below shows the number of packets of chips sold from a vending machine over a 2 week
period. 15, 17, 18, 18, 14, 16, 17, 6, 16, 18, 16, 16, 20, 18 Display this information in a dot plot. c Explain why a dot plot is only suitable for discrete data. 12 Explain why it would not be suitable to represent each of the following data sets on a dot plot. a A cricketer’s scores in a season were 4, 65, 82, 5, 19, 56, 23, 153. b The heights of 8 bushes were 1.93 m, 1.76 m, 1.55 m, 1.86 m, 1.97 m, 1.13 m, 1.05 m, 2.06 m. 13 Tina wants to draw graphs to represent the following data: • The average daily sales in her shop over a week • The most frequently sold items in her shop • The amounts of money spent by her customers Which of these would be most suitable to display on a radar chart? Explain your choice. 14 The data below gives the head circumference (to the nearest centimetre) of 16 fouryearold girls. 48 49 47 52 51 50 49 48 50 50 53 52 43 47 49 50
diGital doC doc10319 drawing graphs using graphics calculators and spreadsheets
a Draw a stemandleaf plot of the data using stems 4 and 5. b Draw the stemandleaf plot with the stems 4 and 5 split into halves. c Jessica decides to draw the stemandleaf plot in fifths, which is five rows for each stem, with 0’s
and 1’s on the first row, 2’s and 3’s on the second row and so on. Complete the stemandleaf plot using this method. 15 Explain why categorical data is unsuitable to be represented by a line graph.
Frequency tables and cumulative frequency 5B
From previous years you should be familiar with compiling a frequency table. In this section we will revise compiling frequency tables, but place an emphasis on the use of the cumulative frequency. When working with quantitative data, each piece of data is known as a score. Quantitative data may be presented as grouped or ungrouped data. Ungrouped data are suitable for discrete data that do not have a wide range of scores. The frequency table will have columns for scores, tally, frequency and cumulative frequency, which is a running total of the frequency column. 146
Maths Quest Preliminary Mathematics General
Worked example 5
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 4 5 3 3 3 3 0 5 1 1 3 3 0 5 2 2 1 3 1 3 4 4 2 0 1 4 0 3 2 0 1 5 5 1 2 3 5 3 4 0 2 0 4 4 4 2 2 4 5 Enter the information in a frequency table. think
Write/draW
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.
Score 0 1 2 3 4 5
Tally             
Frequency 7 7 9 11 9 7
When data are continuous or spread over a wide range it is useful to group the scores into groups or classes. When summarising raw data in a frequency table the group size is important. In general we try to have between 5 and 10 classes. Later we will be looking at some of the calculations that are done using a frequency table. For this reason we need to have a single score to represent every score in the group and so an extra column is created called the class centre. Worked example 6
The height of 40 students was measured and the results are shown below. 146 159 152 164
141 143 148 148
155 152 152 168
166 156 142 169
168 146 162 146
158 146 159 162
169 161 141 151
164 150 151 150
141 141 169 143
154 153 169 140
Put the above results into a frequency table.
Chapter 5 • Displaying single data sets
147
think 1
Write/draW
The data range from 140 cm to 168 cm. Choose a group of 5 cm beginning at 140 cm.
2
Calculate the class centres.
3
Draw a frequency table with four columns and room for six classes.
4
Enter a tally mark in the appropriate class as each height is read.
5
140–<145
Class centre 142.5
Tally  
145–<150
147.5


6
150–<155
152.5


9
155–<160
157.5

160–<165 165–170
162.5

167.5

Height
Complete the frequency table by counting the tally marks.
Frequency 8
5 5 
7
In analysing statistical data the most useful graph that can be drawn is the cumulative frequency histogram and polygon (ogive). The cumulative frequency histogram is drawn without the half column space before the first column, while the ogive is drawn to the top righthand corner of each column. Worked example 7
The frequency table below shows the heights of people in a basketball squad. Height (cm)
Class centre
Frequency
170–<175
172.5
3
175–<180
177.5
6
180–<185
182.5
12
185–<190
187.5
10
190–<195 195–200
192.5
8
197.5
1
a Add a cumulative frequency column to the table. b Draw a frequency histogram and polygon.
148
Maths Quest Preliminary Mathematics General
3
3
175–<180
177.5
6
9
180–<185
182.5
12
21
185–<190
187.5
10
31
190–<195 195–200
192.5
8
39
197.5
1
40
7.5
19
2.5
40 35 30 25 20 15 10 5 0
7.5
Draw the cumulative frequency polygon by joining the lines to the top righthand corner of each column.
172.5
19
3
170–<175
2.5
Draw the columns for the cumulative frequency histogram.
Cumulative frequency
18
2
b
Frequency
7.5
horizontal axis and cumulative frequency on the vertical axis. Show the class centres for the height.
Class centre
18
b 1 Draw the axes with height on the
Height (cm)
2.5
Complete the column by keeping a running total of the frequencies.
17
2
a
Cumulative frequency
a 1 Add a fourth column to the table.
Write/draW
17
think
Height (cm)
It is important to see that the increase in each column of the cumulative frequency histogram represents the frequency of each class. This enables us to reconstruct a frequency table from the cumulative frequency graph. Worked example 8
Cumulative frequency
Use the cumulative frequency histogram below to complete a frequency table of the data. Cumulative frequency
70 60 50 40 30 20 10
0 155 165 175 185 195 205 Class centre
think
Write/draW
1
The difference in class centres is 10 so this is the class size.
2
The classes will be ‘5’ either side of each class centre.
3
Find each frequency by subtracting the cumulative frequencies. The 150−<160 frequency is the same as the cumulative frequency. (8) The 160−<170 frequency = 22 − 8 = 14 The 170−<180 frequency = 30 − 22 = 8 The 180−<190 frequency = 44 − 30 = 14 The 190−<200 frequency = 55 − 44 = 11 The 200−<210 frequency = 60 − 55 = 5
Class
Class centre
Cumulative frequency
150−<160
155
8
8
160−<170
165
14
22
170−<180
175
8
30
180−<190
185
14
44
190−<200
195
11
55
200−210
205
5
60
Frequency
Frequency tables and cumulative frequency exercise 5B
1 We5 Twenty households were surveyed to find the number of people in that household. The results
are shown below. 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. Score 2 3 4 5 6 7
Tally
Frequency
2 The marks of 25 students on a spelling test are shown below.
4 4
5 7
8 5
5 6
10 7
Put this information into a frequency table.
7 5
6 7
9 8
7 4
6 6
5 8
7 7
6
diGital doC Spreadsheet doc1559 Frequency tables
Chapter 5 • Displaying single data sets
149
3 The scores of 50 professional golfers in a round of golf are shown below.
72 77 77
70 72 75
69 73 72
75 72 72
78 72 72
68 72 71
66 74 73
68 70 72
67 71 70
72 73 74
72 72 72
71 77 71
68 74 73
73 76 68
72 68 67
71 69 67
74 68
Display this information in a frequency table. 4 We6 A class of 30 students sat for a Mathematics test. Their results out of 100 are shown below.
68 69
72 73
58 41
45 42
69 73
92 80
38 50
51 60
70 49
65 65
69 94
73 88
52 85
76 53
48 60
Use these results to copy and complete the frequency table below. Score
Class centre
Tally
Frequency
30–39 40– 49 50–59 60–69 70–79 80–89 90–99 5 A farmer measures the heights of his tomato plants.
The results, in metres, are shown below. 0.93 1.82 0.80 0.94 1.50 1.52 2.13
1.21 1.77 2.14 1.23 1.41 1.39
2.03 1.65 1.53 1.72 1.74 1.76
1.40 0.63 2.07 1.34 1.86 1.67
1.17 1.24 1.96 0.75 1.55 1.28
1.53 1.99 1.05 1.17 1.42 1.43
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 11.4 12.2 12.8 11.6
12.0 11.0 12.0 12.4 11.7
11.9 10.9 12.7 11.7 12.2
12.0 11.7 12.9 10.8 12.7
12.6 11.2 11.3 13.3 13.0
11.7 11.8 11.2 11.7 12.2
Construct a frequency distribution table for the data. Use a class size of 0.5 seconds. 150
Maths Quest Preliminary Mathematics General
7 We7 The following data show the number of registered
cars normally kept at each of 30 households. No. of cars 0 1 2 3 4 5
Frequency 2 8 11 6 2 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 48 49 50 51 52 53
Frequency 2 10 32 9 5 2
a Add a cumulative frequency column to the table. b Draw a cumulative frequency histogram and
polygon. 9 The following frequency table gives the number of oysters of different lengths from a tray in a marine
farm. Length (cm)
Class centre
4–<5
Frequency 6
5–<6
10
6–<7
60
7–<8
58
8–<9 9–10
8
Cumulative frequency
4
a Copy and complete the table. b Show the information in the form of a cumulative frequency histogram and polygon.
Chapter 5 • Displaying single data sets
151
10 The following frequency table gives the results of testing the lives of 200 torch batteries.
Lifetime (hours) 20–<25
Frequency 6
25–<30
25
30–<35
70
35–<40
61
40–<45 45–50
30 8
a Redraw the table, including a column for class centre and cumulative frequency. b Draw a cumulative frequency histogram and polygon. 11 Complete the frequency and class columns for the frequency table below.
Class
Class centre 3 8 13 18 23
Frequency
Cumulative frequency 5 12 20 30 35
Cumulative frequency
12 Use the cumulative frequency histogram below to construct a frequency table for the data. Cumulative frequency
100 90 80 70 60 50 40 30 20 10 0 3
4
5
6 7 8 Score
9
10
13 mC Consider the cumulative frequency histogram below. Cumulative frequency
Cumulative frequency 40 35 30 25 20 15 10 5 0 5
Which class has the highest frequency? a 0−<10 B 20−<30
15 25 35 45 Class centre C 30−<40
d 40−<50
Cumulative frequency
14 We8 Use the cumulative frequency histogram below to construct a frequency table for the data. 50 45 40 35 30 25 20 15 10 5 0 5
152
Maths Quest Preliminary Mathematics General
15 25 35 45 Class centre
Further development 15 A survey is conducted where 40 people were asked the number of hours of television that they watched
each week. a Is the data discrete or continuous? b The results are as shown below. 10, 13, 7, 12, 16, 11, 6, 14, 6, 11, 5, 14, 12, 8, 27, 17, 13, 8, 14, 10 13, 7, 15, 10, 16, 8, 18, 14, 21, 28, 9, 12, 11, 13, 9, 13, 29, 5, 24, 11 Why is it more practical in this case to use class groupings rather than individual scores? c Complete the frequency table below.
Class
Tally
Class centre
Frequency
5–<10 10–<15 15–<20 20–<25 25–30 16 The number of phone calls made, on average, per week in a sample of 56 people is listed below.
21, 50, 8, 64, 33, 58, 35, 61, 3, 51, 5, 62, 16, 44, 56, 17, 59, 23, 34, 57, 49, 2, 24, 50, 27, 33, 55, 7, 52, 17, 54, 78, 69, 53, 2, 42, 52, 28, 67, 25, 48, 63, 12, 72, 36, 66, 15, 28, 67, 13, 23, 10, 72, 72, 89, 80 a Is the data discrete or continuous? b Explain why classes of 1–10, 10–20, . . . etc. are not appropriate in this case. c Complete a frequency table using a class size of 10. 17 The following frequency table shows the time taken (in seconds) for 60 people, involved in a
psychology experiment, to complete a simple manipulative puzzle. Time taken (s)
Frequency
6–<8
1
8–<10
4
10–<12
15
12–<14
18
14–<16
12
16–<18
8
18–20
2
a Copy the table and add a cumulative frequency column to it. b Prepare an ogive of the data. 18 The cumulative frequency histogram below shows the marks in a class on an exam. (All marks are
Cumulative frequency
whole numbers.) 30 25 20 15 10 5 0 54.5 64.5 74.5 84.5 94.5 Class centre
Represent the data in a frequency table.
diGital doC WorkSHEET 5.1 doc10320
Chapter 5 • Displaying single data sets
153
range and interquartile range, deciles and percentiles 5C
interaCtiVitY int2362 measures of centre and spread
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 9
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
Write
1
The lowest number of matches played is 1.
Lowest score = 1
2
The highest number of matches played is 24.
Highest score = 24
3
Calculate the range by subtracting the lowest score from the highest score.
Range = 24 − 1 = 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 10
The frequency distribution table below shows the heights of boys competing for a place on a basketball team. Find the range of these data. Height 170–<175
Frequency 3
175–<180
6
180–<185
12
185–<190
10
190–<195 195–200
8
think
154
1 Write
1
The lowest score is at the bottom of the 170–175 class.
Lowest score = 170
2
The highest score is at the top of the 195–200 class.
Highest score = 200
3
Range = highest score − lowest score.
Range = 200 − 170 = 30
Maths Quest Preliminary Mathematics General
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 11
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
b Median =
4th score and the 5th score.
87 + 88 2 = 87.5
The interquartile range is the difference between the upper quartile and the lower quartile. To find the 1 lower and upper quartiles we arrange the scores in ascending order. The lower quartile is 4 of the way 3 through the distribution and the upper quartile is 4 of the way through the distribution. 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 12
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
Write/diSplaY
Method 1 1
Write the data in ascending order.
2, 4, 4, 5, 6, 8, 9, 9, 10, 12
2
Divide the data into two equal halves.
2, 4, 4, 5, 6
3
The lower quartile is the median of the lower half.
Lower quartile = 4
4
The upper quartile is the median of the upper half.
Upper quartile = 9
5
The interquartile range is the upper quartile minus the lower quartile.
Interquartile range = 9 − 4 =5
8, 9, 9, 10, 12
Chapter 5 • Displaying single data sets
155
Method 2 1 From the MENU select STAT.
156
2
Delete any existing data from all lists and then enter the scores into List 1.
3
Press 2 for CALC and then 6 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 w to return to the previous screen and then press 1 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 = 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 =5
Maths Quest Preliminary Mathematics General
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.
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.
Cumulative frequency
Worked example 13
50 40 30 20 10 0
think
a 1 Draw a vertical axis showing the
percentiles.
Write/draW
a
100% 75% 50%
2
Draw a line for the 50th percentile to the ogive and estimate the median.
b 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.
5 5 5 5 5 5 0.2 0.7 1.2 1.7 2.2 2.7 Volumes of concrete
50 40 30 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. Chapter 5 • Displaying single data sets
157
range and interquar tile range, deciles and percentiles exercise 5C
1 We9 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 2 We10 Use the frequency distribution tables below to find the range for each of the following sets of
scores. a
Score 1 2 3 4 5
Frequency 2 6 12 10 7
c
Score 89 90 91 92 93 94 95
Frequency 12 25 36 34 11 9 4
3 For the grouped dispersions below, state the range. a Class Frequency
c
51–60 61–70 71–80 81–90 91–100
2 8 15 7 1
Class 40–43
Frequency 48
44–47
112
48–51
254
52–55
297
56–59
199
60–63
84
b
Score 38 39 40 41 42 43
Frequency 23 46 52 62 42 45
b
Class 150–<155 155–<160 160–<165 165–<170 170–<175 175–180
Frequency 12 25 38 47 39 20
4 The scores below show the number of points scored by two AFL teams over the first 10 games of the
season. Sydney: 110 Collingwood: 125
95 112
74 89
136 111
48 96
168 113
120 85
85 90
99 87
65 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? 158
Maths Quest Preliminary Mathematics General
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: Machine B:
100, 99, 99, 101, 100, 101, 100, 100, 101, 108 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. 6 We11 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 7 We12 For each of the data sets in question 6, calculate the interquartile range. Cumulative frequency
8 We13 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.
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. Chapter 5 • Displaying single data sets
159
10 mC For the frequency table below, what is the range?
a 4
Score
Frequency
25
14
26
12
27
19
28
25
29
19
B 5
C 6
d 17
11 mC 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
12 mC 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 mC The distribution below shows the ranges in the heights of 25 members of a football squad.
Height (cm)
Class centre
Frequency
Cumulative frequency
140–<150
145
2
2
150–<160
155
5
7
160–<170
165
10
17
170–<180
175
7
24
180–190
185
1
25
Which of the statements below is correct? The range of the distribution is 40. The range of the distribution is 49. The range of the distribution is 9. The interquartile range can only be estimated using a cumulative frequency polygon.
a B C d
14 The frequency distribution table below shows the marks obtained by a group of people on an IQ test.
IQ score
Frequency
75–84
12
85–94
25
95–104
50
105–114
24
115–124
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 i interquartile range ii top decile 160
Maths Quest Preliminary Mathematics General
iii 65th percentile.
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.
Fur ther development 17 The salaries of the 40 employees of a small manufacturing company are represented by the
accompanying frequency table.
a b c d e f g h
Salary (× $1000)
Frequency
15–<20
6
20–<25
12
25–<30
8
30–<35
7
35–<40
5
40–<45
1
45–50
1
Copy the table and add a cumulative frequency column to it. Prepare an ogive of the data. How many employees are earning less than $22 000? How many employees are earning less than $31 000? Find the 75th percentile of the data and write a sentence explaining what it means. Find the 50th percentile of the data and write a sentence explaining what it means. Find the 25th percentile of the data and write a sentence explaining what it means. The management decides to award pay rises to its highest earning employees. The top 10% of employees will all get a pay rise. How much salary would an employee need before qualifying for a pay rise? Chapter 5 • Displaying single data sets
161
18 A manufacturer of surf clothing needs to know how many clothes of different sizes to produce. The
manager organises a survey of young people which provides the following data:
a b c d e f g
Waist size (cm)
Frequency
70–<75
13
75–<80
28
80–<85
46
85–<90
30
90–<95
17
95–<100
8
100–<105
7
105–110
1
Copy the table and add a cumulative frequency column to it. Prepare an ogive of the data. How many young people had a waist size of less than 82 cm? How many young people had a waist size of greater than 94 cm? Find the 90th percentile of the data and write a sentence explaining what it means. Find the 50th percentile of the data and write a sentence explaining what it means. The manager decides that production costs can be minimised by only making garments fitting sizes between 78 cm and 100 cm. What percentage of the population will not be catered for by this manufacturer?
19 A biologist who counts the number of seeds produced in each of 60 pumpkins presents his findings on
60
100%
50 40 30
50%
20 10 10 20 30 40 50 60 70 Number of seeds
a b c d e f g h
Cumulative frequency (%)
Cumulative frequency
the ogive below.
How many pumpkins contained 30 or fewer seeds? How many pumpkins contained more than 50 seeds? What percentage of pumpkins had fewer than 45 seeds? What percentage of pumpkins had fewer than 20 seeds? Find the 90th percentile of the data and write a sentence explaining what it means. Find the 75th percentile of the data and write a sentence explaining what it means. Find the 50th percentile of the data and write a sentence explaining what it means. The worst 20% of pumpkins (those with the fewest seeds) are to be kept aside for further investigation. Find the maximum number of seeds for any pumpkin in this group.
20 A time trial is a race in which each competitor rides separately, racing ‘against the clock’.
The following are the times (in seconds) of 20 competitors in a 1 km cycling time trial. 75 73
162
Maths Quest Preliminary Mathematics General
72 75
68 82
78 90
75 92
68 75
77 73
80 72
85 70
82 83
a Copy and complete the frequency table below.
Time (s)
Cumulative frequency
Frequency
65–<70 70–<75 75–<80 80–<85 85–<90 90–95 Prepare an ogive of the data. How many riders finished with a time of 82 seconds or better? Find the 90th percentile and write a sentence which explains what it means. The top 20% of riders are to be selected for a special training squad. What time would be needed to qualify for the squad? 21 The following data, collected from a maternity hospital, gives the birth weights (in kg) of 30 babies. b c d e
3.7 4.2 3.1
3.2 2.5 2.8
3.8 2.7 2.9
4.1 3.9 3.2
2.9 3.6 3.1
3.3 3.2 3.8
3.6 3.0 3.9
3.1 2.9 3.3
3.6 3.4 4.4
3.9 3.0 3.4
Using classes 2.4–<2.8, 2.8–<3.2, . . . etc. complete a frequency distribution table. Prepare an ogive of the data. What percentage of babies had a birth weight of 3.8 kg or less? Find the 50th percentile and explain what it means. Babies that weigh 2.6 kg or less are given special attention by medical staff. What percentage of babies are given special attention? 22 The ogive below shows the temperatures (in degrees Celsius) at which paint starts to blister. a How many paint samples blistered at a temperature of 85 °C or less? Ogive of temperature at which paint blistered b What percentage of the samples blistered at 85 °C or less? c What percentage of the samples could withstand a temperature of 88 °C? 100% 30 d Find the 90th percentile and explain what it means. 20 50% e The manufacturer wishes to guarantee that the paint will not blister at 10 high temperatures. What is the highest temperature at which the paint could be rated, if management wants no more than 20% of the paint 80 90 100 Temperature (°C) returned because of blistering?
Cumulative frequency (%)
Cumulative frequency
a b c d e
Chapter 5 • Displaying single data sets
163
5d interaCtiVitY int2788 parallel boxplots int0802 Boxplots and fivenumber summary
Fivenumber summaries
Once the median and quartiles have been calculated, we are able to summarise a data set using five numbers. This fivenumber 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 14
For the set of scores below, develop a fivenumber summary. 12 15 46 9 36 85 73 29 think
64
50
Write
1
Rewrite the list in ascending order.
9
12
15
29
36
46
50
64
73
2
Write the lowest score.
Lower extreme = 9
3
Calculate the lower quartile.
Lower quartile = 15
4
Calculate the median.
Median =
5
Calculate the upper quartile.
Upper quartile = 64
6
Write the upper extreme.
Upper extreme = 85 Fivenumber summary = 9, 15, 41, 64, 85
85
36 + 46 2 = 41
In most cases you will need to calculate the fivenumber summary from an ogive. Worked example 15
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 fivenumber summary. 1
Draw the 25th, 50th and 75th percentiles on the ogive.
Write/draW
60 Cumulative frequency
think
50 40 30
Maths Quest Preliminary Mathematics General
50%
20 10 0
164
100%
10 15 20 25 30 35 40 45 50 55 60 65 70 Number of seeds
0%
2
Write the lower extreme.
Lower extreme = 10
3
Use the ogive to estimate the lower quartile.
Lower quartile = 31
4
Use the ogive to estimate the median.
Median = 38
5
Use the ogive to estimate the upper quartile.
Upper quartile = 47
6
State the upper extreme.
Upper extreme = 70 Fivenumber summary = 10, 31, 38, 47, 70
It is important that you are able to construct a five number summary regardless of what form the data is presented in. In the example below we are looking at the data in the form of a stemandleaf plot. Worked example 16
The stemandleaf plot shown below shows the ages of 25 people who attend a French speaking course. Stem Leaf 1 8 8 9 2 0 2 7 9 9 3 1 3 3 5 6 6 7 9 4 0 2 2 6 8 5 5 Produce a fivenumber summary of the data. think
Write
1
The lowest score is 18.
Lowest score = 18
2
The highest score is 55.
Highest score = 55
3
There are 25 scores so the middle score will be the 13th score.
Median = 36
4
There are 12 scores in the lower half. The lower quartile will be the average of the 6th and 7th score.
Lower quartile = 28
5
There will be 12 scores in the upper half and so the upper quartile will be the average of the 19th and 20th scores.
Upper quartile = 39.5
6
Write the fivenumber summary.
18
28
36
39.5
555
Once a fivenumber summary has been developed, it can be graphed using a boxandwhisker plot (boxplot), a powerful way to display the spread of the data. The boxandwhisker 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
Boxandwhisker plots are always drawn to scale. This can be drawn with the fivenumber summary attached as labels: 15
4
21 23
28
or with a scale presented alongside the boxandwhisker plot.
0
5
10 15
20 25
30
Scale
Chapter 5 • Displaying single data sets
165
Worked example 17
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 boxandwhisker plot. think 1
draW
Draw a scale from 70 to 120 using 1 cm = 10 km/h. 70
2
Draw the box from 84 to 95.
3
Mark the median at 89.
4
Draw the whiskers to 82 and 114.
exercise 5d
80 90 100 110 120 Speed (km/h)
Fivenumber summaries
1 We14 Write a fivenumber summary for the data set below.
shown at right. Write a fivenumber summary of the data set.
4 A cumulative frequency histogram and polygon is shown
at right. Write a fivenumber summary of the data set.
100 90 80 70 60 50 40 30 20 10 0
Cumulative frequency
3 We15 A cumulative frequency histogram and polygon is
Cumulative frequency
15 17 16 8 25 18 20 15 17 2 For each of the data sets below, write a fivenumber 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
14
1 60 50 40 30 20 10 0
2
3
4 5 Score
6
45 55 65 75 85 95 Score
5 We16 The data below shows the number of race starts that have been had by 24 horses who are
running in a Melbourne Cup. Key: 1  5 = 15 Stem Leaf 1 5 9 2 1 3 3 4 6 7 8 8 9 3 0 1 1 4 5 5 7 4 2 2 4 4 5 5 0 Present the data in the form of a fivenumber summary. 166
Maths Quest Preliminary Mathematics General
7
6 The stemandleaf plot below shows the time taken for 13 athletes to run 200 m.
Key: 19  6 = 1 9.6 sec Stem Leaf 19 6 8 9 9 20 0 1 1 2 2 2 3 5 8 Present the data in the form of a fivenumber summary. 7 The stemandleaf plot below shows the scores by 35 professional golfers in a round of a tournament.
Key: 6  4 = 64 6*  6 = 66 Stem Leaf 6 4 6* 6 8 8 9 9 9 7 0 0 1 1 1 1 2 2 2 2 3 3 4 4 7* 5 5 5 6 6 7 7 8 8 9 8 0 0 0 1 Present the data in the form of a fivenumber summary. 8 We17 A fivenumber 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 boxandwhisker plot of the data.
9 The boxandwhisker plot at right shows the distribution of
final points scored by a football 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?
50
70 90 110 130 150 No. of final points
10 The boxandwhisker 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 No. of honey bears
Questions 11, 12 and 13 refer to the boxandwhisker plot drawn below.
5
10
15 20
25 30 Scale
11 mC The median of the data is: a 20
B 23
C 35
d 31
C 5
d 20 to 25
12 mC The interquartile range of the data is: a 23
B 26
13 mC Which of the following is not true of the data represented by the boxandwhisker plot? a B C d
Onequarter of the scores is between 5 and 20. Onehalf of the scores is between 20 and 25. The lowest quarter of the data is spread over a wide range. Most of the data are contained between the scores of 5 and 20. Chapter 5 • Displaying single data sets
167
14 The data below show monthly rainfall in millimetres.
Jan.
Feb.
Mar.
Apr.
May
June
July
Aug.
Sept.
Oct.
Nov.
Dec.
10
12
21
23
39
22
15
11
22
37
45
30
a Provide a fivenumber summary of the data. b Draw a boxandwhisker plot of the data. 15 The following data detail the number of hamburgers sold by a fast food outlet every day over a 4week
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 boxandwhisker plot of the data.
Further development 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 boxandwhisker plot of the data. d Describe the distribution in words. What does the distribution say about the age that mothers have their first baby? 17 Employees on a factory assembly line are timed as they assemble a particular product.
The results are below (in minutes). 18 19 a b c d e
40 39
31 18
37 28
18 37
46 24
27 32
20 16
54 43
35 39
14 37
48 46
23 24
38 35
Prepare a fivenumber summary of the data by first putting the data in a stemandleaf plot. Find the range. Find the interquartile range. If the slowest 25% of workers were to be dismissed, what would be the cut off time used? If you took 21 minutes to assemble an item, write a sentence to convince your employer that you are a valuable employee.
18 Two classes sat for a Mathematics test. Their results have been summarised by the fivenumber
summary below. Class A: 25 40 52 75 95 Class B: 20 35 56 75 85 a Find the range of marks for each class. b What is the median for each class? c Calculate the interquartile range for each class. d Which class has been the most consistent? Explain your answer. 168
Maths Quest Preliminary Mathematics General
19 The stem plot at right details the age of 25 offenders who were
Stem Leaf 1 8 8 9 9 9 2 0 0 0 1 1 3 4 6 9 3 0 1 2 7 4 2 5 5 3 6 8 6 6 7 4 20 Explain what the boxplot below tells you about the distribution of the data it represents. caught during random breath testing. a Prepare a fivenumber summary of the data. b Draw a boxplot of the data.
x
appropriate graphs and misuse of graphs 5e
When displaying data it is important that the right graph be chosen. To choose the correct graph the type of data needs to be identified. If the data is categorical, usually a comparison needs to be made and so a sector graph of a divided bar graph may be the best choice. Worked example 18
Jindabing High School has 36 rooms. They can be classified as follows: 21 classrooms, 3 kitchens, 3 art rooms, 4 laboratories, 3 manual arts rooms, 1 computer room and 1 library. Represent this information as a sector graph. think
Write/draW
1
Calculate the angle for each room. Since there are 360° in a circle and there are 36 rooms, divide 360 by 36.
360 ÷ 36 = 10 Each room can be represented by an angle of 10°.
2
Calculate the sector angle for each category of room.
Sector angle for each category of room: Classrooms: 21 × 10° = 210° Kitchens: 3 × 10° = 30° Art rooms: 3 × 10° = 30° Laboratories: 4 × 10° = 40° Manual arts rooms: 3 × 10° = 30° Computer room: 1 × 10° = 10° Library: 1 × 10° = 10°
3
Use a pair of compasses to draw a circle and a ruler to mark in the radius.
4
With a protractor, measure an angle of 210° from the radius to form the sector for the classrooms.
5
Repeat step 4 for each of the other categories.
6
Label each sector and include a title for the completed sector graph.
Jindabing High School use of rooms Library
Computer room Manual arts rooms 10°
10°
30° Laboratories 40° Art rooms 30° 30° s n e tch Ki
210°
Classrooms
Chapter 5 • Displaying single data sets
169
If the data represents the change in a quantity over time, a line graph will best display this information, while if the data is cyclical (e.g. months of the year) a radar chart may the best option. In general, however, if the data is continuous and quantitative a histogram best displays data. Graphs are often misrepresented in the media depending upon the purpose of the graph. It may be politicians wishing to magnify their achievements or a company wanting to accentuate their profits.
Vertical axis and horizontal axis
Declared roads as at 30 June 1979 Sealed
It is a truism that the steeper the graph the better the growth appears. A ‘rule of thumb’ for statisticians is that for the sake of appearances, the vertical axis should be twothirds to threequarters the length of the horizontal axis. This rule was established in order to have some comparability between graphs. The figure at right illustrates how distorted the graph appears when the vertical axis is disproportionately large.
13 000
Paved Formed
12 000
Unconstructed 11 000
State highways
10 000 Developmental roads Main roads
Length of road (km)
9000
8000
7000
6000
5000
4000
3000
2000
1000
Urban and subarterials
Source: Dept of Mapping & Surveying (1980), Queensland resources atlas, 2nd rev. ed. (Courtesy Dept of Lands)
Changing the scale on the vertical axis The following table gives the holdings of a corporation during a particular year.
170
Maths Quest Preliminary Mathematics General
Secondary roads
Quarter J–M A–J J–S
Holdings in $m 200 200 201
O–D
202
300
Holdings ($’000 000)
Here is one way of representing these data:
But it is not very spectacular, is it? Now look at the following graph.
x
200
x
x
x
100
J–M
A–J J–S Quarter
O–D
The shareholders would be happier with this one.
Holdings ($’000 000)
202
x
201.5 x
201 200.5 x
200
omitting certain values
x
J–M
A–J Quarter
J–S
O–D x
Holdings ($’000 000)
202
If one chose to ignore the second quarter’s value, which shows no increase, then the graph would look even better.
201.5 x
201 200.5 x J–M
200
Foreshortening the vertical axis
J–S Quarter
O–D
Look at the figures below. Notice in graph (a) that the numbers from 0 to 4000 have been omitted. In graph (b) these numbers have been inserted. The rate of growth of the company looks far less spectacular in graph (b) than in graph (a). x x x
5000
x
4500
x x
4000 2003
x
x x
x x 04
05
06 Year (a)
07
Number of employees in company
Number of employees in company
5500
08
5000 4000
x x
x
x
x
x
x x xx x
3000 2000 1000 2003
04
05
06 Year (b)
07
08
Foreshortening the vertical axis is a very common procedure. It does have the advantage of giving extra detail but it can give the wrong impression about growth rates. Net value of production
Visual impression $m
In this graph, height is the property that gives the true relation, yet the impression of a much greater increase is given by the volume of each money bag.
400 300 200 100 S
2006
S 2007 Year
S 2008
Chapter 5 • Displaying single data sets
171
a nonlinear scale on an axis or on both axes 500
500
400
300
Particles / unit area
Particles / unit area
Consider the following two graphs.
300 200 100 2004 05 06 Year (a)
07
200 100
2004 05 06 Year (b)
08
07
08
Both of these graphs show the same numerical information. But graph (a) has a linear scale on the vertical axis and graph (b) does not. Graph (a) emphasises the everincreasing rate of growth of pollutants while graph (b) suggests a slower, linear growth. Worked example 19
The following data give wages and profits for a certain company. All figures are in millions of dollars. Year Wages % increase in wages Profits
1990 6 25 1
% increase in profits
20
1995 9 50 1.5
2000 13 44 2.5
2005 20 54 5
50
66
100
20 18 16 14 12 10 8 6 4 2
Wages
Profits
Wages and profits (% increase)
Wages and profits ($m)
Now consider the graphs:
100 Profits
75 50
Wages
25
1990 1995 2000 2005 Year (a)
1990 1995 2000 2005 Year (b)
Consider the following questions. a Do the graphs accurately reflect the data? b Which graph would you rather have published if you were: i an employer dealing with employees requesting pay increases? ii an employee negotiating with an employer for a pay increase? think
Write
a 1 Look at the scales on both axes. All scales are 2
b i
172
linear. Look at the units on both axes. Graph (a) has yaxis in $ while graph (b) has yaxis in %. 1
Compare wage increase with profit increases.
2
The employer wants high profits.
Maths Quest Preliminary Mathematics General
a Graphs do represent data accurately.
However, quite a different picture of wage and profit increases is painted by graphing with different units on the yaxis. b i The employer would prefer graph
(a) because he/she could argue that employees’ wages were increasing at a greater rate than profits.
ii
1
Consider again the increases in wages and profits.
2
The employee doesn’t like to see profits increasing at a much greater rate than wages.
ii The employee would choose graph (b),
arguing that profits were increasing at a great rate while wage increases clearly lagged behind.
appropriate graphs and misuse of graphs exercise 5e
Wheat production in SouthEast Asia Othe r
1 The sector graph at right shows the production of wheat
in our region. a Which country produces the most wheat? b Of the three countries shown, which one produces the least wheat? c What reasons would you give for Australia being able to produce more wheat than New Zealand? d From this graph, are you able to tell how much wheat is produced in each country?
Indonesia
New Zealand
Australia
2 The sector graph at right shows the results of a survey
on how students travel to school. a How many degrees are there in a circle? Bus b If 24 students were surveyed, by what angle (taken at the centre of the graph) will each student be Bike represented? c Use a protractor to determine the angles of each sector Car in the graph. Walking d Copy and complete the table at right: Train (Hint: To find the frequencies, divide each sector angle by your answer to part b.) e What fraction of the students travel by bike? f What is the ratio of walkers to bus travellers? Transport to school g Students travelling by which mode of Type of transport Frequency transport are represented by 12.5% of the Bus sector graph? Bike Walking Train Car 24 3 We18 The following table shows the number of people unemployed in each state and territory of
Australia.
State/territory New South Wales Victoria Queensland Western Australia South Australia Tasmania Australian Capital Territory Northern Territory
Unemployed 213 000 152 000 141 000 67 000 52 000 21 000 9 000 8 000
Source: Australian Bureau of Statistics.
Find the sector angle for each state and draw a sector graph to show this information. Chapter 5 • Displaying single data sets
173
4 The figure and table below show various time periods during the week, and the number of fatal
accidents in New South Wales during those periods. Use the figure and table to answer the questions. Day of week
Time
Monday
Midnight
Tuesday
Wednesday
Thursday
Friday
Sunday
J
I
3 am
Saturday
A
B
9 am
C
D
E
3 pm
F
H
G
9 pm
J
I
Midnight
I
For example, time period I is from 9 pm on Sunday, Monday, Tuesday and Wednesday nights to 3 am the following mornings.
Fatalities in a particular year Time period
A
B
C
D
E
F
G
H
I
J
Fatal accidents
58
48
115
25
18
73
52
68
42
64
a What time of the day is represented by letter D? b i How many hours per week are represented by the C period? ii How many hours are there, overall, in the week? iii What percentage of the total number of hours in a week is represented by the C period (correct
to 1 decimal place)? How many hours per week are represented by the J period? What was the total number of fatal accidents for this particular year? Why is this number different from the number of fatalities for this year (620)? How many fatal accidents were there during time period F? During what time period were there 18 fatal accidents? Which time period was the worst for the number of fatal accidents? What percentage of the total number of fatal accidents occurred during the C period (correct to 1 decimal place)? j Why do you think the F and G periods were kept separate as 2 different periods? k What would be the most likely cause of fatal accidents in the I time period? c d e f g h i
5 The average number of phone calls made per week by a sample of 56 people is listed below.
21, 50, 8, 64, 33, 58, 35, 61, 3, 51, 5, 62, 16, 44, 56, 17, 59, 23, 34, 57, 49, 2, 24, 50, 27, 33, 55, 7, 52, 17, 54, 78, 69, 53, 2, 42, 52, 28, 67, 25, 48, 63, 12, 72, 36, 66, 15, 28, 67, 13, 23, 10, 72, 72, 89, 80 a Organise the data into a grouped frequency distribution table using groupings of 0−9, 10−19 and so on. b Display the data as a combined histogram and frequency polygon. 6 Construct the following: a a frequency distribution table using intervals 12−15, 16−19, . . . and b a combined histogram and frequency polygon for the following data.
Class sizes at Mathsville High 38 25 21 12
174
24 16 19 26
20 29 23 22
Maths Quest Preliminary Mathematics General
23 26 18 25
27 15 30 14
27 26 20 21
22 19 23 25
17 22 16 21
30 13 24 31
26 25 18 25
7 We19 This graph shows the dollars spent on research in a company for 2000, 2004 and 2008. Draw
another bar graph that minimises the appearance of the fall in research funds. 1.6
Research funds ($m)
1.5 1.4 1.3 1.2 1.1 2000
2004 Year
2008
8 Examine this graph of employment growth in a company.
6 5 4 3 2 1 0 1974
1981
1988
1993
Why is this graph misleading? 9 Examine this graph. a Redraw this graph with the vertical axis showing percentage of total arrivals starting at 0. b Does the change in visitor arrivals appear to be as significant as the original graph suggests?
1998
2003
2008
International visitor arrivals, by month of visit – 2005 % of total arrivals
Total employment (hundreds)
Growth of total employment, 1974–2008
11 10 9 8 7 6 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Source: Overseas Arrivals and Departures, Australia (3401.0).
10 This graph shows the studenttoteacher ratio in
Australia for the years 1995 and 2005. Students to teaching staff (a), by category of school Government primary Catholic primary Independent primary Total primary Government secondary Catholic secondary Independent secondary Total secondary Total government Total nongovernment 0
1995 2005
5
10
15 20 Ratio
25
30
(a) Number of fulltime equivalent students divided by the number of fulltime equivalent teaching staff. Note: This graph should not be used as a measure of class size. Source: ABS data available on request, National Schools Statistics collection.
a Describe what has generally happened to the ratio of students to teaching staff over the 10year
period. b A note says that the graph should not be used as a measure of class size. Explain why. Chapter 5 • Displaying single data sets
175
11 You run a company that is listed on the Stock Exchange. During the past year you have given
substantial rises in salary to all your staff. However, profits have not been as spectacular as in the year before. The following table gives the figures for the mean salary and profits for each quarter. Draw two graphs, one showing profits, the other showing salaries, that will show you in the best possible light to your shareholders. 1st quarter 6 4
Profits $m Salaries $m
2nd quarter 5.9 5
3rd quarter 6 6
4th quarter 6.5 7
12 You are a manufacturer and your plant is discharging heavy metals into a waterway. Your own chemists
do tests every 3 months and the following table gives the results for a period of 2 years. Draw a graph which will show your company in the best light. 2007 Date Concentration (parts per million)
2008
Jan.
Apr.
July
Oct.
Jan.
Apr.
July
Oct.
7
9
18
25
30
40
49
57
13 This pie graph shows the breakup of national health expenditure in 2005–06 from three sources:
Australian Government, State and local government, and nongovernment. (This expenditure relates to private health insurance, injury compensation insurers and individuals.) Breakup of national expenditure Australian Government State and local governments Nongovernment
Expenditure source Australian Government State and local government Nongovernment
($m) 37 229 21 646
% 45 25
28 004
30
a Comment on the claim that $87 000 m was spent on health from these three sources. b Which area contributes least to national health expenditure? Comment on its quoted percentage. c Which area contributes the next greatest amount to national health expenditure? Comment on its
quoted percentage. d The Australian Government contributes the greatest amount. Comment on its quoted percentage. e Consider the pie chart. i Based on the percentages shown in the table, what should the angles be? ii Based on the actual expenditures, what should the angles be? iii Measure the angles in the pie chart and comment on their values. 14 This graph shows how the $27 that a buyer pays for a CD is distributed among the departments
involved in its production and marketing. Where your $27 goes Other recording costs 65c Record company Distribution 56c sales process $1.27
Record company administration costs $1.54 Mechanical royalties $1.57
diGital doC WorkSHEET 5.2 doc10321
176
Record shop $7.40 Record company profit $1.54 Advertising $1.94
Sales tax $3.27
Production $3.40
Royalties and costs to artist $3.86
You are required to find out whether or not the graph is misleading, to explain fully your reasoning, and to support any statements that you make. Also, a comment on the shape of the graph and how it could be obtained. b Does your visual impression of the graph support the figures?
Maths Quest Preliminary Mathematics General
Summary types of graphs and stemandleaf plots
• • • • •
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.
Frequency tables and cumulative frequency
• 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.
range and interquartile range, deciles and percentiles
• 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). • A percentile shows what percentage of scores are below the given score. • A decile shows which band of 10% a score lies in.
Fivenumber summaries
• A fivenumber summary of a data set is the lower extreme, lower quartile, median, upper quartile and upper extreme. • A fivenumber summary can be graphed using a boxandwhisker plot. • A boxandwhisker plot shows the spread of a data set on a scale.
appropriate graphs and misuse of graphs
• View graphical presentations carefully to detect signs of impression. • Check that the scales on both axes are not lengthened or shortened to create a false impression. • Check that picture graphs are not represented by volumes.
Chapter 5 • Displaying single data sets
177
Chapter review m U lt ip l e C h oiCe
1 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 Which of the following would be greatly affected by the addition of an extreme score to the data set? a B C d
The median The range The interquartile range All would be greatly affected
a 12
100%
25 20 15
50%
10 5 0
0% 0 5 10 15 20 25 30 35 40 45 Weight
B 17
Cumulative frequency %
Cumulative frequency
3 For the cumulative frequency polygon below, the interquartile range of the data is:
C 24
d 12 to 24
Questions 4 and 5 refer to the boxandwhisker plot shown below.
10
15
20 25
30 35
40
45 Scale
4 The upper quartile of the data is: a 24
B 28
C 36
d 42
B 28
C 14 to 42
d 24 to 36
5 The interquartile range is: a 12 Sh ort anS Wer
1 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. 2 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. 178
Maths Quest Preliminary Mathematics General
3 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 3 2 2 3 1 0 Show this information in a frequency table.
4 1
1 1
1 1
2 2
0 0
0 1
4
4 The following data give the amount of cut meat (in kg) obtained from 20 lambs.
4.5 5.9
6.2 5.8
5.8 5.0
4.7 4.3
4.0 4.0
3.9 4.6
6.2 4.8
6.8 5.3
5.5 4.2
6.1 4.8
Show these data in a frequency table using a class size of 0.5 kg. 5 The table below shows the number of sales made each day over a month in a car yard.
Number of sales 0 1 2 3 4 5 6
Frequency 2 7 12 6 2 0 1
Show this information in a frequency histogram and polygon. 6 The frequency table below shows the crowds at football matches for a team over a season.
Class 5000–9999
Class centre
Frequency 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 30–39 40– 49 50–59 60–69 70–79 80–79
Class centre
Frequency 3 6 12 15 18 10
Cumulative frequency
a Copy and complete the frequency table. b Show the information on a cumulative frequency histogram and polygon. 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 Chapter 5 • Displaying single data sets
179
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.
Cumulative frequency
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.
Cumulative frequency
9 For each of the data sets in question 8 calculate: i the median ii the lower quartile iii the upper quartile iv the interquartile range.
300 250 200 150 100 50 0
12 Display the following scores in a stemandleaf plot.
45 36 40
21 21 41
38 38 48
46 45 39
42 44 34
41 40 38
42 29 45
49 28 28
35 35 23
29 35 29
24 33 30
45 40 35 30 25 20 15 10 5 0
0 1 2 3 4 Number of goals scored in a soccer match
.5 .5 .5 .5 .5 .5 34 44 54 64 74 84 Number of apples on a tree
28 38 40
13 Use the stemandleaf plot drawn in the previous question to find: a the range b the median
c the interquartile range.
14 For the data set below, give a fivenumber summary.
24
53
91
57
29
69
29
15
84
6
Cumulative frequency
15 Use the ogive to develop a fivenumber summary for the data set below. 80 70 60 50 40 30 20 10 0
5
15
25 35 Score
45
16 For the boxandwhisker plot below: a state the median b calculate the range c calculate the interquartile range.
0
5
10 15 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 boxandwhisker plot. 180
Maths Quest Preliminary Mathematics General
18 This table shows the number of students in each year level from Years 8 to 12.
Year 8 9 10 11 12
Number of students 200 189 175 133 124
Draw two separate graphs to illustrate the following: a The principal of the school claims a high retention rate of students in Years 11 and 12 (that is,
most of the students from Year 10 continue on to complete Years 11 and 12). b The parents claim that the retention rate of students in Years 11 and 12 is low (that is, a large
number of students leave at the end of Year 10). 1 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 Using classes 10–14, 15–19, 20–24, etc., show this information in a frequency table. b Draw a frequency histogram and polygon for the data. c Draw a cumulative frequency histogram and polygon. d State the range of the data set. e Use the ogive to find: i the median ii the interquartile range. 2 The boxandwhisker plots below show the sales of two different brands of washing powder at a supermarket each day.
ex ten d ed r eS p o n S e
Brand A Brand B 0 a b c d e
5
10
15
20 25
30 35
40
45
50
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.
Scale
diGital doC Test Yourself doc10322 Chapter 5
Chapter 5 • Displaying single data sets
181
ICT activities 5a
types of graphs and stemandleaf plots
diGital doC • Drawing graphs (doc10319): Draw graphs using graphics calculators and spreadsheets. (page 146)
5B
Frequency tables and cumulative frequency
diGital doC • Spreadsheet (doc1559): Use an Excel spreadsheet to interact with frequency tables. (page 149) • WorkSHEET 5.1 (doc10320): Apply knowledge of data to questions. (page 153)
5C
range and interquartile range, deciles and percentiles
interaCtiVitY • int2362: Measures of centre and spread: interact with distributions. (page 154)
5d
Fivenumber summaries
interaCtiVitY • int2788: Parallel boxplots — interact with distributions and boxplots. (page 164) • int0802: Boxplots and fivenumber summary — consolidate your understanding of boxplots. (page 164)
5e
appropriate graphs and misuse of graphs
diGital doC • WorkSHEET 5.2 (doc10321): Apply knowledge of data to questions. (page 176)
Chapter review diGital doC • Test Yourself Chapter 5 (doc 10322): Take the endofchapter test to check your progress. (page 181)
To access eBookPLUS activities, log on to www.jacplus.com.au
182
Maths Quest Preliminary Mathematics General
Answers CHAPTER 5 8 Key 14  3 =
diSplaYinG SinGle data SetS
Stem 13* 14 14* 15 15*
exercise 5a types of graphs and stemandleaf plots
1
Average monthly temperature (°C) January 30 February December 25 20 March 15 November 10 5 0 October April September
2
9 D 10 A 11 a A quick graphical way of displaying
information and seeing key information such as the distribution of the data
May
August
14*  8 = 14.8
14.3
Leaf 8 9 0 2 3 3 5 6 6 7 7 8 8 1 2 2 5 6 7 9
3
b
June July
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2
Percentage of televisions being watched 12.00 am 80% 2.00 am 10.00 pm 60% 8.00 pm
4.00 am
40% 20% 0%
6.00 pm
6.00 am
4.00 pm
8.00 am
2.00 pm
10.00 am 12.00 pm
3
Number of packets of chips
c A dot plot can display only single scores
and hence is best suited to discrete data.
12 a There is too great a range of scores. b The data is continuous. 13 The average daily sales in her shop over a
week, since it is a cyclical pattern 14 a Stem Leaf 4 3 7 7 8 8 9 9 9 5 0 0 0 0 1 2 2 3 Key 4  3 = 43 cm b
0 1 2 3 4 5 6 7 8 9 10 Scores
4 25 26 27 28 29 30 31 32 33 34 35 Maximum temperature (°C)
c
5 Key 0  6 = 6
Stem 0 1 2 3 4 5
Leaf 6 3 5 7 8 0 0 5 6 6 7 8 9 1 2 2 8 3 6 2
6 Key 3  6 = 36
Stem Leaf 3 6 7 8 8 4 0 0 1 2 5 0 2 2 2 6 6 8 8 7 1 2 5 8 2 7 Key 10  1 = Stem Leaf 8* 6 8 9 2 9* 5 5 5 10 2 2 2 10* 7 7 7 11 0 1 2 11* 5 12 12* 0 1 1 13 0
9 9 2 2 2 2 3 5 6 7 7 8 8 8 3 4 5 7 8 9
10*  6 = 106
6 6 6 8 9 3 8 8
4
Score 66 67 68 69 70 71 72 73 74 75 76 77 78
Tally     
Class
Frequency 3 5 5 7 3 1 1
   
