General Maths Preliminary Course Textbook

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I N S I G H T

General Mathematics Preliminary Course John Ley Michael Fuller

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NSW General Mathematics

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1 253 Normanby Road, South Melbourne, Australia Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Athens Auckland Bangkok Bogotá Buenos Aires Calcutta Cape Town Chennai Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Paris Port Moresby São Paulo Singapore Taipei Tokyo Toronto Warsaw with associated companies in Berlin Ibadan OXFORD is a registered trade mark of Oxford University Press in the UK and in certain other countries © John Ley 2000 First published 2000 This book is copyright. Apart from any fair dealing for the purposes of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission. Enquiries to be made to Oxford University Press. Copying for educational purposes Where copies of part or the whole of the book are made under Part VB of the Copyright Act, the law requires that prescribed procedures be followed. For information, contact the Copyright Agency Limited. National Library of Australia Cataloguing-in-Publication data: Ley, John, 1958– . Insight general mathematics: preliminary. Includes index. ISBN 0 19 551312 6. 1. Mathematics — Problems, exercises, etc. 2. Mathematics. I. Fuller, Michael. II. Title. 510 Typeset & Illustrated by Raksar Nominees Pty Ltd Cover design by Judith Grace Printed through Bookpac Production Services, Singapore 100

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PREFACE 100

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This text is written with consideration for the broad range of abilities of students attempting this course. All exercises are graded from easy to more difficult. It is not expected that every question in each exercise should be attempted, rather that the teachers use their professional judgement on the number of questions that particular students attempt. Teachers need to be aware that in order to complete the course in the required amount of time it is necessary to average approximately two weeks per chapter. The first chapter on modelling is not intended to be completed as a distinct chapter but rather that it should be integrated into the whole of the course. Each chapter has a diagnostic test in multiple choice format with each question linked to the appropriate exercise in the chapter. There are four review exercises at the end of each chapter. It is suggested that two of these are completed at the conclusion of the chapter and that the other two are completed throughout the course as revision. There are three cumulative reviews that may be used for revision and consolidation. It is intended that computers and graphics calculators are used to enhance the learning process. There are literacy sections at the end of each chapter along with research activities throughout the text. The text has been written by experienced New South Wales mathematics teachers and adheres closely to the syllabus outcomes as presented in its final form. The content as presented in the text provides the learning experiences needed by students to be successful in the course. The students are provided with comprehensive worked examples and structured exercises and reviews that consolidate the knowledge of students challenged by the course and extend competent students. As a text this stand-alone resource enables teachers to confidently and completely teach the course. The authors wish to acknowledge that this course introduced in 2000 contains interesting and challenging material for teachers and students. We trust that you will find the text an invaluable resource in the successful implementation of the General Mathematics course.

ACKNOWLEDGEMENTS The authors John Ley and Michael Fuller wish to thank the following people and institutions for their contributions to this book. 100

Sandra, Susan and Bob Haese for their commitment to this series, their patience and helpful suggestions, Sandra Haese and Maree Armstrong for proof reading, Louise for her answers, Peter and Joanna Poturaj and John Martin for artwork and diagrams, John Sherwell for Netball photograph, Jennifer and Maree for their patience, love, support and assistance.

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TABLE OF CONTENTS

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MATHEMATICAL MODELLING A.

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E. F. G. H. I. J. K.

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Rounding numbers Scientific notation Units of measurement Investigation 1: Modelling Percentages Investigation 2: Modelling Error in measurement Greatest possible error (absolute error) and percentage error Calculations involving measurements Rates Concentrations Ratio Solving problems using the unitary method Diagnostic Test Review Set 2A Review Set 2B Review Set 2C Review Set 2D

20 23 28 31 31 36 37 38 41 43 48 56 60 63 64 66 67 68

EARNING MONEY A. B. C. D. E. F. G. H. I. J.

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The modelling process

MEASUREMENT A. B. C.

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Earning an income Investigation 1: Time periods Allowances and loadings Commission Bonus and holiday loading Piecework and royalties Incomes from the government Investigation 2: Pensions and allowances Deductions Financial institutions Investigation 3: Account charges Budgeting Bills Diagnostic Test Review Set 3A Review Set 3B Review Set 3C Review Set 3D

70 74 74 78 82 84 86 90 90 94 95 96 100 105 107 108 109 110

ALGEBRAIC MODELLING A. B. C. D. E.

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Linear number patterns Substitution Adding and subtracting terms Multiplication and division of algebraic terms Expansions

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TABLE OF CONTENTS 100

F. G. H.

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One step equations Two step equations Further equations Investigation 1: Graphics calculators Substitution and evaluation of formulae Diagnostic Test Review Set 4A Review Set 4B Review Set 4C Review Set 4D

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DATA COLLECTION AND SAMPLING A. B. C. D. E. F. G. H. I. J. K. L. M.

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Questioning Classification of data Identifying populations being investigated Bias in sampling Tables of random numbers Calculator generation of random numbers Generating random numbers on a computer The random sample Investigation 1: The “capture-recapture” method Stratified random sample Systematic sample Suitability of sample types The effect of the sample size Diagnostic Test Review Set 5A Review Set 5B Review Set 5C Review Set 5D

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Cumulative Review (Chapters 2 – 5)

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6.

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APPLICATION OF AREA AND VOLUME A. B. C. D. E. F. G.

Areas involving triangles and rectangles Investigation 1: Modelling task Solids, nets and cross-sections Investigation 2: Octahedra Drawing 3D objects Investigation 3: Perspective drawings Surface area Calculating the volume of a solid Investigation 4: Volumes of solids and pyramids Further volume Capacity Diagnostic Test Review Set 6A

188 192 194 198 198 200 203 210 213 213 215 219 221

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TABLE OF CONTENTS

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Review Set 6B Review Set 6C Review Set 6D

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DISPLAYING DATA A.

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B. C. D. E. F. G. H. I. J. K. L. M.

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Frequency distribution tables Investigation 1: Travelling to school Cumulative and relative frequency Grouped frequency distribution tables Stem and leaf plots Sector graphs Divided bar graphs Dot plots Column and bar graphs Line graphs Radar charts Misleading graphs Frequency histograms and polygons Cumulative frequency graphs Diagnostic Test Review Set 7A Review Set 7B Review Set 7C Review Set 7D

228 229 229 231 234 237 241 243 245 248 249 252 256 259 261 263 264 264 265

INVESTING MONEY A. B. C. D. E. F. G.

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Simple interest calculations Simple interest graphs Compound interest Compound interest graphs Shares Investments Inflation and appreciated value Diagnostic Test Review Set 8A Review Set 8B Review Set 8C Review Set 8D

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PROBABILITY A. B. C. D. E. F. G. H. I.

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The language of chance The sample space The sample space for multi-stage events Methods of counting Estimating chance Theoretical probability Comparing experimental and theoretical probability Complementary events Probability for multi-stage events Diagnostic Test Review Set 9A

298 298 300 303 305 309 311 312 316 320 321

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Review Set 9B Review Set 9C Review Set 9D

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10. TAXATION

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A. Taxable income B. Calculating tax C. GST and VAT D Graphing tax rates Diagnostic Test Review Set 10A Review Set 10B Review Set 10C Review Set 10D

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11. SUMMARY STATISTICS

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A. B. C. D. E. F. G. H. I. J. K. L. M. N. O. P.

The mean Using your calculator to find the mean. The mean from a frequency distribution table The mean for grouped data The mode The median Median from a frequency distribution table The median for grouped data Using a stem and leaf plot Relative merits of mean, mode and median Investigation 1: The range Interquartile range Deciles Box-and-whisker plots Standard deviation Another type of standard deviation Investigation 2: Major statistical project Diagnostic Test Review Set 11A Review Set 11B Review Set 11C Review Set 11D

350 350 352 354 355 357 359 360 362 363 367 368 369 374 375 378 381 383 384 385 386 386 387

12. SIMILAR FIGURES A. B. C. D. E. F.


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The enlargement transformation Properties of similar figures Investigation 1: Sides and angles of similar figures Finding sides of similar figures Problems involving similar figures Scale drawings Building plans Investigation 2: Modelling Diagnostic Test


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TABLE OF CONTENTS

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Review Set 12A Review Set 12B Review Set 12C Review Set 12D

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13. MODELLING LINEAR RELATIONSHIPS A. B.

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Sketching graphs Gradient and intercept Investigation 1: Road signs Linear graphs Direct variation Other linear functions Simultaneous equations Line of best fit Diagnostic Test Review Set 13A Review Set 13B Review Set 13C Review Set 13D

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14. RIGHT-ANGLED TRIANGLES A. B. C. D. E. F. G. H. I. J.