Frequency 1 3 6 2 3 5 14 5 4 2 1 3 1
     
Class centre Tally Frequency
30–39 40–49 50–59 60–69 70–79 80–89 90–99
Stem Leaf 4 4 3 4 4 7 7 4 8 8 9 9 9 5 0 0 0 0 1 5 2 2 3 5 5 5 Key 4  3 = 43 cm
34.5 44.5 54.5 64.5 74.5 84.5 94.5

   
1 5 5 8 6 3 2
5
Class
15 Categorical data does not show changes in
a quantity over time. The midpoints on a line graph would have no meaning.
Frequency tables and cumulative frequency 1
Tally 
4
Stem Leaf 4 3 4 7 7 8 8 9 9 9 5 0 0 0 0 1 2 2 3 5 Key 4  3 = 43 cm
exercise 5B
101
Score 4 5 6 7 8 9 10
Score
Tally
Frequency
2

3
3

4
4

6
Class centre Tally Frequency
0.6–<0.8
0.7

2
0.8–<1.0
0.9

3
1.0–<1.2
1.1

1.2–<1.4
1.3

6
3
1.4–<1.6
1.5

9
1.6–<1.8
1.7

6
1.8–<2.0 2.0–2.2
1.9

4
2.1

4
6
Class
Class centre
Frequency
10.5–<11.0
10.75
2
11.0–<11.5
11.25
5
11.5–<12.0
11.75
8
5

3
12.0–<12.5
12.25
8
6

3
5

1
12.5–<13.0 13.0–13.5
12.75
7
13.25
2
Chapter 5 • Displaying single data sets
183
b
0
2
2
1
8
10
2
11
3
Lifetime Class Cumulative (hours) centre Frequency frequency 6
6
5–<10

27.5
25
31
21
10–<15

30–<35
32.5
70
101
6
27
15–<20
35–<40
37.5
61
162
4
2
29
40–<45
42.5
30
192
5
1
30
45–50
47.5
8
200
Cumulative frequency
200 180 160 140 120 100 80 60 40 20 0
12
Class
50
32
44
1–5
3
5
5
51
9
53
6 –10
8
7
12
52
5
58
11–15
13
8
20
53
2
60
16–20
18
10
30
21–25
23
5
35
Score
Frequency
3
5
3
60
76
7–<8
7.5
58
134
8–<9
8.5
8
142
9–10
9.5
4
146
Cumulative Frequency 5
Class centre
Frequency
0–9
4.5
6
10–19
14.5
7
20–29
24.5
8
30–39
34.5
5
40–49
44.5
4
50–59
54.5
12
60–69
64.5
8
70–79
74.5
4
80–89
84.5
2
Time taken (s)
Frequency
Cumulative frequency
6–<8
1
1
8–<10
4
5
10–<12
15
20 38
15
12–<14
5
15
30
14–<16
12
50
6
10
40
16–<18
8
58
7
20
60
18–20
2
60
8
25
85
9
10
95
10
5
100
b
13 B 14
Class
Class Cumulative centre Frequency frequency
0–<10
5
10
10
10–<20
15
8
18
Ogive of task times 60 40 20
6
10 14 18 Time (s)
18
Class Cumulative centre Frequency frequency
20–<30
25
14
32
Class
30–<40
35
8
40
50–59
54.5
2
2
50
60–69
64.5
8
10
70–79
74.5
5
15
80–89
84.5
8
23
90–99
94.5
7
30
40–50
4.5 5.5 6.5 7.5 8.5 9.5 Length of oysters (cm)
17 a
Class
10
6
6.5
Cumulative frequency
27.5
4
Class Cumulative centre Frequency frequency
6–<7
184

18
48 49 50 51 52 53 No. of jelly beans
140 120 100 80 60 40 20 0
25–30
Class Cumulative Centre Frequency Frequency
12
16
5
16 a Discrete b The data is discrete and so classes need
for the frequency table below.
10
10
17.5
2
11 Complete the frequency and class columns
49
5.5
19
22.5
Lifetime of battery (hours)
2
5–<6
12.5
to be separate, i.e. 0–9, 10–19 etc.
2
6
11

c
48
4.5
7.5
20–<25
22 .5 27 .5 32 .5 37 .5 42 .5 47 .5
0 1 2 3 4 5 No. of cars
4–<5
b
Class centre Frequency
22.5
9 a
Length (cm)
Tally
25–<30
No. of jelly Cumulative beans Frequency frequency
60 50 40 30 20 10 0
Class
20–<25
b 30 25 20 15 10 5 0
c
10 a
Cumulative frequency
8 a
Cumulative frequency
b
Cumulative No. of cars Frequency frequency
Cumulative frequency
7 a
45
10
15 a Continuous b There are too many scores to consider
as single scores, and continuous data should always be in classes.
Maths Quest Preliminary Mathematics General
4
5–<10
22
26
10–<15
95
121
15–<20
124
245
20–<25
94
339
25–<30
19
358
30–<35
5
363
35–40
2
365
400 350 300 250 200 150 100 50 0
Frequency
0
26
26
1
31
57
Salary (× $1000)
2
22
79
15–<20
16 Check with your teacher. Answers depend
Cumulative Frequency frequency 6
6
8
87
20–<25
12
18
4
3
90
25–<30
8
26
30–<35
7
33
35–<40
5
38
40–<45 45–50
1
39
1
40
2 3 Score
b
4
d 2 11 B 13 B
Class Cumulative IQ score centre Frequency frequency 75–<85
80
12
12
85–<95
90
25
37
95–<105
100
50
87
105–<115
110
24
111
115–125 140 120 100 80 60 40 20 0
c 50 d i 10
120
13
124
80 90 100 110 120 IQ Score
ii 115
iii 104
Ogive of salaries
50%
20 30 40 50 Salary ($ × 1000)
c 11 f $26 500 18 a
d 26 g $21 500
25
d 16 f 84 cm b d f h
11 8% 46 30 or less
Time
Frequency
Cumulative frequency
65–<70
2
2
70–<75
5
7
75–<80
6
13
80–<85
4
17
85–<90
1
18
90–95
2
20
Ogive of time 20 trial results
100%
15 10
50%
5 0 65 70 75 80 85 90 95 Time (s)
90 seconds or less, so 18 riders finished with a time of 90 seconds or less. e To qualify you need a time under 72 seconds.
100%
40 35 30 25 20 15 10 5
21 a
e $33 000 h $38 500
Cumulative Waist size Frequency frequency 70–<75
13
13
75–<80
28
41
80–<85
46
87
85–<90
30
117
90–<95
17
134
95–<100
8
142
100–<105 105–110
7
149
1
150
Cumulative frequency
Time
Frequency
2.4–<2.8
2
2
2.8–<3.2
9
11
3.2–<3.6
7
18
3.6–<4.0
9
27
4.0–<4.4
2
29
4.4–4.8
1
30
b Cumulative frequency
1
50%
50
c 6 d 90% of riders finished with a time of
Cumulative frequency (%)
0
b
Cumulative frequency
on class size.
17 a
75
60 96 cm 25% 12 68% 55 37
e 0–11
3
90 80 70 60 50 40 30 20 10 0
c e g 19 a c e g 20 a
c Lower quartile = 13, upper quartile = 22,
interquartile range = 9
100
70 80 90 100 110 Size (cm)
2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 Maximum temperature (°C)
d Median = 17.5
Cumulative frequency (%)
4
100%
125
Cumulative frequency (%)
0–<5
Score
Cumulative frequency
Cumulative frequency
Ogive of baby weights 100% 30 25 20 15 50% 10 5 2.4
3.2 4.0 4.8 Weight (kg)
Cumulative frequency (%)
b
No. of days
Ogive of waist size 150 Cumulative frequency
Maximum temperature (°C)
Cumulative frequency
c 1 10 A 12 D 14 a
b
b
Cumulative frequency
b
Cumulative frequency
9 a
15 a
Cumulative frequency
exercise 5C range and interquartile range, deciles and percentiles 1 a 5 b 9 c 1 d 6.94 e 89 2 a 4 b 5 c 6 3 a 49 b 30 c 23 4 a Sydney — 120 Collingwood — 40 b Collingwood, because the range is lower. 5 a 9 b 8 c The range for machine A is large, only because of one extreme score. 6 a 5 b 25 c 53 d 15 e 74 7 a 5 b 9 c 2 d 4 e 32 8 a 2 b 1 c 2 d 1 e 4
Chapter 5 • Displaying single data sets
185
c 75% d 3.45 kg, which represents the centre or e 22 a b c d e
middle of the data. 3% 19 63% 26 samples or 75% 93 °C, which means that 10% of temperature lies above this value. 78 °C
Class Cumulative centre Frequency frequency
Class 15–19
17
7
7
20–24
22
15
22
25–29
27
4
26
30–34
32
3
29
37
0
29
40–44
42
0
29
1 2
45–49
47
1
30
3 4 5 6 7 8
35
40
45 50
15
20 25
30 35
10 5 17 22 27 32 37 42 47 Age
15
20
25 30
35 40
45
b 10
Class Cumulative centre Frequency frequency 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
30
50
70 Age
median.
exercise 5e
appropriate graphs and misuse of graphs 1 a Australia b New Zealand c More land available d No. We would need a total of all wheat produced to work this out. 2 a 360° b 15° c Bus 150°, Bike 30°, Walking 120°, Train 15°, Car 45° d
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
Cumulative frequency
120–129
No. of hamburgers sold
c i 155 d
ii 140
120 130 140 150 160 170 180 190
iii 182 e
Northern Territory
Western Australia
New South Wales
South Australia
Queensland Victoria
50
Check with your teacher. 14, 21.5, 33.5, 39, 54 40 17.5 39 minutes You are in the fastest 50% of employees. Class A = 70, Class B = 65 Class A = 52, Class B = 56 Class A = 35, Class B = 40 Class A as they have a lower interquartile range. 19 a 18, 20, 26, 43.5, 74 20 The scores are tightly bunched around the
Class
186
15
d 17 a b c d e 18 a b c d
15 a
b
20
0
45
Tasmania
Type of transport
Sector angle
Sector graph New South Wales Victoria Queeensland Western Australia South Australia Tasmania Australian Capital Territory Northern Territory
25
c
40
Australian Capital Territory
30
55 60
9 a 148 b 56 c 90 d 92 e 28 10 a 58 b 31 c 43 d 27 e 8 11 B 12 C 13 D 14 a 10, 13.5, 22, 33.5, 45 b 10
b
Cumulative frequency
35–39
exercise 5d
Fivenumber summaries 8, 15, 16.5, 18, 25 a 23, 44, 81.5, 83.5, 92 b 1, 2, 4, 6, 7 c 8, 29, 45, 72, 93 1, 3, 4, 5, 7 40, 65, 72, 78, 100 15 25 30.5 39.5 50 19.6 19.9 20.1 20.25 20.8 64 70 73 77 81
Unemployment in Australia by state
3
16 a
Frequency
213 000 152 000 141 000 67 000 52 000 21 000 9000 8000
116° 83° 77° 36° 28° 11° 5° 4°
Total 663 000
Total 360°
4 a 9 am to 3 pm on Saturday b i 30 h ii 168 h iii 17.9% c 18 h d 563 e Multiple fatalities in accidents f 73 g E (9 am to 3 pm on Sunday) h C (9 am to 3 pm from Monday to
Friday)
i 20.4% j G includes Thursday night shopping and
Friday as a popular night for people to go out. k Fatigue 5 a
Class Class interval centre
Tally
Frequency
0–9
4.5

6
10–19
14.5

7
20–29
24.5

8
30–39
34.5
40–49
44.5
50–59
54.5
5 
4
Bus
10
Bike
2
Walking
8
60–69
64.5
Train
1
70–79
74.5

4
Car
3
Total
24
80–89
84.5

2
Total
56
1 12
Maths Quest Preliminary Mathematics General
f 4:5
g Car
 
12 8
Chapter reVieW
6 a
Class
mUltiple ChoiCe
4.5
24.5 44.5 64.5 84.5 14.5 34.5 54.5 74.5 Number of phone calls
1 C 3 A 5 A
2 B 4 C
Short anSWer
1
6 a
Class Class interval centre 12–15
13.5
ABC (54°)
Tally
Frequency

4
Channel 7 (96°)
7
Channel 9 (144°)
16–19
17.5
20–23
21.5

11
24–27
25.5

13
28–31
29.5

4
32–35
33.5
—
0
36–39
37.5

1

Total b
SBS (10°)
2
Friday
Frequency
Score
Wednesday
Frequency 6
1
8
7 Check with your teacher. 8 Horizontal axis uses same division for 5
2
5
3
3
9 a Check with your teacher. b No 10 a Student to teacher ratios have improved
4
2
5
0
6
1
Class
Class centre
Frequency
3.5–4.0
3.75
1
and 7 year periods
slightly.
b Country schools have smaller class sizes.
c
d
e
14 a b
4
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
b
4.0–4.5
4.25
4
4.75
5
5.0–5.5
5.25
2
5.5–6.0
5.75
4
6.0–6.5
6.25
3
6.5–7.0
6.75
1
00 00 00 00 00 00 75 12 5 17 5 22 5 27 5 32 5
Class Cumulative centre Frequency frequency 34.5
3
3
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
34.5 44.5 54.5 64.5 74.5 84.5 English exam mark
8 a 7 b 159 c 1.4 9 a i 25 ii 24 iii 27.5 b i 62.5 ii 43 iii 84 c i 1.1 ii 0.7 iii 1.5 10 a 2 b Lower = 1, upper = 3 c 2 11 a 61 b Lower = 54, upper = 70 c 16 12 Key 2  1 = 21
Stem 2 3 4
8 6 4 2 0 1 2 3 4 5 6 Number of sales
iv 3.5 iv 41 iv 0.8
Leaf 1 1 3 4 8 8 8 9 9 9 0 3 4 5 5 5 6 8 8 8 8 9 0 0 0 1 1 2 2 4 5 5 5 6 8 9
13 a 28 c 12.5 14 6, 24, 41, 69, 91 16 a 43 c 14 17
10
0
10 8 6 4 2 0
30–39
b
4.5–5.0
12 Frequency
b
5
Class
Tuesday
0
21.5 29.5 37.5 13.5 17.5 25.5 33.5 Class size
context ($86 879 m actually). State and local governments. The stated 25% is correct (rounded up from 24.9%). Nongovernment organisations. The stated 30% is rounded down from 32.2%. The percentages being quoted seem to be rounded to the nearest 5%. The quoted percentage (45%) has been rounded up from 42.2%. This could be considered misleading in some contexts. i 162°, 90°, 108° ii 154°, 90°, 116° iii 154°, 78°, 128°. Even though the pie chart gives a rough picture of the relative contributions of the three sectors, it has not been carefully drawn. It is a circle viewed on an angle to produce an ellipse. No, because it causes some angles to be larger and others to be smaller.
1
No. of people at a football match
Thursday
3
11 Check with your teacher. 12 Check with your teacher. 13 a The claim is accurate enough in the
7500 12 500
7 a
40
Mathsville class sizes 12 10 8 6 4 2 0
Frequency
5000–9999
Channel 10 (56°)
Monday 70 60 50 40 30 20 10 0
Class centre
10 000–14 999
Frequency
Frequency
Number of phone calls made per week 12 10 8 6 4 2 0
Cumulative frequency
b
b 38 15 0, 21, 26, 32, 50 b 43
0 2 4 6 8 1012 14 16 18 20
18 Check with your teacher.
Chapter 5 • Displaying single data sets
187
b
Class
Class Cumulative centre Frequency frequency
10–14
12
7
7
15–19
17
4
11
20–24
22
7
18
25–29
27
5
23
30–34
32
4
27
35–39
37
1
28
40–44
42
2
30
188
Frequency
1 a
7 6 5 4 3 2 1 0
12 17 22 27 32 37 42 No. of admissions to hospital in a day
Maths Quest Preliminary Mathematics General
c
Cumulative frequency
extended reSponSe
30 25 20 15 10 5 0
12 17 22 27 32 37 42 No. of admissions to hospital in a day d e 2 a d
34 i 23 45 7
ii 13 b 15 c 35 e Check with your teacher.
Chapter 6
Summary statistics Chapter ContentS 6a 6b 6C 6d
Calculating the mean Median and mode Standard deviation Best summary statistics
6a
Calculating the mean
inveStigate: 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.
interaCtivitY int0084 Measures of centre
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 location 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 ∑ (sigma) means sum or total, x represents each individual score in a list and ∑ x is therefore the sum of the scores. The sum is divided by n, which represents the number of scores. Chapter 6 • Summary statistics
189
Worked exaMple 1
Find the mean of the scores 17, 16, 13, 15, 16, 20, 10, 15. think
Write/diSplaY
Method 1 1
Find the total of all scores.
Total = 17 + 16 + 13 + 15 + 16 + 20 + 10 + 15 = 122
2
Divide the total by 8 (the number of scores).
Mean =
122 8 = 15.25
Method 2 1
From the MENU select STAT.
2
Delete any existing data and enter the scores into List 1.
3
Press 2 for CALC, then 6 for SET. For 1Var Xlist, enter List 1 by pressing 1. This means that the scores are stored in List 1. For 1Var Freq, enter 1 by pressing 1. This means every score entered has a frequency of 1.
4
Press J to return to the previous screen. Press 1 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: x=
∑f ×x ∑f
Worked exaMple 2
Complete the frequency table at right, then calculate the mean.
190
Maths Quest Preliminary Mathematics General
Score (x) 4 5 6 7 8 9
Frequency ( f ) 3 7 11 13 10 6 ∑f =
f×x
∑f × x =
think
Write/diSplaY
Method 1 1
Complete the f × x column by multiplying each score by the frequency.
2
Sum the frequency and f × x columns.
3
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
∑ f = 50
∑ f × x = 338
∑f ×x ∑f 338 = 50 = 6.76
x=
Method 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 2 for CALC, then 6 for SET. For 1Var Xlist, enter List 1 by pressing 1. For 1Var Freq, enter List 2 by pressing 3. This means the entries in List 2 are the frequencies corresponding to the entries in List 1.
4
Press J to return to the previous screen. Press 1 for 1Var to display all summary statistics. The mean (x) is the first summary statistic displayed.
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. Chapter 6 • Summary statistics
191
Worked exaMple 3
Complete the frequency distribution table and use it to estimate the mean of the distribution. Class 25–29 30–34 35–39 40– 44 45–50
Class centre (x)
think
Frequency ( f ) 4 9 13 12 7 ∑f =
∑f × x =
Write
1
Calculate the class centres.
2
Multiply each class centre by the frequency to complete the f × x column.
3
Sum the frequency and the f × x column. Class 25–29 30–34 35–39 40– 44 45–50
4
f×x
Use the formula to calculate the mean.
Class centre (x) 27 32 37 42 47
Frequency ( f) 4 9 13 12 7 ∑ f = 45
f×x 108 288 481 504 329 ∑ f × x = 1710
∑f ×x ∑f 1710 = 45 = 38
x=
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. 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
192
Score 67 68 69 70 71
Frequency 10 23 35 28 12
Maths Quest Preliminary Mathematics General
think
Write
a 1 Put your calculator on to statistics mode and
a
clear the memory. 2
Press each score followed by M+.
3
Get the mean by pressing x.
b 1 Set your calculator to statistics mode and clear
Mean = 30.2 b
the memory. 2
Press each score × frequency then M+.
3
Get the mean by pressing x.
exercise 6a
Mean = 69.1
Calculating the mean
1 We 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 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
c 65, 92, 56, 84
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. 3 An oil company surveys the price of petrol in eight
Sydney suburbs. The results are below. Manly 132.9 c/L Cronulla 129.9 c/L Wentworthville 125.5 c/L Campbelltown 125.9 c/L Lakemba 121.9 c/L Liverpool 119.9 c/L Epping 128.9 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 We2 The marks out of 10 on a spelling
test are recorded in the frequency table at right. a Copy and complete the table. ∑ f ×x b Use the formula: x = to calculate ∑f the mean.
Score 4 5 6 7 8 9 10
Frequency 2 4 5 9 3 5 2 ∑f =
f×x
∑f × x =
Chapter 6 • Summary statistics
193
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 16 17 18 19 20 21 22 23 24 25
digital doC Spreadsheet doc1586 Mean
digital doC Spreadsheet doc1587 Mean (diY)
No. of weeks 4 4 3 6 7 12 8 2 4 2 ∑f =
f×x
∑f × 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 the day. The results are shown in the histogram
Frequency
below. 12 10 8 6 4 2 0
8
10 12 14 16 18 Score
Find the mean of this distribution. Heights of 17 year old boys
40 35 30 25 20 15 10 5
14
0
5 c 15 m 5 c 16 m 5 c 17 m 5 18 cm 5 c 19 m 5 cm
Frequency
10 The graph below shows the distribution in the heights of one hundred 17yearold boys.
Class centre a b c d 194
What is the class size? What is the modal class? Which class has a frequency of 20? Estimate the mean height of 17 year old boys.
Maths Quest Preliminary Mathematics General
11 MC 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 12 MC 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 13 We3 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 ∑f =
f×x
∑f × x =
a Copy and complete the frequency distribution table. b Use the table to calculate the mean class mark.
Frequency
14 The graph below shows the times swum in the 100 m freestyle at an international swimming meet. 40 35 30 25 20 15 10 5 0
50.5 51.5 52.5 53.5 54.5 55.5
Class centre a Use the graph to complete the frequency table below
Class (time)
Class centre
No of swimmers
Cumulative frequency
50.5 51.5 52.5 53.5 54.5 55.5 b Use the table to estimate the mean time. Chapter 6 • Summary statistics
195
15 A cricketer played 50 innings in test cricket for the following scores.
23 25 16 69 0
65 105 70 104 49
8 74 22 57 0
112 54 40 1 33 21 78 158 14 28
0 15 8 0 52
84 33 34 51 21
12 45 36 16 3
21 21 5 6 3
4 47 7 16 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 We4 Use the statistics function on your calculator to find the mean of each of the following scores,
correct to 1 decimal place. 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
Frequency
17 Use your calculator to find the mean from each of the following. a b Score Frequency 30 25 3 7 20 15 4 10 10 5 5 18 0
6 7 8 9 10
19 38 27 10 5
28 29 30 31 32 33 34 Score
18 The table below shows the heights of a group of people.
Height (cm) 150–<155
Class centre 152
Frequency 7
155–<160
157
14
160–<165
162
13
165–<170
167
23
170–<175 175–180
172
24
177
12
Calculate the mean of this distribution. 196
Maths Quest Preliminary Mathematics General
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) Number
12–13 13
13–14 17
14–15 25
15–16 15
16–17 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 51 99 45 39
39 59 62 58 84
31 33 72 1 92
31 51 6 9 43
50 27 42 79 71
43 62 83 41 98
70 30 19 2 8
45 90 49 33 97
57 3 11 97 18
71 30 6 71 89
18 97 63 52 21
26 59 4 97 9
3 33 53 69 4
52 44 20 83 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.
Further development 21 The mean of 5 scores is 12.6. a What is the total of the five scores? b An extra score of 19.2 is added to the data set. What is the new total of the scores? c Find the mean of the six scores. 22 The mean of 9 scores is 58. A tenth score of 19 is added to the data set. Find the new mean of the data
set. 23 The data below shows the ages of 10 people working out at the gym.
23
24
19
59
23
22
16
18
25
a Find the mean of the data set. b One score which is vastly different to all other scores is called an outlier. What is the outlier in
this data set? c Calculate the mean of the nine remaining scores with the outlier omitted. d Write a sentence describing the effect that the outlier has on the mean. 24 The mean of a data set containing 8 scores is 63. After a ninth score is added the mean falls to 60. What
was the ninth score? 25 Livinia has an average score of 14 for 6 essays that she has written. What score will she need to achieve
for her next essay in order to lift her mean to 15.5? 26 Describe the effect on the mean if a score: a greater than the mean is added to the data set b less than the mean is added to the data set. Chapter 6 • Summary statistics
197
Median and mode
6b interaCtivitY int2352 Measures of centre
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 one score is much greater than all the others. For such data sets, we need to use a different measure of location. The median is 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 5
Calculate the median of the scores 3, 5, 8, 4, 4, 6, 9, 1, 6. think
Write
1
Rewrite the scores in ascending order.
1, 3, 4, 4, 5, 6, 6, 8, 9
2
The median is the middle score.
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 6
Find the median of the scores 13, 13, 16, 12, 19, 18, 20, 18. think
Write
1
Write the scores in ascending order.
12, 13, 13, 16, 18, 18, 19, 20
2
There is an even number (8) scores, so average the two middle scores.
Median =
16 + 18 2 = 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
Cumulative 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
The 17th–24th scores are 7.
8
4
28
The 25th–28th scores are 8.
9
2
30
The 29th and 30th scores are 9.
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. 198
Maths Quest Preliminary Mathematics General
Worked exaMple 7
Find the median for the frequency distribution at right.
think
Score
Frequency
34
3
35
8
36
12
37
9
38
8
39
5
Write/draW
Method 1 1
Redraw the frequency table with a cumulative frequency column.
2
There are 45 scores and so the middle score is the 23rd score.
3
Look down the cumulative frequency column to see that the 23rd score is 36.
Score
Frequency
Cumulative frequency
34
3
3
35
8
11
36
12
23
37
9
32
38
8
40
39
5
45
Median = 23rd score = 36
Method 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 2 for CALC, then 6 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 J to return to the previous screen, then 1 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
Chapter 6 • Summary statistics
199
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 reorder 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 14
12 12
14 14
12 12
16 12
10 8
12 18
14 16
16 12
18 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 evennumbered 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 8
Find the mode of the scores below. 4, 5, 9, 4, 6, 8, 4, 8, 7, 6, 5, 4 think
Write/diSplaY
Method 1 The score 4 occurs most often and so it is the mode.
Mode = 4
Method 2 1
From the MENU select STAT.
2
Delete any existing data and enter the scores into List 1.
3
Press 2 for CALC, then 6 for SET. Set the calculator up for a list of scores as shown earlier and as shown at right.
4
Press J to return to the previous screen, then 1 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. To find the mode from a frequency distribution table, we simply give the score that has the highest frequency. 200
Maths Quest Preliminary Mathematics General
Worked exaMple 9
For the frequency distribution below, state the mode. Score Frequency
14 3
15 6
16 11
17 14
think
18 10
19 7
Write
Mode = 17
The highest frequency is 14, which belongs to the score 17 and so 17 is the mode.
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.
exercise 6b
Median and mode
1 We5 The scores of seven people on a spelling test are given below.
5 Calculate the median of these marks.
6
5
8
5
9
8 digital doC Spreadsheet doc1588 Median
2 We6 Below are the scores of eight people who played a round of golf.
75 80 81 Calculate the median for this set of scores.
76
84
83
81
82
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
digital doC Spreadsheet doc1589 Median (diY)
4 A factory has 80 employees. Over a twoweek 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.
digital doC Spreadsheet doc1590 Mode
digital doC Spreadsheet doc1591 Mode (diY)
Chapter 6 • Summary statistics
201
5 We7 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 threeweek 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 50day period. Calculate the median number of errors made by the machine each day.
Cumulative frequency
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
Score
Frequency
1
12
2
13
3
8
4
7
5
5
8 MC There are 25 scores in a distribution. The median score will be the: a b C d
12th score 12.5th score 13th score average of the 12th and 13th score.
9 MC For the scores 4, 5, 5, 6, 7, 7, 9, 10 the median is: a b C d
5 6 6.5 7
10 MC Consider the frequency table at right. The median
of these scores is: a 2 b 3 C 8 d 13 202
Maths Quest Preliminary Mathematics General
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
Cumulative frequency
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 We8 For each of the following sets of scores find the mode. a 2, 5, 3, 4, 5 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 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 15 We9 Use the tables below to state the mode of the distribution. a
Score
Frequency
1
b
Score
Frequency
2
5
2
4
3
c
Score
Frequency
1
38
2
6
3
39
4
5
7
5
40
1
4
6
8
8
41
5
5
3
9
5
42
6
10
3
43
3
44
6
45
2
Frequency
16 Consider the histogram drawn below. 40 35 30 25 20 15 10 5 0
12 13 14 15 16 17 18 19 20 Score
a State the mode of the distribution. b Draw a cumulative frequency histogram and ogive of the distribution. c Use your graph to find the median of the distribution. Chapter 6 • Summary statistics
203
17 For each of the following grouped distributions, state the modal class. a
Class
Frequency
1– 4
b
Class
Frequency
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 below 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
19 The weekly wage (in dollars) of 40 people is shown below.
376 223 556 543
592 295 419 532
299 232 226 435
501 325 494 415
375 311 205 540
366 513 307 260
204 348 417 318
359 235 204 593
382 329 528 592
274 203 487 393
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.
Heights of 17 year old boys
40 35 30 25 20 15 10 5
14
0
5 c 15 m 5 c 16 m 5 c 17 m 5 18 cm 5 c 19 m 5 cm
Frequency
20 The graph below shows the distribution in the heights of one hundred 17yearold boys.
Class centre a State the modal class. b Use the graph to draw a cumulative frequency histogram and ogive of the distribution. c Use your ogive to estimate the median height of 17yearold boys. 204
Maths Quest Preliminary Mathematics General
Further development 21 Consider the stemandleaf plot below:
Stem 60 61 62 63 64 65 66 67
Leaf 2 5 8 1 3 3 6 7 8 9 0 1 2 4 6 7 8 8 9 2 2 4 5 7 8 3 6 7 4 5 8 3 5 4
Key 60  3 = 603
a Find the median of the data set. b Find the mode of the data set. c Explain the advantage of a stemandleaf plot when trying to find the median and mode. 22 Consider the following data set 23, 24, 20, 21, 25, 26, 28, 26. a Find the median of the data set. b An extra score of 56 is added to the data set. Find the new median. c Describe the effect that the extra (outlier) score had on the median. 23 Will the addition of an outlier have any effect on the mode? 24 Consider the following set of 20 scores.
64, 34, 67, 22, 59, 72, 34, 93, 20, 82 30, 45, 27, 70, 44, 82, 71, 65, 45, 66 a Find the mode of the data set. b Daryl put the data into a table with a class size of 10 beginning with 20–29. Find the modal class. c Explain why the modal class will have more meaning than the mode. 25 Explain what will be meant by the term median class. Why do you think this term is seldom used? 26 When the median of a grouped distribution is found using an ogive the result will only be an estimate
of the median. Explain.
6C
digital doC WorkSHEET 6.1 doc10323
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.
interaCtivitY int0144 Standard deviation
Worked exaMple 10
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
Write/diSplaY
Method 1 1
Enter the data set into your calculator.
2
Retrieve the mean using the x function.
x = 74.05
3
Retrieve the standard deviation using the σn function.
σn = 13.07
Chapter 6 • Summary statistics
205
Method 2 1
From the MENU select STAT.
2
Delete any existing data and enter the scores into List 1.
3
Press 2 for CALC, then 6 for SET. Set the calculator up for a list of scores as shown earlier and as shown at right.
4
Press J to return to the previous screen, then 1 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 11
Ian surveys twenty Year 11 students and asks how much money they earn from parttime 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
Write/diSplaY
Method 1 1
Enter the statistics into your calculator.
2
Retrieve the mean using the x function.
x = $77.70
3
Retrieve the standard deviation using the σn − 1 function, as a sample has been used.
σn − 1 = $15.56
Method 2
206
1
From the MENU select STAT.
2
Delete any existing data and enter the scores into List 1.
Maths Quest Preliminary Mathematics General
3
Press 2 for CALC, then 6 for SET. Set the calculator up for a list of scores as shown earlier and as shown at right.
4
Press J to return to the previous screen, then 1 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 12
The table below shows the scores of a class of thirty Year 3 students on a spelling test. Score 4 5 6 7 8 9 10
Frequency 1 2 4 9 6 7 1
Calculate the mean and standard deviation. think
Write
1
Enter the data into your calculator using score × frequency.
2
Retrieve the mean using the x function.
x = 7.4
3
Retrieve the standard deviation using the σn function, as the whole population is included in the statistics.
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. A higher standard deviation means that scores are less clustered around the mean and less dependable. For example, consider the following two students: σn = 5 Student A: x = 60 σn = 15 Student B: x = 60 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. Chapter 6 • Summary statistics
207
Worked exaMple 13
Two brands of light globe are tested to see how long they will burn (in hours). Brand X: 850 950 1400 875 1200 1150 1000 900 850 825 Brand Y: 975 1100 1050 1000 975 950 1075 1025 950 900 Which of the two brands of light globe is more reliable? think
Write
1
Enter both sets of data into your calculator.
2
Choose the sample standard deviation because a sample of each light globe brand has been chosen.
3
Write down the sample standard deviation for each brand.
Brand X: σn − 1 = 190.4 Brand Y: σn − 1 = 62.4
4
The brand with the lower standard deviation is the more reliable.
Brand Y is the more reliable as it has a lower standard deviation.
exercise 6C
Standard deviation
1 We10 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. b 11, 8, 7, 12, 10, 11, 14 d 5.2, 4.7, 5.1, 12.6, 4.8
a 3, 5, 8, 2, 7, 1, 6, 5 c 25, 15, 78, 35, 56, 41, 17, 24 e 114, 12, 3.6, 42.8, 0.5
2 We11 For each of the sets of scores below, calculate the sample standard deviation, correct to
2 decimal places. 25, 36, 75, 85, 6, 49, 77, 80, 37, 66 4.8, 9.3, 7.1, 9.9, 7.0, 4.1, 6.2 112, 25, 56, 81, 0, 5, 178, 99, 41 0.3, 0.3, 0.3, 0.4, 0.5, 0.6, 0.8, 0.8, 0.8, 0.9, 1.0 56, 1, 258, 45, 23, 58, 48, 35, 246
a b c d e
3 For each of the following, state whether it is appropriate to use the population standard deviation or the
sample standard deviation.
208
Maths Quest Preliminary Mathematics General
a b c d e
A quality control officer tests the life of 50 batteries from a batch of 1000. The weight of every bag of potatoes is checked and recorded before being sold. The number of people who attend every football match over a season is analysed. A survey of 100 homes records the number of cars in each household. 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? 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? 6 We12 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
Score 3 4 5 6 7
Frequency 12 24 47 21 7
b
Score 45 46 47 48 49 50
Frequency 1 16 39 61 52 36
c
Score 75 76 77 78 79 80 81
Frequency 22 17 8 10 12 21 29
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 10–12 13–15 16–18 19–21 22–24
c
Class 0– 4 5–9 10–14 15–19 20–24 25–29
Class centre
Class centre
b
Frequency 12 16 25 28 13
Class 31– 40 41–50 51–60 61–70 71–80 81–90 91–100
Class centre
Frequency 15 28 36 19 8 7 2
Frequency 15 24 31 33 29 17
Chapter 6 • Summary statistics
209
8 We13 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 MC From Year 11, 21 students are chosen to complete a test. The scores are shown in the table below.
Class
Frequency
10–19
1
20–29
6
30– 39
9
40–49
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 MC 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–<500 500–<1000 1000–<1500 1500–<2000 2000–<2500 2500–<3000 3000–3500 2
4
8
10
7
Find the mean and standard deviation of the price paid for these wrecks. 210
Maths Quest Preliminary Mathematics General
6
3
12 Times (to the nearest tenth of a second) for the heats in the open 100 m sprint at the school sports are
given below.
Stem 11 11* 12 12*
Leaf 0 2 3 4 4 5 6 6 8 8 9 0 1 2 2 3 4 4 6 9
Calculate the standard deviation for this set of data correct to 2 decimal places. 13 The number of outgoing phone calls from an office each day over a 4week period is shown on the
stem plot below.
Stem 0 1 2 3 4 5
Leaf 8 9 3 4 7 9 0 1 3 7 7 3 4 1 5 6 7 8 3 8
Calculate the standard deviation for this set of data and express your answer correct to 2 decimal places. 14 The dot plot drawn below shows the number of days absent that a class of students have had in a term.
0 1 2 3 4 5 6 7 8 9 10 Number of days absent
Find the standard deviation of the data set. 15 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. 16 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? Chapter 6 • Summary statistics
211
Further development 17 Consider the following two groups of people. Group B 160 170 170 110 230 170 180
Height (cm)
Group A 160 170 170 170 170 170 180
a b c d
Calculate the mean height of each group. Are the groups really the same? In which group would you expect the greater standard deviation? Calculate the standard deviation to confirm your answer.
18 Consider the set of scores 3, 5, 8, 2, 7, 4, 5, 6. a Find the mean of the data set. b Find the standard deviation of the data set. c A score of 9 is added to the data set. What is the difference between this score and the mean? d What is the standard deviation of the data set once the extra score is added? 19 Consider the data set 25, 15, 78, 35, 56, 41, 17, 21. a Find the mean and standard deviation of the data set. b An extra score is added to the data set. Copy and complete the table below to explore the effect
that adding an extra score has on the standard deviation. Extra score 8 30 90 50
Difference from mean
Standard deviation after score added
Standard deviation (increase or decrease)
20 A data set has a mean of 48 and a standard deviation of 23. A score of 55 is added to the data set. a What effect will adding the extra score have on the mean? Explain your answer. b What effect will adding the extra score have on the standard deviation? Explain your answer. 21 MC A data set has a mean of 36 and a standard deviation of 8. A score of 12 is added to the data set.
What will be the effect on the mean and the standard deviation? a The mean will decrease and the standard deviation will decrease. b The mean will decrease and the standard deviation will increase. C The mean will increase and the standard deviation will decrease. d The mean will increase and the standard deviation will increase. 22 Describe in your own words, how adding an extra score to a data set will affect the standard deviation.
6d
best summary statistics
Having now examined all three measures of centre (the mean, the median and the mode), 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. 212
Maths Quest Preliminary Mathematics General
Worked exaMple 14
Below are the wages of ten employees in a small business. $420 $430 $490 $475 $465 $450 $1700 $420 $420 $440 a Calculate the mean wage. b Calculate the median wage. c Calculate the mode wage. d Does the mean, median or mode give the best measure of a typical wage in this business? think
Write
a Total = $5710
a 1 Total all the wages. 2
Mean = $5710 ÷ 10 = $571
Divide the total by 10.
b 1 Write the wages in ascending order.
b $420 $420 $420 $430 $440 $450 $465
$475 $490 $1700 2
Average the 5th and 6th score to find the median.
c $420 is the score that occurs most often and so this
is the mode.
$440 + $450 2 = $445
Median =
c Mode = $420
d The mean is larger than what is typical because of
d The median is the best measure of the typical
one very large wage: the mode is the lowest wage and so this is not typical. Therefore, the median is the best measure.
wage as the mode is the lowest score, which is not typical, and the mean is inflated by the $1700 wage.
When considering the best measure of location and spread we need to consider what the effect an outlier will have on a data set. Consider the following set of scores: 90, 80, 85, 75, 85. For this data set Mean = 83, Median = 85, Mode = 85 Now consider the effect that an extra score will have if it is an outlier. If a score of 20 is added to the data set then Mean = 72.5, Median = 82.5, Mode = 85 We can see that the addition has a significant effect on the mean, small effect on the median and no effect on the mode. Now consider the effect that the outlier has on the measures of spread. Before the addition of the outlier Range = 15, Interquartile range = 10, Standard deviation = 5.7 After the addition of the outlier to the data set these measures become: Range = 70, Interquartile range = 10, Standard deviation = 26.2 The outlier had a large impact on the range, a significant impact on the standard deviation but very little impact on the interquartile range. As the number of scores in the data set becomes larger the impact of an outlier on the standard deviation is decreased, but it only takes a single outlier to have a huge impact on the range regardless of the number of scores in the data set. For each of these examples you will need to think carefully about the relevance of each summary statistic in terms of the particular example. It is important to consider that when a sample is taken the summary statistics that are found can only be considered to be estimates of the entire population. Two or more samples that have been taken from the same population may in fact produce different results. The larger the samples that are taken, the less likely that this is to occur; however, samples may be subject to some variation depending on whether any outliers may (by chance) be included. Chapter 6 • Summary statistics
213
Worked exaMple 15
Tegan and April each sample 10 apple trees from an orchard and record the number of pieces of fruit on each. The results they obtain are given below. Tegan 45, 38, 44, 56, 50, 55, 62, 59, 41, 42 April 12, 84, 56, 42, 68, 32, 41, 42, 70, 30 a For each data set find the mean and standard deviation. b Describe the differences between Tegan and April’s results. c Give possible reasons for these differences. think
Write
a 1 Put Tegan’s data into your calculator and
a Tegan
find the mean and standard deviation. 2
Put April’s data into your calculator and find the mean and standard deviation.
b Discuss the similarity in the mean but the
large difference in the standard deviation.
April
x = 49.2 sn = 8.4 x = 47.7 sn = 21.7
b The mean for both sets of data is similar although
April’s is slightly smaller. The two samples have a greatly different standard deviation as April’s sample included outliers at both extremes of the data set.
c Think of possible reasons why there may be a c The difference in standard deviations may be
difference in the standard deviations.
exercise 6d
coincidental as April may have by chance included a number of outliers in her sample. It is also possible that Tegan selected all her samples from one particular part of the orchard, hence obtaining similar results because these plants would be subject to similar conditions.
best summary statistics
1 We14 There are ten houses in a street. A realestate agent values each house with the following results. digital doC GC program — Casio doc1592 Uv stats
$350 000 $390 000 $375 000 $350 000 $950 000 $350 000 $365 000 $380 000 $360 000 $380 000 a Calculate the mean house valuation. b Calculate the median house valuation. c Calculate the mode house valuation. d 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. Size 4 5 6 7 8 9 10
digital doC GC program — TI doc1593 Uv stats
a b c d
214
Frequency 5 7 19 24 16 8 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?
Maths Quest Preliminary Mathematics General
3 A coffee shop records the number of coffees sold per hour over a 12 hour period. The results are shown
in the dot plot below.
10 11 12 13 14 15 16 17 18 19 20 Number of coffees sold a Calculate the mean, median and mode of the data set. b Find the range and interquartile range of the data set. c Explain why the range is not a good measure of spread in this data set. 4 The stemandleaf plot below shows the number of times that a fire engine was called out each week
over a three month period. Stem 1 2 3 4 a b c d
Leaf 5 8 9 1 6 9 9 0 0 1 2 8 1
Find the mean, median and mode of this data set. Is the mean a satisfactory measure of the typical score in this data set? Find the range and interquartile range of this data set. Find the standard deviation of the data set.
5 Consider the data set below
10, 10, 11, 22, 23, 27, 28, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 47, 49. a b c d
What is the mode of the data set? Display the data set in a frequency table using class groupings 10–19, 20–29, etc. What is the modal class of the data set? Explain why the modal class is a more useful measure than the mode.
6 Consider the set of scores
9, 51, 51, 52, 54, 54, 56, 56, 56, 60. a b c d
There is an outlier in the data set. Explain what is meant by the term outlier. Find the mean, median and mode of the data set. Find the mean, median and mode if the outlier is removed. Explain the effect on each of the above measures if the outlier is removed.
7 The table below shows the crowds at football matches over a season.
a b c d e f
Crowd
Class centre
Frequency
10 000–19 999
15 000
95
20 000–29 999
25 000
64
30 000– 39 999
35 000
22
40 000–49 999
45 000
15
50 000–59 999
55 000
3
60 000–69 999
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? Chapter 6 • Summary statistics
215
8 MC 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. 9 Thirty men were asked to reveal the number of hours they spent doing housework each week. The
results are given below.
a b c d
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 Represent the data in a frequency distribution table. (Use classes 0– 4, 5–9, 10–14 etc.) Find the mean number of hours that the men spend doing housework. Find the median class for hours spent by the men at housework. Find the modal class for hours spent by the men at housework.
10 We15 Lewis and Jim each sample 10 tyres that come off a production line. They use a machine to
measure the distance the tyres can travel before the tread is worn down. The results they obtain are given below in thousands of kilometres. Jim Lewis
38, 30, 32, 26, 30, 35, 32, 39, 29, 42 22, 24, 36, 22, 28, 22, 21, 32, 30, 30
For each data set a find the mean and standard deviation. b describe the differences between Jim and Lewis’s results. c give possible reasons for these differences. 11 Carmen and Jade each take a sample of the number of customers who enter a shop over 8 selected
hours in a week. Their results are shown below. Carmen Jade
28, 20, 22, 26, 32, 25, 72, 29, 29, 42 22, 24, 26, 22, 8, 29, 29, 32, 20, 36
a For each data set find the mean and standard deviation. b Give a possible reason why Carmen has obtained a much greater mean and standard deviation. 12 The batting scores for two cricket players over six innings are as follows:
Player A: 31, 34, 42, 28, 30, 41 Player B: 0, 0, 1, 0, 250, 0 a b c d e
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?
13 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
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. 216
Maths Quest Preliminary Mathematics General
14 Consider the following two groups of people.
Group A 160 170 170 170 170 170 180
Group B 160 170 170 110 230 170 180
a Calculate the mean height, median height and mode height for each group. b Describe the difference between the two groups. c For each group find the range, interquartile range and standard deviation. 15 The following data give the number of fruit that have formed on each of 30 trees in an orchard.
45 81 60
48 73 53
52 46 54
36 48 58
38 44 41
72 39 44
36 52 47
74 58 76
56 57 68
46 65 55
a Complete a frequency distribution table for the data with the class intervals being 30–39, 40–49 b c d e f g
and so on. 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. Use your table to estimate the mean and standard deviation of the data. Use the scores given in this list to find the mean and standard deviation of the data. Explain the difference in your answers to parts e and f.
16 MC The addition of a single outlier to a data set is most likely going to have the greatest effect on a b C d
the mean the median the mode all of the above
17 MC The addition of a single outlier to a data set is most likely going to have the greatest effect on a b C d
the standard deviation the range the interquartile range all of the above
18 A data set is given as 12, 12, 16, 17, 22, 24, 24, 30. This data set is bimodal. a Explain what is meant by the term bimodal. b Greg correctly identifies the two modes as being 12 and 24. However, Greg then averages these
two modes, leading to a result of 18. Explain why this result has no meaning.
Further development 19 The data below shows the age of 25 patients admitted to the emergency ward of a hospital.
18 6 16 75 24 23 82 75 25 21 43 19 84 76 30 78 24 20 63 79 a b c d
Find the mean age of the patients. Find the median age of the patients. What is the mode age? Do any of the measures of central tendency give a clear representation of the typical age of an emergency ward patient? Give a reason for your answer.
20 A small business pays the following salaries (in thousands of dollars) to its employees:
38, 38, 38, 38, 46, 46, 46, 55, 100 (the manager) What is the salary of most workers? What is the mean salary? What is the median salary? The union is negotiating a salary rise for the workers. What measure would be used by: i the union in negotiations ii the employer in negotiations. Explain each answer. a b c d
Chapter 6 • Summary statistics
217
21 The contents of 20 randomly selected boxes of matches were counted. The following data shows the
number of matches in each box: 138, 140,
139, 141,
139, 141,
141, 139,
137, 141,
140, 138,
137, 139,
141, 140,
139, 141,
142 138.
a Find the mean, median and mode of the distribution. b Which of the three measures best supports the manufacturer’s claim that there are 140 matches
per box? c Is this claim by the manufacturer valid? 22 A class of mathematics students got a median mark of 54 for their end of semester test; however,
digital doC WorkSHEET 6.2 doc10324
noone actually scored this result. a Explain how this is possible. b How many students must have scored below 54? 23 a What is the effect of an outlier on the mean of a data set? b When this occurs what is usually the best measure of central tendency? c Give an example of when the mode is the most relevant measure of central tendency. 24 a Explain why the range is an unreliable measure of spread. b Does a single outlier have any effect on the interquartile range and standard deviation? inveStigate: 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?
218
Maths Quest Preliminary Mathematics General
inveStigate: 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.
Chapter 6 • Summary statistics
219
Summary Calculating 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.
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.
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) or sn, is used.
best 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. • We need to consider the impact that outliers have on the summary statistics.
220
Maths Quest Preliminary Mathematics General
Chapter review 1 For the following data set, which of the statements is correct?
M U ltip l e C ho iC e
3, 4, 8, 7, 3, 6, 5, 3, 4, 7 a The mean is 5. C The mode is 5.
b The median is 5. d all of the above
2 For which of the following data sets is the median greater than the mean? a 2, 6, 14, 14, 15, 16, 18 C 12, 15, 15, 15, 15, 18
b 12, 13, 14, 14, 14, 18, 22 d 1, 4, 9, 16, 25, 36, 49
3 For the data set below, which statement is correct?
25, 45, 64, 48, 66, 85, 45, 27 a The mean is 50.625. C The population standard deviation is 18.98.
b The sample standard deviation is 20.29. d all of the above
4 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 For the statistical analysis in question 4 which summary statistic would be the most appropriate? a mean C mode
b median d standard deviation
S ho rt a n S W er
1 Calculate the mean of each of the following sets of scores. a 4, 9, 5, 3, 5, 6, 2, 7, 1, 10 b 65, 67, 87, 45, 90, 92, 50, 23 c 7.2, 7.9, 7.0, 8.1, 7.5, 7.5, 8.7 d 5, 114, 23, 12, 25 2 Copy and complete the tables below and then use them to calculate the mean. a b
Score (x)
Frequency ( f )
11
9.2
36
6
15
9.3
48
7
24
9.4
74
8
21
9.5
65
9
9.6
51
9.7
32
9.8
14
9.9
2
Score (x)
Frequency ( f )
5
9
∑f =
f×x
∑f × x =
∑f =
f×x
∑f × x =
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 ∑f =
f×x
∑f × x =
Chapter 6 • Summary statistics
221
4 Use the statistics function on your calculator to find the mean of each of the following sets of scores. 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
Score 10 20 30 40 50
Class 10–12 13–15 16–18 19–21 22–24 25–27 28–30
Frequency 23 47 68 56 17
Class centre 11 14 17 20 23 26 29
b
Score 24 25 26 27 28 29
Frequency 45 89 124 102 78 46
Frequency 18 32 34 40 28 14 6
6 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 7 Copy and complete each of the following frequency tables and then use them to find the median. a b Cumulative
Score 0 1 2 3 4 5
Frequency 2 6 11 7 6 3
Score 66 67 68 69 70 71 72
Frequency 8 10 12 14 7 5 4
c
222
Maths Quest Preliminary Mathematics General
Cumulative frequency
Cumulative frequency
Score 54 55 56 57 58 59 60
Frequency 2 5 14 11 6 1 1
frequency
8 a Copy and complete the frequency distribution table below.
Class
Class centre
Cumulative frequency
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. 9 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 10 For each of the frequency tables below, state the mode. b a
Score
Frequency
Score
Frequency
1
23
14
9
2
35
15
15
3
21
16
8
4
19
17
12
5
8
18
15
19
7
20
1
11 Use the frequency table below to state the modal class.
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
12 The marks of 30 students in a Geography test are shown below.
66 67 46
47 87 87
43 75 49
80 72 70
42 42 82
92 60 92
92 86 93
90 53 71
92 95 62
77 78 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. Chapter 6 • Summary statistics
223
13 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 1
2 1
3 2
3 2
1 2
2 3
1 1
2 2
4 2
1 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. 14 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
26
d
Class
Class centre
Frequency
12
10–14
12
8
27
25
15–19
17
12
28
29
20–24
22
32
29
28
25–29
27
45
30
14
30–34
32
40
35–39
37
19
40– 44
42
6
15 Below are the number of goals scored by a netball team in ten matches in a tournament.
25 a b c d
26
19
24
28
67
21
22
28
18
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. e x t ended r e SponS e
1 The table below shows the gross annual income for a sample of 100 company executives.
Income
a b c d e f 224
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
Cumulative frequency
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?
Maths Quest Preliminary Mathematics General
2 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. Text A 44 52 95 72 35 48 Text B 65 72 48 58 59 64 a b c d e
76 13 94 83 72 55 81 22 25 64 56 59 84 98 84 21 35 69 28 63 68 59 68 62 75 79 81 72 64 53 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.
digital doC Test Yourself doc10325 Chapter 6
Chapter 6 • Summary statistics
225
ICT activities 6a
Calculating the mean
digital doCS • Spreadsheet (doc1586): Mean (page 194) • Spreadsheet (doc1587): Mean (DIY) (page 194) interaCtivitY • Measures of centre (int0084): Interact with the mean, median and mode. (page 189)
6b
Median and mode
digital doCS • Spreadsheet (doc1588): Median (page 201) • Spreadsheet (doc1589): Median (DIY) (page 201) • Spreadsheet (doc1590): Mode (page 201) • Spreadsheet (doc1591): Mode (DIY) (page 201) • WorkSHEET 6.1 (doc10323): Apply statistical measures to questions. (page 205) interaCtivitY • Measures of centre (int2352): Interact with measures of centre. (page 198)
226
Maths Quest Preliminary Mathematics General
6C
Standard deviation
interaCtivitY • int0144: Standard deviation (page 205)
6d
best summary statistics
digital doCS • GC program — Casio (doc1592): UV stats (page 214) • GC program — TI (doc1593): UV stats (page 214) • WorkSHEET 6.2 (doc10324): Apply your knowledge of statistics to questions. (page 218)
Chapter review digital doC • Test Yourself Chapter 6 (doc10325): Take the endofchapter test to check your progress. (page 225)
To access eBookPLUS activities, log on to www.jacplus.com.au
Answers CHAPTER 6 SUMMarY StatiStiCS exercise 6a
9
Calculating the mean
1 a 5 b 26.5 c 74.25 d 7.72 e 376 2 72.6% 3 125.7 c/L 4 1.81 m 5 Yes, mean mass is 45.035 g. 6 a
Score (x)
Frequency ( f)
f×x
8 10 12 14 16 18
2 7 11 6 2 2
16 70 132 84 32 36
∑ f = 30
∑ f × x = 370
1
Score
Frequency
f×x
Mean = 12 3
4
2
8
5
4
20
6
5
30
10 a 10 c 180–190 cm 11 D 13 a
7
9
63
8
3
Class
24
9
5
45
10
2
20
∑ f = 30
∑ f × x = 210
b 7 7 a
b 170–180 cm d 171.4 cm 12 C
Class centre Frequency (x) ( f)
5.5
12
11–20
15.5
6
21–30
25.5
5
31– 40
35.5
7
41–50
45.5
9
51–60
55.5
9
71–80
75.5
5
81–90
85.5
5
91–100
95.5
7
35.5
41–50
45.5
3
136.5
51–60
55.5
4
222.0
61–70
65.5
7
458.5
71–80
75.5
11
830.5
81–90
85.5
2
171.0
91–100
95.5
2
191.0
∑ f = 30
∑ f × x = 2045
No. of weeks
f×x
16
4
64
17
4
68
18
3
54
19
6
114
50.01–51.00 50.5
4
20
7
140
51.01–52.00 51.5
12
618.0
21
12
252
52.01–53.00 52.5
23
1207.5
22
8
176
53.01–54.00 53.5
38
2033.0
23
2
46
54.01–55.00 54.5
15
817.5
24
4
96
55.01–56.00 55.5
3
166.5
25
2
50
∑ f = 52
∑ f × x = 1060
b 68.17 14 a
f×x 202.0
∑ f = 95 ∑ f × x = 5044.5
b 46.4 21 a 63 b 82.2 c 13.7 22 54.1 23 a 25 b 59 c 21.2 d The outlier greatly increases the mean. 24 36 25 24.5 26 a The mean will increase. b The mean will decrease. exercise 6b
Median and mode
1 6 2 81 3 a 5 b 5.4 c 62 d 102 4 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
b 53.1 15 a
Score
Frequency
Cumulative frequency
17
4
4
18
9
13
f×x
19
6
19
0 –19
9.5
20
190
20
12
31
0
20 –39
29.5
12
354
21
8
39
9
9