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Pythagoras’ theorem Naming the sides of right-angled triangles Defining the trigonometric ratios Finding values of trigonometric ratios Trigonometric ratios by calculator Using the calculator to find the angle Finding angles given two sides Finding sides of triangles Angles of elevation and depression Miscellaneous problems Diagnostic Test Review Set 14A Review Set 14B Review Set 14C Review Set 14D

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ANSWERS

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CHAPTER

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Mathematical modelling AREA OF STUDY

This chapter deals with the application of mathematics to ever yday situations. After completing this chapter you should be able to: 8 construct a model of a real situation 8 find and interpret mathematical solutions 8 evaluate and modify a model 8 present a report.

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MATHEMATICAL MODELLING (Chapter 1)

It is not intended that this chapter be treated as a separate unit of work. Rather it is intended to be an introduction to the process of mathematical modelling which may be referred to throughout the course.

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A

THE MODELLING PROCESS

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Real world problem

Make assumptions

Formulate mathematical problem

Verify the model

Interpret the solution

Solve the mathematical problem

Produce report, explain predict, etc.

Modelling is the basic process by which mathematics is applied to real life problems. The process, as shown in the flow chart, involves the following steps: Take a real world problem, analyse it, simplify it, make assumptions in order to express the mathematical aspects of the problem in terms of variables, graphs, tables, diagrams etc. We then use the skills, processes and content of mathematics to find solutions to the mathematical problem. These solutions need to be interpreted in terms of the real problem and the adequacy of them verified, i.e., are they useful in terms of the context of the problem, or do we need to modify the model?

EXAMPLE

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How can we calculate the number of tiles needed to tile the floor of a room?

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Step 1:

Step 2:

(Analyse, make assumptions.) The factors which are relevant to the solution of this problem are the area of the floor and the area of a tile. To investigate this problem we will assume that the floor is a rectangle, 2000 £ 1550 mm say, and that the tiles are 300 £ 300 mm, a common size for ceramic floor tiles. (Formulate a mathematical problem.) We will assume that the number of tiles can be found by dividing the area of the floor by the area of one tile.

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Step 3:

(Obtain the mathematical solutions of the problem.) Area of floor = 2100 £ 1550 = 3 255 000 mm2

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Area of a tile = 300 £ 300 = 90 000 mm2

3 255 000 90 000 = 36:166::::

) number of tiles needed =

(Interpret and relate the solution to the real problem.) Since we can only buy whole tiles, we need 37 tiles. If we now relate our mathematical solution back to the real problem and check how the tiles will be laid, by drawing a diagram we can see that 37 tiles will be sufficient if the two remaining tiles are cut and the offcuts of these tiles are also cut.

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(Evaluate and modify, if necessary, the model used.) It is not practical to use tiles which have been cut more than once - imagine what the end result would look like if all the small pieces were put next to each other, especially if the tiles have a pattern!

The model we have used (the number of tiles can be found by dividing the area of the floor by the area of one tile) is not satisfactory from a practical view-point. Note that the mathematics in step 3 is perfectly correct; it is the model of the real problem that we have used which needs to be modified. We will now go back to step 2 and change our model: Step 2:

Calculate the number of tiles by multiplying the number of full tiles needed to cover the length of the floor by the number of full tiles needed to cover the width of the floor.

Step 3:

Using the diagram in step 4 above, Number of tiles needed = 7 £ 6 = 42:

Step 4:

The 42 tiles could be laid in 5 rows of 7 complete tiles with each of the 7 tiles in the 6th row being cut to fit the remaining space.

Step 5:

If the floor was 2150 £ 1550 mm then the number of tiles needed = 8 £ 6 = 48: Using a diagram the answer would seem to be correct. By checking the method with different sized rectangular floors our model appears to be satisfactory.

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(Report, explain, predict.) We first calculated the number of tiles needed to cover a rectangular floor by dividing the area of the floor by the area of one tile. In general this method will not work as it requires using different sized offcuts to completely cover the floor. Note however that this method is satisfactory if the dimensions of the floor are multiples of the dimensions of the tiles, for example if the floor was 2100 £ 1500 mm. A method which seems to work for all cases is to multiply the number of tiles needed to cover the length of the room by the number of tiles needed to cover the width of the room.


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Note: The value of our model depends heavily on the assumptions and simplifications we made at step 1. The model may need to be modified if we further consider factors such as: what if the floor is not a rectangle, what if the tiles are not square, should we allow for the gap left between tiles, are the tiles to be laid in a particular pattern, should we allow for breakage, etc.?

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EXAMPLE How long does it take a train to pass through a tunnel? Model 1 (Arithmetic) Step 1:

(Analyse, make assumptions.) The factors which affect the time taken are the length of the tunnel and the length and speed of the train. Assume the tunnel is 100 m long and the train is 80 m long and travelling at 60 km/h.

Step 2:

(Formulate a mathematical problem.) We are required to find how long it is from the time that the front of the train enters the tunnel until the rear of the train leaves the tunnel. The distance travelled by the train in this time is 100+ 80 = 180 m. Its speed is 60 km/h.

Step 3:

(Obtain the mathematical solutions.)

Time =

distance : speed

Since the distance is in metres then we need to convert the speed to metres/second. Speed = 60 km/h 60 £ 1000 = m/s 60 £ 60 = 16:6_ m/s.

180 seconds 16:6_ = 10:8 seconds.

Time taken =

Step 4:

(Interpret the solution.) For a train of length 80 m travelling at 60 km/h, the time it takes to pass through a tunnel of length 100 m is 10:8 seconds.

Step 5:

(Evaluate the model.) The model finds a satisfactory solution for this particular case, but it would obviously be of benefit if we had a model which would provide solutions for other cases.

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Model 2 (Algebraic)

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(Make assumptions.) Let the length of the tunnel be L metres, the length of the train be l metres and the speed of the train be v km/h.

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Step 2:

(Formulate a mathematical problem.) We are required to find the time taken for a train travelling at v km/h to travel a distance of (L + l) metres.

Step 3:

(Obtain the mathematical solution.)

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Speed = v km/h v £ 1000 = m/s 60 £ 60

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Step 4:

(Interpret the solution.) We have found a general formula to find the time taken by a train to pass through a tunnel, given the length of the tunnel and the length and speed of the train.

Step 5:

(Evaluate the model.) Check the solution found in Model 1 by substitution into our formula: 18(100 + 80) time = = 10:8 seconds. 5 £ 60

Step 6:

(Report, explain, predict.) To solve this problem we first investigated a particular case (Model 1). We were then able to use this model to help us find a general (algebraic) solution of the problem. A spreadsheet could now be used to find the times for varying values of the lengths and the speed.

EXAMPLE How much does it cost to run a car?

Step 1:

(Analyse, make assumptions.) We need firstly to make a list of all the costs to be considered. Let us suppose we consider the following items: registration, insurance, loan interest, depreciation, petrol, maintenance and repairs, tyres. In order to cost these items let us make the question more specific: Find the average weekly cost to run a car which is bought for $16 000 using a loan of $10 000. We will assume that the loan interest rate is 13:95% p.a., the annual depreciation rate for this model car is 18%, the car travels 15 000 km per year, the petrol consumption is 11:2 L/100 km, it needs a set of 5 new tyres every 30 000 km. By investigation we find out that the average cost of a service and repairs every 10 000 km is $240, petrol costs 76:4 c/L, tyres cost $95 each, registration is $610 and insurance is $650.

Step 2:

(Formulate a mathematical problem.) We will now use the above information to calculate the total cost for one year of all the above individual items and hence find the average weekly cost.

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Step 3:

(Obtain the mathematical solution.) Loan interest = 13:95% of $10 000 = $1395 Depreciation = 18% of $16 000 = $2880

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Petrol =

15 000 £ 11:2 £ $0:764 = $1283:52 100

15 000 £ $240 = $360 10 000 15 000 Tyres = £ $95 £ 5 = $237:50 30 000 Registration = $610 Insurance = $650 Total annual cost = $7416:02 $7416:02 = $142:62 Average weekly cost = 52

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Maintenance and repairs =

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Step 4:

(Interpret the solution.) This is the average weekly running cost of a car based on the above assumptions.

Step 5:

(Evaluate the model.) This model gives the average weekly running cost for the first year of ownership. The amount of loan interest paid each year will depend on whether the interest is a flat or reducible rate and the amount of depreciation will vary as the car gets older.

Step 6:

(Report, explain, predict.) The model appears to take into account most of the variables involved in the solution of this problem. However the result is only as accurate as the assumptions made. We could refine our solution by investigating further interest rates, the depreciation rates for different models of cars, petrol consumption, insurance -under 25/over 25 years of age, no claim bonuses, how maintenance and repair costs vary with age of the car, etc.

EXAMPLE

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A farmer has 100 m of fencing and wants to fence off a corner of an existing rectangular field. What is the maximum area she can enclose?

Step 1:

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Step 2: existing fence (100 - x) m xm

(Formulate a mathematical problem.)

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If we let the length of the rectangle be x metres, then the width will be (100 ¡ x) metres.

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Hence the area of the rectangular corner is A = x(100 ¡ x):

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We need to find the maximum value of A.



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Step 3:

(Obtain the mathematical solution.)

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Draw the graph of A = x(100 ¡ x):

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Note: x and A must always be positive.

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From the graph, the maximum value of A occurs when x = 50. Hence the maximum value of A = 50(100 ¡ 50) = 2500 m2 :

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Step 4:

(Interpret the solution.) The maximum area which can be enclosed is a square of area 2500 m2 :

Step 5:

(Evaluate the model.) If the corner does not have to be rectangular then it may be possible to enclose a larger area by adding 3, 4, 5 ... sides, or a circular arc, to the existing corner.

Step 6:

(Report, explain, predict.) If the area to be enclosed is a rectangle, using the existing fence for two of the sides, then the maximum area occurs when the length and width are equal, i.e., the corner is a square. We have not investigated whether it is possible, or practical, to enclose a larger area by constructing other shapes.

Note:

It may not always be clear to which step of the process a statement belongs, however we will not overly concern ourselves with this as it is the total process through which models are created and modified that is important.

EXERCISE 1A Use the modelling process to help you solve the following problems. 1 At the end of a netball competition the first four teams all finished with the same number of competition points (from wins, draws and losses). The number of goals for and against each team is shown in the table alongside.

Hawks Eagles Parrots Galas

For

Against

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280 385 350 441

In the semi-finals the first placed team plays the fourth placed team and second plays third. Decide who should play whom in the semi-finals. 100

2 At the inter-school cross-country athletics championships each of the six schools in the district entered five competitors in each age race. The placings in each race were 12 years KHS CHS DHS

13 years MHS CHS CHS

14 years FHS PHS MHS

15 years DHS PHS KHS

16 years FHS KHS CHS

Open PHS DHS MHS

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3 A building supplies company offers the following successive discounts, in any order you request them, on purchases at their store: 15% for tradesmen, 8% for orders over $1000 and 5% for paying cash. Bill the builder buys over $1000 worth of materials and is prepared to pay cash for them. In what order should Bill request the discounts so that he will pay the least amount for his materials? 4 Elizabeth is about to retire from work. She has been paying into a superannuation fund and must decide whether to take a once only lump sum of $300 000 or a pension of $600 per week for the rest of her life. What would you advise her to do? 5 A new car is bought for $25 000 and is sold 8 years later for $5000. What would have been the value of the car after 1, 2, 3, .... 7 years? Predict its value after 9, 10, .... years. 6 $10 000 is invested in an account which pays 8% p.a. interest. How much will the investment be worth in 4 years’ time? 7 Jenny has noticed that the price of liquid petroleum gas (LPG) is much cheaper than petrol and wonders whether she should convert her car to run on LPG. What would you advise her to do? 8 Is it better to rent accommodation or to buy your own home? 9 Is it better to lease or buy a car? 10 George has 18 metres of edging and wants to make a rectangular vegetable garden in his back yard using an existing fence as one side. What dimensions will give the greatest area of garden and what is this maximum area? 11 Suniti has a sheet of cardboard which is 25 cm by 18 cm. She cuts four squares of equal size, one from each corner, and folds up the edges to make an open rectangular shaped container. What is the largest container she can make? (i.e., What is the maximum volume of the container?) 12 A can manufacturer wants to make a can to hold 375 mL of vegetable juice. What dimensions will need the least amount of sheet metal? 13 How many days would it take for Lake Eyre to dry out? 14 How many cats are there in Australia? Scheduling In scheduling problems we usually have a series of tasks to do on a fixed number of “machines” and our problem is to find the best order in which to do them so that the time taken to complete all tasks is a minimum. 100

15 An office has two photocopying machines and receives jobs which will take 7, 9, 5, 10, 11, 4, 4, 7, 10, 9, 10, 4, 7, 10, 11, 4, 5, 7 minutes each. In what order should the jobs be done so that they take the minimum possible time?

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Strategies to try: a Put the jobs on the first available machine in the order in which they were received. b Rearrange the jobs in order from longest to shortest and do the longest jobs first on the first available machine. c Rearrange the jobs in order and do the shortest jobs first on the first available machine.

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Can you find a better strategy for this problem? What is the minimum time needed to complete these tasks? 16 Two typists are given jobs which will take 20, 30, 30, 25, 15, 10, 25, 20, 35, 30, 20, 25, 20, 15, 15 minutes each. What is the shortest time in which the jobs can be completed? How long will it take if a third typist is made available to help? Bin Packing In these types of problems we are given a series of tasks to do and our problem is to find the best order in which to fill the “bins” so that the number of “bins” needed to complete all the tasks is a minimum. 17 What is the mininum number of 1:4 Mb floppy discs needed to store files of the following sizes: 0:6, 0:3, 0:4, 0:2, 0:2, 0:3, 0:4, 0:1, 0:3, 0:5, 0:6, 0:4 Mb? In this problem our “bins” are floppy discs of size 1:4 Mb. We need to find the best order in which to store the files so that the number of discs needed is a minimum. Strategies to try: a Fill each disc with files in the order in which they are received, i.e., load the files onto the first disc in the order in which they are received. When the next file will not fit onto the disc, start a new disc. For example: disc 1: 0:6, 0:3, 0:4 (cumulative total = 1:3 Mb). The next file (0:2 Mb) will not fit on this disc so start disc 2. Disc 2: 0:2, 0:2, etc. b Since the total size of the files is 4:2 Mb then it would seem reasonable to assume that the number of discs needed would be three (1:4 Mb £ 3 = 4:2 Mb) or four, depending on the sizes of the files. Try starting with three discs and load files, in the order in which they are received, onto the first disc with enough space available. c Try starting with three discs and load the largest files first onto the first disc with enough space available. Try your own strategies. 18 Copper hot water pipe was cut into 1 metre lengths. What is the least number of 1 m lengths a plumber will need to use to complete jobs which require the following lengths of pipe to be cut: 600, 450, 300, 280, 850, 250, 325, 195, 560, 240 millimetres?

Queuing 19 Just after a supermarket opens four people arrive at the checkouts and thereafter two people arrive every minute. If one person is served every minute, describe what happens to the queue length if 1, 2, 3, ... checkouts are opened. How many checkouts would you advise the manager to open? Try using tables like the one alongside (for Time (minutes) 0 1 2 3 4 5 6 one checkout open): No. arriving 4 2 2 2 2 2 2 Investigate this problem for different numNo. served 1 1 1 1 1 1 bers of customers and serving times. Queue length 4 5 6 7 8 9 10 You could simulate these numbers by rolling a die or drawing a card from a pack, i.e., roll a die (or draw a card) to give the number of people arriving each minute and then roll the die again (or draw a card) to give the number of people being served each minute. How long does each customer have to wait in the queue?



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20 While a bridge is being repaired it is necessary to close one lane to traffic. Traffic flow on the remaining lane is to be controlled by temporary traffic lights at both ends of the bridge. Investigate the timing sequence of the lights so as to provide for the efficient flow of traffic during repairs. What maximum speed and distance between vehicles should be maintained to ensure the safe flow of traffic?

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CHAPTER

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Measurement

AREA OF STUDY

This chapter deals with metric units of measurement, ratio, rate and errors in measurement. The main mathematical ideas investigated in this chapter are: 8 approximations 8 scientific notation 8 percentages 8 metric units of measurement 8 the accuracy of measurements 8 rates 8 concentrations 8 ratio 8 the unitary method of solving problems.

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MEASUREMENT (Chapter 2)

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ROUNDING NUMBERS

There are many occasions when an exact answer is not necessary or practical. For example,

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If 61 348 people attended a football game, an approximation of 61 000 would be satisfactory for most purposes.

²

If a piece of timber 4 m long has to be cut into 3 equal pieces, then by calculation, each piece should be 1:33333::: m long. It is impossible to cut the timber to this degree of accuracy so we would approximate by making each piece 1:33 m or 1:333 m.

²

If the families of the 24 students in your class have a total of 55 children then the average number of children per family is 55 ¥ 24 = 2:291666... An approximate answer of 2.3 children per family would be more sensible.

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EXAMPLE a c

1

Is 73 closer to 70 or 80? Is 3.5 closer to 3 or 4?

b

Is 483 closer to 400 or 500?

a 73 is closer to 70 than 80, i.e., 73 + 70, rounded to the nearest 10. b 483 is closer to 500 than 400, i.e., 483 + 500, rounded to the nearest 100. c 3:5 is midway between 3 and 4. By convention we round up, hence 3:5 + 4, rounded to the nearest whole number. RULES FOR ROUNDING

To approximate numbers by rounding off, use the following: ² ²

If the digit after the one being rounded off is less than 5 then we round down. If the digit after the one being rounded off is 5 or more then we round up.