40 –59
49.5
8
396
22
5
44
2
18
36
60 –79
69.5
5
347.5
23
4
48
3
10
30
80 –99
24
2
50
Score (x)
Frequency ( f)
f×x
0
4
1
b 2.3
1–10
5
1
Class centre Frequency (x) ( f)
Frequency
65.5
35.5
Class No. of centre swimmers
Class centre
61–70
31– 40
Time
Class
f×x
No. of televisions sold
b 20.4 8 a
16 a 14.1 b 3.4 c 44.4 17 a 6.6 b 30.67 18 166.25 19 a 12.5, 13.5, 14.5, 15.5, 16.5 b 14.4 20 a
4
5
20
5
4
20
∑ f = 50
∑ f × x = 115
Score
89.5
1
89.5
100 –119 109.5
3
328.5
120 –139 129.5
0
0
140 –159 149.5
1
149.5
∑ f = 50
∑ f × x = 1855
b 37.1
b 20 6 7 8 9 10
3 1 C C A
Chapter 6 • Summary statistics
227
11 a
Days sickness
Frequency
Cumulative 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
160 140 120 100 80 60 40 20 0
b 148, 151 d 72
12 13 14 15 16 17 18 19 20 Class centres
50–<100
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
Cumulative frequency
$500–$549 524.5
6
36
$550–$599 574.5
4
40
40 35 30 25 20 15 10 5 0
5 5 5 5 5 5 5 5 22 27 32 37 42 47 52 57 Weekly wage ($)
90 80 70 60 50 40 30 20 10 0
9 10 11 12 13 14 15 16
375
2
23 24
25
58 26
exercise 6C 25 75 125 175 225 275 325 375 Depth of snow (cm)
Class centres
173 cm 628 613, 628 and 632 The stemandleaf plot arranges all the scores in order, making it easier to find the centre and the most frequent score. a 24.5 b 25 c The outlier had only a very small effect. No, as a single outlier will not change the score that occurs the most often. a 82 and 45 b 60–69 c The modal class shows the group that has the most scores. In this case, since the mode is the highest score and with a frequency of just two, it is insignificant compared to the rest of the data. The middle class in a grouped distribution. The median is estimated in grouped distributions using an ogive, which is more useful. This is an estimate as the individual scores are lost once put into a grouped distribution.
c 21 a b c
22
8
228
28 30
b 8
8
c 152
4 2
Median wage = $360
25
60 50 40 30 20 10 0
$400–$449 424.5 $450–$499 474.5
20 a 170–180 cm b 100
0–<50
b
18 24
Cumulative frequency
b 8 d 0.4
Class Cumulative centre Frequency frequency
350– 400
6 6
b $350–$399 c
c 16 17 a 17–20 b 22–28 18 a
Depth (cm)
$300–$349 324.5 $350–$399 374.5
15 cm 5 16 cm 5 17 cm 5 18 cm 5 19 cm 5 cm
16
8 12
1 a 2.29 c 20.17 e 42.44
Maths Quest Preliminary Mathematics General
Standard deviation b 2.19 d 3.07
26.94 b 2.14 57.51 d 0.26 96.04 Sample Population Population Sample Population 616.6 b Sample 270.97 1.44 b Population 0.48 million x = 4.9, σn = 1.0 x = 48.2, σn = 1.2 x = 78.3, σn = 2.3 x = 17.45, σn = 3.69 x = 56.02, σn = 14.26 x = 14.95, σn = 7.49 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 C x = $1825, σn − 1 = 797 θn = 0.51 θn = 15.10 θn = 2.51 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. a Group A = 170 group B = 170 b No, there appears to be greater spread in group B. c Group B d Group A = 5.35 group B = 32.5 a 5 b 1.9 c 4 d 2.2 a x = 36, σn = 20.4
2 a c e 3 a b c d e 4 a c 5 a c 6 a b c 7 a b c 8 a
5
15
8 4
14
14
$200–$249 224.5 $250–$299 274.5
7 12 17 22 27 32 No. of sick days
2
10 5 11 110 17 No mode 2.6 4 42, 44 17 Cumulative frequency
13
c a c e a c e a c a b
45 40 35 30 25 20 15 10 5 0
Class Cumulative centre Frequency frequency
Class
Cumulative frequency
Cumulative frequency
b 10–14 12 a 2, 7, 12, 17, 22, 27, 32 b
19 a
17
18 19
b
Standard Standard deviation deviation Extra Difference after score increase or score from mean added decrease 8
28
21.1
Increase
30
6
19.4
Decrease
90
54
25.7
Increase
50
14
19.8
Decrease
20 a The mean will increase since the added
score is greater than the mean.
b The standard deviation will decrease as
the difference between the mean and the score is less than the standard deviation.
21 B 22 The standard deviation will increase if the
difference between the mean and the score is greater than the standard deviation and decrease if this difference is less than the standard deviation.
best summary statistics
1 a $425 000 b $370 000 c $350 000 d The median, as the mean is inflated
3 a b c 4 a b c d 5 a b
Class
Class centre
Frequency
10–19
14.5
3
20–29
24.5
4
30–39
34.5
6
40–49
44.5
7
c 40–49 d As the mode is the only score that
b c d
7 a b c d
Cumulative frequency
6 a
occurs more than once. The modal class tells us where most scores are gathered. A single score far removed from all others Mean = 49.9, median = 54, mode = 56 Mean = 54.4, median = 54, mode = 56 The removal of the outlier increases the mean and makes it a more typical representation of the data set. 23 550 20 000–30 000 10 000–20 000
c
11 a b
12 a b c d e
13 a
b
14 a
b c
200
15 a
150 100
Class
50 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 f Median
8B 9 a
Class
b
6.8 0–4 0–4 Jim x = 33.3, sn = 5.06, Lewis x = 26.7, sn = 5.21 Lewis’s sample has a much lower mean and both data sets have similar standard deviations. This could be because Lewis and Jim took their samples from different machines. Check other possible answers with your teacher. Carmen x = 32.5, sn = 15.1, Jade x = 24.8, sn = 7.7 By chance it would appear that Jade has included an outlier in her sample. This may be because in this hour the shop was running some specials or a promotion. Player A: 34.3 Player B: 41.8 Player B Player A: 32.5 Player B: 0 Player A Player A would be more useful as he is much more consistent. The mean includes any outliers and so is affected by very large or very small data values. 65 out of 80 earn less than the mean and another 10 only slightly more than the mean. Hence it is misleading. The median and the mode would be useful in your submission as they are not distorted by the larger salaries as the mean is. Group A — mean = 170, median = 170, mode = 170 Group B — mean = 170, median = 170, mode = 170 Group B has a greater spread of scores Group A — range = 20, IQR = 0, std dev. = 5.77 Group B — range = 120, IQR = 20, std dev. = 35.12
Class Cumulative centre Frequency frequency
0– 4
2
16
16
5–9
7
6
22
10–14
12
4
26
15–19
17
2
28
20–24
22
1
29
25–29
27
1
30
Class Cumulative centre Frequency frequency
30–39
34.5
4
4
40–49
44.5
9
13
50–59
54.5
9
22
60–69
64.5
3
25
70–79
74.5
4
29
80–89
84.5
1
30
b
Cumulative frequency
2 a d
by one large score and the mode is the lowest price. 7.1 b 7 c 7 The mode, as it is the size that sells the most. Mean = 15.5, median = 15.5, mode = 15, 18 Range = 10, IQR = 4 There are outliers at both ends of the distribution. Mean = 27.6, median = 29, mode = 29, 30 Yes Range = 26, IQR = 11.5 7.67 10
b c d 10 a
c Median = 53, lower quartile = 44, d e f g
upper quartile = 60 16 Mean = 53.5, std dev. = 13.48 Mean = 54.07, std dev. = 12.50 When scores are put into a table the individual scores are lost, affecting both the mean and standard deviation.
16 A 17 B 18 a There are two modes; that is, two scores
that occur equally most often.
b The mode is about what score occurs
19 a b c d
20 a b c d
21 a b c
22 a b 23 a b c 24 a b
most. A mode of 18 is meaningless as this score does not occur most often. 44.05 27.5 24 and 75 No measure gives a typical score. Patients of all different ages come to the hospital for all different reasons, so there is no typical age. $38 000 $49 444 $46 000 i The mode, as the lowest measure and the amount that more people get paid, makes a stronger case for a pay rise. ii The mean, as the highest measure, makes the case for a pay rise weaker. Mean = 139.6, median = 139.5, mode = 141 Mode as it is the only measure above 140. No, the mean is below 140, therefore half the boxes have fewer than 140 matches. The median was calculated by taking the average of the two middle scores. 12 An outlier makes the mean larger, especially in a small data set. Median Answers will vary, example would be the most frequently sold clothing size. A single outlier can change the range greatly. It has only a very small effect.
Chapter revieW MUltiple ChoiCe
1 2 3 4 5
A A D B B
Short anSWer 30 25 20 15 10 5 0
34 .5 44 .5 54 .5 64 .5 74 .5 84 .5
exercise 6d
Class centre
1 a b c d 2 a b
5.2 64.875 7.7 35.8 7.025 9.46
Chapter 6 • Summary statistics
229
4 5 6
c
f×x
Class
Class centre
Frequency
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
37– 40
38.5
29
1116.5
41– 44
42.5
25
1062.5
45– 48
46.5
19
883.5
49–52
50.5
10
505.0
∑ f = 143
∑ f × x = 5461.5
Mean = 38.2 a 31.1 a 29.9 a 27 d 5.5 a 2
b b b e b
7 8 a
23.2 26.4 6 128 56
180 160 140 120 100 80 60 40 20 0
d 58
.5 .5 .5 .5 .5 .5 .5 34 44 54 64 74 84 94 Score
a 2 a 2
b 23, 27 b 15, 18
c No mode
46– 49 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 15 a 27.8 b 24.5 c 28 d Median 16 Check with your teacher. extended reSponSe
c 0.445 c 18.6 c 3.2
1 a
c 68.5
Class
Class centre
Frequency
Cumulative frequency
30–39
34.5
18
18
40– 49
44.5
34
52
50–59
54.5
39
91
60–69
64.5
45
136
70–79
74.5
29
165
80–89
84.5
10
175
90–99
94.5
5
180
b 50–59
230
9 10 11 12 13 14
Cumulative frequency
3
Maths Quest Preliminary Mathematics General
Income
Class centre
Frequency
Cumulative 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
b d f 2 a b c d e
$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.
ChapTer 7
Measurement ChapTer ConTenTS 7a 7B 7C 7d 7e
Measurements as approximations Units of measurement and scientific notation Ratios Rates Percentage change
7a
measurements as approximations
How far is it from your house to school? If you live close to school you may give your answer in metres, but if you are further away you would probably answer in kilometres. In either case your answer would not be exact. In fact no measurement is exact. Measurements can only be taken to the degree of accuracy that the instruments used allow. All measurements are approximations. No measuring instrument is perfect and different people can sometimes obtain a slightly 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 may repeat the same measurement a number of times and average the result. Worked example 1
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
WriTe
1
Find the total of the 8 readings.
Total = 169 + 169 + 168 + 170 + 169 + 169 + 168 + 168 = 1350
2
Divide the total by 8 to find the average.
Average = 1350 ÷ 8 = 168.75 cm
As no measurement can ever be exact each measurement taken will be given to some degree of accuracy.
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. ChapTer 7 • Measurement
231
• 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 = π r 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 correct to 3 significant figures • 1024 km is correct to 4 significant figures • 78.54 cm2 is also correct to 4 significant figures. Significant figures are the number of nonzero 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 lightyears 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 lightyear is the distance that light will travel in 1 year. Therefore: 1 lightyear = 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 nonzero digits and round off using the same rules as for decimal places. We then fill out the remaining places with zeros. Worked example 2
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
a 1 Rounding off to 2 significant figures, so we 2
b 1 2
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.
WriTe
a
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. 232
Maths Quest Preliminary Mathematics General
Worked example 3
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 1 Rounding off to 3 significant figures, so we look at the 2
b 1 2
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.
exercise 7a
a
0.005 25 b
0.015
measurements as approximations
1 We1 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. 2 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. 3 At a swimming carnival there are 4 timekeepers to record the winning time. In the 50 m freestyle they
take the time of the winner to be 36.4 secs, 36.25 secs, 36.3 secs and 36.15 secs. What time should be allocated to the winner of the race? 4 In an international diving competition there are 7 judges. Each judge awards the competitor a score out
of 10. The scores given to a particular diver were 6.8, 6.9, 6.7, 6.6, 7.9, 6.7, 6.2. a What is the average score of the 7 judges? (Answer correct to 1 decimal place.) b What is most noticeable about this set of scores? c To eliminate bias the rules of diving state both the highest and lowest scores are to be disregarded and the remaining five scores should be averaged. What is the final score given to this diver? (Answer correct to 1 decimal place.) 5 We2 Round each of the following off to the number of significant figures indicated. a 24 587 258 d 4587 (1)
(2)
b 236 500 258 (1) e 654 200 (1)
c 8 782 568 (3) f 287.35 (3)
6 We3 Round each of the following off to the number of significant figures indicated. a 0.032 579 81 (2) d 6.256 677 158 (4)
b 0.003 658 (1) e 68.254 (3)
7 Write the distance 146 565 992 km correct to: a 1 significant figure c 3 significant figures
c 0.001 498 758 (3) f 0.000 201 47 (1)
b 2 significant figures d 5 significant figures.
8 mC When rounded to 2 significant figures, 0.035 81 is equal to: a 0.03
B 0.04
C 0.035
d 0.036
9 mC 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
10 mC 45.5698 = 45.57 when it is rounded to which degree of accuracy? a 2 decimal places C Both A and B
B 4 significant figures d Neither A nor B
Further development 11 By first rounding each number correct to one significant figure, estimate the value of each of the
following calculations. a 183 + 58 d 1010 ÷ 98
b 78 × 11 e 17 × 19
c 632 + 169 f 476 ÷ 11 + 52 ChapTer 7 • Measurement
233
g (51 + 68) × 12 j 42 × 8 + 18 × 5
h 68 + 19 × 9 k 176 ÷ 18 + 689
i l
5 × (78 − 59) 473 × 276
12 mC The number 6.831 is rounded to 6.83. The number has been rounded correct to: a 2 decimal places C both A and B
B three significant figures d neither A or B
13 mC A number rounded to three significant figures is 4.80. The number could have been: a 4.79
B 4.794
C 4.798
d 4.81
14 The length of a block of land is measured in mm by 8 people who obtained the following results.
21 568, 21549, 21598, 21 572, 21 566, 21 581, 21 576, 21 570 a Find the average of the eight measurements and give your answer correct to four significant
figures. b First round each measurement correct to four significant figures and then take the average of the
eight measurements giving your answer correct to four significant figures. c Find the average of the original measurements if we first discard both the highest and the lowest
measurement. Give your answer correct to four significant figures.
Units of measurement and scientific notation 7B
By now you should be familiar with all of the main units of measuring length, mass and capacity.
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) and kilometre (km). 10 millimetres = 1 centimetre 100 centimetres = 1 metre 1000 metres = 1 kilometre The flow chart at right shows how to convert units of measurement.
× 1000
kilometres
÷ 1000
metres ÷ 100
× 100 centimetres × 10
millimetres
÷ 10
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). 1000 grams = 1 kilogram 1000 kilograms = 1 tonne Remembering these conversions can be aided by a flow chart.
× 1000
tonnes
÷ 1000
kilograms × 1000
÷ 1000 grams
Units of capacity Capacity is the measure of liquid volume. The three common kilolitres units used to measure capacity are: millilitres (mL), litres (L) × 1000 ÷ 1000 and kilolitres (kL). litres 1000 millilitres = 1 litre × 1000 ÷ 1000 1000 litres = 1 kilolitre millilitres The flow chart for converting these units is similar to that for mass. With these units of measurements it is important to recognise the use of prefixes. The prefix ‘kilo’ means 1000 times larger: 1 kilometre = 1000 metres, 1 kilogram = 1000 grams and so on. Similarly, the prefix ‘milli’ means 1000 times smaller: 1000 millimetres = 1 metres, 1000 millilitres = 1 litre. 234
Maths Quest Preliminary Mathematics General
Worked example 4
Complete each of the following: a 30 mm = ____ cm c 0.65 t = ___ kg
b 3.2 kg = ___ g d 0.8 L = ___ mL.
Think
WriTe
a Changing millimetres to centimetres: divide by 10.
a 30 mm = 30 ÷ 10 cm
b To change kilograms to grams: multiply by 1000.
b 3.2 kg = 3.2 × 1000 g
c To change tonnes to kilograms: multiply by 1000.
c 0.65 t = 0.65 × 1000 kg
d To change litres to millilitres: multiply by 1000.
d 0.8 L = 0.8 × 1000 mL
= 3 cm
= 3200 g
= 650 kg
= 800 mL
More complex are the units used for area. Area is expressed in square units with the exception of the hectare. Consider the case of a square with a side length of 1 cm which is the same as 10 mm.
1 cm
10 mm
The area can be calculated as 1 cm2 (using a side length of 1 cm) or 100 mm2 (using a side length of 10 mm). As they are in fact the same sized square it can be concluded that 1 cm2 = 100 mm2. Now consider a square of side length 1 m or 100 cm.
1m
100 cm
The area can be calculated as 1 m2 (using a side length of 1 m) or 10 000 cm2 (using a side length of 100 cm). As they are also the same size it can be concluded that 1 m2 = 10 000 cm2. The area conversions are the square of the linear conversions. One hectare is defined to be equivalent to a square of side length 100 m. Therefore 1 ha = 10 000 m2. × 1 000 000
km2
100 m
÷ 10 000
hectares × 10 000
÷ 10 000 m2
× 10 000
÷ 10 000 cm2
× 100
÷ 100 mm2
ChapTer 7 • Measurement
235
Similarly we can find the conversion between units of volume by considering equal sized cubes.
1 cm
10 mm
The volume can be calculated as 1 cm3 (using a side length of 1 cm) or 1000 mm3 (using a side length of 10 mm). As they are in fact the same sized cube it can be concluded that 1 cm3 = 1000 mm3.
1m
100 cm
m3
The volume can be calculated as 1 (using a side length of 1 m) or 1 000 000 cm3 (using a side length of 100 cm). As they are also the same size it can be concluded that 1 m3 = 1 000 000 cm3. ÷ 103 Cubic millimetres (mm3)
÷ 1003 Cubic metres (m3)
Cubic centimetres (cm3)
× 103
÷ 10003
× 1003
Cubic kilometres (km3)
× 10003
Worked example 5
Complete each of the following. a 4 cm2 = ___ mm2 b 54.2 ha = ___ km2 Think
c 8000 mm3 = ___ cm3 WriTe
a 1 cm2 = 100 mm2 so multiply by 100.
a 4 cm2 = 400 mm2
b 1 km2 = 100 ha so divide by 100.
b 54.2 ha = 0.542 km2
c 1 cm3 = 1000 mm3 so divide by 1000.
c 8000 mm3 = 8 cm3
d 1 m3 = 1 000 000 c m3 so multiply by 1 000 000.
d 4.74 m3 = 4 740 000 cm3
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. 236
d 4.74 m3 = ___ cm3
Maths Quest Preliminary Mathematics General
Worked example 6
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 1 Look at the fourth significant figure. 2 3
b 1 2 3
c 1 2
3
a
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. 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. 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.
2.55 × 1010 b
4.58 × 10−6 c
5.00 × 102
Units of measurement and scientific notation exercise 7B
1 We 4 Copy and complete each of the following. a 70 mm = d 9 cm =
cm
b 600 cm = e 12 m =
mm
m cm
c 5000 m = f
9 km =
i
2400 m =
km
2.2 cm =
mm
g 86 mm =
cm
h 9.2 km =
6.4 cm =
mm
k 11.25 m =
cm
l
b 3000 kg =
t
c 7t= f
j
m
km m
2 Copy and complete the following. a 8000 g =
kg
d 5 kg =
g
e 9500 kg =
t
g 5.5 t =
kg
h 4.84 kg =
g
kg
2350 g =
kg
3 Copy and complete the following. a 2000 mL = d 15 L = g 7.9 kL =
L
b 11 000 L =
kL
mL
e 4800 L =
kL
L
h 12.8 L =
mL
c 4 kL = f
L
8650 mL =
L
4 Copy and complete the following. a 240 s =
min
d 5 days =
h
g 2 years = j
36 h =
days days
b 360 min = e 7h=
h min
c 72 h = f
3h=
h 3 years =
months
i
4 years =
k 1 week =
h
l
450 min =
days min
diGiTal doC Spreadsheet doc1459 Converting metric units
weeks 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? ChapTer 7 • Measurement
237
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 We5 Complete each of the following. a 23 400 m2 = _____ km2
b 0.06 cm2 = ____ mm2
c 5 500 000 cm2 = ____ m2
d 0.008 m2 = ____ cm2
e 0.73 km2 = ____ m2
f
g
49 000 m2
= ___ ha
h 5 ha = ____
200 mm2 = ____cm2
km2
9 Complete the following. a 3000 mm3 = ___ cm3
b 4.2 m3 = ___ cm3
c 40 cm3 = ___ mm3
d 6000 cm3 = ___ m3
e 82 500 mm3 = ___ cm3
f
1 m3 = ___ mm3
10 We6 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 11 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. 12 Copy and complete each of the following measurement
conversions. a 2.4 × 106 mm = c 4.6 ×
cm = 7 e 3.45 × 10 t = g 8.9 × 107 L = i
106
7.1 × 103 L =
m mm kg kL
b 9.1 × 108 m = d 4.9 × 105 kg =
8.11 × 106 kg = h 1.09 × 1010 kL = f
km t g L
mL
13 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.) 14 A lightyear 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 lightyear, correct to 3 significant figures. (1 year = 365.26 days)
Further development 15 Builders generally work in millimetre measurements. A hardware store sells plasterboard in the
following sizes. Convert each measure to metres. a 1800 mm × 900 mm b 2400 mm × 900 mm
c 2700 mm × 1200 mm
16 A marathon is an athletic event run over a distance of 42.2 km. At the Olympic Games the event starts
and finishes with a lap of the stadium which has a circumference of 400 m. Calculate in metres the distance that is run outside of the stadium. 17 Mario goes to the timberyard to buy three lengths of timber, 2100 mm, 65 cm and 4.25 m. Calculate the
cost of the timber if it sells for $7.60 per metre. 18 Write each of the following numbers in scientific notation, correct to three significant figures. a 4378 b 56 450 867 c 87 444 000 000 d 230 098 e 0.01225 f 0.000 785 4 g 0.528 874 24 h 27.897 469 8 19 A rectangle measures (3.54 × 106) mm by (4.987 × 109) mm. a Find the perimeter of the rectangle correct to two significant figures. b Find the area of the rectangle correct to two significant figures. 20 A pipeline is built using sections measuring (5.82 × 104) mm. The total length of the pipeline is diGiTal doC WorkSHEET 7.1 doc10326
238
(1.84 × 106) mm. a How many full sections are needed? b What will be the length of the final section needed to complete the pipeline?
Maths Quest Preliminary Mathematics General
7C
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. 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 7
Fully simplify the following ratios. a 24 cm : 32 cm b $3.60 : $2.10 c 3 m : 80 cm Think
a 1 Write down the question, ignoring units. 2
Divide both numbers by the HCF (8).
b 1 Write down the question. 2
Write both money quantities as cents.
3
Divide by the HCF (30).
c 1 Write the question. 2
Change metres to centimetres.
3
Divide by the HCF (20).
WriTe
a 24 : 32
3:4 b $3.60 : $2.10
360 : 210 12 : 7 c 3 m : 80 cm 300 : 80 15 : 4
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 8
Jane and Brooke’s heights are in the ratio 9 : 10. If Jane is 162 cm tall, how tall is Brooke? Think 1
Model the problem by comparing the ratio to the known information.
2
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.
3 4 5
WriTe
Jane : Brooke 9 : 10 162 cm : ??? 9 shares = 162 cm 1 share = 18 cm 10 shares = 180 cm Brooke is 180 cm tall.
Returning to the problem at the start of this section, we can use ratios to divide a quantity into unequal parts. ChapTer 7 • Measurement
239
Worked example 9
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 Write down the whole ratio. 2
a $25 000 : $30 000
Divide by the HCF (5000).
b 1 Sum the parts and make this equal to $33 000. 2
Divide $33 000 by 11 to find 1 share.
3
Zhong receives 5 shares so multiply 1 share by 5.
4
Hasam receives 6 shares so multiply 1 share by 6.
5
Give a written answer.
exercise 7C
5:6 b 11 shares = $33 000
1 share = $3000 5 shares = $15 000 6 shares = $18 000 Zhong’s share is $15 000 and Hasam’s share is $18 000.
ratios
1 We 7 Fully simplify each of the following ratios. a $20 : $4 b 50 c m : 45 c m d 560 km : 240 km e 35 t : 21 t g 15c : 80c h 4 w eeks : 52 w eeks j 1250 mL : 300 mL k 80 c m : 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 2 mC The ratio 3 h : 45 min fully simplified is: a 3 : 45
diGiTal doC Spreadsheet doc1467 ratios
B 1 : 15
3 mC The ratio 80 cm : 2 km fully simplified is: a 40 : 1
B 1 : 2500
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
C 180 : 45
d 4:1
C 2:5
d 1 : 125
4 We 8 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? 240
Maths Quest Preliminary Mathematics General
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 mC 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? B 63%
a 49%
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? 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 Sydney Olympic 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 mC 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 We 9 Sandra and Kevin purchase a Lotto entry. The entry costs $24.80. Sandra puts in $15.50 and
Kevin the rest. 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? 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?
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?
ChapTer 7 • Measurement
241
Further development 21 The ratio of Michael’s height to Neville’s height is 4 : 5 and the ratio of Neville’s height to Raymond’s
height is 4 : 5. Find the ratio of Michael’s height to Raymond’s height. 22 The ratio of Vicky’s earnings to Kerrie’s earnings is 6 : 5. The ratio of Kerrie’s earnings to Sonya’s
earnings is 8 : 5. Given that Vicky earned $720 calculate the amount that Kerrie earned. 23 Fully simplify each of the following ratios. a 0.8 : 0.36 e 65% : 1
1 2
24 To make a sauce,
b 3% : 0.5% f
2 3
1
:14
1 4
c
1 3
:
1
1 4
d 0.9 : 1 4
g 50% : 1.4 :
7 8
h 0.07 : 49%
1 22
of a cup of concentrate is to be mixed with cups of water. This mixture is in a recipe that serves 4 people. 2 a In a restaurant a large mixture is being made that uses of a cup of concentrate. How much water 3 needs to be added? b How many people can be served using the mixture?
25 Two stroke petrol of varying blends is made by mixing petrol and oil in different ratios. a Find the amount of petrol to be added to 600 mL of oil in a 16 : 1 blend. b A second blend is 32 : 1. Find the amount of oil in 6.6 L of this fuel. c One litre of each blend is mixed. Find the ratio of the combination fuel. d How much 32 : 1 blend must be added to one litre of the 16 : 1 blend to make a 20 : 1 blend? 26 Given a : b = 4 : 5 and b : c = 4 : 7 find the ratio c : a.
7d
rates
It is the last day of a test cricket match between Australia and India. To win the match, Australia needs to make 280 runs in 80 overs. How many runs per over do they need to score?
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. 242
Maths Quest Preliminary Mathematics General
Worked example 10
What quantities are being compared in each of the following rates? a 60 km/h b $2.50/kg c 1500 kL/year Think
a A kilometre is a measure of distance. An hour is a
WriTe
a 60 km/h compares distance with time.
measure of time. b $2.50 is an amount of money. Kilograms is a
b $2.50/kg compares money with mass.
measure of mass. c A kilolitre is a measure of capacity. Years are a
c 1500 kL/year compares capacity with time.
measure of 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 11
Simplify each of the following rates fully. a 240 km in 3 hours b $29.96 for 40 litres Think
a 1 Rewrite the original rate.
c 280 runs in 80 overs WriTe
a 240 km in 3 hours
2
Divide the first quantity by the second quantity. (240 ÷ 3)
= 80 km in 1 hour
3
Write the answer as a simplified rate.
= 80 km/h
b 1 Rewrite the original rate.
b $29.96 for 40 litres
2
Divide the first quantity by the second quantity. (29.96 ÷ 40)
= $0.749 for 1 litre
3
Write the answer as a simplified rate.
= 74.9 c/L
c 1 Rewrite the original rate.
c 280 runs in 80 overs
2
Divide the first quantity by the second quantity. (280 ÷ 80)
= 3.5 runs in 1 over
3
Write the answer as a simplified rate.
= 3.5 runs/over
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 12
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 1 case, each ingredient would need to be multiplied by 1 2 . ChapTer 7 • Measurement
243
Worked example 13
Below are the ingredients to make a banana pudding for 6 people. 4 bananas 10 mL lemon juice 20 g castor sugar 250 g coconut Janice is having 9 people to dinner. Modify this recipe to serve 9. Think
2 eggs 20 g apricot jam
WriTe
Multiplication factor = 9 ÷ 6 1 = 12
1
Divide 9 by 6 to calculate the multiplication factor.
2
Multiply each ingredient by 1 2 .
1
bananas = 4 × 1 12 =6 eggs = 2 × 1 1 2 =3 1 coconut = 250 g × 1 2 = 375 g
1
lemon juice = 10 mL × 1 2 = 15 mL castor sugar = 20 g × 1 12 = 30 g 1 apricot jam = 20 g × 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. Worked example 14
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. Give a written answer.
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 15
Convert the speed 20 m/s into km/h. Think 1 2 3
244
WriTe
Convert 20 m/s to m/min by multiplying 20 m/s = 1200 m/min by 60. Convert 1200 m/min to m/h by multiplying = 72 000 m/h by 60. Convert 72 000 m/h to km/h by dividing = 72 km/h by 1000.
Maths Quest Preliminary Mathematics General
exercise 7d
rates
1 We10 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 We11 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 mC 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 12 hours labour. At what rate has she been charged for labour? 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 mC 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 We12 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? ChapTer 7 • Measurement
245
12 A patient in hospital is placed on an intravenous drip. The medication is given to the patient at a rate of 13
14
15
16 17 18 19 20 diGiTal doC Spreadsheet doc1464 Speed converter
15 drips/min. Each drip is 0.25 mL. How much medication will the patient receive in 4 hours? We13 Below are the ingredients for seafood mornay. 600 g of rice 300 g of pink salmon 1 egg 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. The ingredients below make 15 chocolate cookies. 300 g of brown sugar 90 g of oil 30 g of cocoa powder 120 g of selfraising flour 120 g of plain flour 90 g of choc bits 60 g of white chocolate Modify the recipe to make 10 chocolate cookies. We14 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, calculate the maximum number of doses a person can take each day. 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. A river is flowing at a rate of 2 m per second. Given that every metre of river contains 3500 litres of water, calculate the amount of water that flows through the mouth of the river each hour. A swimming pool has a volume of 150 m3. Given that 1 m3 = 1000 L of water, calculate the length of time that it will take to fill a pool using a hose with a flow rate of 40 L/min. We15 Convert a speed of 15 m/s to km/h. Convert each of the following rates. a 90 km/h to m/s b 2.5 m/s to km/h c 8 mL/m to L/km d 8 km/L to L/100 km
21 mC Which of the following is the most economical fuel consumption? a 10 km/L C 12 km/L
B 10 L/100 km d 12 L/100 km
22 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. 23 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?
Further development 24 A car can travel at an economy rate of 14.5 L/100 km of city driving and 9.6 L/100 km of country
driving. a Calculate the petrol used on a journey of 460 km of which 330 km was country driving. b Find the overall economy rate for the journey. 25 A Year 7 science class makes two salt water solutions. The first solution has 20 g of salt per litre and the second solution has 50 g of salt per litre. If 2 litres of the first solution is mixed with 5 litres of the second solution find the number of grams per litre of salt in the mixture. 26 7500 cars can pass along a toll road each hour. The toll for the road is $4.50. a Express the maximum earnings for the road as a rate per hour. b Given that the road operates at capacity for 4 hours per day, 75% capacity for 6 hours per day and 40% capacity for 12 hours per day calculate the daily earnings for the road. 27 The instructions on a container of weed killer say to mix 10 grams of the powder with 5 litres of water. This mixture will cover an area of 30 m2. a Express the mixture as a rate in simplest form. b Express the coverage of the solution as a rate in simplest form. c How much powder should be added to 14 litres of water? d What area will the solution in part c cover? e The powder can be bought in a 125 g jar. Once mixed what area will this cover? 246
Maths Quest Preliminary Mathematics General
28 A certain bathroom cleaner mixes 25 grams of concentrate with 2 litres of water. a Calculate the amount of water to be mixed with 20 grams of the concentrate. b For a particular bathroom mould, the instructions say to triple the concentrate. To treat this mould
how much of the concentrate should be added to 500 mL of water? 29 A farm irrigation system requires 100 megalitres of water per year to water a 24 hectare farm. The
water supply for the farm can provide 400 000 L of water per day. a Express the water requirement as a rate in litres per hectare per day. (Give your answer correct to three significant figures.) b The farmer hopes to expand his farm but can access no further water. Calculate the maximum area of the farm that can be irrigated using the existing system. Give your answer to the nearest hectare.
7e
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 16
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 1
Increase $120 by 20%.
2
Decrease $144 by 20%.
WriTe
120% of $120 = 120 ÷ 100 × $120 = $144 80% of $144 = 80 ÷ 100 × $144 = $115.20
ChapTer 7 • Measurement
247
exercise 7e
percentage change
1 We16 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. 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. 9 Increase $280 by 10% and decrease the result by 5%. 10 Decrease $13.50 by 20% and then increase the result by 20%.
Further development 11 Find the single percentage increase that is equivalent to successive increases of 15% and 5%. 12 An item is increased in price by 10%. Find the percentage discount that is applied to reduce the item
back to its original price.
248
Maths Quest Preliminary Mathematics General
13 Tori claims that if an item is discounted by 5% and then 10% it does not matter what order the
discounts are applied. Is Tori correct? Use calculations to justify your answer. 14 Find the single discount equivalent to successive discounts of 20% and 10%. 15 If there is a percentage increase and decrease of the same percentage, is the overall effect an increase or decrease? Does it matter the order in which the operations are applied. 16 How many price increases of 10% are needed to double the original price of an object?
diGiTal doC WorkSHEET 7.2 doc10327
ChapTer 7 • Measurement
249
Summary measurements as approximations
• All measurements are approximations. • Every measuring instrument is limited in the degree of accuracy that it allows. • To obtain greater accuracy, a measurement should be repeated several times. • Rounding to a given number of significant figures is an alternative to rounding to a given number of decimal places, and is more appropriate for large numbers. • Zeros at the beginning or end of any number are not significant but are used to fill place values. • The method of rounding off for significant figures is the same as rounding for decimal places. Look at the first digit after the cut off point. If this digit is 5 or greater the previous digit is increased by 1, other digits are ignored, and zeros are put in their place if necessary to fill place values.
Units of measurement and scientific notation
• 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 • Measures of area: 1 cm2 = 100 mm2, 1 m2 = 10 000 cm2, 1 ha = 10 000 m2, 1 km2 = 100 ha • Measures of volume: 1 cm3 = 1000 mm3, 1 m3 = 1 000 000 cm3 • Large numbers will be written in scientific notation and rounded to a set number of significant figures. • For such numbers move the decimal point between the first two significant figures and multiply by the corresponding power of 10 to the number of places moved. This power of 10 will be positive for numbers greater than 1 and negative for numbers between 0 and 1. • When writing the result to such a question do not write all digits, round to the given or an appropriate number of decimal places.
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.
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.
250
Maths Quest Preliminary Mathematics General
Chapter review 1 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
m U lTip l e C ho iC e
d unknown
2 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
C 1.5 km/min
d 2500 km/day
3 Which of the following is the greatest speed? a 100 km/h
B 30 m/s
4 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
1 The girth of a tree is measured by five people to be 152 cm, 160 cm, 158 cm, 155 cm, 156 cm. What is
the best measurement that should be given for the tree’s girth?
S ho rT a n S W er
2 A group of surveyors measure the height of a bridge to be 32.6 m, 32.7 m, 32.5 m, 35.6 m, 32.7 m, 32.4 m. a Find the average of their readings. b It is suggested that one surveyor made a large error in her measurement and the measurement should
be discarded. What would be the average reading if the greatest outlying score is disregarded? 3 Write each of the following correct to the number of significant figures stated in the brackets. a 3 458 258 [2] b 49 718 564 [4] c 0.000 097 252 [2] d 1.587 362 458 5 [ 4] 4 Copy and complete each of the following. a 90 mm = d 4800 m = g 9000 g =
11 000 L = m 300 s = j
cm km kg kL min
b 6m=
c 6.7 km =
e 6.9 cm =
f
cm mm h 9500 kg = t k 4550 mL = L
i l
11.25 m = 4.84 kg = 12.8 L =
m cm g mL
5 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? 6 Write each of the following measurements in scientific notation. a 60 000 000 km b 400 000 mm d 643 000 t e 0.8739 t g 0.002 874 mL h 0.005 874 g
c 147 000 000 m f 0.000 574 g
7 Copy and complete each of the following. a 5.2 × 105 cm =
mm
b 9.1 × 107 g =
kg
8 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 3.45 × 107 t =
kg
c 90 m : 150 m f 500 m : 3 km
9 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? 10 Divide $2000 in the ratio 3 : 7. 11 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? 12 Simplify each of the following rates. a $2.50 for 10 L b 80 km in 2 h
c $42 for 5 h
d 3 h for 2 kg
13 A car is travelling at 90 km/h. How far will it travel in 7 hours at this rate? 14 Eric earns $12.45/h. How many hours does he need to work to earn more than $400? 15 If petrol costs $1.27/L, how much petrol can be bought for $40? 16 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. ChapTer 7 • Measurement
251
e x T ended r e SponS e
1 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
diGiTal doC Test Yourself doc10328 Chapter 7
252
164 000 000 mm 2. Give this measurement in scientific notation. 2 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.
Maths Quest Preliminary Mathematics General
ICT activities 7B
Units of measurement and scientific notation
diGiTal doCS • Spreadsheet (doc1459): Converting metric units (page 237) • WorkSHEET 7.1 (doc10326): Apply your knowledge of measurement to questions. (page 238)
7C
ratios
diGiTal doC • Spreadsheet (doc1467): Ratios (page 240)
7d
rates
7e
percentage change
diGiTal doC • WorkSHEET 7.2 (doc10327): Apply your knowledge of measurement to questions. (page 249)
Chapter review diGiTal doC • Test Yourself Chapter 7 (doc 10328): Take the endofchapter test to check your progress. (page 252)
To access eBookPLUS activities, log on to www.jacplus.com.au
diGiTal doC • Spreadsheet (doc1464): Speed converter (page 246)
ChapTer 7 • Measurement
253
Answers CHAPTER 7 meaSUremenT exercise 7a
measurements as approximations 1 750.4 mL 2 79.25 km 3 36.275 secs 4 a 6.8 b One score is much greater than all others. c 6.7 5 a 25 000 000 b 200 000 000 c 8 780 000 d 5000 e 700 000 f 287 6 a 0.033 b 0.004 c 0.001 50 d 6.257 e 68.3 f 0.0002 7 a 100 000 000 km b 150 000 000 km c 147 000 000 km d 146 570 000 km 8 D 9 D 10 C 11 a 260 b 800 c 800 d 10 e 400 f 100 g 1200 h 270 i 100 j 420 k 710 l 150 000 12 C 13 C 14 a 21 570 mm b 21 570 mm c 21 570 mm exercise 7B Units of measurement and scientific notation 1 a 7 cm b 6m c 5 km d 90 mm e 1200 cm f 9000 m g 8.6 cm h 9200 m i 2.4 km j 64 mm k 1125 cm l 22 mm 2 a 8 kg b 3t c 7000 kg d 5000 g e 9.5 t f 2.35 kg g 5500 kg h 4840 g 3 a 2L b 11 kL c 4000 L d 15 000 m L e 4.8 kL f 8.65 L g 7900 L h 12 800 m L 4 a 4 min b 6h c 3 days d 120 h e 420 min f 180 min g 730 or 731 days 1 h 36 months i 208 weeks j 1 days 2 1 k 168 h l 7 h 2
5 5200 mm long, 2400 mm wide and
2500 mm high
6 150 kg 7 1800 L 8 a 0.0234 km2 b 6 mm2 c 550 m2 d 80 cm2 e 730 000 m2 f 2 cm2 g 4.9 ha h 0.05 km2 9 a 3 cm3 b 4 200 000 cm3 c 40 000 mm3 d 0.006 m3 e 82.5 cm3 f 1 000 000 000 mm3 10 a 4.7 × 108 km b 7.4 × 1012 mm 4 c 1.3 × 10 t d 2.4 × 10−3 g e 2.2 × 10−1 L f 3 × 10−3 s 11 a No. It is accurate to the nearest hundred
thousand kilometres. b 4 c 230 000 000 km 12 a 2.4 × 103 m b 9.1 × 105 km
254
4.6 × 107 mm 3.45 × 1010 kg 8.9 × 104 kL 7.1 × 106 mL 2.05 × 10−7 mm 9.44 × 1012 km a 1.8 m × 0.9 m b 2.4 m × 0.9 m c 2.7 m × 1.2 m 41 400 m $53.20 a 4.38 × 103 c 8.74 × 1010 e 1.23 × 10−2 g 5.29 × 10−1 a 1.0 × 1010 a 31 c e g i
13 14 15
16 17 18
19 20
exercise 7C
d 4.9 × 102 t f 8.11 × 109 g h 1.09 × 1013 L
b d f h b b
5.65 × 107 2.30 × 105 7.85 × 10−4 2.79 × 101 1.8 × 1016 35 800 mm
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 13 14 15 16 17 18 19 20 21 22 23
14 hours. 10 25 000 B a 2:5 b 600 a $9.30 b 5:3 c Sandra — $400 000, Kevin — $240 000 Sand 75 kg, cement 50 kg and gravel 25 kg a 4:5:2 b 12 a 4:5 b Vicky pays $80 and Monica pays $100. 16 : 25 $375 a 20 : 9 b 6:1 c 4:3 d 18 : 25 e 13 : 30 f 8 : 15 g 20 : 56 : 35 h 1:7 2
24 a 6 3
b 10
25 a 9.6 L c 24 : 1 26 35 : 16
b 200 mL d 647 mL
exercise 7d
1 a c e g
rates Distance/time Money/time Goals/games Distance/time
Maths Quest Preliminary Mathematics General
b d f h
Money/mass Mass/capacity Temperature/time Mass/area
i Money/capacity j Time/mass k Capacity/distance l Distance/time 2 a c e g i k 3 4 5 6 7 8 9 10 11 12 13
14
15 16 17 18 19 20 21 22 23 24 25 26 27
28 29
90 km/h 5 runs/over $16.25/h 7.5 m/s 7.7 km/L 12.5 kL/day
b d f h j l
$4/kg 5°/h 22.5 m2/kg $44.60/day 2.875 goals/game 77.6 c/L
C $27.40/h 130 beats/min 87 km/h a 196.9 s b 16.3 m/s 7.2 km/L 12 L/100 km C $471.66 900 mL 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 200 g of brown sugar 60 g of oil 20 g of cocoa powder 80 g of selfraising flour 80 g of plain flour 60 g of choc bits 40 g of white chocolate 8 1.9 kg 25 200 000 L 62 hours 30 minutes 54 km/h a 25 m/s b 9 km/h c 8 L/km d 12.5 L/100 km C No, she has enough lawn food for only 40 m2. a 9.6 km/L b 33.75 L a 50.53 L b 11 L / 100 km 41.4 g / L a $33 750 b $448 875 a 2g/L b 6 m2 / L (or 0.167 L /m2) c 28 g d 84 m2 e 375 m2 a 1.6 L b 18.75 g a 11 400 L / ha / day b 35 ha
exercise 7e
percentage change
1 $742.50 2 a 24.9375 km b 547.4 kg c 3 h 21 min 36 s 3 a 1 h 54 min b 2 h 16 min 48 s 4 $181.50 5 225.625 kg 6 Yes, each equals 66 L. 7 $555.75 8 a $250 b 20% 9 $292.60
10 $12.96 11 20.75% 12 9.09% 13 Tori is correct. Check with your teacher to
see if appropriate calculations have been used. 14 28% 15 The overall effect is a decrease and the order does not matter. 16 8
ChapTer reVieW mUlTiple ChoiCe
1A
2 B
3 B
4 C
ShorT anSWer
1 156 cm 2 a 33.1 m 3 a 3 500 000 c 0.000 097
b 32.6 m b 49 720 000 d 1.587
4 a 9 cm d 4.8 km g 9 kg j 11 kL m 5 min 5 112 kg
b e h k
600 cm 69 mm 9.5 t 4.55 L
6 × 107 km 1.47 × 108 m 8.739 × 10−1 t 2.874 × 10−3 mL 7 a 5.2 × 106 mm b 9.1 × 104 kg c 3.45 × 1010 kg 8 a 3:4 c 3:5 e 25 : 4 g 2:9 9 $360 6 a c e g
b d f h
c f i l
6700 m 1125 cm 4840 g 12 800 m L
4 × 105 mm 6.43 × 105 t 5.74 × 10−4 g 5.874 × 10−3 g
10 $600 : $1400 11 a 7 : 3 b Yasmin $175 000, Carrie $75 000 12 a 25 c/L b 40 km/h c $8.40/h d 1.5 h/kg 13 630 km 14 33 h 15 31.5 L 16 $891 exTended reSponSe
b d f h
4:3 8:5 1:6 3 : 10
1 a b c d 2 a b c
20.5 m 50 mm 0.24% 1.64 × 108 55 800 000 km 0.09% 3200 km/h
ChapTer 7 • Measurement
255
Chapter 8
Perimeter, area and volume Chapter ContentS 8a 8B 8C 8D 8e
Perimeter of plane shapes Area of plane shapes Field diagrams Applications of area Volume of prisms
8a
perimeter of plane shapes
The perimeter of a plane figure is the total length of the outside boundary of that figure. For some figures, a formula can be used to calculate the perimeter. In others, the lengths of each side need to be added. Shape
Square
Rectangle
Circle
l r
l
Formula for calculating perimeter
P = 4l, where l is the side length
b
P = 2(l + b), where l is the length and b is the breadth or width
P = Circumference C = 2π r, where r is the radius As 2r = d (diameter), C = πd
All other shapes have their perimeters calculated by adding their side lengths. WorkeD exaMple 1
Find the perimeter of each of the following shapes. (Where appropriate, state your answer correct to 2 decimal places.) a
c
b
12 cm
7 cm
5 cm
5 cm
think
a The shown shape is a parallelogram. Add the side
lengths.
Write
a P = 5 + 12 + 5 + 12
= 34 cm
Chapter 8 • Perimeter, area and volume
257
b 1 The given shape is a circle, whose radius is
given. Write the formula for the circumference that contains the radius.
b C = 2π r
2
Identify the value of r.
r=7
3
Substitute the values of r and π into the formula and use a calculator to evaluate, correct to 2 decimal places.
C=2×π×7 = 43.98 cm
c The shape is a regular hexagon. There are 6 sides
of equal length, so the perimeter is 6 times the side length.
c P=6×5
= 30 cm
Some figures will require a calculation to be made in relation to one or more sides. WorkeD exaMple 2
Find the perimeter of the following shape, correct to 2 decimal places.
7 cm
think 1
2
120° Write
The perimeter consists of two straight sides and an arc. The length of the arc is 120 of the 360 circumference of a circle with radius 7 cm.
Curved length = 120 ×C 360
Add the arc length and the length of the straight sides.
P = 14.66 + 7 + 7 = 28.66 cm