EXAMPLE

2

Round off 2563.8 to the nearest a whole number b 10

c

100

d

a

Find the digit in the units column - 2563:8. The digit after this one is greater than 5. Hence, rounding off to the nearest whole number, 2563:8 + 2564.

b

Find the digit in the tens column - 2563.8. The digit after this one is less than 5. Hence, rounding to the nearest 10, 2563:8 + 2560.

1000

The remaining places, up to the decimal point, have to be filled with zeros.

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MEASUREMENT (Chapter 2) 100

c Find the digit in the hundreds column - 2563.8. The digit after this one is greater than 5. Hence, rounding to the nearest 100, 2563:8 + 2600.

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d Find the digit in the thousands column - 2563.8. The digit after this one is 5. Hence, rounding to the nearest 1000, 2563:8 + 3000.

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EXERCISE 2A 1 Round off to the nearest 10 a 76 b 473

c

25

d

42 589

e

4

2 Round off to the nearest 100 a 391 b 34 520

c

1954

d

80

e

48

3 Round off to the nearest 1000 a 5670 b 24 299

c

763 500

d

287

e

560

ROUNDING DECIMAL NUMBERS

Rounding to a number of decimal places indicates the number of digits to be used after the decimal point.

EXAMPLE

3

Round off 9.362 to a 1 decimal place b 2 decimal places a Find the digit in the first place after the decimal point - 9.362. The digit after this one is greater than 5. Hence, rounding to 1 decimal place, 9:362 + 9:4 b Find the digit in the second place after the decimal point - 9.362. The digit after this one is less than 5. Hence, rounding to 2 decimal places, 9:362 + 9:36

4 Round off to 1 decimal place a 13:65 b 3:7264

c

0:28

d

5:091

e

16:037

5 Round off to 2 decimal places a 6:726 b 0:0894

c

32:708

d

285:116 28

e

5:001

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SIGNIFICANT FIGURE ROUNDING

The significant figures in a number are the important or meaningful figures. A crowd of 61 348 is approximated to 61 000 to indicate that only the first 2 figures (digits) are important. It is impossible to cut a piece of timber to a length of 1:333 333::::: m. After the 4th figure, the digits are completely meaningless in this case.

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EXAMPLE

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Round off to i 1 a 293 568

ii 2 iii 3 significant figures b 0:07604

The first significant figure in a number is the first non zero digit starting from the left. a i The first non-zero digit is 2 which is in the 100 000’s column. Hence, rounding off, 293 568 + 300 000 to 1 significant figure (or to the nearest 100 000, in this case).

Fill in the remaining places with zeros.

ii The second significant figure is 9 which is in the 10 000’s column. Hence, rounding off, 293 568 + 290 000 to 2 significant figures (or to the nearest 10 000, in this case). iii The third significant figure is 3 which is in the 1000’s column. Hence, rounding off, 293 568 + 294 000 to 3 significant figures (or to the nearest 1000, in this case). b

i The first non-zero digit is 7 which is in the second column after the decimal point. Hence, rounding off, 0:07604 + 0:08 to 1 significant figure (or to the nearest hundredth, in this case). ii The second significant figure is 6 which is in the third column after the decimal point. Hence, rounding off, 0:07604 + 0:076 to 2 significant figures (or to the nearest thousandth, in this case). iii The third significant figure is 0 which is in the fourth column after the decimal point. Hence, rounding off, 0:07604 + 0:0760 to 3 significant figures (or to the nearest ten thousandth, in this case).

6 Round off to 1 significant figure a 42 600 b 380 e 108 f 0:6529 i 0:00038 j 0:025

c g k

59 0:034 990

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4:6 0:0082 0:099

7 Round off to 2 significant figures a 290 365 b 487 e 2653 f 8:63 i 0:000 162 8 j 0:003 97

c g k

3 960 0:823 0:295

d h l

24:9 0:0487 0:999

8 Round off to 3 significant figures a 3688 b 4 562 800 e 813:4 f 14:294 i 0:6508 j 0:03914

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20 657 1:0482 1:999

9 Round off to i 1 significant figure ii 2 significant figures a 17:256 b 0:450 72 c 521 500 d



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154 299 0:003 508 1 0:9999

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SCIENTIFIC NOTATION

Scientific (or standard) notation is a convenient way of writing very large and very small numbers. To express a number in scientific notation it is written as the product of a number between 1 and 10 and a power of 10, i.e., it is put in the form A £ 10n where A lies between 1 and 10.

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State whether or not the following numbers are expressed in scientific notation.

?

a

5:3 £ 107

78 £ 105

a b c d e

Yes, as the first number is between 1 and 10 and it is multiplied by a power of 10. No, as the first number is not between 1 and 10. No, as the second number is not expressed as a power of 10. Yes, as the first number is between 1 and 10 and it is multiplied by a power of 10. No, as it is not written as a product.

b

4:9£10 000

c

d

3 £ 10¡4

e

294 000

EXERCISE 2B 1 State whether or not the following numbers are written in scientific notation. b 5:2 £ 10 000 c 21 £ 105 d 2:87 £ 10¡6 a 3:6 £ 105 1 1 000 000 7

e

6:07 £

i

41 £ 10

f

594 £ 10¡5

j

7:8503 £ 1015

g

70 £ 108

h

3:06 £ 10¡9

2 Copy and complete the following tables a

b

100 1

101 10

10¡1

10¡2

1 101

1 102

0:1

EXAMPLE

102

104 10 000

1000 10¡3

105 1 000 000

10¡4 1 104

0:001

1 105

0:000 01

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Express the following numbers in scientific notation

a

0:000 001

60 000 = 6 £ 10 000 = 6 £ 104

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a 60 000

b

75

b 7130

7130 = 7:13 £ 1000 = 7:13 £ 103

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3 Express the following numbers in scientific notation a 80 000 b 200 000 c 5000 d 4 000 000 e 46 000 f 4300 g 790 000 h 350 000 i 75 400 j 2 354 000

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EXAMPLE

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Express the following numbers in scientific notation 0:0003 = 3 £ 0:0001 = 3 £ 10 1000

a

Positive powers for large numbers!

=3£

a 0.0003 b

b 0:005 28

0:00528 = 5:28 £ 0:001 1 = 5:28 £ 1000

1 104

= 5:28 £ 1013 = 5:28 £ 10¡3

= 3 £ 10¡4

4 Write in scientific notation a 0:004 b d 0:09 e g 0:005 17 h j 0:000 295

EXAMPLE

0:000 07 0:000 53 0:000 085

c f i

0:0002 0:062 0:005 62

Negative powers for small numbers!

4

Write the basic numeral for (i.e., as ordinary decimal numbers) a 4 £ 106 a

b 5:3 £ 104

4 £ 106 = 4 £ 1 000 000 = 4 000 000

b

c 2:96 £ 105

5:3 £ 104 = 5:3 £ 10 000 = 53 000

5 Write the basic numeral for a 5 £ 106 b 3 £ 104 4 e 4:3 £ 10 f 5:6 £ 103 4 i 3:57 £ 10 j 8:72 £ 106

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c

6 £ 103 2:9 £ 105 2:08 £ 103

2:96 £ 105 = 2:96 £ 100 000 = 296 000

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8 £ 105 9:4 £ 106 7:05 £ 105

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EXAMPLE

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Write the following as ordinary decimal numbers a 7 £ 10¡3 b 3:6 £ 10¡5

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7 £ 10¡3 = 7 £ 1013

a

75

=7£

3:6 £ 10¡5 = 3:6 £ 1015

b

1 1000

= 3:6 £

= 7 £ 0:001 = 0:007

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1 100 000

7:29 £ 10¡4 = 7:29 £ 1014 = 7:29 £

= 3:6 £ 0:000 01 = 0:000 036

1 10 000

= 7:29 £ 0:0001 = 0:000729

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6 Write the following as ordinary decimal numbers a e i

6 £ 10¡5 4:2 £ 10¡5 2:49 £ 10¡4

b f j

3 £ 10¡2 8:3 £ 10¡3 6:29 £ 10¡3

c g k

5 £ 10¡4 3:4 £ 10¡2 1:67 £ 10¡5

9 £ 10¡3 5:8 £ 10¡4 7:06 £ 10¡2

d h l

Here is a time-saving method for writing numbers in scientific notation and vice versa:

EXAMPLE

6

a 138 000

Write in scientific notation a

b 0:000 486

Move the decimal point so that it is positioned between the first and second digits. (This always produces a number between 1 and 10.) In this case we get 1.38000 Count the number of places back to the original position of the decimal point in the number: 1:38000

Number of places = 5 to the right = +5.

This becomes the power of 10, i.e., 138000 = 1:38000 £ 105 = 1:38 £ 105 (we now leave off the zeros) Move the decimal point so that it is positioned between the first and second digits. In this case we get 4.86 Count the number of places back to the original position of the decimal point in the number:

b

00004:86

Number of places = 4 to the left = ¡4

This becomes the power of 10, i.e., 0:000 486 = 4:86 £ 10¡4 100

7 Use the method of Example 6 to write the following numbers in scientific notation a e i

526 000 28 000 000 72 000 000 000 000

b f j

28 000 603 000 000 365 900 000 000

c g

7 000 000 910 000

d h

49 800 13 200 000 000

d h

0:000 029 0:000 06

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8 Use the method of Example 6 to write in scientific notation a e i


0:000 43 0:065 0:000 000 79

b f j

0:008 21 0:000 387 0:000 000 000 065 8


c g

0:000 007 0:000 008 2

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EXAMPLE

a 4:83 £ 107

Write as an ordinary number

b 9:2 £ 10¡6

Reversing the process of Example 5: a Since the power of 10 is +7, then the decimal point is moved back 7 places to the right and so, 4:83 £ 107 = 48 300 000

i.e., 48300000

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b Since the power of 10 is ¡6, then the decimal point is moved back 6 places to the left and so, 9:2 £ 10¡6 = 0:000 009 2

i.e., 00000092

9 Express as an ordinary number b 8:3 £ 108 a 3:4 £ 106 e 5:26 £ 105 f 3:02 £ 1012 i 7 £ 1010 j 4:835 £ 109

c g

2:94 £ 107 2:9 £ 107

d h

2:58 £ 105 8:75 £ 108

10 Write the basic numeral for a 5:9 £ 10¡4 b 3:2 £ 10¡6 ¡7 e 8 £ 10 f 2:64 £ 10¡5 ¡4 i 3:512 £ 10 j 6:05 £ 10¡10

c g

7:1 £ 10¡8 8:67 £ 10¡9

d h

2 £ 10¡3 2:97 £ 10¡6

11 Express the following numbers in scientific notation a The distance of Mars from the Sun is approximately 229 000 000 km. b The diameter of the hydrogen atom is 0.000 000 000 025 4 metres. c The sun produces the same amount of light as 3 000 000 000 000 000 000 000 000 000 candles. d The number of hairs on a person’s head is approximately 130 000. e The number of cells in the human body is 10 000 000 000 000. 12 Change the following to ordinary numbers a b c d e

There are approximately 3:16 £ 107 seconds in a year. The number of different hands of the card game Poker is approximately 2:6 £ 106 . Swarms of locusts have been known to be as large as 3 £ 1010 . The size of the influenza virus is approximately 2:6 £ 10¡4 mm. The diameter of a molecule is 8:9 £ 10¡7 mm. 100

EXAMPLE

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Using your calculator to find a (3:5 £ 107 ) £ (2:4 £ 109 ) p c 2:4 £ 1010 a Press 3.5 EXP 7


×

2.4 EXP 9


b

(6:4 £ 108 ) ¥ (2:5 £ 10¡6 )

d

(1:5 £ 107 )3

=

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Answer: 8:4 £ 1016

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b Press 6.4 EXP 8

÷

c Press Ö` 2:4 EXP 10

2:5 EXP ¡6

Note:

Answer: 1:55 £ 105

=

d Press 1.5 EXP 7 x y 3 25

Answer: 2:56 £ 1014

=

to 3 sig figs

Answer: 3:375 £ 1021

=

If the answer is not displayed in scientific notation you could use the SCI function on the calculator to express the answer in this form.

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13 Calculate, giving the answer in scientific notation, correct to 3 significant figures a c e

(2:6 £ 108 ) £ (4:1 £ 107 ) (9:2 £ 1024 ) £ (3:5 £ 10¡8 ) (8:4 £ 1018 ) ¥ (2:5 £ 107 )

b d f

g

(1:82 £ 10¡6 ) ¥ (2:9 £ 10¡10 ) p (6:8 £ 1017 ) (8 £ 10¡10 )6

h

(5:8 £ 109 ) £ (8:2 £ 1012 ) (5:8 £ 10¡6 ) £ (2:4 £ 10¡9 ) (5:25 £ 1012 ) ¥ (4:2 £ 10¡8 ) p (5:76 £ 1016 )

j l

(3:1 £ 108 )5 (2:8 £ 108 )4 £ (1:6 £ 106 ) ¥ (2:1 £ 1015 )

i k

14 Light travels at 300 000 kilometres/second. a Express this number in scientific notation. b How far does light travel in i 1 minute ii 1 hour iii 1 day iv 1 year? c If it takes light 4.1 minutes to reach Earth from Mars, what is the distance from Earth to Mars? 15 The radius of the Earth is approximately 6400 km. a Calculate the area of the Earth’s surface, to 2 significant figures. (Use A = 4¼r2 ) b Calculate the volume of the Earth, to 2 significant figures. (Use V = 43 ¼r3 ) 16 The radius of the Earth’s orbit around the sun is approximately 1:49 £ 108 km. Assuming that the orbit is circular, calculate the distance travelled by the Earth in one orbit, to 2 significant figures. (Use C = 2¼r) 17 The human brain contains about 1010 cells. a Write this as an ordinary number. b Each of these cells is about 2.8 £ 10¡5 m long. If all the brain cells could be placed next to each other, in a straight line, how long would this line be?

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18 Measure your pulse to determine the number of times your heart beats in a minute. If you live to 75 years of age, how many times will your heart beat in this time?

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UNITS OF MEASUREMENT

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RESEARCH PROJECT: UNITS OF MEASUREMENT Prefix Symbol Value (£ base unit) giga mega kilo k 1000 hecto deca base unit 1 qA_p_ deci centi c milli micro

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The metric system uses base units for each quantity (length, mass, capacity, etc.) with prefixes which indicate the factor by which the base quantity is multiplied. Research and complete thr following table.

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EXERCISE 2C 1 Copy and complete the following conversion diagram for length ´¡10

´¡1000 kilometres (km)

metres (m)

centimetres (cm)

millimetres (mm) ¸¡10

2 Convert a 3.6 km to m d 0.56 km to m g 0.658 m to cm j 3.69 cm to mm m 0.08 km to m 3

a b

i i

b e h k n

8.4 m to mm 2.9 m to cm 45.2 cm to mm 16.37 m to mm 1.065 m to cm

Complete 1 km = ¤ cm. Complete 1 km = ¤ mm.

4 Change a 7000 m to km d 6000 mm to m g 800 cm to m j 14 300 m to km m 94 mm to cm p 23 000 mm to m s 3600 cm to m

b e h k n q t

ii ii

c f i l o

34.82 0.964 15.68 4.265 0.075

m to cm m to mm km to m km to m m to mm

Express this answer in scientific notation. Express the answer in scientific notation.

594 cm to m 40 mm to cm 328 mm to cm 86 cm to m 70 mm to m 14 960 mm to m 72 945 mm to m

c f i l o r

8930 m to km 85 m to km 620 mm to m 630 m to km 24 895 m to km 16 270 cm to m

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5 What would be a convenient unit (mm, cm, m, km) to use to measure the a width of your classroom b length of the text book c height of a student d length of a baby

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distance from Sydney to Brisbane length of a match length of a driveway

length of your foot length of material for a dress distance travelled between railway stations

f h j

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6 Copy and complete the following conversion diagram for mass ´¡1000

´¡1000

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tonnes (t)

kilograms (kg)

grams (g)

milligrams (mg)

0

¸¡1000

7 Convert a 2.7 t to kg d 0.34 t to kg g 1.538 g to mg j 23.49 t to kg m 0.875 kg to g 8

a b c

i i i

b e h k n

4.5 g to mg 5.6 kg to g 23.49 t to kg 0.8 g to mg 2.05 t to kg

Complete 1 t = ¤ g. Complete 1 t = ¤ mg. Complete 1 kg = ¤ g.

9 Change a 4000 kg to t e 750 g to kg i 400 kg to t m 9 mg to g

b f j n

ii ii ii

8500g to kg 45 mg to g 950 mg to g 5 kg to t

10 What would be a convenient unit to a man b d bag of sand e g vitamin pill h j bag of potatoes

c f i l o

23.92 kg to g 1.758 t to kg 0.058 kg to g 4.05 kg to g 0.05 g to mg

Write the answer in scientific notation. Write the answer in scientific notation. Write the answer in scientific notation.

c g k o

1650 mg to g 1480 g to kg 90 g to kg 200 g to kg

d h l

8365 kg to t 1900 kg to t 80 kg to t

use (t, kg, g, mg) to measure the mass of a jar of jam c packet of biscuits truck f elephant knitting needle i paper clip

11 Copy and complete the conversion diagram for capacity ´¡1000 kilolitres (kL)

litres (L)

millilitres (mL)

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12 Convert a 35 kL to L e 0.06 L to mL i 2.804 L to mL

b f j

15.9 L to mL 1.08 kL to L 400 L to mL

c g

1.65 L to mL 0.015 L to mL

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d h

0.85 kL to L 0.005 kL to L

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13 How many millilitres are there in 1 kilolitre? Express your answer in scientific notation.