= 120 × 2πr 360 = 120 ×2×π×7 360 = 14.66 cm
This calculation may extend to the use of Pythagoras’ theorem to find one of the side lengths before finding the perimeter. WorkeD exaMple 3
Find the length of side PQ in triangle PQR, correct to 3 significant figures. Find the perimeter of triangle PQR, correct to 3 significant figures. think
258
Write
1
Write Pythagoras’ theorem (a2 + b2 = c2).
a2 = c2 − b2
2
Substitute the lengths of the known sides.
a2 = 162 − 92
3
Evaluate the expression.
4
Find the answer by finding the square root.
a = 175 = 13.2 m
5
Add all sides together to find the perimeter.
P = 16 + 9 + 13.2 = 38.2 m
Maths Quest Preliminary Mathematics General
= 256 − 81 = 175
R 16 m P
a
9m Q
exercise 8a We1
perimeter of plane shapes
Where appropriate in this exercise, give answers correct to 2 decimal places.
1 Find the perimeter of each of the following shapes. a b 10 cm 15 mm
c 9 cm
3 cm
d
e
2 cm
0
5 mm
f 3 cm
8 cm 6 cm
5 cm
4 cm
g
h
2 cm
1 cm
12 cm
2 cm 1 cm 5 cm
2.5 cm 1.5 cm
5 cm
3 cm
4 cm
2 Find the perimeter of each of the following: a a rectangle 20 cm by 12 cm b a regular hexagon of a side length 15 mm c an equilateral triangle of a side length 12 cm d a circle of diameter 25 cm e an isosceles triangle with its base 12 mm and equal sides of length 16 mm each. 3 We2, 3 Find the perimeter of each of the following shapes. a b
c
10 cm 135°
30 cm 21 cm d
e
14 cm
100 m
f
20 m
14 cm
30 cm
g
10 cm
7 cm
h
i
8 cm
28 mm
20º 22 cm 15 cm
60 mm
Chapter 8 • Perimeter, area and volume
259
4 Find the perimeter of the shapes below. a 35 cm
b
14 cm 7cm
25 cm 2cm
8cm
8 cm
c
14 cm
4cm
5 A rectangular paddock 38 m by 27 m is fenced with 5 rows of wire. What is the total length of wire needed? 6 MC Which of the following shapes has the same perimeter to the nearest cm as
the circle shown? a
28 cm
B
30 cm 14 cm
29 cm
C
D
30 cm
8 cm 6 cm
15 cm
10 cm
7 MC Which of the following measurements cannot represent a perimeter?
Note: There may be more than one correct answer. a 70 mm3 B 3m C 4.8 km
D 12 cm2
8 The length of a rectangular pool is twice its breadth (or width). If the perimeter of the pool is 81 m, find
its dimensions. 9 A length of masking tape, 100 cm long, is wrapped once around the edges of a rectangular block. How
long is the block if its width is 15 cm? 10 A circle has a circumference of 81.64 cm. What is its radius? 11 An athletic track has four lanes. The lanes are 1 m apart. If the inside lane is exactly 400 m, with 100 m
straights, find the lengths of the other three lanes. 12 An equilateral triangle has side lengths given as (x + 2) cm. If the perimeter is 42 cm, what is the
value of x? 13 A figure has a perimeter of 64 mm. Azi suggests it is a square of side length 16 mm; Robyn suggests
it is a rhombus of side length 16 mm; Lauren suggests it is a square of side length 8 mm and Simon suggests it is a 20 mm by 12 mm rectangle. Comment on the suggestions made by each person. 14 A bundle of rope is cut into pieces to lay on the ground to mark a soccer field along those lines. The
layout of the pieces is shown below. The marks on each end of the field are identical. What is the total length of rope required? 110 m 16.2 m 5.4 m 9m
260
Maths Quest Preliminary Mathematics General
40 m 18 m
60 m
15 The track for a school crosscountry competition is shown below. Checkpoints are labelled by the
letters A, B, C etc. What is the total distance in this crosscountry competition? A
Scale
E
B
50 m
D
Start Finish
N
F
C
150 m 100 m
0
G
Trees Bushes Plantation Bridge Oval Building Road
H
Fur ther development 16 A timber gate is 2 m wide by 900 mm high. The gate, which consists of a rectangular frame, is to have a
single diagonal brace. Calculate the length of timber needed to build the gate, correct to the nearest metre.
15
20 mm
17 Find the perimeter of each of the following figures. a b 16 mm
10
c
21.6
60 mm 9.2
12
12.4 f
x 610 cm
e
d
28 m
12 m 6m 16 m
24 m
920 cm
18 Calculate the perimeter of each of the following figures. a b
b
12.4
a
430 cm
34
6.2 21.2
20
m m 12.6 mm
30
9. 6
c
c
d
4.6 9.2
3.4 d
10.6 mm 19 Consider the figure at right. Find the length of: a AC b AD c AE d AF
Leave each answer in square root form.
F E A 1m B
D C
Chapter 8 • Perimeter, area and volume
261
area of plane shapes
8B
Area is a measure of the amount of space within a closed shape. As we have seen in chapter 7, area is measured in square units. Most common shapes have a formula we can use to find the area of that shape. Square
A = s2 (s = side length)
Rectangle
A = l × b (l = length, b = breadth)
Triangle
A = 12 bh (b = base, h = height) or using Heron’s formula A = s(s − a)(s − b)(s − c) a + b + c (a, b, c are side lengths) s = semiperimeter s = 2
Parallelogram
A = bh (b = base, h = height)
Rhombus
A = 12 × D × d (D = long diagonal, d = short diagonal)
Trapezium
A = 12 (a + b)h (a, b = parallel sides, h = height)
WorkeD exaMple 4
Find the area of the figures below. a
c
b 19 mm
9.4 cm
62 mm
12.8 cm
8.5 m think
Write
a A = s2
a 1 Write the formula. 2
Substitute the side length.
= 8.52
3
Calculate the area.
= 72.25 m2 bA=l×b
b 1 Write the formula. 2
Substitute the length and the breadth.
= 62 × 19
3
Calculate the area.
= 1178 mm2 c A=
c 1 Write the formula.
1 2
×b×h
2
Substitute the base and the height.
= 12 × 12.8 × 9.4
3
Calculate the area.
= 60.16 cm2
WorkeD exaMple 5
Find the area of each of the following shapes. b 9m
26 14 m
m
18 m
a
c
5.9 cm 7.2 cm 11.4 cm
262
Maths Quest Preliminary Mathematics General
think
Write
aA=b×h
a 1 Write the formula. 2
Substitute the base and height.
= 14 × 9
3
Calculate the area.
= 126 m2
b 1 Write the formula.
bA=
1 2
×D×d
1 2
× 18 × 26
2
Substitute the diagonal lengths.
=
3
Calculate the area.
= 234 m2
c 1 Write the formula.
c A=
1 2
× (a + b) × h
1 2
× (5.9 + 11.4) × 7.2
2
Substitute the sides and height.
=
3
Calculate the area.
= 62.28 cm2
Circles The area of a circle can only be found exactly in terms of π. The area of a circle is found using the formula A = π r 2. To get a numerical answer an approximation needs to be made.
area of composite figures A composite figure is one that is made up of a composition of simple figures. By breaking the figure into these simple shapes we can find the area of the total shape by adding the area of each simple figure. WorkeD exaMple 6
Find the area of each of the following composite shapes. C AB = 8 cm EC = 6 cm FD = 2 cm A
E
F
B
D think
Write
1
ACBD is a quadrilateral but there is no formula to get its area. The figure can be split into two triangles: ABC and ABD.
Area ACBD = Area ABC + Area ABD
2
Write the formula for the area of a triangle containing base and height.
Atriangle = 2 bh
3
Identify the values of b and h for ABC.
ABC: b = AB = 8, h = EC = 6
4
Substitute the values of the pronumerals into the formula and, hence, calculate the area of ABC.
Area of ABC =
Identify the values of b and h for ABD
ABD: b = AB = 8, h = FD = 2
5
1
1 2 1 2
× AB × EC
= ×8×6 = 24 cm2
Chapter 8 • Perimeter, area and volume
263
exercise 8B
area of plane shapes
1 We4a Find the area of each of the squares below. b a DiGital DoC Spreadsheet doc1472 area converter (DiY)
8 cm d
DiGital DoC GC program — Casio doc1473 Mensuration
DiGital DoC GC program — TI doc1474 Mensuration
e
c
3.6 km
29 mm 3.7 cm
f
12.5 cm
2.9 m 2 We4b Find the area of each of the rectangles below. b a 3m 27 mm 9m 38 mm d e 2.2 km 3.85 m 49.7 km
c
f
47 cm 62 cm
6.4 m 34 m 6.3 m 3 We4c Find the area of each of the triangles below. b a
DiGital DoC Spreadsheet doc1475 perimeter and area
c
6.2 cm
12 m
9.4 cm
76 mm
9m
82 mm e
d
6.3 km
f
8.4 km
4.2 m
3.7 m
9.7 m
4 We5a Find the area of each of the parallelograms below. a b 6m 36 cm 12 m
c 7.8 m 9.3 m
17 cm d
e
38 mm
12.8 km
f 80 cm
87 mm
8m
16.9 km
Maths Quest Preliminary Mathematics General
m
4
7m
m
264
cm
c
7.7
15
mm
m 9c
31
5 We5b Find the area of each of the rhombuses below. a b
8.8
m
e
14 .25 km
d
f 20.9 m 10.2 m
3m 39 cm
25
.25
km
6 We5c Find the area of each of the trapeziums below. a b 26 mm 3m 4m
5.6 m 3.2 m
97 mm
7m d
c
58 mm e
1m
8.4 m f
2.8 m 3.65 m
12 m 9m
3.6 cm 9.5 cm
5.4 cm
0.4 m
7 Find the area of each of the following circles correct to 2 decimal places. a b c 7.3 cm
41.8 cm
11.6 cm
8 Look at the figure below. 3 m 12 m
7m
20 m 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. 9 Find the shaded area in each of the following. a 14 m
b
8m
5 cm
10 m
5 cm 9 cm
16 m c
d 8m
3m 5m 9m 10 m
8m 8m
12 m
12 m
Chapter 8 • Perimeter, area and volume
265
10 Find the area of the following shapes. a b
c
30°
70°
17 mm
18 cm
12 cm 345°
11 We6 Find the area of the following composite shapes. a b 40 m 20 cm
c
8 cm 3 cm
28 m
15 cm
d
2 cm
4 cm
e
f 28 cm
3.8 m
2.1m
18 cm
5 cm 12 cm 12 MC The area of the triangle at right is: a B C D
36 cm2 54 cm2 108 cm2 1620 cm2
15 cm 12 cm 9 cm
13 MC 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 14 MC The area of the figure at right is: a B C D
54 m2 165 m2 225 m2 255 m2
17 m 15 m 7m
15 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. 266
Maths Quest Preliminary Mathematics General
15 m
Further development 16 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. Draw a diagram of the garden and the path. Find the area of the garden. What are the dimensions of the rectangle formed by the path? Find the area of concrete needed for the path. 17 Find the area of each of the following composite figures. a b c d a
b
4m 6m 100 mm c
d
42 cm 40 cm
36 cm e
f
240 m 160 m
3m 6m 2.4 m
g
h 80 m 16 cm
120 m 18 Find the area of glass in the window shown.
Give your answer correct to 2 decimal places.
0.9 m
0.8 m 1.8 m
Chapter 8 • Perimeter, area and volume
267
19 A National Park has an area of 500 000 ha. a Calculate the area in square kilometres. b A mining company wants to claim 4% of the National Park for their industry. Calculate the
number of square metres they are claiming.
8C
Field diagrams
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. 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. C
100 interaCtiVitY int2407 Field diagram
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.
268
Maths Quest Preliminary Mathematics General
WorkeD exaMple 7
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
a 1 Draw a 100 mm line. 2
Draw in all offsets at the appropriate points on the traverse line.
3
Join all corners of the field.
4
Write all measurements on your diagram.
Write/DraW
a
A2
C
A3
25 45
B
40
5 50
D A4
A1 30 20
E A5
A b 1 Calculate the area of the four triangles and
the trapezium.
b A1 =
= = A4 =
2
Add the areas together.
1 2 1 2
×b×h × 75 × 45
1687.5 m2
=
1 2 1 2
=
1750 m2
=
1 2 1 2
=
562.5 m2
A2 =
× (a + b) × h × (40 + 30) × 50
× b × h A3 = × 25 × 45 A5 = =
= 1 2 1 2
1 2 1 2
×b×h × 30 × 40
= 600 m2 ×b×h × 20 × 30
= 300 m2
Area = 1687.5 + 562.5 + 600 + 1750 + 300 = 4900 m2
WorkeD exaMple 8
Consider the field book entry at right. a Draw a scale diagram of the field. b Use your diagram to find the perimeter of the field. think 1
Draw a 50 mm line.
2
Draw in all offsets at the appropriate points in the traverse line.
3
Join all corners of the field.
50 15 30
Write/DraW
15 20 0
20 15 4
Write all measurements on your diagram. 20 15
5
Measure each side length.
6
Add the side lengths to find the perimeter.
P = 34 + 25 + 40 + 25 = 124 m
Chapter 8 • Perimeter, area and volume
269
When you draw a scale diagram of the block of land, you can use measurement to find the perimeter. inVeStiGate: 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.
exercise 8C
Field diagrams
C
50
1 We7 At right is a surveyor’s field diagram of a block of land. a Draw a scale diagram of the block of land, using the
B
25 40
scale 1 mm = 1 m.
b Calculate the area of the block of land.
15 20 0
D
A D
80
2 We8 For the field diagram shown at right: a draw a scale diagram of the block of land using the
42 65
C
scale 1 mm = 1 m b calculate the area of the block of land c use measurement to find the perimeter.
40 28
E
35 10 0
B
A
3 Use the field diagrams below to calculate the area of each block of land. b c a C D D 100 70 75 C 23 70 C 30 90 B
40 40 30 15
B
40 30
D
20 20
B
E
50 30
E
30 40 20
F
0
0
0
A
A
A
Further development
C
4 For the diagram at right sketch the surveyor’s field diagram. B
25 m 35 m
5 Consider the field diagram drawn below. D 100
45 m 25 m
C 20 75 B 25 55 20 45 E 0 A
D
20 m A
Is the shortest path from A to D via E or via B and C? DiGital DoC WorkSHEET 8.1 doc10329
270
6 For the field diagram in question 5 determine if the area ABCD or AED is larger. 7 A traverse line is 100 metres long from point A to point B. Alan is to set an offset at C 25 metres at
right angles to the traverse line. Where should Alan set the offset to maximise the area of ABC?
Maths Quest Preliminary Mathematics General
8D
applications of area
In an earlier section we found the area of plane shapes. This needs to be applied to a wide range of practical situations. In many cases this will require drawing a diagram. WorkeD exaMple 9
A clock has a minute hand that is 6 cm long and an hour hand that is 3 cm long. In one full revolution of each band, the minute hand would sweep out a larger circle than the hour hand. What is the difference in the area they cover (to the nearest square centimetre)? think
6 cm
Write
R r 3 cm
1
The area required is the area between two circles. Write down the appropriate formula.
A = outer area − inner area = π R2 − π r2
2
Identify the value of R (radius of larger circle) and the value of r (radius of smaller circle).
R = 6, r = 3
3
Substitute the values of the pronumerals into the formula and evaluate.
A = π × 62 − π × 32 = 113.097 − 28.274 = 84.823 cm2
4
Write an answer sentence with the value rounded to the nearest square centimetre.
The difference in area covered by the two hands is approximately 85 cm2.
exercise 8D
applications of area
1 We9 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 2
3 4 5
3m
12 m
7m
20 m
inner rectangle from the area of the outer rectangle. A sheet of cardboard is 1.6 m by 0.8 m. The following shapes are cut from the cardboard: • a circular piece of radius 12 cm • a rectangular piece 20 cm by 15 cm • 2 triangular pieces of base 30 cm and height 10 cm • a triangular piece of side length 12 cm, 10 cm and 8 cm. What is the area of the remaining piece of cardboard? A rectangular block of land, 12 m by 8 m, is surrounded by a concrete path 0.5 m wide. Find the area of the path. Concrete slabs 1 m by 0.5 m are used to cover a footpath 20 m by 1.5 m. How many slabs are needed? A city council builds a 0.5 m wide concrete path around the garden as shown below. 12 m 5m 8m
3m
Find the cost of the job if the workman charges $40.00 per m2. 6 A yacht race consists of 12 laps around a triangular course. The triangle is equilateral with each side 810 m. Find the total length of the race, in kilometres. Chapter 8 • Perimeter, area and volume
271
7 A box is built to contain 12 tennis balls, as shown in the
figure at right. If the radius of each tennis ball is 4.6 cm, find the perimeter of the smallest possible rectangle that will contain the balls. 8 A block of land is in the shape of a square with an equilateral triangle on top. Each side of the block of land is 50 m. a Draw a diagram of the block of land. b Find the perimeter of the block of land. c Find the area of the block of land. 9 An athletics track consists of a rectangle with two semicircular ends. The dimensions are shown in the diagram below. 70 m 90 m
82 m
The track is to have a synthetic running surface laid. Calculate the area which is to be laid with the running surface, correct to the nearest square metre. 10 A garden is to have a concrete path laid around it. The garden is rectangular in shape and measures 40 m by 25 m. The path around it is to be 1 m wide. a Draw a diagram of the garden and the path. b Calculate the area of the garden. c Calculate the area of the concrete that needs to be laid. d If the cost of laying concrete is $17.50 per m2, calculate the cost of laying the path. 11 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.
Further development 12 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. 13 A familysize pizza is cut into 8 equal slices. If the diameter of the pizza is 33 cm, find (to the nearest square centimetre) the area of the top part of each slice. 14 The collectable plate shown at right is 22 cm in diameter and has a golden ring that is 0.5 cm wide. Find (to 1 decimal place) the area of the golden ring if its outer edge is 1 cm from the edge of the plate. 15 Earlymodel vehicles had a single
windscreenwiper blade to remove water from the windscreen. (The bus at right has two single blades of this type.) Using the dimensions given in the diagram: a what area (to the nearest whole number) did the blade cover? b what percentage (to 1 decimal place) of the windscreen was cleared? 272
Maths Quest Preliminary Mathematics General
120 cm 60 cm
140° 45 cm
1 cm 0.5 cm 22 cm
8e
Volume of prisms
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.
interaCtiVitY int1150 Maximising the volume of a cuboid
inVeStiGate: 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 10
Calculate the volume of the prism at right.
think
Write
A = 63 cm2
5 cm
V=A×h
1
Write the formula.
2
Substitute the area of the base and the height.
= 63 × 5
3
Calculate the volume.
= 315 cm3
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. This becomes the formula used for the volume of a cube. V = s3
s
WorkeD exaMple 11
Find the volume of the cube at right.
think
Write
V = s3
1
Write the formula.
2
Substitute the side length.
= 6.83
3
Calculate the volume.
= 314.432 cm3
6.8 cm
Chapter 8 • Perimeter, area and volume
273
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 b
l
WorkeD exaMple 12
Calculate the volume of the rectangular prism at right. 12 mm think
Write
47 mm
29 mm
V=l×b×h
1
Write the formula.
2
Substitute the length, breadth and height.
= 47 × 29 × 12
3
Calculate the volume.
= 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 the formula V=A×h r V = π r 2h 2 since for a circle A = π r . 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 13
Find the capacity of a cylinder with a radius of 1.3 m and a height of 7.8 m. think
Write
V = π r 2h
1
Write the formula.
2
Substitute the radius and the height.
= π × (1.3)2 × 7.8
3
Calculate the volume in m3.
≈ 41.412 m3
4
Calculate the capacity by multiplying the volume by 1000.
Capacity = 41.412 × 1000 = 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. 274
Maths Quest Preliminary Mathematics General
h
WorkeD exaMple 14
Calculate the volume of the triangular prism at right. 7.9 cm
think 1
5.6 cm
Write
A=
Calculate the area of the triangular base.
=
1 2 1 2
1.2 cm
×b×h × 5.6 × 7.9
= 22.12 cm2 V=A×h
2
Write the volume formula.
3
Substitute the area and the height.
= 22.12 × 1.2
4
Calculate the volume.
= 26.544 cm3
exercise 8e
Volume of prisms
1 We10 Calculate the volume of each of the solids below. b a A = 24 cm2
c
5 cm A = 19 cm2
A = 57 cm2 4 cm
12 cm f
e
d
DiGital DoC Spreadsheet doc1481 Volume
18 mm 9.2 m A = 27.9
A = 15.93 mm2
A = 77.7 cm2
m2
7.7 cm
2 A prism has a base area of 74.5 m2 and a height of 3.1 m. Calculate the volume. 3 We11 Calculate the volume of each of the cubes below. b a
c
5 cm 2.4 m d
13 m f
e 29 mm
8.2 m 5.64 m 4 We12 Find the volume of each of the rectangular prisms below. b a
c 3.7 m
3 cm 6 cm
4 cm
42 mm
6.3 m
4.5 m
13 mm 9 mm
Chapter 8 • Perimeter, area and volume
275
d
e
f
20.5 m
16.5 m
12.5 m
50 mm 3.2 m 4.2 m 9 mm
9 mm
5 We13 Calculate the volume of each of the cylinders below, correct to 1 decimal place. a b c 27 cm 6 cm 13 cm
12 m
12 cm 3m
d
e
15 cm
f
18.5 cm
9 cm
3 mm
25 cm
47 cm
6 We14 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
3 cm c
12 cm
8 cm
5 cm d
3.4 m
12.5 m 2.7 m
3.2 m 7.8 m
1.5 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 10 m 15 m 20 m 5m c
d
3.1 cm
19 m 12 m
10 m 1.7 cm
276
Maths Quest Preliminary Mathematics General
2.4 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 16 cm
8m
10 cm
9m
c
d 15 cm
3m
8 cm 12 cm 20 cm
12 m
6m 6 cm
18 m
12 m
9 MC The shape at right could be described as a: a B C D
cube square prism rectangular prism both B and C
10 MC 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 C 124.12 cm2
B 38.27 cm3 D 124.812 cm3
11 MC 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 semitrailer 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 below.
2m
12 m
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). 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 concrete needed for 15 m 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. Chapter 8 • Perimeter, area and volume
277
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.
Further development 17 Find the volume of each of the following shapes. a Shaded area = 33 cm2
CSA = 116 mm2
10.5 mm
>
>
3.5 cm
b
c
d
>
0.5 m
Shaded area = 42 cm2
55 cm2
>
> 8 cm
18 The figure below shows a concrete paver in the shape of a trapezoidal prism. 60 cm
50 cm 50 cm
1.2 m
Calculate the number of pavers that will have a total volume of 10 m3.
40 cm
19 An apple has a volume of 512 cm3. 160 apples are packed into the box drawn below.
60 cm
38 cm
Calculate the amount of wasted space in the box. 20 Find the volume of the following figure.
1.4 m
60 cm
2m DiGital DoC WorkSHEET 8.2 doc10330
278
1.8 m
21 A cube has a volume of 778.688 cm3. Find the side length. 22 A rectangular prism has a square base, and the height is twice the length of the other two sides. Find the
dimensions of the prism given that it has a volume of 600 cm3.
Maths Quest Preliminary Mathematics General
Summary perimeter of plane shapes
• The perimeter of a plane shape is the total length of the boundary forming the figure. • There is a formula for the perimeter of a square (P = 4l ), a rectangle (P = 2(l + b)) and for the circumference of a circle (C = π d, C = 2π r). All other shapes will have their perimeter found by simply adding all the side lengths. • To find the perimeter of some irregular shapes you will need to calculate a side length using a known formula or Pythagoras’ theorem.
area of plane shapes
• Area formulas that you will need to remember – Square A = s2 – Rectangle A=l×b 1 – Triangle A = 2 bh – Parallelogram A = bh 1 – Rhombus A=2×D×d 1
– Trapezium A = 2 × (a + b) × h – Circle A = π r2 • Composite areas are calculated by breaking the plane shape into smaller regular figures. Field diagrams
• To calculate the area and perimeter of an irregular shape a traverse survey is used to draw a field diagram from which a scale drawing is made. • Areas are found by dividing the irregular area into trapeziums and triangles. • Perimeters can be found using the scale diagram
applications of area
• Practical area questions often require you to first draw a diagram. • A worded question may also require a worded answer
Volume of prisms
• 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 1 Cone V = 3 π r 2h 4
Sphere V = 3π r 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.
Chapter 8 • Perimeter, area and volume
279
Chapter review M U lt ip l e C h oiCe
1 Examine the diagram at right. a The circles cover an area of approximately:
16 cm
a 50 cm2 C 201 cm2
B 101 cm2 D 402 cm2
b The shaded area is approximately: a 55 cm2 C 155 cm2
B 146 cm2 D 206 cm2
2 A rectangular garden measures 5 m by 4 m. A path 1 m wide is to be built around the garden. The area
of the concrete is: a 10 m2 B 20 m2 2 C 22 m D 35 m2 3 A cylindrical tank has a radius of 2.5 m. When 43 mm of rain falls the amount of water collected by the tank is closest to: a 8L B 84 L C 840 L D 8440 L Sh ort anS Wer
1 Find the perimeter of the following shapes. a b 14 mm
13 cm
20
°
20 mm
c 12 cm
7 cm 5 cm 9 cm
e
cm
3.2 m
16 cm
21 .3
24
cm
d
4 cm 18 cm
1.5 m
5.5 m
2.5 m 2 Find the area of each of the figures drawn below. a b
c 6.3 cm
32 mm
17.9 cm 74 mm
5.2 cm
26
d
e
f
35 cm
80 cm
m
mm
m
43
15 cm 3m 70 cm
3 Find the area of each of the triangles drawn below. a 9 cm
b
40 cm 15.5 cm 18.2 cm
280
Maths Quest Preliminary Mathematics General
d
c
40 m 35 m
12.5 cm 12.5 cm
C
95
4 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.
70 25
B
D
36 30 0 A
82
5 Draw a scale diagram to find the area and perimeter of the represented by the field
46 45
diagram at right.
32 37 15 26 0 6 A rectangular block of land 2.8 m × 25 m is surrounded by a concrete path 1 m wide. a Find the area of the path. b Find the cost of concreting at $45 per square metre. 7 Use the formulas to calculate the volume of each of the following cubes, rectangular prisms and
cylinders. a
c
b
3.8 m 11.6 m
29 mm
6.5 cm
f
e
d
4.6 m
18 mm 13 cm
41 cm
32 mm
3 cm 3 cm
8 cm
8 A prism has a base area of 45 cm2 and a height of 13 cm. Calculate the volume. 1 A silo is in the shape of a cylinder with a diameter of 5 m and a height of 20 m. a The silo is to have a square fence built around it. The fence is to come no closer than 1 m at any
ex ten D eD r eS p o n S e
point to the silo. Calculate the perimeter of the fence. b The area between the fence and the silo is to be grassed. Calculate the grassed area correct to the
nearest square metre. c The silo is half full of grain. Given that each cubic metre of grain weighs 165 kg, calculate the
weight of the grain in the silo in tonnes. 2 A cylinder has a radius of ‘r’ and a height of ‘h’. Gary has a cylinder with the same radius but is twice as tall. Gwenda has a cylinder of the same height but double the radius. Determine whether Gary or Gwenda has the cylinder with the greater volume. Justify your answer.
DiGital DoC Test Yourself doc10331 Chapter 8
Chapter 8 • Perimeter, area and volume
281
ICT activities 8B
area of plane shapes
DiGital DoCS • Spreadsheet (doc1472): Area converter (DIY) (page 264) • GC program — Casio (doc1473): Mensuration (page 264) • GC program — TI (doc1474): Mensuration (page 264) • Spreadsheet (doc1475): Perimeter and area (page 264)
8C
Field diagrams
interaCtiVitY • int2407: Field diagram (page 268) DiGital DoC • WorkSHEET 8.1 (doc10329): Apply your knowledge of measurement to questions. (page 270)
8e
Volume of prisms
interaCtiVitY • int1150: Maximising the volume of a cuboid (page 273)
282
Maths Quest Preliminary Mathematics General
DiGital DoCS • Spreadsheet (doc1481): Volume (page 275) • WorkSHEET 8.2 (doc10330): Apply your knowledge of measurement to questions. (page 278)
Chapter review DiGital DoC • Test Yourself Chapter 8 (doc10331): Take the endofchapter test to check your progress. (page 281)
To access eBookPLUS activities, log on to www.jacplus.com.au
Answers CHAPTER 8 periMeter, area anD VolUMe exercise 8a
perimeter of plane shapes
1 a 26 cm b 40 mm c 56.55 cm d 24 cm e 15 cm f 31 cm g 10 cm h 30 cm 2 a 64 cm b 90 mm c 36 cm d 78.54 cm e 44 cm 3 a 77.12 cm b 74.99 cm c 43.56 cm d 95.99 cm e 262.83 m f 71.98 cm g 42.43 cm h 174.55 cm i 163.98 cm 4 a 148.54 cm b 47.14 cm c 54.27 cm 5 650 m 6A 7 A, D 8 27 m × 13.5 m 9 35 cm 10 12.99 cm 11 406.28 m, 412.57 m, 418.85 m 12 12 cm 13 Azi’s, Robyn’s and Simon’s suggestions
14 15 16 17
18
are correct as their figures have a perimeter of 64 mm. Lauren is incorrect as her suggested figure has a perimeter of 32 mm. 658.95 m Approx. 2250 m. Answers may vary. 8m a 50 b 144.33 mm c 60.74 d 60 m e 84.93 m f 2767.92 cm a 124.69 b 65.13 mm c 39.57 d 25.44 m
19 a 2 c 2 exercise 8B
1 a c e 2 a c e 3 a c e 4 a c e 5 a c e 6 a c e 7 a c 8 a c 9 a c
b d
3 5
area of plane shapes 64 cm2 b 841 mm2 12.96 km2 d 8.41 m2 13.69 cm2 f 156.25 cm2 27 m2 b 1026 mm2 2914 cm2 d 109.34 km2 130.9 m2 f 40.32 m2 54 m2 b 29.14 cm2 3116 mm2 d 20.37 m2 26.46 km2 f 6.845 m2 72 m2 b 612 cm2 72.54 m2 d 216.32 km2 3306 mm2 f 6.4 m2 67.5 cm2 b 728.5 mm2 33.88 m2 d 179.91 km2 106.59 m2 f 0.585 m2 20 m2 b 4074 mm2 22.4 m2 d 60 m2 5.84 m2 f 26.82 cm2 167.42 cm2 b 422.73 cm2 1372.30 cm2 140 m2 b 36 m2 104 m2 144 m2 b 68.5 cm2 10 m2 d 80 m2
10 a 37.70 cm2 c 819.96 cm2 11 a 123.29 cm2 c 52 cm2 e 78 cm2 12 B 13 A 14 B 15 a 34.56 m2 16 a 1.5 m
b 870.09 mm2
9 2513 m2 10 a
1m
b 1427.88 m2 d 30.4 m2 f 2015.50 cm2
25 m 40 m b c d 11 a b 12 a
b $960.77
12 m 15 m
1000 m2 134 m2 $2345 34.56 m2 $960.77 1.5 m 12 m
b 180 m2 c 18 m long and 15 m wide d 90 m2 17 a 13 927 mm2 b 30.2832 m2 c 2936.64 cm2 d 2313.88 cm2 2 e 28 346.9 m f 21.6 m2 g 4573.45 m2 h 402.12 cm2 2 18 2.71 m 19 a 5000 km2 b 200 000 000 m2 exercise 8C
1 a B
Field diagrams C 10
b 1125
15 m
b 180 m2 c 18 m long and 15 m wide d 90 m2
13 107 cm2 14 65.2 cm2 15 a 2474 cm2 b 34.4% exercise 8e
m2
25 25 20 15
D
A
2 a b 3 a c 4
Check with your teacher. 3727.5 m2 c 234 m 1925 m2 b 2667.5 m2 3650 m2 C 90
B 35 65
20 25 D 0 A
5 It will be shorter to travel via B and C. 6 The area AED is larger. 7 50 metres from A exercise 8D
applications of area b 36 m2
1 a 140 m2 c 104 m2 2 11 707.92 c m2 3 21 m2 4 60 5 $840 6 29.16 km 7 128.8 cm 8 a
50 m c 3582.5 m2
b 250 m
Volume of prisms
1 a 120 cm3 b 228 cm3 c 228 cm3 d 256.68 m3 e 286.74 mm3 f 598.29 cm3 2 230.95 m3 3 a 125 cm3 b 13.824 m3 c 2197 m3 d 24 389 mm3 e 179.406 144 m 3 f 551.368 m3 4 a 72 cm3 b 4914 mm3 c 104.895 m3 d 56.448 m3 e 4050 mm3 f 4228.125 m3 5 a 1357.2 cm3 b 339.3 m3 c 29 772.9 c m3 d 3817.0 cm3 e 26 880.3 c m3 f 13.3 cm3 6 a i 12 cm2 ii 60 cm3 b i 24 cm2 ii 288 cm3 c i 4.59 m2 ii 6.885 m3 d i 12.48 m2 ii 156 m3 7 a 187.5 m3 b 875 m3 c 2280 m3 d 6.324 cm3 8 a i 200 cm2 ii 2000 cm3 b i 99 m2 ii 792 m3 c i 204 cm2 ii 1224 cm3 d i 153 m2 ii 1836 m3 9D 10 D 11 D 12 a 504 000 c m3 b 504 L 13 101.25 m3 14 a 150.796 m3 b 150 800 L 15 a 175 m2 b 17.5 m3 c $796.25 16 a 2.8 m3 b 2800 L c 24 429 L d 27.5 cm 17 a 115.5 cm3 b 1218 mm3 c 2750 cm3 d 336 cm3 18 44.45 19 9280 cm3 20 6.12 m3 21 9.2 cm 22 6.694 cm × 6.694 cm × 13.389 cm
Chapter 8 • Perimeter, area and volume
283
Chapter reVieW
4 a
C
MUltiple ChoiCe
1 a C 2 C 3 C
25 25
b A
40 B
Short anSWer
1 a c e 2 a c e 3 a c
284
48 mm 54 cm 13.4 m 27.04 cm2 2368 mm2 787.5 cm2 180 cm2 78.125 cm2
b 16.44 cm d 49.3 cm b d f b d
112.77 cm2 559 mm2 2.4 m2 141.05 cm2 700 m2
36 30 A
b 2897.5 m2 5 Area = 3417.5 m2 6 a 59.6 m2 b $2682 7 a 274.625 cm3 b 24 389 m m3 c 202.768 m3
Maths Quest Preliminary Mathematics General
D
d 984 cm3 e 367.57 cm3 f 57 905.84 m m3
8 585 cm3
extenDeD reSponSe
1 a 28 m b 29 m2 c 32.4 t 2 Gary
V = π × r 2 × 2h = 2πr2h Gwenda V = π × (2r)2 × h = 4πr2h Gwenda’s cylinder is larger.
ChapTer 9
Similar figures and trigonometry ChapTer ConTenTS 9a 9B 9C 9d 9e 9F
Similar figures and scale factors Solving problems using similar figures Calculating trigonometric ratios Finding an unknown side Finding angles Applications of rightangled triangles
9a
Similar f igures 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.
inTeraCTiviTy int2403 Similarity
Worked example 1
In the figure at right, show that !ABC  !XYZ.
X A
B Think
C
Y
Z
WriTe
1
∠BAC and ∠YXZ are equal.
∠BAC = ∠YXZ
2
∠ACB and ∠XZY are equal.
∠ACB = ∠XZY
3
∠ABC and ∠XYZ are equal.
∠ABC = ∠XYZ
4
Make a conclusion.
!ABC  !XYZ (3 pairs of equal angles)
ChapTer 9 • Similar figures and trigonometry
285
Worked example 2
Show that the triangles LMN and PQR are similar.
P L 4 cm
M 3 cm N Think
15 cm
12 cm 5 cm Q
9 cm
R
WriTe
1
Simplify the ratio LM : PQ.
LM : PQ = 4 : 12 = 1:3
2
Simplify the ratio MN : QR.
MN : QR = 3 : 9 = 1:3
3
Simplify the ratio LN : PR.
4
Make a conclusion.
LN : PR = 5 : 15 = 1:3
!LMN  !PQR (3 pairs of sides in equal ratio)
To determine if other figures are similar, we need to examine the ratio of sides. Worked example 3
Determine if the rectangles ABCD and PQRS are similar.
P
Q
B
A 4m D
15 m 10 m
C S
Think
6m
R
WriTe
1
Simplify the ratio of corresponding sides AD and RS.
AD : RS = 4 : 6 = 2:3
2
Simplify the ratio of corresponding sides CD and QR.
CD : QR = 10 : 15 = 2:3
3
Make a conclusion.
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
The two figures at right are similar. a What is the ratio of their sides? b What is the scale factor?
7.5 cm 3 cm
6 cm 15 cm
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Maths Quest Preliminary Mathematics General
Think
WriTe
a 1 The 6 cm side corresponds to the 15 cm side.
Write this as a ratio and simplify.
a 6 : 15 = 2 : 5
2
The 3 cm side corresponds to the 7.5 cm side. Check that this simplifies to the same ratio.
3 : 7.5 = 30 : 75 = 2:5
3
Make a conclusion.
The similar figures are in the ratio 2 : 5.
b 1 The scale factor is written by comparing one
unit on the first figure with the second. 2
Make a conclusion.
1
b 2:5 = 1:2 2 1
The scale factor is 2 2.
A special case of similarity occurs when the scale factor is 1. These shapes are identical and are called congruent figures.
exercise 9a
Similar f igures and scale factors
1 We1 Prove that !ABC  !ZYX.
X
A
B
Y
Z
C
A
2 Prove that !ABC  !EDC.
B C
D
E
3 We 2 Prove that !LMN  !PQR.
P L 5 cm M
4 Prove that !ABC  !FED.
4 cm 6 cm
10 cm
N Q
A
8 cm
D
16 cm 12 cm F
12 cm
B 9 cm C
5 Prove that !LMN  !RST.
12 cm 16 cm E
R
L
M
R
12 cm
N S
T
ChapTer 9 • Similar figures and trigonometry
287
6 Prove that !VWX  !VYZ.
V
W
X
Y
Z
7 We 3 Determine if the rectangles ABCD and
WXYZ are similar.
A 5 cm B 2 cm D C
8 Determine if the rectangles KLMN and
L 3 cm K
PQRS are similar.
9 We 4 The figures at right are similar. a What is the ratio of sides? b What is the scale factor?
A 4 cm E
10 The figures at right are similar. a What is the ratio of sides? b What is the scale factor?
15 cm
W 6 cm
7 cm
Z
Y
C
8 cm
P
M
S
R
P
R Q
6 cm D
16 cm
Q
4 cm
N
B 8 cm
X
T
S
12 cm 40 cm
4 cm
10 cm
6 cm 15 cm 11 In the figure below, !MNO  !MPQ.
Calculate the ratio of sides. 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.
Further development 14 Scale factors can be given as comparative distances or as ratios. Convert each of these scale factors,
given as comparative distances, to ratios. a 1 mm to 1 m b 2 cm to 16 m d 40 cm to 1 m e 20 cm to 10 cm 288
Maths Quest Preliminary Mathematics General
c 4 cm to 25 m f 375 mm to 1 m
15 Write each of the ratios in question 14 as a rule with the real life (RL) as the subject of a formula in
terms of scale length (SL). For example part (a) RL = 1000 × SL.
16 A scale plan for the construction of a television unit is drawn. The scale of the plan is 3 cm to 20 cm.
The height of the unit on the plan is 15 cm. a What will be the real life height of the unit? b The real life width of the unit is to be 125 cm. What will be the scale width of the unit on the plan? 17 In a furniture catalogue you decide to buy a bookcase that is 74 cm high. Below is a diagram of the
bookcase with the measurements as shown on a scale diagram. 1.85 cm 4.35 cm
0.55 cm
a What is the scale used? b Find the length and width of the bookcase. 18 The photo at right is of a 30 metre tall tree. a The tree is 5 cm tall in the photo. Find the
scale factor as a ratio. b The diameter of the tree is 1.2 m. Find the
diameter of the tree in the photo. 19 State whether each of the following statements are
true or false. a All squares are similar. b All rectangles are similar. c All triangles are similar. d All equilateral triangles are similar. e All circles are similar. inveSTigaTe: enlarging a f igure
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 A called the centre of enlargement. 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 Repeat step 2 for the vertices B and C.
C A'
A P B
C B'
C'
ChapTer 9 • Similar figures and trigonometry
289
4 Join the points A′, B′ and C′.
A' A P B
C B'
C'
inveSTigaTe: 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. inveSTigaTe: 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. A 1 Draw the !ABC shown (it is not drawn to scale). Draw !XYZ, 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 ZX 60° 50° and . Are these ratios constant (within the limits B C CA of the accuracy of the constructions)? Does it appear that !XYZ is a true enlargement of !ABC? Repeat the process, drawing !XYZ smaller than !ABC. Is !XYZ similar to !ABC? T 2 Construct the two triangles shown where 2 cm D !TVW is twice the size of !DEF. 1 cm The ratio of their corresponding sides is V 4 cm E 2 cm TV VW WT constant as = = = 2. Measure 1.5 cm DE EF FD 3 cm F their corresponding angles. Are the two triangles similar? W 3 Construct !GHJ and !QRS to the Q measurements shown at right. G 6 cm 3 cm Find the ratio of their corresponding sides (as in part 1) and measure all H 30° J 30° 2 cm R S angles. What do you conclude? 4 cm 4 Draw the rightangled triangles KLM N and NPU to the dimensions given. Again, find the ratio of their corresponding sides (as in part 1) K and measure all angles. What do you 7.5 cm conclude? 5 cm 5 Summarise the results of your investigation. What are the minimum requirements to ensure the similarity P U L M 3 cm 4.5 cm of two triangles?
290
Maths Quest Preliminary Mathematics General
9B
Solving problems using similar f igures
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 1metre ruler casts a shadow 60 cm long. Calculate the height of the tree. h Think
WriTe
1m
1
The triangles are similar because all three angles are equal.
2
Write the shadow lengths as a ratio and simplify.
7.5 m : 60 cm = 750 : 60 = 25 : 2
3
Write a proportion and solve to find the height of the tree.
h = 25 1 2 h = 12.5 m
7.5 m
60 cm
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
Multiply 6.2 cm by the given scale.
3
Give a written answer.
WriTe
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. 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 1
Multiply the measurement by the scale.
2
Change the units from cm to m.
3
Give a written answer.
WriTe
8.5 cm × 150 = 1275 cm = 12.75 m The front of the house is 12.75 m long.
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. ChapTer 9 • Similar figures and trigonometry
291
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
a 1 Measure the length and width of the house on
the plan. 2
Multiply each of these measurements by 100.
3
Write your answer.
b 1 Measure the length and width of the lounge
room on the plan.
WriTe
a Length of house on plan = 12 cm
Width of house on plan = 10 cm
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. b Length of lounge room on plan = 6 cm
Width of lounge room on plan = 6 cm
2
Multiply each of these measurements by 100.
Actual length of lounge room = 6 cm × 100 = 600 cm =6m Actual width of lounge room is also 6 m.
3
Calculate the area of the lounge room.
A = 62 A= 36 m2
4
Write your answer.
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. 292
Maths Quest Preliminary Mathematics General
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
WriTe
a 1 Measure the height on the plan for the lower
side of the house.
a Height on the plan = 3.5 cm
2
Multiply the plan measurement by 100.
Actual height = 3.5 cm × 100 = 350 cm = 3.5 m
3
Write your answer.
The height of the eaves is 3.5 m.
b 1 Measure the angle that the slope of the roof
makes with the horizontal. 2
b Angle to horizontal = 45°
Write your answer.
exercise 9B
The angle of the pitch of the roof is 45°.
Solving problems using similar f igures
1 We 5 A tree casts a shadow 2.5 m long. At the same time
a 1metre ruler casts a shadow 40 cm long, as shown in the figure. Calculate the height of the tree. 2 A building casts a shadow 9.5 m long. At the same time
a 1metre ruler casts a shadow 25 cm long. a Draw a diagram to represent this situation. b Calculate the height of the building.
1m 2.5 m
3 Kerry is 170 cm tall and her shadow measures 50 cm.
40 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
1m
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. 100 m 5 mC A tree casts a 6 m shadow. At the same time a 1metre 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 ChapTer 9 • Similar figures and trigonometry
293
6 We 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 mC 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 mC 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 c m B 12.8 cm
C 320 cm
d 12.8 km
11 We 7 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. 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. 13 We 8 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? 294
Maths Quest Preliminary Mathematics General
b What are the dimensions of the house?
14 A house plan is shown below.
Lounge
Bed 3
Bed 4
Family
Pantry Kitchen Bed 1
Bed 2
Toilet
Bathroom Laundry Scale 1:150
a Calculate the dimensions of the house. b What are the dimensions of the lounge room? 15 We 9 Below is the front elevation of a house, drawn to scale.
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.
Further development 16 Solve each of the following by drawing a pair of similar triangles. a A low bridge casts a shadow that is 1.44 metres long at the opening. A handyman’s truck is
2.5 metres high. To determine if the truck will pass under the bridge the handyman gets out of ChapTer 9 • Similar figures and trigonometry
295
17
18
19
20
>
>
the truck and finds that a 1.5 m rod casts a 90 cm shadow. Will the truck pass under the bridge? Explain your answer. b Steve Hooker is Australia’s pole vault gold medallist from the Beijing Olympics. Steve’s 1.8 m pole casts a 96 cm shadow on the ground at the same time as the bar he is about to vault casts a 2.4 m shadow. How high is the bar on the pole vault? c The fire brigade need to rescue people from atop a building. Their 15 m ladder casts a 20 m shadow while the building casts a 24 m shadow. What length ladder is needed? Andrew is 1.8 m metres tall and plays basketball. He casts a 60 cm shadow on the court at the same time as the basketball pole casts a 1 metre shadow. If the basketball ring is 45 cm below the top of the pole, how high must Andrew reach to slam dunk the basketball? A triangle with sides 18 cm, 36 cm and 48 cm is similar to a triangle that has a measure of 8 cm down the longest side. a What is the scale factor? b What is the perimeter of the smaller triangle? A data projector is to be set up to project onto a screen that is 1.5 m high. A The image is projected from a point behind the lens such that B AB = 10 cm and the image height is 5 cm. What is the maximum Lens distance that the projector can be placed from the screen? Slide A rectangle has dimensions 8 cm by 12 cm. a What is the area of the rectangle? Projector b A similar rectangle is drawn with scale factor 2 : 5. screen What will be the dimensions of the larger rectangle? c Find the area of the larger rectangle. d What is the ratio of the two areas? e What do you notice when you compare your answer to part d to the scale factor?
inveSTigaTe: 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. 21 The figure below is a survey plan of a street. Lot X
30.0
a b c d e f
296
5.0
4
189
.91
607
7.219 13 .21 9
38
23.0
38.
110
630 m2
21.0
21.0
187
27.499
23.0
796 m2 15
.0
What is the area in square metres of Lot 110? What is the length and breadth of Lot 110? What scale has been used to draw the survey plan? Redraw Lot 110 using a 1 : 500 scale. The area of Lot 187 is not shown. Find this area. Before the metric system was introduced, the area of house blocks was measured in perches (1 perch = 25.3 m2). i A block of land with an area of 42 perches is advertised for sale at $160 000. Convert the area to square metres and find the price per square metre. ii One lot is 850 m2 and another is 28 perches. Which is the larger?
Maths Quest Preliminary Mathematics General
g Lot 110 is for sale at $119 700, and Lot 189 is for sale at $159 500. i Which lot represents the best value per square metre? ii What features of a block of land might attract a purchaser even though its dollar value per square
metre may be higher than surrounding blocks? (Comparing the positions of Lots 110 and 189 can assist in your answer, but include as many other features as possible.) 22 The plan at right shows a simple design of
a kitchen. a What is the scale used in this plan? b Find the dimensions of the fridge c Find the length of the side marked x. d Find the area of cupboard space (i.e. the space not taken up by the fridge or cook top).
250 cm Cupboard space
Fridge
x cm Cook top
inveSTigaTe: 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.
9C
Calculating trigonometric ratios
Trigonometry is the study of triangles. Trigonometry allows us to calculate the lengths of sides and size of angles in different types of triangles. In this chapter we will be looking at rightangled triangles. The hypotenuse side is the longest side of a rightangled triangle. In trigonometry, we need to be able to name the two shorter sides Hypotenuse as well. We do this with reference to a given angle, and label them Opposite opposite and adjacent. They are the sides opposite and adjacent θ to the given angle. The diagram shows the sides labelled with Adjacent respect to the angle, θ. inveSTigaTe: looking at the tangent ratio
The tangent ratio is a ratio of sides in similar rightangled triangles, such as those in the diagram. ∠BAC is common to each triangle and is equal to 30°. We are going to look at the 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 ‘Tangent’ from the Maths Quest General Mathematics Preliminary Course ebook. Complete each of the following measurements and calculations.
I
digiTal doC Spreadsheet doc1626 Tangent
G E C
H
F
D
B
A
ChapTer 9 • Similar figures and trigonometry
297
1 a BC = _______ mm
b AB = _______ mm
2 a DE = _______ mm
b AD = _______ mm
3 a FG = _______ mm
b AF = _______ mm
4 a HI = _______ mm
b AH = _______ mm
c BC = _______ AB DE = _______ c AD FG c = _______ AF HI = _______ 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? 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: tan θ =
opposite side adjacent side
In the investigation above 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 b 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 10
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 b and enter 60.
a tan 60° = 1.732
b Enter 15, press * and b, enter 75.
b 15 tan 75° = 55.981
c Enter 8, press / and b, enter 69.
c
8 = 3.071 tan 69°
Method 1 d Press b, enter 49, press
DMS
, enter 32, press
DMS
.
Method 2
298
1
From the MENU select RUN.
2
To set the calculator up to accept degrees, minutes and even seconds, press OPTN , 6 for more options, then 5 for ANGL. The screen should appear as shown.
Maths Quest Preliminary Mathematics General
d tan 49°32′ = 1.172
3
Now to find the trigonometric ratio, enter tan 49 and press 4 for degrees; enter 32 and press 4 for minutes; then press w.
The tangent ratio is used to solve problems involving the opposite side and the adjacent side of a rightangled 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. inveSTigaTe: looking at the sine ratio
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 rightangled triangles used in the tangent investigation on pages 297–8. Complete each of the following measurements and calculations by using your calculator or the spreadsheet ‘Sine’ from the Maths Quest General Mathematics Preliminary Course eBookPLUS. 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. 1 a BC = _______ mm
b AC = _______ mm
c BC = _______ AC
2 a DE = _______ mm
b AE = _______ mm
c
DE = _______ AE
3 a FG = _______ mm
b AG = _______ mm
c
FG = _______ AG
4 a HI = _______ mm
b AI = _______ mm
c
HI = _______ AI
digiTal doC Spreadsheet doc1627 Sine
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). In any rightangled 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 11
Find, correct to 3 decimal places: a sin 57°
b 9 sin 45°
c
18 sin 44°
d 9.6 sin 26°12′.
Think
WriTe/diSplay
a Press s and enter 57.
a sin 57° = 0.839
b Enter 9, press * and s, enter 45.
b 9 sin 45° = 6.364
c Enter 18, press / and s, enter 44.
c
d Enter 9.6, press * and s, enter 26, press
enter 12, press
DMS
DMS
,
18 = 25.912 sin 44°
d 9.6 sin 26°12′ = 4.238
.
ChapTer 9 • Similar figures and trigonometry
299
A third trigonometric ratio is the cosine ratio. This ratio compares the length of the adjacent side and the hypotenuse. inveSTigaTe: looking at the cosine ratio digiTal doC Spreadsheet doc1628 Cosine
Look back to the rightangled triangles used in the tangent investigation on pages 297–8. 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 eBookPLUS. AB 1 a AB = _______ mm b AC = _______ mm c = _______ AC 2 a AD = _______ mm
b AE = _______ mm
c
AD = _______ AE
3 a AF = _______ mm
b AG = _______ mm
c
AF = _______ AG
4 a AH = _______ mm
b AI = _______ mm
c
AH = _______ 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. 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 12
Find, correct to 3 decimal places: a cos 27°
b 6 cos 55°
c
21.3 cos 74°
d
4.5 . cos 82°46′
Think
WriTe/diSplay
a Press c and enter 27.
a cos 27° = 0.891
b Enter 6, press * and c, enter 55.
b 6 cos 55° = 3.441
c Enter 21.3, press / and c, enter 74.
c
21.3 = 77.275 cos 74°
d Enter 4.5, press * and c, enter 82, press
d
4.5 = 35.740 cos 82°46′
DMS
, enter 46, press
DMS
.
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 13
Find θ, correct to the nearest degree, given that sin θ = 0.738. Think
300
WriTe/diSplay
[sin−1] and enter 0.738.
1
Press
2
Round your answer to the nearest degree.
2nd F
Maths Quest Preliminary Mathematics General
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. Worked example 14
Given that tan θ = 1.647, calculate θ to the nearest minute. Think
WriTe/diSplay
Method 1 [tan–1] and enter 1.647.
1
Press
2
Convert your answer to degrees and minutes by pressing DMS .
2nd F
θ = 58°44′
Method 2 1
From the MENU select RUN.
2
As with a scientific calculator, press ! [tan−1] and enter 1.647, then press w.
3
Display the angle options by pressing K, 6 for more choices, and then 5 for ANGL.
4
The function for getting the answer displayed in degrees, minutes and seconds is accessed by pressing 5.
exercise 9C
Calculating trigonometric ratios
1 We10 Calculate the value of each of the following, correct to 3 decimal places. a tan 57°
c
b 9 tan 63°
8.6 tan 12°
d tan 33°19′
2 We11 Calculate the value of each of the following, correct to 3 decimal places. a sin 37°
c
b 9.3 sin 13°
15 sin 72°
d
48 sin 67°40′
3 We12 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°
0.5 tan 20°
i
15 sin 72°
g
4.5 cos 32°
h
ChapTer 9 • Similar figures and trigonometry
301
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′
g
19 tan 67°45′
h
49.6 cos 47°25′
i
0.84 sin 75°5′
6 We13 Find θ, correct to the nearest degree, given that sin θ = 0.167. 7 Find θ, correct to the nearest degree, given that: a sin θ = 0.698 b cos θ = 0.173
c tan θ = 1.517.
8 We14 Find θ, correct to the nearest minute, given that tan θ = 17.169. 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.
Further development 10 Find the value of each of the following trigonometric ratio pairs. Give your answers correct to
4 decimal places. a sin 40°, cos 50° b sin 70°, cos 20° c sin 13°, cos 77° d sin 84°, cos 6° 11 What did you notice about the relationship between sin and cos in question 10? Use this to complete each of the following. a sin 30° = cos ___ b cos 75° = sin ___ c sin 28° = ____ d cos 45° = sin ___ 12 Find: a sin 23° b cos 23° sin 23° c cos 23° d tan 23°. 13 Find: a sin 67°
b cos 67°
c (sin 67°)2 + (cos 67°)2.
14 Use your answer to question 13 to find (sin 34°)2 + (cos 34°)2. Check your answer with your calculator. digiTal doC WorkSHEET 9.1 doc10332
15 Fred tries to solve sin θ = 1.2 on his calculator, however an error statement is returned. a Explain why there is no solution to this question. b What is the only trigonometric ratio that can possibly equal 1.2?
9d
Finding an unknown side
We can use the trigonometric ratios to find the length of one side of a righthyp angled triangle if we know the length of another side and an angle. Consider x opp the triangle at right. 30° In this triangle we are asked to find the length of the opposite side and 14 cm adj 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 our calculator adjacent 14 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 302
Maths Quest Preliminary Mathematics General
Worked example 15
Use the tangent ratio to find the value of h in the triangle at right, correct to 2 decimal places.
h 55°
Think 1
17 m
WriTe
Label the sides of the triangle opp, adj and hyp. hyp
h opp
55° 17 cm adj 2
Write the tangent formula.
3
Substitute for θ (55°) and the adjacent side (17 m).
4
Make h the subject of the equation.
5
Calculate.
tan θ =
opp adj
tan 55° = h 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. 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 16
Find the length of the side marked x, correct to 2 decimal places. 24 m Think 1
WriTe/diSplay
x
50°
Label the sides of the triangle. hyp 24 m
opp x
50° adj 2
x is the opposite side and 24 m is the hypotenuse, therefore use the sin formula.
3
Substitute for θ and the hypotenuse.
sin θ = sin 50° =
opp adj x 24 ChapTer 9 • Similar figures and trigonometry
303
4
Make x the subject of the equation.
5
Calculate.
x = 24 sin 50° = 18.39 m
Method 2
inTeraCTiviTy int2405 SohCahToa
1
From the MENU select EQUA.
2
Press 3 for Solver.
3
Delete any existing equation and enter x sin 50° = by pressing sin 50 ! = X ÷ 24 w. 24 Do not worry about a different value of X in the display at this stage as it is a previously stored value.
4
Press 6 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 cos θ = tan θ = opp sin θ = hyp hyp adj 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 17
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
Draw a diagram and show information.
WriTe
f 15° 100 m
304
2
Label the sides of the triangle opp, adj and hyp.
3
Choose the cosine ratio because we are finding the hypotenuse and have been given the adjacent side.
Maths Quest Preliminary Mathematics General
cos θ =
adj hyp
4
Write the formula.
5
Substitute for θ and the adjacent side.
6
Make f the subject of the equation.
7
Calculate.
100 f f cos 15° = 100 100 f= cos 15° = 103.5 m
8
Give a written answer.
The cable is approximately 103.5 m long.
exercise 9d
cos 15° =
Finding an unknown side
1 Label the sides of each of the following triangles, with respect to the angle marked with the
pronumeral. a
b
c
α
θ
γ
2 We 15 Use the tangent ratio to find the length of the side marked x
(correct to 1 decimal place). x 71° 51 mm 3 Use the sine ratio to find the length of the side marked a
(correct to 2 decimal places).
13 m
a
23° 4 Use the cosine ratio to find the length of the side marked d
(correct to 3 significant figures). 35 cm d
31°
5 We 16 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
c 49°
x
13 cm
12.5 km
48 m
68°
41°
y
z
6 Find the length of the side marked with the pronumeral in each of the following
(correct to 3 significant figures). a
b
x 76°
c
9° m
a
67°
8.5 km
116 mm
2.3 m
ChapTer 9 • Similar figures and trigonometry
305
d
e
d
f
g 20° 15.75 km
44.3 m 11°
16.75 cm
g
83° x m
51'
h
2.34 m
i
q 32'
r 26.8 cm
84.6 km
32'
7 mC Look at the diagram at right and state which of the following is correct. a x = 9.2 sin 69°
B x=
9.2 sin 69°
x
C x = 9.2 cos 69°
d x=
9.2 cos 69°
69° 9.2
8 mC Study the triangle at right and state which of the following
ϕ
is correct. 8
B tan ϕ = 8
15
d cos ϕ = 17
a tan ϕ = 15 C sin ϕ = 17
17
15
8
15 8
9 mC Which of the statements below is not correct? a B C d
The value of tan θ can never be greater than 1. The value of sin θ can never be greater than 1. The value of cos θ can never be greater than 1. tan 45° = 1
10 mC Study the diagram at right and state which of the statements
is correct. a w = 22 cos 36°
22 B w= sin 36°
C w = 22 cos 54°
d w = 22 sin 54°
w
22 mm 36°
11 We 17 A tree casts a 3.6 m shadow when the sun’s angle of elevation is 59°. Calculate the height of
the tree, correct to the nearest metre. 12 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)? 13 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)? 14 A rectangle 13.5 cm wide has a diagonal that makes a 24° angle
Port
60°
A
23 km
with the horizontal. a Draw a diagram of this situation. b Calculate the width of the rectangle, correct to 1 decimal place. 15 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 width of the gate (correct to 4 decimal places). c Use Pythagoras’ Theorem to find the length of the diagonal brace. 306
Maths Quest Preliminary Mathematics General
B
16 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 height of the flagpole (correct to 2 decimal places). 17 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.
Further development 18 Find the length of the side marked ‘c’ in the triangle at right. 5m 37° c
3m 19 In the diagram at right find the size of angle θ (to the nearest degree)
and the side lengths ‘x’ and ‘y’ correct to 1 decimal place. 8m
θ 60°
20° x
y
20 Find the value of the unknown sides in each of the following: a b c 6.5 cm 15 cm 110° 27 m x 20 cm 35° x 65° x 21 A boat is moored in calm water with a depth of 29 m. If the anchor line makes a 28° angle with the
surface of the water, what is the length of the anchor line? 22 A person hopes to swim across a river of width 43 m. When she swims across the current pushes
her downstream to a point such that the swim line makes a 67° angle with the river bank. Calculate: a how far she swam b how far she finished from the planned finishing point. 23 Peter says that whenever finding the hypotenuse of a rightangled triangle, the solution will involve
dividing by the trigonometric ratio. Is Peter correct? Explain your answer. ChapTer 9 • Similar figures and trigonometry
307
9e
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 of the angle marked θ. 10 cm opp 5 cm Using the formula sin θ = we know that in this triangle hyp θ 5 sin θ = 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 18
Find the size of angle θ, correct to the nearest degree, in the triangle at right.
4.3 m
θ Think
6.5 m
WriTe/diSplay
Method 1 1
Label the sides of the triangle and choose the tan ratio.
hyp
4.3 opp
θ 6.5 adj
tan θ = opp adj 2
Substitute for the opposite and adjacent sides in the triangle and simplify.
3
Make θ the subject of the equation.
4
Calculate.
Method 2
308
1
From the MENU select EQUA.
2
Press 3 for Solver.
3
Delete any existing equation and enter tan 4.3 x = 6.5 by pressing tan X! [=] 4.3 / 6.5 w.
Maths Quest Preliminary Mathematics General
= 4.3 6.5 = 0.6615
θ = tan−1(0.6615) = 33° (to the nearest degree)
4
Press 6 for SOLV to solve this equation.
The same methods can be used to solve problems. As with finding sides, we set up the question by drawing a diagram of the situation. Worked example 19
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.
tan θ =
= 10 4 = 2.5
θ = tan−1(2.5)
4
Make θ the subject of the equation.
5
Calculate and give your answer correct to the nearest degree.
6
Give a written answer.
exercise 9e
opp adj
= 68° The ladder makes an angle of 68° with the ground.
Finding angles
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
c 25 mm γ
7m 12 m
θ
162 mm
11 m
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 degree. a
13 m
24 m
θ
b
c
4.6 m
6.5 m
θ
α
9.7 km
5.6 km
ChapTer 9 • Similar figures and trigonometry
309
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 degree. a
b
c 27.8 cm 2.6 m
4.6 m
15 cm
β 9 cm
θ
19.5 cm
4 We18 In the following triangles, you will need to use all three trigonometric ratios. Find the size of
the angle marked θ, correct to the nearest degree.
a
b
c
θ 15 cm
11 cm
θ
7 cm d
14 cm
θ
9 cm
8 cm e
3.6 m
f 196 mm
θ
32 mm
14.9 m
26.8 m
9.2 m
θ 5 In each of the following, find the size of the angle marked θ, correct to the nearest degree. a b c 0.6 m θ 30 m
θ
63 cm
θ 10 cm
19.2 m d
2.5 m
e
3.5 m
f
θ
8.3 m
18.5 m
θ
6.3 m
16.3 m 18.9 m
θ 6 mC 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° 7 mC Consider the triangle drawn at right. θ is closest to: a 41°55′ B 41°56′ C 48°4′ d 48°5′ 8 mC The exact value of sin θ = a 30° C 60°
3 . 2
A 5 cm
10 cm
θ
C
θ 9.3 m
B 12.5 m
The angle θ = B 45° d 90°
kite
9 We19 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 degree.
310
Maths Quest Preliminary Mathematics General
40 m
10 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. 10 nm
1.5 nm 7m
Find the error in the compass reading, correct to the nearest degree. 12 The diagram at right shows a footballer’s shot at goal.
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 degree.
Further development 14 mC The figure below shows a BMX bicycle
ramp. All measurements are shown in metres.
5
3
θ 4
The correct expression for the angle of elevation, θ, of the ramp is: 4 B cos−1 4 a sin−1 5 5 C tan−1 4
5
d cos−1 3
5
15 mC A flagpole that is 2 metres tall casts a shadow that is 0.6 metres long. The angle of the sun to the
ground is: a 70° C 72°
B 71° d 73°
16 A javelin that is 1.95 m long is thrown and sticks 20 cm into the ground. Given that the sun is directly
overhead and that the javelin casts a 90 cm shadow, find the angle that the javelin makes with the ground. 17 A hot air balloon is hovering in strong winds 10 vertical metres above the ground. The balloon is being
held in place by a rope that is 15 m long. What angle does the rope make with the ground? 18 A cable car follows a direct line from a mountain peak (altitude 1250 m) to a ridge (altitude 840 m). If
the horizontal distance between the peak and the ridge is 430 m, calculate the angle through which the cable car descends. 19 A ramp joins two points 1.2 metres apart. One point is 25 cm higher than the other. a Find the length of the ramp. b Find the angle of inclination of the ramp.
ChapTer 9 • Similar figures and trigonometry
311
9F
applications of rightangled triangles
In earlier sections we have seen some examples of practical problems that can be solved using trigonometry. One of the major examples that you will need to understand is with 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 20
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
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.
3
Write the formula.
4
Substitute for θ and the adjacent side.
5
Make h the subject of the equation.
6
Calculate.
7
Give a written answer.
h 40° 50 m
WriTe
h opp
hyp 40° 50 m adj
opp adj h tan 40° = 50 h = 50 tan 40° tan θ =
= 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 21
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
Label the sides opp, adj and hyp.
2
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
10 m
hyp
10 m adj
312
Maths Quest Preliminary Mathematics General
64°
64° 1.6 m
tan θ =
opp adj
3
Write the formula.
4
Substitute for θ and the adjacent side.
5
Make h the subject of the equation.
6
Calculate h.
7
Add the eye height.
20.5 + 1.6 = 22.1
8
Give a written answer.
The height of the building is approximately 22.1 m.
h 10 h = 10 tan 64°
tan 64° =
= 20.5 m
A similar method for finding the solution is used for problems that involve an angle of depression. Worked example 22
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. 2 km 10° h
Think
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.
adj 2 km 10° opp h
hyp
opp adj h tan 10° = 2000 tan θ =
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
Give a written answer.
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 rightangled triangle to represent a situation. Worked example 23
A 5.2 m building casts a 3.6 m shadow. Calculate the angle of elevation of the sun, correct to the nearest degree.
5.2 m
θ 3.6 m
ChapTer 9 • Similar figures and trigonometry
313
Think 1
Label the sides opp, adj and hyp.
2
Choose the tan ratio because we are given the length of the opposite and adjacent sides.
WriTe
opp 5.2 m
hyp
θ 3.6 m adj
opp adj
3
Write the formula.
tan θ =
4
Substitute for opposite and adjacent.
tan θ = 5.2 3.6
5
Make θ the subject of the equation.
6
Calculate.
7
Give a written answer.
θ = tan−1 5.2 3.6 = 55° The angle of elevation of the sun is approximately 55°.
To capture the top of the building in this photo the photographer had to tilt the camera upwards, hence, increase the angle of elevation.
314
Maths Quest Preliminary Mathematics General
exercise 9F
applications of rightangled triangles
1 We 20 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. 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 60° metre.
15° 100 m
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. 4 We 21 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°
x
37° 50 m
1.5 m
5 Richard is flying a kite and sights the kite at an angle of elevation of 65°. 40 m x 65°
The length of the string is 40 m and Richard’s eyes are at a height of 1.8 m. Calculate the height at which the kite is flying, correct to 1 decimal place.
1.8 m 6 We 22 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 70 m of depression of 35°. Calculate the distance of the ship from shore, to the nearest metre. 7 From an aeroplane flying at an altitude of 4000 m, the runway 15° 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? 9 We 23 A 12 m high building casts a shadow 15 m long. Calculate the angle of elevation of the sun, to the nearest degree. 12 m
35°
4000 m
69° 10 m
θ 15 m
10 An aeroplane that is at an altitude of 1500 m is 4000 m from a ship in a horizontal direction, as shown
below. Calculate the angle of depression from the aeroplane to the ship, to the nearest degree. θ
4000 m
1500 m
Further development 11 A lifesaver sits in a tower 2 m above sea level. He sees a swimmer having difficulty at an angle of
depression of 12°. How far is the swimmer from the tower? 12 From the top of a lookout 50 m above the ground, the angle of depression to a campsite is 37°. How far is the camp from the base of the lookout? ChapTer 9 • Similar figures and trigonometry
315
13 Two buildings 50 m and 75 m tall are separated by 70 m. Find the angle of elevation from the top of the digiTal doC WorkSHEET 9.2 doc10333
shorter building to the top of the taller building. 14 Wayne says that the angle of elevation from A to B will be equal to the angle of depression from B to A. Is Wayne correct? Explain your answer. inveSTigaTe: 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.
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 drawing a diagram 6 km 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. By measurement, the height of the tree is approximately 10 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.
h 40° 12 m
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. inveSTigaTe: 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, crosscountry running courses and orienteering events. inveSTigaTe: Using proportional diagrams
Plan a track for a crosscountry 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.
316
Maths Quest Preliminary Mathematics General
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.
Calculating trigonometric ratios
opp adj opp • sin θ = hyp adj • cos θ = hyp • tan θ =
• SOHCAHTOA — this acronym will help you remember trigonometric formulas. Finding an unknown side
• • • • •
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.
Finding angles
• • • • •
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.
applications of rightangled triangles
• 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 rightangled 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.
ChapTer 9 • Similar figures and trigonometry
317
Chapter review m U lT ip l e C h oiCe
1 Which of the triangles below are similar? I II 9 cm 3 cm
6 cm
4 cm
2 cm
4 cm a C
III
I and II II and III
B I and III d I, II and III
2 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 C Statements I and II 3 Look at the triangle at right.
Statement 1. cos θ = Statement 2. tan θ =
9 41 9 40
B Statement II d Neither statement
41 m
9m
40 m
θ
Which of the above statements is true? a 1 only C both 1 and 2
B 2 only d neither statement
4 Which of the following statements is correct?
Sh orT anS Wer
a cos 30° = tan 60°
B cos 30° = sin 60°
C cos 30° = sin 30°
d cos 60° = sin 60°
1 Prove that !MNO  !PQO.
M 2 cm 18 cm
9 cm Q
P 2 ABCD and WXYZ are rectangles. a Prove that the two rectangles are similar. b State the ratio of sides in the two similar figures.
W A 2 cm D
O
N 4 cm
6 cm
X
B C
15 cm
Z 5 cm Y 3 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
4 When a 1metre ruler casts a shadow 75 cm long, a building casts a 15 m shadow. Calculate the height
of the building. 5 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? 318
Maths Quest Preliminary Mathematics General
6 Calculate θ, correct to the nearest degree, given that: a cos θ = 0.5874 b tan θ = 1.23
c sin θ = 0.8.
7 Calculate θ, correct to the nearest minute, given that: a cos θ = 0.199 b tan θ = 0.5
c sin θ = 0.257.
8 Find the length of each side marked with a pronumeral, correct to 1 decimal place. c a b 6.8 m 6 cm 81° 3.9 m 65° 78°
q
d
g
e z 30'
x
138 mm
t
f
38.5 m
k
8'
2.9 m 42' g 63 km 12' m
9 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. 10 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)? 11 Find the size of the angle marked θ in each of the following, giving your answer correct to the nearest degree. a
2.3 m
b
c
43 cm 16 m
4.6 m
θ
θ
116 cm
19 m
θ
12 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. 13 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, correct to the nearest metre.
h 40° 27 m
14 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 1.57 m
50 m
ChapTer 9 • Similar figures and trigonometry
319
e x T ended r e SponS e
digiTal doC Test Yourself doc10334 Chapter 9
320
1 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. 2 A hotair balloon takes off and after 30 minutes of flying reaches an altitude of 2000 m. At that time, the angle of depression to its launch pad is 15°. a Calculate the horizontal distance that the balloon has travelled in that halfhour (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).
Maths Quest Preliminary Mathematics General
4.2 km
B 3.1 km
θ C
15° 2000 m
A
ICT activities 9a
Similar f igures and scale factors
inTeraCTiviTy • int2403: Similarity (page 285)
9C
Calculating trigonometric ratios
digiTal doCS • Spreadsheet (doc1626): Tangent (page 297) • Spreadsheet (doc1627): Sine (page 299) • Spreadsheet (doc1628): Cosine (page 300) • WorkSHEET 9.1 (doc10332): Apply your knowledge to questions. (page 302)
9d
9F
applications of rightangled triangles
digiTal doC • WorkSHEET 9.2 (doc10333): Apply your knowledge of trigonometry to problems. (page 316)
Chapter review • Test Yourself Chapter 9 (doc10334): Take the endofchapter test to check your progress. (page 320)
To access eBookPLUS activities, log on to www.jacplus.com.au
Finding an unknown side
inTeraCTiviTy • int2405: SOHCAHTOA (page 304)
ChapTer 9 • Similar figures and trigonometry
321
Answers CHAPTER 9 Similar FigUreS and TrigonomeTry exercise 9a
Similar figures and scale
factors 1 Corresponding angles equal 2 Corresponding angles equal 3 Corresponding sides in equal ratio 4 Corresponding sides in equal ratio 5 Corresponding angles equal 6 Corresponding angles equal 7 They are similar. 8 They are not similar. 1
9 a 2:3
b 12
10 a 2 : 5
b 22
1
11 2 : 3 12 200 14 a 1 : 1000 c 4 : 2500 = 1 : 625 e 2:1 15 a RL = 1000 × SL
16 17 18 19
2500 SL 4 SL RL = 2 100 cm 1 : 40 1 : 600 True False True
13 2 000 000
b d f b
c RL =
d
e
f
a a a a c e
b b b b d
1 : 800 2:5 3:8 RL = 800 × SL 5 SL RL = 2 8 SL RL = 3 18.75 cm 174 cm and 22 cm 2 mm False True
exercise 9B Solving problems using similar figures 1 6.25 m
2 a
h 1m 9.5 m
25 cm
b 38 m 3 a
h 170 cm 3m
50 cm c 10.2 m
b 1:6 4 10 m 5 B 6 60 km 7 a 20 km b 36 km c 26 km d 51.2 km e 3.6 km f 1.6 km 8 a 4 cm b 9 cm c 4.4 cm d 68 cm e 1.6 cm f 7.5 cm 9 D 10 B 11 20 m 12 4 cm 13 a 20 m × 25 m b 10 m × 15 m 14 a 18 m × 12 m b 6.75 m × 4.5 m
322
15 a 8.5 m b 3.5 m c 40° 16 a The truck will not pass under the bridge
which is 2.4 m high.
b 4.5 m
c 18 metres 2.55 m a 6:1 b 17 cm 2.9 metres a 96 cm2 b 20 cm × 30 cm c 600 cm2 d 4 : 25 e It is the square of the scale factor. 21 a 630 m2 b 30 m, 21 m c Approx. 1 : 1500 d Rectangle is 6 cm by 4.2 cm. e 632 m2 f i 1063 m2, $151/m2 ii 850 m2 is larger g i Lot 110 ii Does it front a main road? Is it low lying? Slope of land, views, aspect. 22 a 1 : 50 b 80 cm × 70 cm c 210 cm d 22 800 c m2 17 18 19 20
exercise 9C
exercise 9d
Finding an unknown side
1 a
hyp
opp
adj
θ
b α adj
hyp opp
adj γ
opp hyp
2 148.1 mm
Maths Quest Preliminary Mathematics General
3 5.08 m
4 30.0 cm
b b e h
55.2 m 2.06 km 5.40 m 42.3 km 8 A 10 C 12 4.2 m
c c f i
9.43 km 18.4 mm 5.39 km 13.0 cm
b 6.0 cm
24° 13.5 cm
b 0.8083 m
15 a 1.4 m 60°
c 1.617 m
16 9.067 m 17 a
b 58 m 15°
60 m
Calculating trigonometric
ratios 1 a 1.540 b 17.663 c 40.460 d 0.657 2 a 0.602 b 2.092 c 15.246 d 51.893 3 a 0.707 b 0.247 c 6.568 d 5.896 4 a 0.5 b 0.9659 c 1 d 548.6 e 64 f 1.301 g 5.306 h 1.374 i 15.77 5 a 0.42 b 1.56 c 0.09 d 5.10 e 2.87 f 0.38 g 7.77 h 73.30 i 0.87 6 10° 7 a 44° b 80° c 57° 8 86°40′ 9 a 42°57′ b 31°21′ c 16°5′ 10 a 0.6428 b 0.9397 c 0.2250 d 0.9945 11 a 60° b 15° c 62° d 45° 12 a 0.3907 b 0.9205 c 0.4245 d 0.4245 13 a 0.9205 b 0.3907 c 1 14 1 15 a For any rightangled triangle, the opposite side must be smaller in length than the hypotenuse, Therefore, the value of sin θ cannot be greater than 1. b Tan
c
5 a 12.1 cm 6 a 5.42 m d 3.20 cm g 0.205 m 7 D 9 A 11 6 m 13 20 km 14 a
c 15.5 m 18 19 20 21 22 23
5.3 m θ = 30°, x = 16.0 m, y = 13.0 m a 24.3 cm b 74.2 m c 4.6 cm 62 m a 46.7 m b 18.3 m Peter is correct. If finding the hypotenuse of the triangle, the ‘unknown side’ will be in the denominator of the formula.
exercise 9e
a 30° a 33° a 53° a 50° d 21° 5 a 40° d 79° 6 A 8 C 10 76° 12 13° 14 B 16 59° 18 44° 19 a 1.23 m 1 2 3 4
exercise 9F
Finding angles b 75° b 45° b 56° b 32° e 81° b 81° e 63° 7 B 9 37° 11 9° 13 5° 15 D 17 42°
c c c c f c f
81° 35° 45° 33° 34° 14° 19°
b 12°
applications of rightangled triangles 1 26.8 m 2 3984 m 3 190 m 4 39.2 m 5 42.1 m 6 100 m 7 15 km 8 Yes, the ladder needs to be only 28 m long. 9 39° 10 21° 11 9.41 m 12 66.35 m 13 20° 14 Wayne is correct. The same sized rightangled triangle will be drawn between the two points.
ChapTer revieW mUlTiple ChoiCe
1C 3B
2 D 4 B
ShorT anSWer
1 Check with your teacher. 2 a Check with your teacher.
20 cm 20 m 22.5 m a 54° a 78°31′ a 37.9 cm d 1.4 m g 5.5 km 9 8.5 m 3 4 5 6 7 8
b 2:5
b b b e
51° 26°34′ 3.8 m 16.8 mm
c c c f
53° 14°54′ 14.6 m 10.6 m
10 11 12 13 14
2.5 km a 57° 39° 23 m 57.5°
b 27°
c 68°
exTended reSponSe
1 a 5.22 km 2 a 7500 m
b 54° b 7°36′
ChapTer 9 • Similar figures and trigonometry
323
ChapTer 10
Probability ChapTer ConTenTs 10a 10b 10C 10d 10e 10F 10g 10h 10i
Multistage events The fundamental counting principle Probability statements Relative frequency Equally likely outcomes The probability formula Writing probabilities as decimals and percentages Range of probabilities Complementary events
10a
multistage events
1st coin
2nd coin
A multistage event is where there is more than one part to the probability Heads experiment. Tree diagrams are used to find the elements in the sample space Heads Tails in a multistage probability experiment. Consider the case of tossing two Heads coins. How many elements are there in the sample space? We draw a tree Tails Tails diagram to develop a system that will list the sample space for us. 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 = {HeadsHeads, HeadsTails, TailsHeads, TailsTails} There are four elements in the sample space; HeadsTails and TailsHeads are distinct elements of the sample space. Worked example 1
A coin is tossed and a die is rolled. List all elements of the sample space. Think 1
Draw the branches for the coin toss.
2
From each branch for the coin toss, draw the branches for the die roll.
WriTe
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, T5, T6}
ChapTer 10 • Probability
325
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. Worked example 2
The numbers 2, 4, 7 and 8 are written on cards and are chosen to form a twodigit number. List the sample space. Think
WriTe
1
Draw the first branch of the tree diagram to show each possible first digit.
2
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.
3
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}
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. When drawing a tree diagram, the tree needs to branch once for every stage of the experiment. When we roll two dice, there are two levels to the tree diagram. If we were to toss three coins, there would be three levels to the diagram, as shown below. 1st coin
2nd coin Heads
Heads Tails Heads Tails Tails
3rd coin Heads Tails Heads Tails Heads Tails Heads Tails
Worked example 3
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? 326
Maths Quest Preliminary Mathematics General
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
b List the sample space by following the paths to
the end of each branch.
c Count the number of elements that contain
3 boys and 1 girl.
exercise 10a
4th child Boy Girl Boy Girl Boy Girl Boy Girl Boy Girl Boy Girl Boy Girl Boy Girl
b S = {BBBB, BBBG, BBGB, BBGG, BGBB,
BGBG, BGGB, BGGG, GBBB, GBBG, GBGB, GBGG, GGBB, GGBG, GGGB, GGGG} c There are four elements of the sample space
which contain 3 boys and 1 girl.
multistage events
1 We1 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 twodigit 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? 5 The five vowels are written on cards and two are selected. a In how many ways can the cards be selected if the same vowel can be used twice? b In how many ways can the cards be selected if the same vowel cannot be used twice? 6 In a game of soccer a win, draw and loss are all equally likely. In three matches of soccer how many
elements does the sample space of outcomes have? 7 We2 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. 8 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. 9 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. 10 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.) ChapTer 10 • Probability
327
11 We3 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? 12 mC Three coins are tossed into the air. The number of elements in the sample space is: a 3
b 6
C 8
d 9
13 mC A twodigit 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
14 mC A twodigit 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
15 A twodigit 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. 16 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 twodigit number, how many elements are in the
sample space? b If three cards are chosen and used to form a threedigit number, how many elements are in the
sample space? c How many fourdigit numbers can be formed using these digits?
Fur ther development 17 A school captain and vicecaptain 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 vicecaptain. 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? 18 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. 19 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.
Total
2
3
4
5
6
7
8
9
10
11
12
No. of elements c How many elements of the sample space have a double number? 20 Vanessa is doing a multiple choice exam. Each question has four options A, B, C and D. Vanessa knows
all but three answers and decides to guess each. a How many elements will the sample space for the three guesses have? b If Vanessa decides not to guess the same letter more than once, how many elements will the sample space have? 21 Theresa is drawing a tree diagram to represent the roll of two standard dice. She does not need to know
the number rolled, only if a six is rolled or not rolled. a Draw the tree diagram to show the outcomes to Theresa’s experiment. b Explain if each outcome in the sample space is equally likely. 22 The numbers 1, 2, 3 and 4 are written on cards. a If two cards are chosen at random and repetition is not allowed, how many fewer ways can they
be selected? b If three cards are chosen at random and repetition is not allowed, how many fewer ways can they
be selected? 23 Explain why a tree diagram is a useful way of displaying the results to a multistage experiment. 328
Maths Quest Preliminary Mathematics General
invesTigaTe: Twostage experiments
1 Toss two coins 100 times. Copy and complete the table below.
Result
No. of times digiTal doC Spreadsheet doc1644 Coin toss lister
2 Heads 1 Head, 1 Tail 2 Tails 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.
Total
2
3
4
5
6
7
8
9
10
11
12
No. of times Percentage
digiTal doC Spreadsheet doc1646 die rolling
Compare your results with your answer to question 19 in the previous exercise.
10b
The fundamental counting principle
A threecourse 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
Icecream 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 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 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 = 24 There are 24 ways in which the threecourse meal can be chosen. This multiplication principle is called the fundamental counting principle. 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.
Entree
Main course Spaghetti
Beef broth
Calamari
Dessert Icecream Banana split Strawberries
Roast chicken
Icecream Banana split Strawberries
Pasta salad
Icecream Banana split Strawberries
Grilled fish
Icecream Banana split Strawberries
Spaghetti
Icecream Banana split Strawberries
Roast chicken
Icecream Banana split Strawberries
Pasta salad
Icecream Banana split Strawberries
Grilled fish
Icecream Banana split Strawberries
ChapTer 10 • Probability
329
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.
Worked example 4
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
WriTe
1
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.
Total outcomes = 20 × 20 × 20 = 8000
2
Give a written answer.
There are 8000 different ways in which the wheels of the poker machine can land.
Worked example 5
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? 330
Maths Quest Preliminary Mathematics General
Think
WriTe
1
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.
Total possible outcomes = 20 × 22 × 18 × 25 = 198 000
2
Give a written answer.
The committee of four people can be chosen in 198 000 different ways.
Sometimes we need to reconsider examples that have some type of restriction placed on the possible selections. Worked example 6
If motor vehicle 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
WriTe
1
There are 3 possible first letters.
2
There are 26 possible second and third letters.
3
There are 8 possible first digits.
4
There are 10 possible second and third digits.
5
Multiply all these possibilities together.
Total number plates = 3 × 26 × 26 × 8 × 10 × 10 = 1 622 400
6
Give a written answer.
There are 1 622 400 possible number plates under this system.
exercise 10b
The fundamental counting principle
1 We4 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? ChapTer 10 • Probability
331
5 Some number plates have two letters followed by 4 numbers. How many of this style of plate are
possible? 6 mC 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 mC 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 8 We5 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? 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 A punter takes a ‘Big 6’ which requires her to select the winner of six races. How many ways can the
Big 6 be selected if there are 15, 12, 7, 8, 18 and 14 runners in the six races? 13 Rhonda takes a mystery holiday. She can go to one of five destinations, fly on one of three airlines and
stay at one of six different hotel chains. How many different mystery holidays are possible? 14 A theme park advertises that every time you ride the Hurricane, the ride will be different. When
the ride begins it can go through two different tunnels after which, they merge before it can go through one of three different tunnels before merging again, and then go though one of two different tunnels. a How many different rides are possible? b Is the claim made by the theme park operators correct? 15 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? 16 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? 17 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? 18 We6 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 than five. How many possible PIN numbers could he try? 19 mC 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 332
Maths Quest Preliminary Mathematics General
C 7000
d 10 000
Further development 20 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
21 22 23
24 25
courses are seasoned with garlic. How many possible choices can she make without choosing a garlic dish? 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? 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? A poker machine with 5 reels has 10 electronic symbols on each reel. a How many combinations are possible on the poker machine? b New software increases the number of symbols on each reel by 20%. How many symbols will now be on each reel? c Connie claims that the number of combinations on the poker machine will increase by 20%. Is Connie correct? Use calculations to justify your answer. A poker machine with 5 reels has 20 symbols on each reel. The machine has the number of symbols on each reel reduced by 25%. By what percentage is the total number of combinations reduced? What is the fundamental counting principle?
10C
probability statements
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 ‘fiftyfifty’ 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. certain 1 almost certain Now let’s consider an impossible situation. In a board game you have one last throw of the die. To win you must roll probable a 7. We know that this cannot be done. We would say that this is impossible. When an event is impossible, the probability of the event is 0. 1 fiftyfifty 2 The chance of any event occurring will often be somewhere between being certain and impossible, and we use a variety of terms to describe where the chance lies in unlikely this range as shown in the figure at right. We use these terms based on our general knowledge of the world, the total possible very unlikely outcomes and how often an event occurs. impossible 0 Worked example 7
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
a There is an equal chance of the coin landing
WriTe
a The chance of tossing a head is fiftyfifty.
Heads and Tails. b There is only one chance in six of rolling a 6.
b It is unlikely that you will roll a 6. ChapTer 10 • Probability
333
c There is only a very small chance of winning
c It is very unlikely that you will win the lottery.
the lottery. d There are more spot cards than picture cards in a deck.
d It is probable that you will select a spot card.
You will need to use these terms to describe events that are more likely to occur than others. Worked example 8
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
There are 5 chances that the baby will be born on a weekday and 2 chances that it will be born on a weekend.
WriTe
It is more likely that Mrs Graham’s baby will be born on a weekday.
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?’ 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 9
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
Brisbane Broncos have won more games than South Sydney.
WriTe
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 10
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
It has rained only 12 times on the last 80 Christmas Days. This is much less than half of all Christmas Days.
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Maths Quest Preliminary Mathematics General
WriTe
It is unlikely that it will rain on Christmas Day this year.
We need to look at many probability statements and think critically about what is being said. Is the source of the statement bias? Has emotive language been used to try to influence opinion? These things and others mean that probability statements need to be looked at critically. The project link will look at examples of critical analysis of probability statements.
exercise 10C
invesTigaTion doc10348 Chance in the media
probability statements
1 We7 Describe the chance of each of the following events occurring, using an appropriate probability
term. a Selecting a ball with a doubledigit 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
(fiftyfifty), 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 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. 4 We8 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 b A coin landing Heads c Rolling a total of 3 with two dice d Winning a raffle made up of 50 tickets e Winning a prize in the Lotto draw
Wet weather Christmas Day A coin landing Tails Rolling a total of 7 with two dice Winning a raffle made up of 200 tickets Not winning a prize in the Lotto draw
6 We9 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? 7 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 semifinal Ben Christie — best time 10.06 s and 3rd in his semifinal Give an explanation for your answer. 8 mC 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 9 mC Which of the following events is the most likely to occur? a b C d
Selecting the first number drawn from a barrel containing 20 numbered marbles Selecting a diamond from a standard deck of cards Winning the lottery with one ticket out of 150 000 Drawing the inside lane in the Olympic 100metre final with eight runners ChapTer 10 • Probability
335
10 mC 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 fiftyfifty 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. 11 We10 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. Describe the chance of the light globe burning for more than 1500 hours. 12 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. 13 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.
Fur ther development 14 Give an example of an event that is: a almost but not quite certain b almost but not quite impossible. 15 Explain why it is usually sufficient to describe the probability of an event rather than assign an exact
numerical value. 16 Explain why the probability of rain tomorrow cannot be exactly defined. 17 Explain why the probability of winning the lottery can be exactly defined. 18 List as many ‘chance’ words as you can think of. 19 Explain why probabilities range between 0 and 1. invesTigaTe: Common descriptions of chance
digiTal doC WorkSHEET 10.1 doc10335
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?
10d
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 number of trials In this formula, a trial is the number of times the probability experiment has been conducted. Relative frequency =
336
Maths Quest Preliminary Mathematics General
Worked example 11
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
WriTe
number of times an event has occurred number of trials
1
Write the formula.
Relative frequency =
2
Substitute the number of fine Christmas Days (32) and the number of trials (40).
Relative frequency = 32
3
Calculate the relative frequency as a decimal.
40
= 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 12
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
WriTe
number of times an event has occurred number of trials
1
Write the formula.
Relative frequency =
2
Substitute 144 (the number of times the event occurred) and 150 (number of trials).
Relative frequency = 144
3
Calculate the relative frequency.
= 0.96
4
Convert the relative frequency to a percentage.
= 96%
150
Relative frequencies can be used to solve many practical problems. Worked example 13
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
≥1500
35
Determine if the batch is unsatisfactory. ChapTer 10 • Probability
337
Think
WriTe
1
Count the number of light globes that burn for less than 1000 hours.
2
Write the formula.
3
Substitute 31 (number of times the event occurs) and 200 (number of trials).
4
Calculate the relative frequency.
= 0.155
5
Convert the relative frequency to a percentage.
= 15.5%
6
Make a conclusion about the quality of the batch of light globes.
exercise 10d
31 light globes burn for less than 1000 hours.
Relative frequency =
number of times an event has occurred number of trials
Relative frequency =
31 200