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14 Change a 15 000 mL to L e 9280 L to kL i 6 mL to L

b f j

8000 L to kL 725 L to kL 2108 mL to L

7600 mL to L 95 mL to L

c g

800 mL to L 40 L to kL

d h

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15 What would be an appropriate unit to use (kL, L, mL) to measure the capacity of a a teaspoon b swimming pool c bucket d fish tank e laundry tub f farm dam g car’s petrol tank h kettle i garbage bin j road tanker

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16 Copy and complete the conversion diagram for time

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´¡24

´¡60

days (d)

hours (h)

minutes (m)

seconds (s) ¸¡60

17 Convert a 3 d to h e 7 min to s i 3 h to s

EXAMPLE

1

Convert to minutes a

b f j

a

5 d to h 15 min to s 12 h to s

c g

2 h and 25 min

2 h and 25 min = 2 £ 60 + 25 min = 145 min

18 Convert a 3 h and 26 min to min c 2 min and 16 s to s e 2 d and 8 h to h g 1 h and 5 min and 40 s to s i 1 d and 4 h and 25 min to min k 4.6 h to min m 3.25 h to min o 1.9 min to s q 4.75 d to h s 5.82 h to s

EXAMPLE

b d f h j l n p r t

4 h to min 2 d to min

12 h to min 7 d to min

d h

b

2.25 h

b

2.25 h = 2:25 £ 60 min = 135 min

5 h and 51 min to min 12 min and 17 s to s 5 d and 23 h to h 4 h and 38 min and 18 s to s 3 d and 16 h and 50 min to min 2.4 h to min 5.2 min to s 7.8 min to s 2.375 d to h 5.24 h to s

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2

Convert 384 minutes to

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a hours

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b hours and minutes

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a 384 min = 384 ¥ 60 h = 6.4 h

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b Method 1: 25

Method 2:

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0.4 h = 0.4 £ 60 min. = 24 min. Hence, 6.4 h = 6 h and 24 min. On your calculator find the degrees/minutes/seconds key. Press 6:4 SHIFT

0

o

, ,, .

The display shows 6o 24o 0o which may be read as 6 h and 24 min.

19 Convert to i hours and ii hours and minutes a 168 min b 192 min c 261 min 20 Convert to i minutes and a 114 s b 216 s

d

ii minutes and seconds c 153 s d

339 min

e

267 min

411 s

e

243 s

INVESTIGATION 1: MODELLING Copper hot water pipe comes in 1 metre lengths. What is the least number of 1 metre lengths a plumber would need to buy to complete jobs which require the following lengths of pipe to be cut: 600, 450, 300, 280, 850, 250, 325, 195, 560, 240 millimetres?

D

PERCENTAGES

EXAMPLE

1

a 1 37

Convert to a percentage:

b 1.385 100% = 1

To convert a fraction or decimal to a percentage, multiply it by 100: 1 37

a =

1 37

b £ 100%

= 142:9%

(1 dec pl)

0:385 = 0:385 £ 100% = 38:5%

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EXERCISE 2D 1 Convert to a percentage b 1 12 a 34 g

0:34

h

0:7

25

c

5 8

i

1:29

d

2 3

j

0:587

e

4 5

f

2 56

k

2:024

l

0:003

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100

EXAMPLE

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75

25

5

2 a 56%

Convert to a fraction

on the calculator:

Using the fraction key 56%

a =

0

=

c 2 34 %

b 135%

135%

b

56 100 14 25

= =

2 34 %

c

135 100 7 1 20

2 34 100 = 2 34 ¥ 100 =

=

11 400

2 Convert to a fraction a

48%

b

180%

c

225%

d

74%

e

95%

f

5 14 %

g

2 12 %

h

12 12 %

i

4 16 %

j

15 35 %

EXAMPLE

3

Convert to a decimal

a

196%

b

196%

a =

17:85% 17:85%

b

196 100

=

= 196 ¥ 100 = 1:96

17:85 100

= 17:85 ¥ 100 = 0:1785

3 Convert to a decimal a

16%

b

80%

c

137%

d

3%

e

208%

f

45:2%

g

8:9%

h

164:5%

i

17:91%

j

0:4%

EXAMPLE Find a

4

37:5% of 4.8 kg

a

37:5% of 4.8 kg = 37:5 100 £ 4:8 kg = 1:8 kg

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b

5 14 %

of $680

75

5 14 % of $680

b =

(5 14 ) 100

25

£ $680

= $35:70

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4 Find a 16:5% of 12 m

75

0

8:4% of $350

c

160% of 27 kg

f

37 12 % of $125:60

c

75 mL of 90 mL

d

0:8% of $69 000

e

2 12 % of 56 m

g

83 13 % of 90 minutes

h

3 34 % of $12 000

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5

b

EXAMPLE

5

What percentage is 84 mm of 96 mm? percentage =

84 96

£ 100%

= 87:5%

5 What percentage is a

17 mm of 25 mm

b

350 g of 400 g

d

$8 of $250

e

5 L of 60 L

EXAMPLE

6

What percentage is

a 487.2 g of 4.2 kg

b 25 minutes of 2 hours

a Changing both quantities to the same units

487:2 4200

b Changing both quantities to the same units

25 120

£ 100% = 11:6%

£ 100% = 20:8% (to 1 d.p.)

6 What percentage is a

960 g of 3 kg

b

$469:20 of $690

d

$66 150 of $540 000

e

23 minutes of 1 hour

EXAMPLE

7

a Increase 180 by 14% a

180 + 14% of 180 = 100% of 180 + 14% of 180 = 114% of 180 = 114 100 £ 180 = 205:2

c

680 cm of 3.6 m

b Decrease 96 by 35% b

100

95

96 ¡ 35% of 96 = 100% of 96 ¡ 35% of 96 = 65% of 96 65 = 100 £ 96 = 62:4

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7 Increase a 160 by 12% d 466 by 20%

b e

240 by 35% 285 by 100%

c

850 by 8%

8 Decrease a 86 by 25% d 115 by 30%

b e

350 by 40% 480 by 24%

c

780 by 16%

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EXAMPLE

8

a Increase $75 by 20% a

b Decrease 3 metres by 12%

$75 + 20% of $75 = 100% of $75 + 20% of $75 = 120% of $75 = 120 100 £ $75 = $90

b

3 m ¡ 12% of 3 m = 100% of 3 m ¡12% of 3 m = 88% of 3 m 88 £3 m = 100 = 2.64 m

9 Increase a $450 by 28% d $6 by 200%

b e

15 m by 75% 40 s by 62 12 %

c

2 t by 1:5%

10 Decrease a 4.8 km by 19% d $456 by 8:5%

b e

120 kg by 13:2% $3000 by 2 14 %

c

57 s by 33 13 %

11 What would be the value of $10 000 worth of shares at the end of 2 years if they a increase in value by 15% in the first year and then increase in value by 12% in the second year b increase in value by 20% in the first year and then decrease in value by 8% in the second year c decrease in value by 14% in the first year and then increase in value by 16% in the second year d increase in value by 10% in the first year and then decrease in value by 10% in the second year e decrease in value by 10% in the first year and then increase in value by 10% in the second year f decrease in value by 25% in the first year and then increase in value by 33% in the second year? 12 A car which cost $28 900 new depreciated in value by 22% in the first year of ownership, 20% in the second year and 18% in the third year. What was its value at the end of 3 years?

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EXAMPLE

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a John’s weight increased from 64 kg to 68 kg. Find the percentage increase in his weight. b The value of a car decreased from $18 500 to $14 900 in one year. Calculate the percentage decrease in its value.

25

Increase = 68 ¡ 64 kg = 4 kg

a Method 1: 5

4 ) percentage increase = 64 £ 100% = 6:25%

0

68 64

Method 2:

£ 100% = 106:25%

The increased weight is 106.25% of the original weight. Hence, percentage increase = 106:25% ¡ 100% = 6:25% Decrease = $18 500 ¡ $14 900 = $3600

b Method 1:

) percentage decrease =

3600 18 500

£ 100%

= 19:5% (to 1 decimal place) 14 900 18 500

Method 2:

£ 100% = 80:5% (to 1 decimal place)

i.e., the decreased value is 80.5% of the original value. Hence, percentage decrease = 100% ¡ 80:5% = 19:5% 13 Find the percentage increase (to 1 decimal place) from a $350 to $425 b 7.2 m to 7.8 m d $80 to $215 e 4.2 kg to 8.4 kg

c

63 kg to 68 kg

14 Find the percentage decrease from a $256 to $190 b d 430 Ha to 385 Ha e

c

15.8 s to 15.5 s

EXAMPLE

55 kg to 51 kg $5400 to $1800

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A stamp collection was bought for $3600. In the first year of ownership its value increased by 5%. In the second year it increased in value by a further 6%. a b c d

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Calculate its value at the end of the first year. Calculate its value at the end of the second year. What is the overall change in its value after 2 years? Find the percentage change in value over the 2 years.