More than 15% of the light globes burn for less than 1000 hours and so the batch is unsatisfactory.
relative frequency
1 We11 At the opening of the ski season, there has been sufficient snow for skiing for 37 out of the past
50 years. Calculate the relative frequency of sufficient snow at the beginning of the ski season. 2 A biased coin has been tossed 100 times with the result of 79 Heads. Calculate the relative frequency of
the coin landing Heads. 3 Of eight Maths tests done by a class during a year, Peter has topped the class three times. Calculate the
relative frequency of Peter topping the class. 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. 5 We12 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 mC Four surveys were conducted
and the following results were obtained. Which result has the highest relative frequency? a Of 1500 Pplate 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. 338
Maths Quest Preliminary Mathematics General
9 mC A study of cricket players found that of 150 players, 36 batted left handed. What is the relative
frequency of lefthanded batsmen? a 0.24 b 0.36
C 0.64
d 0.76
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 oneday 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 We13 A car maker recorded the first time that its cars
came in for mechanical repairs. The results are in the table below. Time taken 0–<3 months
No. of cars 5
3–<6 months
12
6–<12 months
37
1–<2 years
49
2–<3 years
62
≥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. 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 0–<20 000
No. of shock absorbers 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 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.
D
ChapTer 10 • Probability
339
Fur ther development 17 Roger is a tennis player. He has
won 165 of his last 200 matches. a What is the relative frequency of Roger: i winning? ii losing? b After Roger has some time off tennis he comes back and wins only 6 of his next 20 matches. For this period of time what is the relative frequency of Roger: i winning? ii losing? c For all 220 matches, what is Roger’s overall relative frequency of: i winning? ii losing? 18 mC A biased coin is tossed a number of times. The relative frequency of tails is found to be 0.2. The coin is tossed again several times. The relative frequency of tails this time is now found to be 0.3. What is the relative frequency of tails, when the results of the two trials are combined? a 0.25 b 0.5 C 0.6 d cannot be determined 19 In 60 rolls of a standard die, the relative frequency of a six is 0.2. In another 100 rolls of the die the
relative frequency of a six is 0.3. Find the overall relative frequency of a six. 20 During a oneyear period, an insurance company finds that in a city of 125 000 homes, there has been
1275 home robberies. a What is the relative frequency of robberies in the city? b The average insurance claim after a robbery is $14 250. When setting home insurance premiums the insurance company calculates the premiums based on this average claim. Allowing for a 25% profit margin, calculate the amount of each premium in the city. 21 In NSW there are approximately 250 000 ‘P’ plate drivers. In total there are approximately
3 000 000 drivers. In one year there was approximately 5 000 traffic accidents, of which 800 involved ‘P’ plate drivers. a What is the relative frequency of ‘P’ plate drivers? b What was the overall relative frequency of drivers being involved in an accident? c What was the relative frequency of ‘P’ plate drivers being involved in accidents? d What was the relative frequency of an accident involving a ‘P’ plate driver? e Explain the difference between parts c and d. 22 Explain why: a the sum of all relative frequencies will be 1 b the relative frequency of one event increases, if the relative frequency of another event decrease. invesTigaTe: 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.
340
Maths Quest Preliminary Mathematics General
10e
equally likely outcomes
Below is the field for the 2011 Melbourne Cup. Melbourne Cup Odds Horse
Odds
Horse
Odds
Americain
4/1
Hawk Island
150/1
Jukebox Jury
14/1
Illo
20/1
Dunaden
15/2
Lost In The Moment
30/1
Drunken Sailor
40/1
Modun
30/1
Glass Harmonium
30/1
At First Sight
9/1
Manighar
40/1
Moyenne Corniche
30/1
Unusual Suspect
30/1
Saptapadi
80/1
Fox Hunt
30/1
Shamrocker
40/1
Lucas Cranach
12/1
The Verminator
100/1
Mourayan
SCR
Tullamore
20/1
Precedence
50/1
Niwot
8/1
Red Cadeaux
30/1
Older Than Time
inTeraCTiviTY int–0089 random number generator
100/1
There were 23 horses in this race after Mourayan 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. 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 14
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
Each team may not be of equal ability and draws occur less often than one of the teams winning.
WriTe
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. ChapTer 10 • Probability
341
Worked example 15
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.
exercise 10e
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.
equally likely outcomes
1 We14 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 Twentysix 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 mC In which of the following is each member of the sample space equally likely to occur? a b C d
Kylie’s softball team is playing a match that they could win, lose or draw. A bag contains 4 red counters and 2 blue counters. One counter is selected from the bag. The maximum temperature on a January day will be between 20 °C and 42 °C. A rose that may bloom to be red, yellow or white is planted in the garden.
7 We15 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. 342
Maths Quest Preliminary Mathematics General
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 threecourse meal.
The menu is shown at right. 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?
Entree
Main course
Dessert
Prawn cocktail Oysters Soup
Seafood platter Chicken Supreme Roast beef Vegetarian quiche
Pavlova Icecream
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.
Fur ther development 11 The weather tomorrow could be either wet or dry. a Explain why these two outcomes are not equally likely. b What factors affect the probability of each? 12 In a tennis match, a player can either win or lose. a Explain why these two outcomes are not equally likely. b What factors affect the probability of each? 13 When rolling a standard die in a game, a player can get either a 6 or not get a 6. a Explain why these events are not equally likely. b Explain why we are able to calculate the probability of each. 14 When two children are born the outcomes can be two boys, two girls or one boy and one girl. Is each of
these outcomes equally likely to occur? Explain your answer. 15 In a given situation, explain what some of the factors are that can indicate that each outcome is not
equally likely. 16 When outcomes are not equally likely: a explain what we use in order to determine the likelihood of the event occurring b explain how we describe the likelihood of the event.
10F
The probability formula
In this chapter we discuss 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: P(event) =
number of favourable outcomes 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. ChapTer 10 • Probability
343
Worked example 16
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
WriTe
1
There are 6 possible outcomes.
S = {1, 2, 3, 4, 5, 6}
2
The favourable outcomes are to roll a 3, 4, 5 or 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 2 method is used to calculate the probability of any single event. Worked example 17
Andrea selects a card from a standard deck. Find the probability that she selects an ace. Think 1
There are 52 cards in the deck (total number of outcomes).
2
There are 4 aces (number of favourable outcomes).
3
Write the probability.
WriTe
P(ace) = =
4 52 1 13
The probability formula is used to calculate the probability of multistage events once the number of elements in the sample space has been calculated. Worked example 18
A coin is tossed and a die is rolled. Calculate the probability of tossing a Tail and rolling a number greater than 4. Think 1
Consider the number of stages in the experiment. There are two.
2
Draw the tree diagram. At the first stage there are two outcomes and at the second stage there are six outcomes.
3
WriTe/draW
Coin
H
List the sample space.
T
344
4
Calculate the probability using the probability formula. There are two favourable outcomes: T, 5 and T, 6.
5
Simplify.
Maths Quest Preliminary Mathematics General
Die 1 2 3 4 5 6 1 2 3 4 5 6
Sample space H, 1 H, 2 H, 3 H, 4 H, 5 H, 6 T, 1 T, 2 T, 3 T, 4 T, 5 T, 6
P(Tail and number greater than 4) =
2 12
=1 6
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 19
The digits 1, 3, 4, 5 are written on cards and these cards are then used to form a fourdigit number. Calculate the probability that the number formed is: a even b greater than 3000. Think
WriTe
a 1 If the number is even the last digit must be even. 2
There are four cards that could go in the final place (total number of outcomes).
3
Only one of these cards (the 4) is even (number of favourable outcomes).
4
Write the probability.
b 1 If the number is greater than 3000, then the
a
P(even) = 1 4
b
first digit must be a 3 or greater. 2
There are four cards that could go in the first place.
3
Three of these cards are a 3 or greater.
4
Write the probability.
exercise 10F
P(greater than 3000) = 3 4
The probability formula
1 We16 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 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 4 We17 A coin is tossed. Find the probability that the coin will show Tails. 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. ChapTer 10 • Probability
345
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
Q
K
K
Q
K
A
Q
K
A
A
2
2
A
8 7 6 5 4 3 2 09 J1
2
8 7 6 5 4 3 2 09 J1
8 7 6 5 4 3 2 09 J1
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. 8 We18 A twodigit number is formed using the digits 2, 4, 6 and 7. No digit may be repeated. Draw a
tree diagram to list all possible numbers that can be formed. 9 A cricket team needs to elect a captain and vicecaptain. The four nominations for these positions are
Belinda, Danika, Kate and Adrienne. Use a tree diagram to list all ways in which these three positions can be filled. 10 The digits 4, 5, 7 and 8 are used to form a threedigit number. If no digit can be used more than once,
list the sample space. 11 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 does not select a pear. 12 We19 The digits 2, 3, 5 and 9 are written on cards. They are then used to form a fourdigit number.
Find the probability that the number formed is: a even c divisible by 5 e greater than 5000.
b odd d less than 3000
13 mC A die is cast. The probability that the number on the uppermost face is less than 4 is: a C
1 6 1 2
b d
1 3 2 3 1 2
14 mC When a die is cast, which of the following outcomes does not have a probability equal to ? a b C d
The number on the uppermost face is greater than 3. The number on the uppermost face is even. The number on the uppermost face is at least a 3. The number on the uppermost face is a prime number.
15 mC A card is chosen from a standard deck. The probability that the card chosen is a court card is: a C
1 52 3 13
b d
1 13 4 13
16 mC When a card is chosen from a standard deck, which of the following events is most likely to
occur? a choosing a seven C choosing a picture card
b choosing a club d choosing a black card
17 A debating team consists of two men, Ashley and Benito, and three women, Carly, Donna and Ella.
From the team, a first, second and third speaker are to be chosen. Calculate the probability that: a Ashley is one of the three speakers b the team is made up of three women c both men are chosen d Carly is the first speaker. 346
Maths Quest Preliminary Mathematics General
18 The Jones family is planning a
Disneyland holiday. They have the choice of Disneyland in California, Disney World in Florida or Euro Disney in France. The holiday can be taken during two seasons, peak season or offpeak season, and they can fly economy, business or first class. a Use a tree diagram to list all combinations of holiday that could be taken by choosing a destination, season and class. b Find the probability that a certain holiday package involves travelling in Business class during the offpeak season. 19 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? c is odd? e is greater than 30 000?
b is even? d is divisible by 5? f is less than 20 000?
20 Write down an example of an event which has a probability of: 1 1 a b 2
c
4
2 . 5
21 A threedigit 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?
Fur ther development 2 5
22 It is known that the probability of selecting a blue ball from a bag is . a How many blue balls are in the bag if it known there is 80 balls in the bag? b How many balls are in the bag if it is known that there are 28 blue balls? 23 The probability that a black ball is selected from a bag is
7 . Another 20
bag contains an equal number 7 of balls. The probability that a black ball is selected from the second bag is 12 . Find the probability of selecting a black ball from the combined contents of the two bags.
24 A card is drawn at random from a standard deck of cards. Find the probability that the card
selected is: a a black ace c not a heart e a jack or a spade.
b a black card d a jack or a queen
25 John has a 12 sided die and Lisa has a 20 sided die. They are playing a game where the first person to
roll a 10 wins. a Find the probability of John rolling a 10. b Find the probability of Lisa rolling a 10. c Is the game fair? Explain your answer. 26 A number is chosen at random from the set {1, 2, 3, . . . . . . ., 25}. Find the probability that the number is: a a multiple of 4 b a multiple of 6 c a multiple of 4 or 6. 27 Events are said to be mutually exclusive if the occurrence of one prevents the occurrence of the other.
State whether each of the following pairs of events are mutually exclusive. a Obtaining a 4 or an even number. b Obtaining an odd number or a 6. c Obtaining a number less than 8 or greater than 5. d Obtaining a factor of 6 or a multiple of 6. ChapTer 10 • Probability
347
invesTigaTe: Comparing probabilities with actual results
digiTal doC Spreadsheet doc1655 Tossing a coin
digiTal doC Spreadsheet doc1656 rolling a die
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 1 a If we toss a coin P(Heads) = . Therefore, if you toss a coin, how many Heads would you expect in: 2 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? 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. Number 1 2 3 4 5 6
Occurrences
Percentage of throws
How close are the results to the results that were expected?
Writing probabilities as decimals and percentages 10g
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 daytoday 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 20
If I select a card from a standard deck, what is the probability of selecting a heart, expressed as a decimal? Think 1
There are a total of 52 cards in the deck (elements of the sample space).
2
There are 13 hearts in the deck (elements of the event space).
3
Write the probability.
4
Convert to a decimal.
WriTe
P(heart) = 13 25
= 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. 348
Maths Quest Preliminary Mathematics General
Worked example 21
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
WriTe
1
There are 20 counters in the bag (elements of the sample space).
2
There are 9 yellow counters in the bag (elements of the event space).
3
Write the probability.
4
Convert to a percentage.
P(yellow counter) =
9 20
× 100%
= 45%
Writing probabilities as decimals and percentages exercise 10g
1 We20 A die is rolled. What is the probability of rolling an even
number, expressed as a decimal? 2 We21 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 d Selecting a black card e Selecting a court card
c Selecting any 2
8 mC 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 mC 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 mC Which of the following does not describe the chance of selecting a diamond from a standard deck
of cards? a
13 52
b 0.13
C 0.25
d 25%
ChapTer 10 • Probability
349
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 5 1 following results (expressed as a decimal)? a A5 4 2 b An even number 3 c An odd number d A number greater than one 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.
Further development 13 A survey of the vehicles in a car park is conducted. The results are shown in the table below.
Vehicle type Number
Bus 30
Car 170
Motor bike 40
4 wheel drive 60
Find the probability (as a decimal) that a vehicle leaving the car park is: a a car b a bus c not a 4 wheel drive. 350
Maths Quest Preliminary Mathematics General
14 The table below shows the origin of visitors to a tourist information centre in Sydney.
Origin Victoria Queensland Other Australian states Europe Asia Total
Male 7 5 3 16 10 41
Female 9 7 2 17 4 39
Total 12 12 5 33 14 80
a Calculate (as a decimal) the relative frequency of: i Victorian male visitors ii European female visitors iii Asian visitors. b If a person is selected at random, find (as a percentage) the probability that the person is: i from another Australian state ii a female Queensland visitor iii a European male visitor. 15 Arlo and Roberta are playing a dice game. Arlo has an eightsided die and Roberta has a sixsided die. a The first person who rolls a 5 wins. Find the probability (as a decimal) that each person rolls a
five. Is this game fair? b The first person who rolls an even number wins. Find the probability (as a percentage) that each
person rolls an even number. Is this game fair? 16 The probability that a red marble is selected from a bag is 0.3. a If there are 80 marbles in the bag, how many are red? b If there are 54 red marbles how many marbles in the bag are not red? 17 A number is selected from the numbers set {1, 2, 3, . . . . . ., 20}. Find, as a percentage, the probability
that the number chosen is: a a factor of 10 c a factor of 10 or 20.
b a factor of 20
18 The functioning time of a car battery is shown in the table below.
Time (months)
6
7
8
9
10
11
12
Number
1
3
7
12
15
18
14
Find as a percentage, to the nearest whole number, that a battery will last for more than 10 months.
10h
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. ChapTer 10 • Probability
351
All probabilities therefore lie in the range 0 to 1. An event with a probability 1 of 2 has an even chance of occurring or not 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
1
probable fiftyfifty
1 2
unlikely very unlikely impossible
0
Worked example 22
For the following probabilities, describe whether the event would be certain, probable, fiftyfifty, unlikely or impossible. 4 18 a b 0 c 9
36
Think
a
4 9
is less than 12 and is therefore unlikely to occur.
b A probability of 0 means the event is
WriTe
a The event is unlikely as it has a probability of less
than 12.
b The event is impossible as it has a probability of 0.
impossible. c
18 36
= 1. Therefore, the event has an even chance 2 of occurring.
c The event has an even chance of occurring as the
probability = 1. 2
Worked example 23
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
There are 400 televisions (elements of the sample space).
2
There are 380 televisions that are not defective (number of favourable outcomes).
3
Write the probability.
WriTe
P(not defective) = 380 400
= 19 20 4
Since the probability is much greater than 12 and very close to 1, it is very probable that it will not be defective.
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 24
State whether the following statements are true or false, and give a reason for your answer. 1 a The probability of correctly selecting a number drawn out of a barrel between 1 and 10 is . 10 1 b The weather tomorrow could be fine or rainy, therefore the probability of rain is . 2 352
Maths Quest Preliminary Mathematics General
Think
WriTe
a Each outcome is equally likely.
a True, because each number is equally likely to be
selected. b Each outcome is not equally likely.
b False, because there is not an equal chance of the
weather being fine or rainy. Earlier we looked at the ways of organising data. Frequency table, frequency histograms and ogives can all be used to estimate the probability that an event will occur based on data collected. Worked example 25
A biased die is rolled 30 times and gives the following results 2 1 4
3 2 6
4 6 2
4 5 5
3 4 1
2 1 2
5 5 4
1 4 2
3 2 3
2 5 4
a Display the results in a frequency histogram. b Use the histogram to estimate the probability of rolling a 6 with this die. Think
WriTe
a 1 Complete a frequency table.
Score 1 2 3 4 5 6
Draw the histogram. Frequency
2
a
Frequency 4 8 4 7 5 2
8 7 6 5 4 3 2 1 0
1 2 3 4 5 6 Number rolled on die
b The probability of a six is estimated by the number
P(six) =
of times a six has been rolled over the total number of trials.
=
exercise 10h
Tally        
2 30 1 15
range of probabilities
1 We22 For each of the probabilities given below, state whether the event would be impossible,
unlikely, even chance, probable or certain. 7 14
b
d 1
e
a
g
19 36
10 13 37 40
h 0
c f i
3 8 25 52 12 25
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 ChapTer 10 • Probability
353
c d e f g h
Rolling a die and getting an even number Selecting a card from a standard deck and getting a red card Selecting a card from a standard deck and getting a spot card Selecting a card from a standard deck and getting an ace Reaching into a moneybox and selecting a 30c piece 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 d unlikely e impossible.
c even chance
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 mC 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 mC 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 8 mC 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 100metre final with 8 runners 9 We23 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’. 354
Maths Quest Preliminary Mathematics General
12 We24 For each of the following determine whether the statement is true or false, giving a reason for
your answer. a The probability of selecting an ace from a standard deck of cards is
4 . 52
b The probability of selecting the letter P randomly from a page of a book is
1 . 26
1 . 30 1 is 30.
c In a class of 30 students, the probability that Sam tops the class in a Maths test is d In a class of 30 students the probability that Sharon’s name is drawn from a hat
13 We25 The data below shows the amount of rain (in mm) that falls each of the 30 days during June.
2 0 0 a b c d
0 0 15
15 0 19
18 9 21
11 39 6
21 32 13
0 21 21
5 16 0
19 4 0
11 0 6
Using classes 0–4, 5–9 etc., put the data into a frequency distribution table. Show the results in a cumulative frequency histogram and ogive. Use your graph to find the probability that there is less than 5 mm of rain on a certain day in June. Use the ogive to complete the following statement.: ‘There is a 50% chance that there will be more than ____ mm of rain on a given day in June.’
14 Forty sample pieces of rope are tested in an effort to determine their breaking strain. The results are Ogive of rope strength
40 35 30 25 20 15 10 5
100%
50%
40 45 50 55 60 65 70 75 Breaking strain (kg)
Cumulative frequency (%)
Cumulative frequency
shown in the ogive below.
Find the probability that a piece of rope will withstand a stain of: b 65 kg. a 50 kg 15 A biologist who counts the number of seeds in each of 60 pumpkins presents his findings on the 60
Cumulative frequency (%)
Cumulative frequency
ogive below. 100%
50 40 30
50%
20 10 10 20 30 40 50 60 70 Number of seeds
a Find the probability that a pumpkin chosen at random will produce more than 50 seeds. b Find the minimum number of seeds such that there is a 75% chance that a pumpkin chosen at
random will have more than this number of seeds.
Fur ther development 16 Describe the following. a The probability of rain tomorrow b Will everyone give the same answer to this question? c Can the rules of probability be applied to this question? Explain why or why not. 17 A barrel contains 100 marbles, of which 50 are black and 50 are white. a What is the probability that a marble selected from the barrel is black? b The marble is not replaced in the bag before a second marble is drawn. Paul says that there is a
fifty–fifty chance that the second ball will be black. Is Paul correct? Explain your answer. ChapTer 10 • Probability
355
18 Explain why probabilities are most easily displayed as a fraction. 19 Luke says that an event with a probability of 0.49 is unlikely to occur. Does this mean that if the event
occurs one should be surprised? 20 The table below shows the Mathematics exam results of a group of Year 11 students.
Mark Number
0–25 11
26–40 16
41–50 17
51–65 34
66–75 57
76–85 54
86–95 21
96–100 4
a Determine the probability that a student from the group passes the exam (i.e. gets a mark greater
than 50). b Describe the chance that the person selected gets over 95. 21 Consider the probability scale on page 352. Give an example of an event that matches each point
labelled on the scale. invesTigaTe: 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.
Complementary events
10i
When tossing a coin, we know there are two elements in the sample space. P(Heads) = 12 and P(Tails) = 12. The total of the probabilities is 1. Now consider a slightly more difficult problem. Worked example 26
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 3 10 7 P(black) = 10 3 7 Total = 10 + 10
a There are 10 counters of which 3 are white.
a P(white) =
b There are 10 counters of which 7 are black.
b
c Add
3 10
and
7 10
together.
c
=1
356
Maths Quest Preliminary Mathematics General
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 27
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. 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.
a The complementary event is that the coin lands
Tails. 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 28
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) = 10 50 1
=5 b This is the complement of selecting a heavy
b P(not heavy metal) = 1 − P(heavy metal)
metal CD. Subtract the probability of selecting a heavy metal CD from 1.
exercise 10i
1
=1−5 1
=5
Complementary events
1 We26 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? ChapTer 10 • Probability
357
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. 3 We27 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 lefthand column with the complementary event in the righthand column.
A coin landing Heads An odd number on a die A picture card from a standard deck A red card from a standard deck Winning 1st prize in a raffle with 100 tickets Making the last 4 teams in a 20 team tournament
A coin landing Tails A spot card from a standard deck Not winning 1st prize in the raffle A team not making the last 4 An even number on a die 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 lefthand 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 oddnumbered 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?
9 mC 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
10 mC The probability of rolling at least one six is
event? a
9 36
b
C Rolling 1 six 11 . What 36
11 36
C
d Rolling at least 1 six
is the probability of the complementary
25 36
d 1
11 We28 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: a orange
b not orange.
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. 358
Maths Quest Preliminary Mathematics General
14 In a golf tournament there are 40 players. Of these, 16 are Australian and 12 are American. If they are
15
16
17
18
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. 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. 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. 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? A basketballer is about to take a shot from the freethrow line. His past record shows that he has a 91% success rate from the freethrow line. What would be the relative frequency (as a percentage) of his: a being successful with the shot? b missing the shot?
Further development 19 Explain whether or not each of the following pairs of events are complementary. a Having WeetBix or Corn Flakes for breakfast. b Walking or driving to school. c Watching TV or surfing the Internet. d Rolling a number less than 3 or rolling a number greater than 3 with a normal die. e Passing or failing a test. 20 Two coins are tossed. Are the events of tossing two heads and tossing two tails complementary? 21
22
23 24
Explain your answer. A race has eight horses in it. The favourite Make Be Diva is given a 22% chance of winning. a What is the probability of Make Be Diva not winning? b Is it possible to calculate the probability that the second favourite Shucking will win the race? Explain your answer. The probability of an event A is given as P(A) = 0.32. The probability of an event B is given as P(B) = 0.15. Given that A and B are mutually exclusive find: a P(A or B) b P(neither A nor B). In question 22 are the events (A or B) and (neither A nor B) complementary? How do you know? In a barrel there are 10 blue balls and 10 red balls. Two balls are selected at random from the bag. Will is interested in the probability of selecting at least one red ball. What event is complementary to selecting at least one red ball?
digiTal doC WorkSHEET 10.2 doc10336
ChapTer 10 • Probability
359
Summary multistage events
• 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.
The fundamental counting principle
• This principle can be used to count the number of elements in a sample space of a multistage experiment. • The total number of possible outcomes is calculated by multiplying the number of ways each stage of the experiment can occur.
probability statements
• 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, fiftyfifty, 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.
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.
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 probability formula
• 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.
360
Maths Quest Preliminary Mathematics General
Chapter review 1 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 17 2 The probability of a missile hitting its target is . The probability of missing the target is: 25 1 8 17 d 1 a b C 25
25
m U lTip l e C ho iC e
25
3 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 fiftyfifty d probable 4 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. 5 A threedigit number is to be formed using the digits 3, 6, 7 and 8. The same number cannot be used
more than once. How many threedigit numbers can be formed? a 4 b 12 C 24 d 64 6 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: C 218 500 d 874 000 a 87 b 348 s ho rT a n s W er
1 Two coins are tossed. Draw a tree diagram to find the sample space. 2 Two dice are rolled. How many elements are in the sample space? 3 A twodigit 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? 4 Mary, Neville, Paul, Rachel and Simon are candidates for an election. There are two positions,
5
6 7 8 9 10
president and vicepresident. 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? 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? A poker machine has five wheels. Each wheel has 15 symbols on it. In how many ways can the wheels land? 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? 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? 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? 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? ChapTer 10 • Probability
361
11 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? 12 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. 13 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 14 Give an example of an event which is: a certain b impossible. 15 The Chen family are going on holidays to Queensland during January. Are they more likely to
experience hot weather or cold weather? 16 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 d a square number e a prime number
c an odd number f a doubledigit number.
17 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. 18 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. 19 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? 20 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.
21 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. 22 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 23 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. 7 24 The probability that a person must stop at a set of traffic lights is . What is the probability of not 12 needing to stop at the lights? 25 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 nonfiction? 362
Maths Quest Preliminary Mathematics General
1 At a school athletics carnival, a relay team must be selected. Below is the list of students who qualified
and the house for which they compete. There must be one member of the relay team from each house. RED Richard Stan
YELLOW Andrew Frank Ned Voula
BLUE Boris Harry Danny
ex Ten d ed r es p o n s e
GREEN Milan
Draw a tree diagram to show all possible relay teams. How many elements are in the sample space? If Ned is to be in the relay team, how many favourable outcomes are there? Describe the chance of Milan being in the relay team. Is each element of the sample space equally likely to occur? Explain your answer. 2 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? 3 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? 4 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. a b c d e
digiTal doC Test Yourself doc10337 Chapter 10
ChapTer 10 • Probability
363
ICT activities 10a
multistage events
digiTal doCs • Spreadsheet (doc1644): Coin toss lister (page 329) • Spreadsheet (doc1646): Die rolling (page 329)
10C
probability statements
digiTal doCs • Investigation (doc10348): Chance in the media (page 335) • WorkSHEET 10.1 (doc10335): Apply your knowledge of probability to problems. (page 336)
10e
equally likely outcomes
inTeraCTiviTY • int0089: Random number generator (page 341)
10F
The probability formula
digiTal doCs • Spreadsheet (doc1655): Tossing a coin (page 348) • Spreadsheet (doc1656): Rolling a die (page 348)
364
Maths Quest Preliminary Mathematics General
10i
Complementary events
digiTal doC • WorkSHEET 10.2 (doc10336): Apply your knowledge of probability to problems. (page 359)
Chapter review digiTal doC • Test Yourself Chapter 10 (doc10337): Take the endofchapter test to check your progress. (page 363)
To access eBookPLUS activities, log on to www.jacplus.com.au
Answers CHAPTER 10 probabiliTY exercise 10a
1 2 3 4 5 6 7 8
9 10 11
12 13 14 15
16 17
18
19
multistage events S = {HH, HT, TH, TT} S = {R1B1, R1B2, R1B3, R2B1, R2B2, R2B3, R3B1, R3B2, R3B3} S = {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG} a 12 b No a 25 b 20 27 S = {13, 14, 18, 31, 34, 38, 41, 43, 48, 81, 83, 84} S = {DZ, DM, DK, DS, ZD, ZM, ZK, ZS, MD, MZ, MK, MS, KD, KZ, KM, KS, SD, SZ, SM, SK} S = {AM, AS, AL, PM, PS, PL, YM, YS, YL} S = {CA, CR, CL, CK, AR, AL, AK, RL, RK, LK} a Check with your teacher. b S = {HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT} c 6 C D B a S = {22, 25, 27, 28, 52, 55, 57, 58, 72, 75, 77, 78, 82, 85, 87, 88} b S = {25, 27, 28, 52, 57, 58, 72, 75, 78, 82, 85, 87} a 12 b 24 c 24 a S = {TJ, TS, TR, TM, JT, JS, JR, JM, ST, SJ, SR, SM, RT, RJ, RS, RM, MT, MJ, MS, MR} b 20 c 2 d 6 e 12 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. a 36 b
Total
2 3 4 5 6 7 8 9 10 11 12
No. of elements 1 2 3 4 5 6 5 4 3 2 1 c 6 20 a 64 21 a First die
b 24 Second die 6
6 not 6 6 not 6 not 6
b They are not equally likely as she is less
likely to get a 6 than not.
22 a 4 b 40 23 A tree diagram is a way of systematically
showing the outcome to each stage.
exercise 10b The fundamental counting principle 1 50 625 2 a 4 b 36 c 12 3 1000 4 7776 5 6 760 000 6 D 7 B 8 1833 9 72 10 a 64 b 144 c 168 d 112 e 384 11 a 3 200 000 b 240 12 2 540 160 13 90 14 a 12 b Not correct. After 12 rides the next ride must have been experienced before. 15 a 5408 b 676 16 36 17 a 100 000 000 b 80 000 000 c 1 000 000 d 2 000 000 18 96 19 C 20 a 1200 b 550 21 20 000 22 307 328 000 23 a 100 000 b 12 c No. There will be 248 832 combinations, an increase of approximately 148%. 24 76.3% 25 The number of ways that a series of independent events can occur is found by multiplying the number of outcomes at each individual stage. exercise 10C
probability statements
1 a Probable b Unlikely c Impossible d Fiftyfifty 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 Australia 7 Carl Bailey because he has better past
performances. A B C Probable Unlikely Fiftyfifty Answers will vary. Common language is used to describe how likely an event is to occur, and unless mathematical calculations are to be made the exact likelihood will not need to be defined. 16 Each outcome for the weather is not equally likely to occur. 8 9 10 11 12 13 14 15
17 The number of chances of winning and the
total number of chances can be counted.
18 Answers will vary. 19 0 represents no chance and 1 represents
100% chance, hence nothing can lie outside this range.
exercise 10d
1 3 4 5 6 7 8 10 11 12 13 14
relative frequency 0.74 2 0.79 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 frequency is 27%. a 2.5% b 51.5% c 17.5% 40 000 km
15 16 a
Result
Number
Win Loss Draw
15 14 11
b Win = 0.375, Loss = 0.35, Draw = 0.275 17 a i 0.825 ii 0.175 b i 0.3 ii 0.7 c i 0.777 ii 0.223 18 D 19 0.2625 20 a 0.0102 b $181.69 21 a 0.0833 b 0.00167 c 0.02 d 0.16 e Part c finds the fraction of ‘P’ plate
drivers that have accidents while part d finds the fraction of accidents that involves ‘P’ plate drivers. 22 a The sum of relative frequencies must account for 100% of all outcomes. b As all relative frequencies add to 1, if one increases another must decrease as the sum must not change. exercise 10e
1 2 3 4 5
6 7 8 9
equally likely outcomes 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.
ChapTer 10 • Probability
365
10 a 10 b No. Each horse has a different rider and
11
12
13
14
15 16
ability. c Yes. Yes. The selection of horse is made randomly. a Tomorrow’s weather depends heavily on the weather today. b Scientific factors affect the likelihood of rain. a The selection of winner is not random. b Ability is the main factor affecting the outcome. a There is one face with a 6 and five faces that are not. b The likelihood of each face is equally likely. The outcome of one boy and one girl can occur in either order making it twice as likely as the other two outcomes. Answers will vary. a Science and experience b Informal chance words
9
Captain
Vicecaptain
Belinda
Danika
Kate
Adrienne
10
1st digit
The probability formula 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 2
5 a d 6 a d g 7 a d 8
1 6 1 2 1 45 23 45 19 45 1 52 1 2
b e b e h b e 1st digit 2
4
6
7
366
1 6 2 3 1 45 1 5 2 9 1 13 3 13
c f c f i c f
1 2 1 3 22 45 1 3 2 15 1 4 3 13
26
7
27
2
42
6
46
7
47
2
62
4
64
7
67
2
72
4
74
6
76
Belinda, Adrienne
Belinda
Danika, Belinda
Kate
Danika, Kate
Adrienne
Danika, Adrienne
b A number greater than 400 22 a 32 b 70
Belinda
Kate, Belinda
23 15
Danika
Kate, Danika
Adrienne
Kate, Adrienne
Belinda
Adrienne, Belinda
Danika
Adrienne, Danika
Kate
Adrienne, Kate
7 8 4 5 8 4 5 7
10
12 a d 13 14 15 16
b
3 4
e
1 2 = 2 4
C C C D 3
1
17 a 5
b 10
3
1
c 10 18 a
d 5 Destination
Season Peak
Disneyland (California) Offpeak Peak Disneyworld (Florida) Offpeak Peak Euro Disney (France) Offpeak
b
1 6
Maths Quest Preliminary Mathematics General
f
4 5 1 5
could go in the last place.
7
24 a
1 26
b
1 2
d
2 13
e
4 13
25 a
1 12
b
1 20
6 25
b
c
c
3 4
c
8 25
Game is not fair. 4 25
Not mutually exclusive Mutually exclusive Not mutually exclusive Not mutually exclusive
exercise 10g Writing probabilities as decimals and percentages 1 0.5 2 37.5% 3 a 0.5 b 50% 4 a 0.17 b 0.5 c 0.33 5 a 16.7% b 50% c 83.3% 6 a 0.02 b 0.25 c 0.08 d 0.5 e 0.23 7 a 1.9% b 25% c 7.7% d 50% e 23.1% 8 A 9 D 10 B 11 a 0.2 b 0.4 c 0.6 d 0.8 12 a 20% b 40% c 40% 13 a 0.567 b 0.1 c 0.8
1 4
c
e
c
20 Check with your teacher. 21 a Because there are two numbers which
27 a b c d
5
1 4 1 4
d
26 a
3rd Sample digit space 457 7 458 8 475 5 478 8 485 5 487 7 547 7 548 8 574 4 578 8 584 4 587 7 745 5 748 8 754 4 758 8 784 4 785 5 845 5 847 7 854 4 857 7 874 4 875 5
11 12 = 6
2nd Sample digit space 24 4 6
Adrienne
4
8
1 5 4 5
Belinda, Kate
8
7
b
Belinda, Danika
7
5
1 5 1 5
Kate
5
exercise 10F
19 a
Danika
2nd digit
4
Sample space
Class
Sample space
Econ Bus First Econ Bus First Econ Bus First Econ Bus First Econ Bus First Econ Bus First
Cal, Peak, Econ Cal, Peak, Bus Cal, Peak, First Cal, Off, Econ Cal, Off, Bus Cal, Off, First Florida, Peak, Econ Florida, Peak, Bus Florida, Peak, First Florida, Off, Econ Florida, Off, Bus Florida, Off, First France, Peak, Econ France, Peak, Bus France, Peak, First France, Off, Econ France, Off, Bus France, Off, First
Roberta — 0.167 The game is not fair. b Arlo — 50% Roberta — 50% The game is fair. 16 a 24 b 126 17 a 25% b 30% c 30% 18 67%
b
30
range of probabilities Even chance b Probable Unlikely d Certain Probable f Unlikely Probable h Impossible Unlikely 0, impossible 1, certain 3 c , even chance 6 d e f
26 , 52 40 , 52 4 , 52
even chance
20 10 5 5 10 15 20 25 30 35 40 Rainfall (mm) 37% d 10 mm 20% b 85% 20% b 48 Answers will vary. No No, since each outcome is not equally likely.
probable
18
unlikely
even chance 3 Check with your teacher. 13 7 9 8 6 4 , , , , 20 13 18 19 25
19
A, D, C, B, E D B D a 0.025 b Unlikely 98 10 , very probable 100 11 a Very unlikely b Even chance c Probable d Unlikely 12 a True, as there are 4 aces from 52 cards in the deck. b False, as each letter does not occur equally often. c False, as each student is not of equal ability. d True, as the name is chosen randomly. 5 6 7 8 9
13 a
Class Frequency Cumulative centre frequency
0–4
2
11
11
5–9
7
4
15
10 – 14
12
3
18
15 – 19
17
6
24
20 – 24
22
4
28
25 – 29
27
0
28
30 – 34
32
1
29
35 – 39
37
1
30
d e 7 a b c d 8 a b c 9 A 10 C 11 a 12 a 13 a d
1
17 a 2 b No, he is not correct as the probability 49
6 , 12
Class
50%
15
c 14 a 15 a 16 a b c
g 0, impossible h
100%
25
exercise 10h
1 a c e g i 2 a b
Ogive of rainfall
Cumulative frequency (%)
iii 0.175 iii 20%
Cumulative frequency
14 a i 0.0875 ii 0.2125 b i 6.25% ii 8.75% 15 a Arlo — 0.125
20 21
will be 99 , slightly less than fifty–fifty. A fraction simply shows the number of chances that the event has of occurring over the total number of possible outcomes. Although the event is technically unlikely it has close to an even chance of occurring and one should not be surprised if it occurs. 170 a b Very unlikely 214 Answers will vary.
exercise 10i
14 a d 15 a 16 a
Complementary events
1 6
c 1 2 a i b 3 a b c 4
7 20
ii
8 20
c 6 a b c
5 20
1 Losing a race Failing a test Your birthday not falling on a Monday
A coin landing Heads An odd number on a die A picture card from a standard deck A red card from a standard deck Winning 1st prize in a raffle with 100 tickets Making the last 4 teams in a 20 team tournament 1 1 3 1 1 1 5 a , , , , , 2 2 13 2 100 5 b
iii
1 1 10 1 99 4 , , , , , 2 2 13 2 100 5
A coin landing Tails An even number on a die A spot card from a standard deck A black card from a standard deck Not winning 1st prize in the raffle A team not making the last four
1 Rolling an odd number Rolling a number less than 4 Rolling a number greater than 2
1 8 1 5 9 20 17 20 2 5 7 10 7 20 3 25
b b b e b e b b
7 8 4 5 11 20 3 5 3 10 3 10 13 20 21 50
c f c
c
3 20 2 5 3 5
29 50
17 0.3 18 a 91% b 9% 19 a Not complementary as this does not
1 a S = {1, 2, 3, 4, 5, 6} b
Not rolling a 6 Rolling a 1 Choosing an evennumbered ball Choosing a ball numbered greater than 19 Choosing a ball that has a number less than 24 Choosing a ball that is not a multiple of 5 Selecting a coloured ball Selecting a black ball Not selecting a pink ball
20 21
22 23 24
cover all possible outcomes. There are other cereals that could be eaten for breakfast. b Not complementary as this does not cover all possible outcomes. Other means of transport can be used. c Not complementary as this does not cover all possible outcomes. Other activities could be undertaken. d Not complementary as this does not cover all possible outcomes. The die could show a 3. e Complementary as they cover all possible outcomes. They are not complementary as there could be one head and one tail. a 78% b No because the probability of the other six horses winning is not known. a 0.47 b 0.53 They are complementary as their probabilities add to 1. Selecting no red balls
ChapTer revieW mUlTiple ChoiCe
1 D 4 C
2 B 5 C
3 B 6 C
shorT ansWer
1 S = {HH, HT, TH, TT} 2 36 3 a S = {56, 57, 59, 65, 67, 69, 75, 76,
79, 95, 96, 97}
b 9
ChapTer 10 • Probability
367
4 a S = {MN, MP, MR, MS, NM, NP, NR,
NS, PM, PN, PR, PS, RM, RN, RP, RS, SM, SN, SP, SR}
b 8 5 a S = {TCB, TCM, TCZ, TDB, TDM,
TDZ, THB, THM, THZ, TIB, TIM, TIZ, KCB, KCM, KCZ, KDB, KDM, KDZ, KHB, KHM, KHZ, KIB, KIM, KIZ}
b 15
759 375 7 20 272 9 160 a 10 000 b 500 a 100 000 000 b 94 109 400 Marcia will probably get a higher card. a Probable b Impossible c Even chance d Unlikely e Unlikely 14 Check with your teacher. 15 Hot weather 6 8 10 11 12 13
16 a d
368
1 25 1 5
b e
1 25 9 25
c f
13 25 16 25
17 a d 18 a 19 a
1 52 1 2 7 20 1 400
b e b b
1 13 3 13 1 10 4 1999
c f c
1 4 10 13 3 4
20 a 0.025 b 0.5 c 0.75 7 8
21 . It is probable that the car will have a
defect. 22 a Tossing a coin that lands Heads b Rolling a die and getting a number
greater than 4
c Not choosing a blue ball 23 a b 24 25
3 10 7 10
5 12 18 25
Maths Quest Preliminary Mathematics General