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Value after 1 year = 105% of $3600 = 105 100 £ $3600 = $3780 Overall change in value = $4006:80 ¡ $3600 = $406:80

b

d

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Value after 2 years = 106% of $3780 = 106 100 £ $3780 = $4006:80 Percentage change in value change in value £ 100% = original value =

406:8 3600

£ 100%

= 11:3%

15 A piece of antique jewellery was bought for $2400. In the first year of ownership its value increased by 15%. In the second year it increased in value by a further 8%. a Calculate its value at the end of the first year. b Calculate its value at the end of the second year. c What is the overall change in its value after 2 years? d Find the percentage change in value over the 2 years. 16 For each part of question 11: i Calculate the overall change in the value of the shares after 2 years. ii Find the percentage change in value over the 2 years. 17 For the car in question 12: a Calculate the overall change in the value of the car after 3 years. b Find the percentage change in value over the 3 years.

INVESTIGATION 2: MODELLING A new car is bought for $ 30000 and 8 years later its value is $ 5000. 1 What would have been its value after 1, 2, 3, ..., 7 years? 2 Can you predict its value after 9, 10, ... years? (You could consider the cases where the value decreases ² by a fixed amount each year or ² by a fixed percentage each year or ² make up your own model. Check your models by finding out the new and used prices of some cars. How do car dealers determine the value of used cars?)

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E

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ERROR IN MEASUREMENT

When physically measuring a quantity there are several sources of possible error: ²

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The zero on the scale of the measuring instrument must coincide with the end of the object or with the pointer on the measuring instrument.

²

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1

2

2

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6 7 8 9

0

3

An error occurs if the end of the measuring instrument has been damaged. In this case start measuring from the 1, say, instead of zero.

3

Parallax error occurs if your eye is not directly above the scale on the measuring instrument.

object

0

² ²

4 5

2 1

object

object

1

²

3

1

2

3

Calibration error can occur if the scale is not accurately marked on the measuring instrument. There is always an error due to the limit of reading of the measuring instrument:

EXERCISE 2E 1

a

b 13

14

15

c

13

14

15

13

14

15

d 13

14

15

The diagrams above show several steel rods being measured with a ruler. Write down the length of each rod, using the scale given on the ruler.

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2 If the length of a rod is measured using the ruler above, and the measurement is recorded as 14 cm a b c d

would this be the exact length of the rod? Between what values would the actual length lie? What is the greatest possible error in stating that the length is 14 cm? How could we find a more accurate value of the length of the rod?

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GREATEST POSSIBLE ERROR (ABSOLUTE ERROR) AND PERCENTAGE ERROR

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The rod in question 2 of Exercise 2E above has been measured to the nearest centimetre because this is the smallest unit on the ruler, i.e., the length is closer to 14 cm than to 13 cm or 15 cm. The greatest possible error is 0.5 cm. This is half of the smallest scale (centimetres) on the ruler. The actual length will lie between 13.5 cm and 14.5 cm, i.e., between (14 ¡ 0:5) cm and (14 + 0:5) cm. To obtain a more accurate measurement we would need to use a more accurate ruler, i.e., one which has smaller units on it. The smallest unit on a measuring instrument is called the limit of reading of the instrument. The greatest possible error in measuring a quantity (sometimes called the absolute error) is equal to half the limit of reading. The smallest and largest values between which the actual measurement lies are called the lower and upper limits of the true measurement.

EXAMPLE

1

For each of the following measurements find i the smallest unit of measurement (the limit of reading) ii the greatest possible error a 18 centimetres

b

254 grams

c

2.4 kilograms

d

12.6 seconds

a The smallest unit of measurement is 1 cm, i.e., the measurement has been made to the nearest centimetre. Hence, i Limit of reading = 1 cm ii Greatest possible error = 12 £ 1 cm = 0.5 cm b The smallest unit of measurement is 1 g, i.e., the measurement has been made to the nearest 1 gram. Hence, i Limit of reading = 1 g ii Greatest possible error = 12 £ 1 g = 0:5 g c The smallest unit of measurement is 0.1 kg, i.e., the measurement has been made to the nearest 0.1 of a kilogram. Hence, i Limit of reading = 0.1 kg ii Greatest possible error = 12 £ 0:1 kg = 0:05 kg

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d The smallest unit of measurement is 0.1 seconds, i.e., the measurement has been made to the nearest 0.1 of a second. Hence, i Limit of reading = 0.1s ii Greatest possible error = 12 £ 0:1 s = 0:05 s

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EXERCISE 2F 1 For each of the following measurements find i the smallest unit of measurement (the limit of reading) ii the greatest possible error 16 centimetres 16 litres 8.1 litres

a e i

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b f j

286 grams 3.6 kilograms 3.76 metres

c g

38 metres 15.3 seconds

d h

14 minutes 2.8 metres

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EXAMPLE For each i ii iii

2

of the following measurements find the smallest unit of measurement (the limit of reading) the absolute error the lower and upper limits of the true measurement

a 16 seconds a

b

c

b 5.7 kilograms

c 9.38 metres

i

The smallest unit of measurement = 1 second i.e., this measurement of time has been made to the nearest second. Limit of reading = 1 second

ii

Greatest possible error = 12 £ 1 second = 0:5 seconds

iii

Lower limit = 16 ¡ 0:5 Upper limit = 16 + 0:5 = 15:5 seconds = 16:5 seconds i.e., the true measurement lies between 15.5 and 16.5 seconds

i

Smallest unit of measurement = 0.1 kg i.e., this measurement of mass has been made to the nearest 0.1 of a kilogram. Limit of reading = 0.1 kg

ii

Greatest possible error = 12 £ 0:1 kg = 0.05 kg

iii

Lower limit = 5:7 ¡ 0:05 Upper limit = 5:7 + 0:05 = 5:65 kg = 5:75 kg i.e., the true measurement lies between 5.65 and 5.75 kg

i

Smallest unit of measurement = 0.01 m i.e., this measurement of length has been made to the nearest 0.01 of a metre. Limit of reading = 0.01 m

ii

Greatest possible error = 12 £ 0:01 m = 0.005 m

iii

Lower limit = 9:38 ¡ 0:005 m Upper limit = 9:38 + 0:005 m = 9:375 m = 9:385 m i.e., the true measurement lies between 9.375 and 9.385 m

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2 For each of the following measurements find i the limit of reading ii the greatest possible error (absolute error) in the measurement iii the lower and upper limits of the true measurement. a e

12 millimetres 18.4 seconds

b f

348 grams 4.9 kilograms

c g

375 millilitres 2.37 metres

8.2 kilometres 5.81 litres

d h

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EXAMPLE The a b c

mass of a car was given as 2300 kg to the nearest 100 kg. Find the limit of reading the greatest possible error (absolute error) in the measurement the lower and upper limits of the true measurement.

a The smallest unit used for this measurement is given as 100 kg i.e., limit of reading = 100 kg b Hence, the absolute error = 12 £ 100 kg = 50 kg c Lower limit = (2300 ¡ 50) kg = 2250 kg

Upper limit = (2300 + 50) kg = 2350 kg

i.e., the true measurement lies between 2250 kg and 2350 kg

3 For each of the following measurements find i the smallest unit of measurement ii the absolute error in the measurement iii the lower and upper limits of the true measurement. a The mass of a can of soup is 420 g, to the nearest 20 g. b The capacity of a drink bottle is 375 mL, to the nearest 10 mL. c The area of a block of land is 1285 m2 , to the nearest 5 m2 . d The crowd at a cricket match was 38 000, to the nearest 1000. e The time taken for a plane flight was 6 12 hours to the nearest 12 hour.

EXAMPLE 100

a Find the greatest possible error for the measurement 18 kg. b Express the greatest possible error as a percentage of the measurement.

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a Smallest unit of measurement = 1 kg ) greatest possible error = 0.5 kg 25

b Percentage error = 0:5 18 £ 100% = 2:8% (to 1 decimal place)

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H:\...\NSWGM\NSWGM_02\040NG02.CDR Mon Feb 21 12:56:38 2000


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4 Find

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a e i

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i ii

the greatest possible error the percentage error for each of the following measurements

10 centimetres 14 minutes 13.5 seconds

b f j

32 seconds 6 litres 12.56 metres

c g

250 grams 2.4 kilograms

d h

75 millilitres 25.8 litres

CALCULATIONS INVOLVING MEASUREMENTS

0

Since there is always some degree of error in a numerical value found by measurement then it follows that the results of any calculations involving this value will also contain a degree of error.

EXAMPLE The a b c

1

length and breadth of a rectangle were measured to be 8 cm and 6 cm respectively. Calculate the perimeter of the rectangle using these measurements. Find the lower and upper limits of its true perimeter. Hence find the maximum error in the answer to a.

a Using the given measurements, perimeter = 8 + 8 + 6 + 6 cm = 28 cm b The greatest possible error of each measurement is 0.5 cm. Hence the length lies between 7.5 cm and 8.5 cm and the breadth lies between 5.5 cm and 6.5 cm. Lower limit of perimeter = 7:5 + 7:5 + 5:5 + 5:5 cm = 26 cm Upper limit of perimeter = 8:5 + 8:5 + 6:5 + 6:5 cm = 30 cm c Hence, the apparent perimeter has a maximum error = 28 cm ¡ 26 cm (or 30 cm ¡ 28 cm) = 2 cm Note that this is the sum of the greatest possible errors of the original measurements.