exTended response
1 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 c d e 2 a b c 3 a b c
24 6 Certain No. Each runner is of different ability. S = {GD, GK, GC, DK, DC, KC} E = {GC, DC, KC} 90 1 5 2 5 3 5
4 a 0.7 b 0.3 c The missile will probably hit its target.
ChapTer 11
Algebraic manipulation ChapTer ConTenTS 11a 11B 11C 11d 11e
Operations with algebraic expressions Further multiplication and division Expanding and simplifying algebraic expressions Substitution Solving linear equations
operations with algebraic expressions 11a
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 1
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.
a m + m + m + m + m + m + m = 7m
b 1 Write p + p + p as a multiplication.
b
2
Write q + q + q + q + q + q as a multiplication.
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. 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. ChapTer 11 • Algebraic manipulation
369
Worked example 2
Simplify: a 5k + 9k − k
b 5b + 2 + 2b − 6
Think
c 4m − 3n + 2m − 5n. WriTe
a Each term uses the same pronumeral so we add and a 5k + 9k − k = 13k
subtract the coefficients. Remember that where no coefficient is written it is assumed to be 1. b 1 Rewrite the expression, grouping like terms. 2
c 1 2
Complete each addition and subtraction separately. Rewrite the expression, grouping like terms.
b 5b + 2 + 2b − 6 = (5b + 2b) + (2 − 6)
= 7b − 4 c 4m − 3n + 2m − 5n = (4m + 2m) + (−3n − 5n)
= 6m − 8n
Complete each addition and subtraction separately.
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 3
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 4
Simplify each of the following. a t5 × t4
b 4m3 × 6m2
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,
c 12a2b × 4a5b3 = 48a7b4
then add the base ‘b’ indices.
370
c 12a2b × 4a5b3
Maths Quest Preliminary Mathematics General
The reverse to the index law for multiplication is the index law for division. Index Law 2: a x ÷ a y = 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 5
Simplify each of the following. a w8 ÷ w3
b 24d4 ÷ 6d
Think
WriTe
a Subtract the indices.
a w8 ÷ w3 = w5
b Divide the coefficients then subtract the indices.
b 24d 4 ÷ 6d = 4d3
exercise 11a
operations with algebraic expressions
1 We1 Simplify each of the following. a t+t b r+r+r d x+x+x+x e q+q+q+q+q+q+q
c w+w+w+w+w+w f n+n+n+n+n+n+n+n
2 Write the following algebraic terms as additions. a 2m b 6n d 8w e 3y
c 9s f 4r
3 We2 Simplify the following. a d g j m
5y + 7y 34j + 13j 14r − 6r −4w + 6w 4j − 2j − j
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 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
c f i l o
15e + 24e 14k − 10k m − 5m 14m + 5m + 3m 5z − 15z + 9z
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
5 We 3 Write each of the following in index form. a 2×2×2×2 d q×q×q×q×q×q×q
b 7×7×7×7×7×7 e p×p×p×p×p
c 9×9 f w×w×w
6 Write each of the following in expanded form. a 36 d m3
b 45 e y2
7 We 4 Simplify each of the following. a q3 × q4 b x5 × x6 d 45 × 42 e a5 × a 4 2 g 3b × 2b h 5d 4 × 2d 4 6 j 9 × 3j k 8k3 × k m 4x4y3 × 5x3y6 n 5m2n4 × 4m3n5 4 3 6 p 6jk × 8j k q 10p4q3 × 5p
c 87 f j9 c f i l o r
y7 × y8 s × s2 7g4 × 5g6 4m × 6m 9a3b7 × 7a2b2 9r3s2t 4 × 6rst 6 ChapTer 11 • Algebraic manipulation
371
8 We 5 Simplify each of the following. a a5 ÷ a e
b b8 ÷ b4
18 g5 3g 3
f
c9 c7 g 56j7k 5 ÷ 7j 4k 3
d e5 ÷ e
c
24h6 ÷ 4h2
h
84 s 7 72r 6 k 7s 5 8r 3 9 Explain why 4xy and 5yx are like terms and 4x2 and 5x are not like terms. i
42p9q4 ÷ 6pq
j
l
64 m 7n3 16m 4 n 2 6t 8 18t 5
Fur ther development 10 Simplify each of the following. a
8 p6 × 3 p 4 16 p5
b
12bb5 × 4 b 2 18b 2
c
5a3 × 6a 4 2a 4 × 5a
d
12m 4 2m 2 × 3m 3
11 Simplify. a
8 p3 × 7r 2 × 2s 6 p × 14 r
b
27a9 × 18b3 × 4c 2 18aa 4 × 12b 2 × 2c
c
81 f 15 × 25g12 × 16h34 27 f 9 × 15g10 × 12h30
12 Gordon says that (5x4)2 = (5x2)4. Explain whether or not Gordon is correct.
11B
Further multiplication and division
Many algebraic calculations may involve the use of fractions. The process of multiplying algebraic fractions is relatively simple. The numerators are multiplied together and the denominators are multiplied together. Worked example 6
Simplify 6m2 × 8m. 5 Think 1
Consider 6m2 as a fraction over 1, then multiply both the numerators and denominators.
2
Simplify the resulting fraction if possible, which in this case it is not.
WriTe
6m2 × 8m = 48m3 1 5 5 8m 48m3 2 6m × = 5 5
When dividing fractions, remember the method is to multiply by the reciprocal of the second fraction. Worked example 7 3 Simplify 2v ÷ 3v . 5 5
Think
372
WriTe
2v ÷ 3v3 = 2v × 10 5 10 5 3v3
1
Rewrite the division by multiplying by the reciprocal of the second fraction.
2
Cancel the fractions by dividing by common factors.
= 2 × 22 1 3v
3
Multiply the numerators and denominators. Look for any further simplifications. In this case there are none.
= 42 3v
Maths Quest Preliminary Mathematics General
exercise 11B
Further multiplication and division
1 We 6 Simplify each of the following. a 2×
5a 3
b
2b × 3 5
c 4×
2 ×9 3e e 4× 5d 2 4 7h g g× h 5h × 2g 10 2 We 7 Simplify each of the following. d
3x ÷ 2 5 3t ÷ t c 4 3 Simplify each of the following. x 20 x 12 a b × × 5 y 4 y a
e
x 25 × 10 2 y
f
3w 7 × 14 x
x 9z 5y x j × × 3x 8 y 3z 2 y 4 Simplify the following expressions. 3 5 2 9 a b ÷ ÷ x x x x i
f
4 c
9 × 2f 3
4y ÷ 2 5 2v d 2÷ 3 b
c
y 16 × 4 x
d
x 9 × 2 2y
g
3 y 8z × 4x 7y
h
y 6z × 3x 7 y
k
20 y 21z × 7x 5y
l
y x × 3w 2 y
c
4 12 ÷ x x
d
20 20 ÷ x 3y
e
1 5 ÷ 5w w
f
7 3 ÷ 2 x 5x
g
3 xy 3 x ÷ 7 4y
h
2 xy 5 x ÷ 5 y
i
6 y 3x ÷ 9 4 xy
j
8 wx 3w ÷ 5 4y
k
2 xy 3 xy ÷ 5 5
l
10 xy 3 xy ÷ 5 5
Fur ther development 5 Simplify each of the following. a
2a 9b × 3 2
b
3 10 x × 5 3y
c
5 3q × 12 p 5
d
4 3b × 15a 4 a
e
x 9 × 3 x
f
4 y × y 12
g
4 m × 3 16
h
n 3 × 9 2m
i
7m 10 × 5 m
j
5 x × 3 x 15
k
20 6 × 3y 5
l
2 x 15 y × 3y 6x
m
4 m 2 9n × 27n 7m
n
2 p 7 pq × × 15 p2 21
o
x 11y 2 6 z × × 22 y 12 z xy
6 Simplify each of the following. a
2a 2 ÷ 5 15b
b
3 3 ÷ 4 8x
c
5 15 ÷ 6y 6
d
9 p 36q ÷ 10 10
e
x x ÷ 3 9
f
4 12 ÷ m m
g
a a ÷ 5 20
h
6 20 ÷ b b
i
3a 2 a ÷ 14 7
j
21b 2 3 ÷ 3 4 b
k
6m 6 2m 3 ÷ 15 3
l
ab ac ÷ 9 24
m
2m 10 m ÷ 3 p 9 pq
n
3 10 12 × ÷ 5 m m
o
3x 3 2 y2 y2 × ÷ 8y 15 4 ChapTer 11 • Algebraic manipulation
373
expanding and simplifying algebraic expressions 11C
To expand an algebraic expression means to remove a set of brackets. This is done by multiplying what is inside the brackets by what is directly outside the brackets. Worked example 8
Expand each of the following expressions. a 5(x + 3) b −a(x − y) Think
a 1 Write the expression.
c x(2x + 3y) WriTe
a 5(x + 3)
2
Expand the grouping symbols.
= 5(x) + 5(3)
3
Multiply out the grouping symbols.
= 5x + 15
b 1 Write the expression.
b −a(x − y)
2
Expand the grouping symbols.
= −a(x) − a(−y)
3
Multiply out the grouping symbols. (Remember that a negative term multiplied by a negative term makes a positive term.)
= −ax + ay
c 1 Write the expression.
c x(2x + 3y)
2
Expand the grouping symbols.
= x(2x) + x(3y)
3
Multiply out the grouping symbols. (Remember that x multiplied by itself gives x2.)
= 2x2 + 3xy
An algebraic expression may be composed of brackets that need to be expanded, as well as other terms. In such cases, after expansion it may be possible to simplify the expression by collecting like terms. Worked example 9
Expand and simplify each of the following expressions. a 7x + 6( y − 2x) b 5(x + 2y) + 6(x − 3y) Think
a 1 Write the expression.
WriTe
a 7x + 6( y − 2x)
2
Expand the grouping symbols.
= 7x + 6( y) + 6(−2x)
3
Multiply out the grouping symbols.
= 7x + 6y − 12x
4
Collect any like terms.
= −5x + 6y
b 1 Write the expression.
b 5(x + 2y) + 6(x − 3y)
2
Expand the grouping symbols.
= 5(x) + 5(2y) + 6(x) + 6(−3y)
3
Multiply out the grouping symbols.
= 5x + 10y + 6x − 18y
4
Collect any like terms.
= 11x − 8y
c 1 Write the expression.
374
c −5xy(1 + 2y) + 6x( y + 4x)
c −5xy(1 + 2y) + 6x( y + 4x)
2
Expand the grouping symbols.
= −5xy(1) − 5xy(2y) + 6x(y) + 6x(4x)
3
Multiply out the grouping symbols.
= −5xy − 10xy2 + 6xy + 24x2
4
Collect any like terms.
= xy − 10xy2 + 24x2
Maths Quest Preliminary Mathematics General
expanding and simplifying algebraic expressions exercise 11C
1 We 8 Expand the following expressions. a 3(x + 2) b 4(x + 3) d 2( p + 5) e 4(x + 1) g −4(y + 6) h −5(a + 1) j −(x − 1) k −(x + 3) m 3(2b − 4) n 8(3m − 2) p −3(9p − 5)
c f i l o
5(m + 4) 7(x − 1) −3( p − 2) −(x − 2) −6(5m − 4)
2 Expand each of the following. a x(x + 2) d c(c + 4) g m(7 − m) j 5p(q + 4) m −b(3 − a) p −4x(7 − 4x)
c f i l o
a(a + 5) y(5 + y) 2x( y + 2) −10p(q + 9) −6a(5 − 3a)
3 We 9 Expand and simplify by collecting like terms. a 2( p − 3) + 4 b 5(x − 5) + 8 d −4(3p − 1) − 1 e 6x(x − 3) − 2x g 3x( p + 2) − 5 h 4y( y − 1) + 7 j 5(x − 2y) − 3y − x k 2m(m − 5) + 2m − 4 m −7a(5 − 2b) + 5a − 4ab n 4c(2d − 3c) − cd − 5c p 5 − 9m + 2(3m − 1)
c f i l o
−7( p + 2) − 3 2m(m + 5) − 3m −4p( p − 2) + 5p −3p( p − 2q) + 4pq − 1 6p + 3 − 4(2p + 5)
4 Expand and simplify the following expressions. a 2(x + 2y) + 3(2x − y) b 4(2p + 3q) + 2( p − 2q) d 5(3c + 4d ) + 2(2c + d ) e −4(m + 2n) + 3(2m − n) g −2(3x + 2y) + 3(5x + 3y) h −5(4p + 2q) + 2(3p + q) j 5(2x − y) − 2(3x − 2y) k 4(2p − 4q) − 3( p − 2q) m 7(2x − 3y) − (x − 2y) n −5( p − 2q) − (2p − q) p 4(3c + d ) − (4c + 3d )
c f i l o
7(2a + 3b) + 4(a + 2b) −3(2x + y) + 4(3x − 2y) 6(a − 2b) − 5(2a − 3b) 2(c − 3d) − 5(2c − 3d) −3(a − 2b) − (2a + 3b)
b e h k n
y( y + 3) x(4 + x) q(8 − q) −3y(x + 4) −7m(5 − n)
5 Expand and simplify the following expressions. a a(b + 2) + b(a − 3) b x( y + 4) + y(x − 2) d p(q − 5) + p(q + 3) e 3c(d − 2) + c(2d − 5) g 2m(n + 3) − m(2n + 1) h 4c(d − 5) + 2c(d − 8) 6a
mC
c c(d − 2) + c(d + 5) f 7a(b − 3) − b(2a + 3) i 3m(2m + 4) − 2(3m + 5)
What is the equivalent of 3(a + 2b) + 2(2a − b)?
a 5a + 6b
B 7a + 4b
C 5(3a + b)
d 7a + 8b
a −4x + 11y
B −4x − 11y
C 4x + 11y
d 4x + 7y
a 3m + 4n − 8
B 5mn + 4m
C 5mn + 10m
d 5mn + 6m
b What is the equivalent of −3(x − 2y) − (x − 5y)?
c What is the equivalent of 2m(n + 4) + m(3n − 2)?
Fur ther development 7 Expand and simplify the following expressions. a 5c(2d − 1) − (3c + cd ) c −4c(2c − 6d ) + d(3d − 2c) e 2p( p − 4) + 3(5p − 2) g −2y(5y − 1) − 4(2y + 3)
b −3a(5a + b) + 2b(b − 3a) d 6m(2m − 3) − (2m + 4) f 7x(5 − x) + 6(x − 1)
diGiTal doC WorkSHEET 11.1 doc10338
ChapTer 11 • Algebraic manipulation
375
11d
Substitution
When the numerical values of pronumerals are known, we can substitute them into an algebraic expression and evaluate it. It can be useful to place any substituted values in brackets when evaluating an expression. Worked example 10
If a = 4, b = 2 and c = −7, evaluate the following expressions. a a−b b a3 + 9b − c Think
a 1 Write the expression.
WriTe
aa−b
2
Substitute a = 4 and b = 2 into the expression.
=4−2
3
Simplify.
=2
b 1 Write the expression.
b a3 + 9b − c
2
Substitute a = 4, b = 2 and c = −7 into the expression.
= (4)3 + 9(2) − (−7)
3
Simplify.
= 64 + 18 + 7 = 89
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 = 43 π r3 is used to calculate the volume of a sphere. Calculate the value of V, correct to 2 decimal places, when r = 4.7. Think
WriTe
V = 43 πr3
1
Write the formula.
2
Substitute 4.7 for r.
= 43 × π × (4.7)3
3
Calculate the value of V and round off to 2 decimal places.
= 434.89
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
Method 1 1 Write the formula.
376
WriTe/diSplaY
v = u + at
2
Substitute 12.8 for u, 9.8 for a and 5 for t.
= 12.8 + 9.8 × 5
3
Calculate the value of v.
= 61.8
Maths Quest Preliminary Mathematics General
Method 2 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 a a U w 9.8 a a A w 5a a T w
3
To evaluate v, enter the expression to which v is equal, that is U + AT. Press w to obtain the value of the expression.
Substitution
exercise 11d
1 We10 If a = 2, b = 3 and c = 5, evaluate the following expressions. a a+b
b c−b
d c − (a − b)
e 7a + 8b − 11c
g abc
h ab(c − b) k −a × b × −c
j
c2
+a
c−a−b a b c + + f 2 3 5 i a2 + b2 − c2 l 2.3a – 3.2b f
2 If d = −6 and k = −5, evaluate the following. a d+k d kd
b d−k e −d(k + 1)
g k3
h
3 If x =
1 3
c k−d f d2
k −1 d
i
3k − 5d
1
and y = 4 , evaluate the following. c xy a x+y b y−x x 9x e x 2y 3 d f y y2 9C 4 We11 The formula F = + 32 converts degrees Celsius to degrees Fahrenheit. Use the formula to 5 convert the following temperatures to degrees Fahrenheit. a 20 °C b 35 °C c −5.3 °C 5 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) 2 p d C = 45 + 65d (d = 1.4) e S = 4π r2 (r = 8.8) f Q= ( p = 6.5) 9.8 6 We12 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 7 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. 1 d Given that A = 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
diGiTal doC Spreadsheet doc1489 Substitution
diGiTal doC Spreadsheet doc1490 Substitution game
ChapTer 11 • Algebraic manipulation
377
5 9
g Given that C = (F − 32), find C when F = 212. 1
h Given that S = ut + 2 at2, find S when u = 8, t = 4 and a = 6. 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. 8 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. 9 mC A formula is given as c = mp. If m = 2 and p = 5 then c is equal to: a 3
1 3
B 7
C 10
d 25
10 The formula V = π r2h is used to calculate the volume of a cone, where r is the 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 , calculate the value of B when: h2 a m = 56 and h = 2 b m = 3.6 and h = 6 yA 12 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 13 Fried’s rule to calculate the infant dosage of a medicine mA is given by the formula D = , where D is the infant 150 dosage, m is the infant’s age in months and A is the adult dosage. Calculate the dosage of medicine given to: a a ninemonthold baby, where the adult dosage is 50 mL b an 18monthold child, where the adult dose is 30 mL c a twoyearold child, where the adult dose is 40 mL. 11 In the formula B =
c m = 1.6 and h = 0.8. c y = 0.24 and A = 96.
yA . In this formula, D is the y + 12 child’s dose, y is the age of the child in years and A is the adult’s dose. Calculate the dosage for a threeyearold child taking a medicine for which the adult dose is 45 mL. kA 15 Clark’s rule for calculating a dosage is D = , where k is the mass of the child in kilograms and A is 70 the adult dose. Calculate the dosage required for a child who weighs 20 kg, where the adult dosage is 35 mL. 16 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. m 17 The Body Mass Index, B, is a measure of how healthy a person is. The formula is B = 2 , where m is a h person’s mass in kilograms and h is a person’s height in metres. 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 14 Young’s rule for the calculation of a child’s dose of medicine is D =
Fur ther development 18 Pythagoras’ formula is c2 = a2 + b2. Use the formula to find c when: a a = 6 and b = 8 b a = 8 and b = 15 378
Maths Quest Preliminary Mathematics General
c a = b = 6.
19 The volume of any prism can be found using the formula V = Ah where A is the crosssectional area
and h is the height of the prism. Find the value of V when: a A = 7 cm2, h = 9 cm b A = 12 cm2, h = 48 cm
c A = 3.6 cm2, h = 2.3.
20 Euler’s formula, E = F + V – 2, is used to find the number of edges on a prism given the number of
faces and vertices. Use the formula to find the number of edges on a prism with: b 7 faces and 10 vertices c 10 faces and 12 vertices.
a 5 faces and 7 vertices.
1
21 The kinetic energy of an object is found using the formula E = 2mv2, where m is the mass and v is the
velocity of the object. Find E when: a m = 4 and v = 4.8 b m = 6 and v = 3.6
c m = 0.2 and v = 20.
22 The volume of a cylinder can be found using the formula V = πr2h where r is the radius of the cylinder
and h is the height. Use the formula to determine which of the following cylinders has the greater volume: Cylinder A: with radius of 6 cm and a height of 5.2 cm Cylinder B: with a radius of 5.2 cm and a height of 6 cm.
23 The surface area of a cylinder can be found using the formula SA = 2πr2 + 2πrh. Use the formula to determine which of the cylinders in question 22 has the greater surface area.
11e
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 onestep equation. The solution to these equations uses only one of the above four steps. Worked example 13
Solve each of the following equations. a x + 48 = 75
b y − 43 = 56
c 7d = 91
d
Think
a 1 Write the equation. 2
Subtract 48 from both sides.
q = 29 13
WriTe
a x + 48 = 75
x = 27 ChapTer 11 • Algebraic manipulation
379
b y − 43 = 56
b 1 Write the equation. 2
y = 99
Add 43 to both sides.
c 1 Write the equation. 2
d 1 2
c
7d = 91
d
d = 13 q = 29 13 q = 377
Divide both sides by 7. Write the equation. Multiply both sides by 13.
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 14
Solve the equations. a 12 + 3x = 45
b
m − 14 = −25 7
Think
WriTe
a 12 + 3x = 45
a 1 Write the equation. 2
Subtract 12 from each side.
3
Divide both sides by 3.
3x = 33
b 1 Write the equation. 2
Add 14 to each side.
3
Multiply both sides by 7.
b
x = 11 m − 14 = −25 7 m = −11 7 m = −77
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. 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 15
9C + 32 is used to convert degrees Celsius to degrees Fahrenheit. Use the 5 formula to convert 68° Fahrenheit to degrees Celsius. The formula F = Think
380
WriTe
1
Write the formula.
2
Substitute 68 for F.
3
Multiply both sides of the equation by 5.
9C + 32 5 9C 68 = + 32 5 340 = 9C + 160
4
Subtract 160 from each side.
180 = 9C
5
Divide both sides by 9.
Maths Quest Preliminary Mathematics General
F=
C = 20
exercise 11e
Solving linear equations
1 We13 Solve each of the following onestep equations. a z + 24 = 67 b w − 34 = 54
y d = 19 14 g 17x = 306
e r + 387 = 435
p = 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 2 mC Which of the following is the exact solution to 7x = 23? 2 7 a x= B x = 37 h
23
C x = 3.28
3 Solve each of the following. a 5x = 23 d −3x = 20
f
t − 253 = 78
i
e + 79 = 45
l
−7a = 84
o d + 8.5 = 13.7 r
4
1
t − 25 = 32
diGiTal doC Spreadsheet doc1496 equation solver
d x = 3.29
b 7x = 45 e 13x = 45
c 6x = 37 f 9x = 2
4 We14 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
c 9q = 162
q 8y −
3 4
=
1 22
c f i l o
3z + 6 = 27 13x − 85 = 227 4c + 70 = 2 9v − 10 = 5 23 − 2b = −1
r
1 − 7h = −65
diGiTal doC Spreadsheet doc1497 Solving equations
diGiTal doC GC program — Casio doc1498 equations
5 We15 Solve each of the following equations.
s v r f + 7 = 12 b −8=9 c + 5 = 11 d − 1 = 12 3 7 8 4 p s k v e + 3 = 11 f − 10 = 2 g + 20 = 27 h −4=0 11 4 15 3 m j p g l − 10 = −4 k + 35 = −4 j − 13 = −11 i − 1 = −1 6 10 7 8 6 Solve each of the following equations. 4y 3p 2q a =8 b =9 c =8 5 2 3 2s 5w 12m d = −8 e = 10 f = −6 5 9 5 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 3 = c 32p = −256 ( p = −8) d (b = 6 4 ) 9 4 e 5t − 98 = 56 (t = 30.8) f 7y + 13 = −65 ( y = −11.1) 3w − 2 2e 2 =3 g (w = 4 3 ) h +1=7 (e = 9) 4 3 a
8 mC For which of the following equations is x = 12 not a solution?
diGiTal doC GC program — TI doc1499 equations
diGiTal doC GC program — Casio doc1500 expanding
diGiTal doC GC program — TI doc1501 expanding
5x x−4 + 1 = 16 B 4x − 7 = 53 − x C 4x − 12 = 48 − x d =4−x 4 2 9 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 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 a
ChapTer 11 • Algebraic manipulation
381
j
6(6g + 5) = 210
m 6(z − 2) = 44
k 4(5g − 1) = −44
l
7(3v − 11) = −161
n 3(6y + 13) = 76
o 5(4u − 9) = 34
10 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. 11 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.
12 In the formula A = lb, calculate: a l, when A = 437 and b = 23
b b, when A = 36.225 and l = 6.3.
13 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.
PRT , find R when I = 500, 100 P = 2500 and T = 2. Given that T = a + 8d, find d when T = 59 and a = 11. Given that P = 2l + 2w, find w when l = 34 and P = 176. 1 Given that A = 2 bh, find h when A = 19.43 and b = 5.8. Given that V = lbh, find b when V = 74.375, l = 2.5 and h = 3.5. D Given that S = , find D when S = 90 and T = 12. T
14 a Given that I = b c d e f
Fur ther development 15 Solve the following linear equations. a 11 – 6x = 59
b 21 – 9x = 3
c 63 – 7x = 21
d 15 – 6x = 2
16 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: b $49.00
a $29.00
382
Maths Quest Preliminary Mathematics General
c $25.25.
17 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: i 1 minute ii 5 minutes iii 8 minutes. c Calculate the length of a call for which the charge is: i $8.10 ii $16.90 iii $34.50.
diGiTal doC WorkSHEET 11.2 doc10339
ChapTer 11 • Algebraic manipulation
383
Summary operations with algebraic expressions
• Like terms are those which use the same pronumeral. • We can only simplify expressions involving addition and subtraction that contain like terms. • When multiplying and dividing algebraic expressions we need to use the index laws. ax Index Law 1: a x × ay = a x+y = ax− y or x y x − y Index Law 2: a ÷ a = a ay • 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.
Further multiplication and division
• To multiply fractions, simplify where possible, then multiply the numerators together and the denominators together. • To divide fractions, multiply the first fraction by the reciprocal of the second fraction.
expanding and simplifying algebraic expressions
• Expansion means to multiply everything inside the brackets by what is directly outside. • After expanding simplify by collecting any 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.
Solving linear 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.
384
Maths Quest Preliminary Mathematics General
Chapter review 1 7x − 5y − 6x + 4y = a x+y
2 5x4 × 3x3 = a 8x7
B x−y
C x + 9y
d x − 9y
B 8x12
C 15x7
d 15x12
C 48
d 144
C x=4
d x=5
m U lTip l e C ho iC e
3 The value of 3m2 when m = −4 is a −144
B −48
4 The solution to the equation 1 − 2x = −9 is a x = −5
B x = −4
1 Simplify each of the following. a y+y+y+y d 15t − 9t
b 8w + 9w e 6q − 5q
c 6r + 9r − r f 9x + 6x − x
2 Simplify each of the following. a 8m + 4n − 3m d 15m − 7 + m + 1
b 6a + 4 − 3a − 9 e 5x + 20 + 3x − 6
c 12k − 5l + 3l − 8k f 12m − 20 − 2m + 4
S ho rT a n S W er
3 Simplify each of the following. a w3 × w5
b a × a6
×p
e
5p3
i
5x5y4 × 6x7y6
c 4x5 × 6x3
5y × 6y
g
j
5ab3 × 4a3b2
k 7g3h2 × 4gh
l
o 45r6 ÷ 5r3
p 63y5 ÷ 7y
m b6 ÷ b2
n f4 ÷ f
q 45r5s2 ÷ 5r4s3
r
36s3t5 ÷ 9st
s
28 p v k5 × k4 ÷ k3 4 12 p 4 Simplify the following expressions. 2 y a 12 a b × × y 8 6 a e a × 4 × 10 f a÷a 12 5 a 2 4 5m 15m 20 a 4 ab j i ÷ ÷ 6 y 3 xy 14c 7c
×
d 9 × 4q3
f
6x2
h 4r4 × 3r4
8x4
64 a6 16a 4
t
12m4n3 × 4mn3 32m 6 8m 4
u
7 x × 2 x 14 g 5 ÷ 30 m m
5 Expand each of the following. a m(m + 3)
b 5p(2p − 6q)
− 3)
4pq(3p2
d
2w3(3w2
e
1 6p × 3p 5 24 8 h ÷ 7y y
c
−
d
c x3(4x5 − 2)
2q4)
f
7a6(3a8 − 9b2)
a , find the value of S when a = 20 and r = 12 . 1− r 3e + 18 7 The formula N = + 70 is used to calculate the number of videorecorders, N, that can be 5 produced by ‘e’ employees. Calculate the number of videorecorders that can be produced by 89 employees. h 8 In the formula A = (a + b), calculate the value of A when h = 5, a = 8.5 and b = 6.2. 2 1 9 Given that S = ut + at2, find S when u = 9.5, t = 5 and a = 5.8. 2 6 In the formula S =
10 Solve each of the following equations.
d = 42 23
a a + 98 = 165
b b − 76 = 84
c 43c = 3827
d
e −8e = −96
f
−f = 19 4
g g + 45 = 12
h 9h = 25
ChapTer 11 • Algebraic manipulation
385
i
12 − i = 23
j
m 45 + 3m = 18
4j − 17 = 47
n 33 − 4n = 7
k 7k + 13 = 76 o
t + 9 = 17 5
−3r = −15 7 11 Solve the following equations by substituting the given values. a A = lw (A = 56, w = 7) b P = 2l + 2w a 5 c C = (F − 32) (C = 25) d S= 9 1− r q
e x T ended r e SponS e
1 In the formula v = ut + a b 2a b
diGiTal doC Test Yourself doc10340 Chapter 11
386
5q = 15 4
c d
l
5l + 43 = −2
p 8−
p = −5 5
r
1 2
at2: calculate the value of v when u = 0.8, t = 12 and a = 6 calculate the value of u when v = 100, t = 4 and a = 1.6. Simplify 4x + 9y − 5x − 8y. 4 x 3 y2 × 6 x 2 y4 Simplify . 3 xy8 (2 x 3 y) Simplify the expression . 8 x 5 y9 Solve the equation 7x + 15 = 113.
Maths Quest Preliminary Mathematics General
(P = 94, l = 16) (S = 6, a = 3)
ICT activities 11C expanding and simplifying algebraic expressions diGiTal doC • WorkSHEET 11.1 (doc10338): Solve problems involving algebra. (page 375)
11d
Substitution
diGiTal doCS • Spreadsheet (doc1489): Substitution (page 377) • Spreadsheet (doc1490): Substitution game (page 377)
11e
Solving linear equations
• • • • •
GC program — Casio (doc1498): Equations (page 381) GC program — TI (doc1499): Equations (page 381) GC program — Casio (doc1500): Expanding (page 381) GC program — TI (doc1501): Expanding (page 381) WorkSHEET 11.2 (doc10339): Solve problems involving algebra. (page 383)
Chapter review • Test Yourself Chapter 11 (doc10340): Take the endofchapter test to check your progress. (page 386)
To access eBookPLUS activities, log on to www.jacplus.com.au
diGiTal doCS • Spreadsheet (doc1496): Equation solver (page 381) • Spreadsheet (doc1497): Solving equations (page 381)
ChapTer 11 • Algebraic manipulation
387
Answers CHAPTER 11 alGeBraiC manipUlaTion exercise 11a operations with algebraic expressions 1 a 2t b 3r c 6w d 4x e 7q f 8n 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 39e d 47j e 25k f 4k g 8r h w i −4m j 2w l 22m k −7s mj n 6p o −z 4 a 10x + 7 b 4h − 4 c 13k d 3p + 7q f 13t e 5 + 4w g 4j + 8k h 3x − 15 i 16b − 26 j 3a − 8 k 17b − 6 l 4r − 17 m 10y − 6 n 3c − b o 13d − 8 p 19e − 4p r 2w q 20t − 23s 5 a 24 b 76 c 92 d q7 e p5 f w3 6 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 7 a q7 b x11 c y15 d 47 6 e a f s3 g 6b6 h 10d8 10 i 35g j 27j 6 4 k 8k l 24m2 m 20x7y9 n 20m5n9 5 9 o 63a b p 48j 4k10 5 3 q 50p q r 54r4s3t10 4 8 a a b b4 c c2 d e4 2 e 6g f 6h4 3 2 g 8j k h 4m3n 8 3 i 7p q j 9r3 t3 k 12s2 l 3 9 The order of terms when multiplied does not matter, they give the same result; hence, 4xy and 5yx are like terms, but 4x2 and 5x will always give different values. 3 p5 8b 5 10 a b 2 3 2 2 c 3a d m 2 4 p rs 9a5bc 11 a b 3 2
c
388
20 f 6 g 2 h 4 3
12 Gordon is not correct. (5x4)2 = 25x8 while
(5x2)4 = 625x8.
exercise 11B
and division 10 a 1 a 3 16 c c
6b 5 18 d 5d f h
3x 2 a 10
b
3
c 4 3 a
4x y
4y x 5x e 4y c
6z 7x 3x i 2y 12 z k x g
3
4 a 5 1
c 3 1
e 25
4 y2 7 8 y2 i 9 g
2
k 3 5 a 3ab c
q 4p
e 3 g
m 12
i 14
8 k y 4m m 21 o
d b d
6f 7h 2 2 2y 5 3 v 3x y 9x 4y
3w 2x 2z h 7x f
j
5 24
l
x 6w
b
2 9
d 3 f h j
35 6
y2 2x b y b d 5a 2 l
f
1 3
h
n 6m
j
1 9
l
5 3
n
2q 45
b 2x
1 c 3y
d
p 4q
e 3
f
1 3
Maths Quest Preliminary Mathematics General
or
2 y2 25 32 xy 15
1 4
6 a 3ab
3a 2 3m 3 k 5 3q m 5 x3 o 5y
b
g 2
h
i
Further multiplication
e 6e
g 4
5 6
j l n
exercise 11C expanding algebraic expressions 1 a 3x + 6 b c 5m + 20 d e 4x + 4 f g −4y − 24 h i −3p + 6 j k −x − 3 l m 6b − 12 n o −30m + 24 p 2 a x2 + 2x b c a2 + 5a d e 4x + x2 f g 7m − m2 h i 2xy + 4x j k −3xy − 12y l m −3b + ab n o −30a + 18a2 p 3 a 2p − 2 b c −7p − 17 d e 6x2 − 20x f g 3px + 6x − 5 h i −4p2 + 13p j k 2m2 − 8m − 4 l m −30a + 10ab n o −2p − 17 p 4 a 8x + y b c 18a + 29b d e 2m − 11n f g 9x + 5y h i −4a + 3b j k 5p − 10q l m 13x − 19y n o −5a + 3b p 5 a 2ab + 2a − 3b b c 2cd + 3c d e 5cd − 11c f g 5m h i 6m2 + 6m − 10 6 a B b A 7 a 9cd − 8c b c −8c2 + 3d2 + 22cd d e 2p2 + 7p − 6 f g −10y2 − 6y − 12 exercise 11d
1 a d g j 2 a d g
5 6 30 27 −11 30 −125
3 10
7b 5 4 8b 3c 1 2
and simplifying
4x + 12 2p + 10 7x − 7 −5a − 5 −x + 1 −x + 2 24m − 16 −27p + 15 y2 + 3y c2 + 4c 5y + y2 8q − q2 5pq + 20p −10pq − 90p −35m + 7mn −28x + 16x2 5x − 17 −12p + 3 2m2 + 7m 4y2 − 4y + 7 4x − 13y −3p2 + 10pq − 1 7cd − 12c2 − 5c 3 − 3m 10p + 8q 19c + 22d 6x − 11y −14p − 8q 4x − y −8c + 9d −7p + 11q 8c + d 2xy + 4x − 2y 2pq − 2p 5ab − 21a − 3b 6cd − 36c c D −15a2 + 2b2 − 9ab 12m2 − 20m − 4 −7x2 + 41x − 6
Substitution b 2 e −17 h 12 k 30 b −1 e −24 h 1
c f i l c f i
0 3 −12 −5 1 36 15
3 a
7 12
d 1
1 3
4 a 68 °F 5 a 30 d 136 6 1.335 7 a 600 d 32.83 g 100 j 25.5 8 a C = 4.5 + 0.6d b i $7.50 9C 10 a 134.04 11 a 14 12 a 13 14 15 16 17
18 19 20 21 22 23
2 3
a 3 mL
2 3 4
e
1 12
1 576
c
1 12
f 48
b 95 °F b 29.2 e 973.14
c 22.46 °F c 6.93 f 4.31
b −8 e 236.6 h 80
c 176 f 75 i −85.4
ii $16.50
iii $34.50
b 18 229.61 b 0.1
c 214.94 c 2.5
b 1.81
c 1.88
b 3.6 mL
c 6.4 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 a 10 b 17 a 63 cm3 b 576 cm3 a 10 b 15 a 46.08 b 38.88 Cylinder A Cylinder A
exercise 11e
1
b −
g e = −2
h d = −2
j r = 11.6
k f = −2
mt = 6
n g = −3
p s = −1.55 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 8D 9 a b=5 d p = 10 g t=5 j g=5 mz = 9
c 12 mL
c c c c
8.5 8.28 cm3 20 40
Solving linear equations a z = 43 b w = 88 c q = 18 d y = 266 e r = 48 f t = 331 g x = 18 h p = 19 573 i e = −34 j k = 22 k b = −16 l a = −12 m u = −180 n f = 40 o d = 5.2 2 3 q v = 30.4 p c=3 r t=6 5 10 B 3 3 1 a x=4 b x=6 c x=6 5 7 6 2 6 2 d x = −6 e x=3 f x= 3 13 9 a a=6 b q = 17 c z=7 d s=4 e w = 17 f x = 24
q y=
30 x + 20 5 b $12 b=7 a l = 19 a l = 14 a R = 10 d h = 6.7 a x = −8
15
c x=6
l v=1 r h=9
v = 119 p = 88 v = 12 p = −273 p=6 w = 18 Incorrect Correct Correct
n=2 k = 26 n=4 g = −2 1 n y=2 18 =
b b b e b
2 3
o b = 12
13 32
c f i l c f c f
3 7
r = 48 s = 48 g=0 m = 36 q = 12 m = −2.5 Correct Incorrect
h = 13 m = 31 r=5 v = −4 o u = 3.95
b e h k
1 3
10 a 11 12 13 14
b e h k b e b e h
i c = −17
6 7
c f i l
25 x + 4 4 b = 5.75 l = 54 d=6 b = 8.5 x=2
d x=2
d
2 5
g
1 6
3a − 5 8x + 14 a7 5p4 12r8 28g4h3 f3 9r q s t 4m2 b e b e h k n
c f c f i l o
4k − 2l 10m − 16 24x8 30y2 30x12y10 48m5n6 9r3
r 4s2t4 u
7 3 p3
b
1 4
c 1
e
2 3
f 2
h
3 7
i
5 2b a m2 + 3m c 4x8 − 2x3 e 12p3q − 8pq5 40 36.75 a a = 67 d d = 966 g g = −33 j j = 16 m m = −9 p p = 65 a l=8 c F = 77
4
x 6
j 5
11
1 6
10p2 − 30pq 6w5 − 6w3 21a14 − 63a6b2 127 120 b = 160 c c = 89 e = 12 f f = −76 i i = −11 h = 27 9 k=9 l l = −9 n = 6.5 o t = 40 q = 12 r r = 35 w = 31 r=1 b d f 7 9
b e h k n q b d
2
exTended reSponSe
1 a 441.6 2 a −x + y c
mUlTiple ChoiCe
3 C
s 4a2 v k6 4 a 2
c l = 2.8 c w = 54 f D = 1080
ChapTer reVieW 2 C
p 9y4
6 8 10
16 a 10 km b 18 km c 8.5 km 17 a C = 1.5 + 2.2m b i $3.70 ii $12.50 iii $19.10 c i 3 min ii 7 min iii 15 min
1 B
2 a 5m + 4n d 16m − 6 3 a w8 d 36q3 g 48x6 j 20a4b5 m b4
1 4 x 2 y8
b 21.8
8x 4 y2 d x = 14 b
4 D
ShorT anSWer
1 a 4y d 6t
b 17w e q
c 14r f 14x
ChapTer 11 • Algebraic manipulation
389
ChapTer 12
Modelling linear relationships ChapTer ConTenTS 12a 12B 12C 12d 12e
Graphing linear functions Gradient and yintercept Drawing graphs using gradient and intercept Simultaneous equations Practical applications of linear functions
12a
Graphing linear functions
Imagine a car travelling at a constant speed of D 60 km/h. The graph at right compares the distance 600 500 travelled with time. 400 This graph can be given by the relation D = 60t. 300 200 This relation is an example of a function. A 100 function is a rule with two variables. In the above 0 0 1 2 3 4 5 6 7 8 9 10 t 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) Wage (W)
10 85
20 170
30 255
40 340
50 425
Draw the graph of wage, W, against hours, H. Think 1
Draw the graph with H on the horizontal axis and W on the vertical axis.
2
Plot the points (10, 85) (20, 170) (30, 255) (40, 340) and (50, 425).
3
Join the points with a straight line.
draW
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. ChapTer 12 • Modelling linear relationships
391
Worked example 2
Draw up a table of values and plot the graph of y = 2 x + 3 and label the line. Think
WriTe/draW
y = 2x + 3
1
Write the rule.
2
Draw up a table and choose simple xvalues.
3
Use the rule to find each yvalue and enter them in the table. When x = −2, y = 2 × −2 + 3 = −1. When x = −1, y = 2 × −1 + 3 = 1. When x = 0, y = 2 × 0 + 3 = 3. When x = 1, y = 2 × 1 + 3 = 5. When x = 2, y = 2 × 2 + 3 = 7.
4
Draw a Cartesian plane and plot the points.
5
Join the points to form a straight line and label the graph.
x
−2
−1
0
1
2
y
−1
1
3
5
7
y 7 6 5 4 3 2 1
y = 2x + 3
0 −5 −4 −3 −2−1 −1 −2 −3 −4 −5
1 2 3 4 5
x
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 1
Draw a table choosing five values for n to substitute.
2
Calculate the values of C for each value of n chosen.
3
392
Draw the axes, plot the points generated and join with a straight line.
Maths Quest Preliminary Mathematics General
WriTe/draW
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
Graphing linear functions
exercise 12a
1 We1 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
diGiTal doC Spreadsheet doc1531 Graph paper
Draw the graph of this function. 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. 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. A
100
Y
8000
200
300
400
diGiTal doC GC program — Casio doc1525 linear
500
16 000 24 000 32 000 40 000
diGiTal doC GC program — TI doc1526 linear
Draw a graph converting Australian dollars to Japanese yen. 4 Complete the following tables of values, plot the
points on a Cartesian plane, and join them to make a linear graph. Label the graphs with the rules. a Rule: y = x + 4 x
−2
−1
0
y
2
3
4
1
−2
y
−6
−2
−1
0
1
2
0
−1
f
0
1
2
y
−2
0
1
−2
2 1
−1
x
−2
y
−7
−1
0
1
2 diGiTal doC GC program — TI doc1528 myrule
−1
Rule: y = −2x x
−2
−1
0
1
2
−2
y
g Rule: y = 4x + 1
x
−1
diGiTal doC GC program — Casio doc1527 myrule
d Rule: y = 2x − 3
−8
y
−2
y
e Rule: y = 3x − 5
x
x
2
c Rule: y = 3x
x
b Rule: y = x − 1
h Rule: y = −5x + 4
0
1 5
2
x y
−2
−1
0
1
2 −6
9
5 We2 Draw up a table of values and plot the graph for each of the following rules. Label each graph. a y=x+2 b y=x−4 c y=x−5 d y=x+6 e y = 5x f y = 7x g y = 4x − 3 h y = 2x + 4 i y = 3x + 2 j y = 2x − 2 k y = −6x + 2 l y = −3x + 2 6 We3 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. 7 Use the graph from question 6 to calculate the cost of a telephone call that lasts for: a 17 minutes b 45 minutes. ChapTer 12 • Modelling linear relationships
393
8 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. 9 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. 10 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. 11 Casey has a job that pays $10 per hour. Draw a graph that will show the money earned against hours
worked. 12 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.
Fur ther development 13 The cost of hiring a tennis court consists of a booking fee and an hourly rate.
a Write an equation for the total cost in terms of the hourly rate. b Sketch a graph of this relationship. c What would be the charge for 3 hours? 14 A singing telegram service charges a $60 appearance fee and $8 per minute sung. a Write an equation for the total cost of a singing telegram in terms of the number of minutes sung. b Sketch a graph of the relationship. c What would be the charge for a 5 minute singing telegram? d How many minutes can be bought with $150? (Whole minutes only) 15 Colleen delivers junk mail and is paid $32 to traverse a particular route and a further 10 cents per leaflet
delivered. a What method of payment is Colleen being paid? b Write an equation for the total payment that she receives. c Sketch a graph of the relationship expressed in b. d What would Colleen’s pay be if she delivered 1650 pamphlets? 16 A pay TV salesperson receives $300 per week plus $20 for every household that he signs up to have
payTV connected. How much does he receive in a week where he signs up 33 households? 17 a Draw a graph of the relationship described in question 16. b What is the point where the graph cuts the vertical axis? c How does this relate to the rate at which he is paid? 18 A person is running at 10 km/h. The speed at which she runs decreases by 1 km/h for every 30 minutes
she has been running. a Draw a graph of the relationship between speed and the time that she has been running. b How long can she run before she will have to stop? 394
Maths Quest Preliminary Mathematics General
inVeSTiGaTe: 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.
12B
Gradient and yintercept
Consider the graph of a car that is travelling D at 60 km/h. Earlier we drew the graph of this 600 as a linear function. 500 400 Two points on this graph are (1, 60) and 300 (2, 120). From the graph we can see that for a 200 100 one unit increase in the independent variable, 0 there is a 60 unit increase in the dependent 0 1 2 3 4 5 6 7 8 9 10 t 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: m=
vertical change in position horizontal change in position
Using this formula, the gradient can be calculated by measurement from a graph by choosing any two points on the graph. The graph at right shows the function C = 2 + 0.5d C C = 2 + 0.5d. (8, 6) 6 On the graph, the two points (2, 3) and (8, 6) 5 3 units 4 are marked. Between these two points the vertical (2, 3) 3 rise = 3 and the horizontal run = 6. Using the gradient 2 6 units 1 formula: 0
gradient =
3 6
=
1 2
0
1
2
3
4
5
6
7
8
9
10 d
Worked example 4
For the linear function drawn below, calculate the gradient. D 20 18 16 14 12 10 8 6 4 2 0
D = 4t
0
1
2
3
4
5
6 t
ChapTer 12 • Modelling linear relationships
395
Think 1
Choose two points on the graph: (1, 4) and (5, 20) for example.
2
Measure the vertical rise and the horizontal run.
WriTe
D 20 18 16 14 12 10 8 6 (1, 4) 4 2 0 0 1
gradient =
D = 4t (5, 20)
16
4 2
3
4
6 t
5
vertical change in position horizontal change in position
3
Write the gradient formula.
4
Substitute for the rise and the run.
=
5
Calculate the gradient.
=4
16 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
For the function drawn at right calculate the gradient.
y 5 4 3 2 1
Think 1
Choose two points on the graph. In this case we choose (0, 4) and (4, 0).
2
Measure the vertical rise and the horizontal run.
−5−4−3−2−1 0 −1 −2 −3 −4 −5
WriTe
1 2 3 4 5
y=4−x
y 5 (0, 4) 4 4 3 2 1 −5−4−3−2−1 0 −1 −2 −3 −4 −5
396
gradient =
x
y=4−x
vertical change in position horizontal change in position
3
Write the gradient formula.
4
Substitute for the rise and the run.
=
5
Calculate the gradient.
= −1
Maths Quest Preliminary Mathematics General
−4 (4, 0) 1 2 3 4 5
−4 4
x
The gradient of −1 in the example above means that for every oneunit increase in x, there is a one unit decrease in y. In the example above, the graph cuts the yaxis at 4. Therefore, for this function the yintercept is 4. During this course you will need to look at using linear graphs in a practical context. In such a context you need to consider which of the variables is the independent variable, which is the dependent variable and the meaning of both the gradient and vertical intercept in context. Consider the scenario in Worked example 3. A preschool hired an C entertainment group in a concert and the cost of hiring the entertainment 800 700 group was given by the equation C = 80 + 3n. 600 The independent variable is the number of students who attend the 500 400 concert. 300 The dependent variable is the cost of staging the concert. This is 200 100 because the cost depends on the number of students who attend. 0 The gradient of this graph is 3. In this situation the cost of staging the 0 50 100 150 200 n concert rises by $3 for every extra student who attends. The vertical intercept of the graph is 80. In this situation this means there is a cost of $80 to stage the concert even if no students attend. This is known as a fixed cost.
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
$600
a Draw a graph of the cost of this excursion. b Calculate the gradient and explain its meaning in this context. c Use your graph to find the intercept on the vertical axis and explain its meaning in this context. Think
a
Cost ($)
a 1 Draw a set of axes and plot the points given.
WriTe/draW
0
20 40 60 80 100 Number of students
Join with a straight line.
b 1 Choose two points on the graph and measure
b
the vertical change in position and horizontal change in position. Cost ($)
2
600 550 500 450 400 350 300 250 200 150 100 50 0
600 550 500 450 400 350 300 250 200 150 100 50 0
100 20
0
20 40 60 80 100 Number of students
ChapTer 12 • Modelling linear relationships
397
2
gradient =
Calculate the gradient.
=
vertical change in position horizontal change in position 100 20
=5 3
The gradient is the increased cost of the excursion per student.
c 1 Find the point where the graph cuts the
vertical axis. 2
10 9 8 7 6 5 4 3 2 1 0
y
b
100 90 80 70 60 50 40 30 20 10 0 01 23 45
01 23 45 6 x
y 10 9 8 7 6 5 4 3 2 1 0
The excursion has a fixed cost of $100, meaning it would cost $100 even if no children attended.
Gradient and yintercept
1 We4, 5 For the functions below, find the gradient. a y
c
c Intercept = 100
The intercept is the fixed cost of running an excursion without considering the number of students.
exercise 12B
diGiTal doC Spreadsheet doc1530 Gradient
A gradient of 5 means that the cost of the excursion increases by $5 for every student who attends.
d
y 10 9 8 7 6 5 4 3 2 1 0
01 23 4 x
x
0 1 2 3 4 5 6 7 8 9 10
2 mC Which of the functions below has a negative gradient? a
B
y
diGiTal doC Spreadsheet doc1531 Graph paper
x
C
x
d
y
x
398
y
Maths Quest Preliminary Mathematics General
y
x
x
3 We6 The table below shows the payment made to a person on a newspaper delivery round.
Deliveries Payment
200 120
400 180
600 240
800 300
1000 360
a Draw the graph of the function. b Find the gradient of the function. c Find the intercept on the vertical axis. 4 The table below shows the profit or loss made by a cinema for showing a movie.
No. of people Profit
20
50 0
−60
100 100
150 200
200 300
a Draw the graph of the function. b Find the gradient of the function. Explain the meaning of the gradient in this context. c Find the intercept on the vertical axis. Explain its meaning in this context. 5 A function is given by the rule y = 5x − 4. a Copy and complete the table below.
x y
0
1
2
b Draw the graph of this function. c Find the gradient and intercept of this function. 6 The graph at right represents the cost of hiring a generator. a Find the gradient and vertical intercept of this function. b What does the vertical intercept mean in this case? c What does the gradient mean in this case?
3
T ($) 3200 1700 200 1
7 The graph at right shows the cost of a singing telegram. a Find the gradient and vertical intercept of this function. b What does the vertical intercept mean in this case? c What does the gradient mean in this case?
2 n (Days)
Cost ($) 76 68 60
Fur ther development
1 2 Time (min)
8 a Sketch two different graphs that have the same gradient. b What can be concluded about lines that have the same gradient? 9 a Sketch a graph that has a gradient of zero. b What can be said about lines with a zero gradient? 10 a Sketch a vertical line. b Why is it not possible to find the gradient of a vertical line? 11 Bughar plans the construction of a proposed driveway on a plan which is below.
Garage
way Drive 2 m
17 m
What is the gradient of the proposed driveway? ChapTer 12 • Modelling linear relationships
399
12 An assembly line is pictured below.
0.85 m 15 m
What is the gradient of the sloping section? Give your answer as a simplified fraction. 13 A passenger jet takes off on the following path.
150 m 110 m Runway 500 m
What is the gradient of the planes ascent? 14 The picture below shows the cost of hiring a tennis court.
Hire Ch
arges
Book in Hourly g fee $5 rate $10
Explain how the booking fee and the hourly rate relate to the function that would represent the hire cost of the court.
drawing graphs using gradient and intercept y 12C
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 yintercept) is 1. Comparing the gradient and yintercept 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 (yintercept) to complete the function. Any linear function can be written in the form y = mx + b where m = gradient and b = yintercept. 400
Maths Quest Preliminary Mathematics General
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 yintercept of: a y = 3x − 4
b y = 7 − 2x.
Think
a 1 The gradient is the coefficient of x (3). 2
a
The yintercept is the constant term (7).
gradient = 3 yintercept = −4
The yintercept is the constant term (−4).
b 1 The gradient is the coefficient of x (−2). 2
WriTe
b
gradient = −2 yintercept = 7
We can use the gradient and yintercept to draw the graph of a function in the form y = mx + b. By plotting the yintercept 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 yintercept 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
Find the gradient (3).
2
Find the yintercept (−2).
3
Mark the yintercept on the axis.
4
Count a rise of 3 and a run of 1 to mark the point (1, 1).
5
From (1, 1) count a rise of 3 and a run of 1 to mark the point (2, 4).
WriTe
gradient = 3 yintercept = −2
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 5 4 3 2 1 −5−4−3−2−1 0 −1 −2 −3 −4 −5
1 2 3 4 5
x
y = 3x − 2
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. ChapTer 12 • Modelling linear relationships
401
Worked example 9 2
Sketch the graph of y = 3 x − 2. Think
WriTe 2
1
Find the gradient ( 3 ).
2
Find the yintercept (−2).
3
Mark the yintercept on the axis.
4
Count a rise of 2 and a run of 3 to mark the point (3, 0).
5
From (3, 0) count a rise of 2 and a run of 3 to mark the point (6, 2).
gradient =
2 3
yintercept = −2
y 5 4 3 2 1 −4−3−2−1 0 −1 −2 −3 −4 −5
6
Join these points with a straight line.
1 2 3 4 5 6
x
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 − 2 x. Think 1
Find the gradient (−2).
2
Find the yintercept (3).
3
Mark the yintercept on the axis.
4
Count a rise of −2 and a run of 1 to mark the point (1, 1).
5
From (1, 1) count a rise of −2 and a run of 1 to mark the point (2, −1).
WriTe/draW
gradient = −2 yintercept = 3
y 5 4 3 2 1 −5−4−3−2−1 0 −1 −2 −3 −4 −5
402
Maths Quest Preliminary Mathematics General
1 2 3 4 5
x
6
Join these points with a straight line.
y 5 4 3 2 1 −5−4−3−2−1 0 −1 −2 −3 −4 −5
1 2 3 4 5
x
y = 3 − 2x
drawing graphs using gradient and intercept exercise 12C
1 We7 For each of the functions below, state the gradient and the yintercept. a y = 2x + 2 b y = 3x − 8 c y = 2 − 4x 3
d y = 4x + 3
e y=
x +1 2
f
3
y = 3 − 2x
diGiTa