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EXERCISE 2G 100

1 The length and breadth of a rectangle were measured to be 9 cm and 5 cm respectively. a b c d

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Calculate the perimeter of the rectangle using these measurements. Find the lower and upper limits of its true length and breadth. Hence find the lower and upper limits of its true perimeter. Find the maximum error in the answer to a.

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2 Two pieces of timber were measured to be 164 cm and 128 cm respectively. a If the 2 pieces were placed end to end what would be their total length, using the measurements given?



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b Find the lower and upper limits of the true length of each piece. c Hence calculate the lower and upper limits of the true total length of these 2 pieces of timber. d Find the maximum error in the answer to a.

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3 The masses of two bags of sand were measured and found to be 47 kg and 52 kg. a b c d

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What is the total mass of the 2 bags? Find the lower and upper limits of the true mass of each bag. Hence calculate the lower and upper limits of the true total mass. What is the maximum error in the answer to a?

4 Repeat question 3 given that the masses of the bags were 47.4 kg and 51.9 kg. 5 A carpenter measures the length of a piece of timber to be 5.6 m. He then measures and cuts from it a piece of length 1.8 m. a b c d

Using the measurements given, calculate the length of the remaining piece. Find the lower and upper limits of the true length of each measured piece. What are the lower and upper limits of the true length of the remaining piece? Find the maximum error for the answer to a.

EXAMPLE The a b c

2

length and breadth of a rectangle were measured to be 8 cm and 6 cm respectively. Calculate the area using the measurements given. Find the lower and upper limits of the true area. 6 cm Hence, find the maximum error in the answer to a. 8 cm

a Area = 8 £ 6 = 48 cm2 b Lower limit of area = 7:5 £ 5:5 = 41:25 cm2

Upper limit of area = 8:5 £ 6:5 = 55:25 cm2

i.e., the true area lies between 41.25 cm2 and 55.25 cm2 c

48 ¡ 41:25 = 6.75 cm2 55:25 ¡ 48 = 7.25 cm2 Hence, the maximum error = 7.25 cm2 100

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6 A rectangular room was measured to be 5 m long by 3 m wide. a b c d

5m

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Calculate the area of the room using these measurements. Find the lower and upper limits of the true length and width. What are the lower and upper limits of the true area? Find the maximum error in the answer to a.

3m 25

5

7 Repeat question 6 given the length is 5.4 m and the width is 3.2 m.



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8 The diameter of a circular pizza tray is measured to be 28.6 cm.

¼d2 ) 4 b What are the lower and upper limits of the true length of the diameter? c Find the lower and upper limits of the true area of the tray. d What is the maximum error in the answer to a? a Calculate the area of the tray using the measurement given. (A =

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PRACTICAL ACTIVITIES

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d e f g h i 2 a b c d e f g

H

1 a Measure the width of a desk. b Write down the limit of reading of your measuring instrument. c Within what range does the actual width of the desk lie? Measure the width of another desk Within what range does this measurement lie? Calculate the combined width of the two desks using the meaurements in a and d. Within what range does the combined width lie? Find the combined width of the two desks by direct measurement. Compare the result of h with the answers to parts f and g. Comment on your findings. Measure the mass of two different items using a set of scales. Write down the limit of reading of your scales. Hence determine the lower and upper limits of each mass. Calculate the combined mass of the two items using the results of a . Within what range does the actual sum of the masses lie? Check this total by actual measurement, i.e., place both items together on the scales. Compare this result with the result of d. Comment on your findings.

RATES A rate is a comparison between quantities of different kinds.

For example, we may compare distance travelled with petrol used, amount of fertiliser needed with area of land, amount of pay with time worked etc. The comparison between quantities is made by dividing one quantity by the other, in the required order. The rate has then been expressed in the form ‘the first quantity per unit of the second quantity’.

EXAMPLE

1

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a A 1.5 kg packet of soap powder costs $6.90 . What is the cost per kg? b Wendy types 600 words in 8 minutes. How many words per minute does she type? c A car used 49 litres of petrol on a trip of 500 km. i Calculate the number of kilometres the car travels per litre of petrol consumed. ii Calculate the number of litres of petrol the car consumes per kilometre of travel.



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100

a This rate is comparing cost with weight, in that order.

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Hence we divide the cost by the weight.

75

$6:90 1:5 kg = $4.60/kg

Rate =

This is the cost per unit of weight.

Remember to write the units!

25

b This rate is comparing number of words with time, in that order. Hence, we divide the number of words by the time.

5

0

So, rate =

600 words 8 minutes

= (600 ¥ 8) words/minute = 75 words/minute This is the number of words per unit of time. c

i This rate is comparing the distance travelled with the amount of petrol consumed, in that order. 500 km Rate = 49 L = (500 ¥ 49) km/L = 10:2 km/L (to 1 d.p.) This is the number of kilometres travelled per unit of petrol consumed, i.e., the car travels 10.2 km for every one litre of petrol used. 49 L 500 km = (49 ¥ 500) L/km = 0:098 L/km

Rate =

ii

This is the amount of petrol consumed per unit of distance travelled, i.e., the car used 0.098 litres of petrol for every one kilometre travelled. Because, in practice, this second rate is often very small, petrol consumption is usually quoted as the amount of petrol used per 100 km, 49 L i.e., rate = 500 km (49 ¥ 5) L = (500 ¥ 5) km = 9:8 L/100 km 100

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EXERCISE 2H 1

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a A 2.5 kg box of soap powder costs $10.90. Find the cost per kg. b Jenny typed 300 words in 5 minutes. What is her typing rate per minute? c A truck used 114 litres of petrol on a trip of 600 kilometres. Express the petrol consumption in L /100 km. d Vicki was paid $85.20 for 6 hours work. Calculate her rate of pay per hour.

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e Howard had to pay $55.20 for 120 telephone calls. What was the cost per call?

100

f The temperature rose 14 degrees in 3 12 hours. At what rate, in degrees per hour, did the temperature rise? g Jeremy spread 24 kg of fertiliser over an area of 60 m2 . Calculate the rate of application in kg per m2 . h Fred’s electricity bill was $174.72 for 1560 kilowatt hours of power (kwh). What was the cost of electricity per kwh? i Peggy drove 195 kilometres in 2 12 hours. Calculate her speed in km/h.

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j Calculate the flow rate per minute if 119 litres of water flow through a pipe in 35 minutes.

0

EXAMPLE

2

Convert 8 tonnes/hectare to

a kg/Ha

a

8 t/Ha = 8000 kg/Ha

c

8 t/Ha =

b kg/m2 b

8000 kg 10 000 m2 = (8000 ¥ 10 000) kg/m2 = 0.8 kg/m2

8 t/Ha =

8 £ 1000 £ 1000 g 10 000 m2 = (8 000 000 ¥ 10 000) g/m2 = 800 g/m2

EXAMPLE

c g/m2

3

Convert 72 litres/hour to millilitres/second. 72 L/h =

72 000 mL (60 £ 60) seconds

= f72 000 ¥ (60 £ 60)g mL/s = 20 mL/s

2 Convert a 16 t/Ha b $3.75/h c $1.80/m d 15%/year e 2.4 kg/L f 45 L/h g $12/kg h 18 km/h i 27 L/h j 7.2 kg/day


to to to to to to to to to to

i i i i i i

kg/Ha cents/h cents/m %/quarter g/L L/min cents/g m/s mL/s g/min


ii ii ii ii ii ii

kg/m2 cents/min cents/cm %/month g/mL mL/min

100

iii

cents/mm

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EXAMPLE Convert

4

a 5 m/s to km/h

b 2 cents/min to $/day

a Multiply the number of metres travelled in 1 second by the number of seconds in an hour. 5 m/s = 5 £ (60 £ 60) m/h = 18 000 m/h fDividing by the number of metres in a kilometreg = (18 000 ¥ 1000) km/h = 18 km/h or

5 m = 5 ¥ 1000 km and 1 s = 1 ¥ 60 ¥ 60 h = 0:005 km = 0:00027 h 0:005 km 0:00027 h = (0:005 ¥ 0:00027) km/h = 18 km/h

Hence 5 m/s =

b Multiply the number of cents/min by the number of minutes in a day 2 cents/min = 2 £ (24 £ 60) cents/day = 2880 cents/day =

2880 100

fDividing by the number of cents in a dollarg

$/day

= 28.8 $/day or $28.80 /day or

Hence,

2 cents = $(2 ¥ 100) and 1 min = 1 ¥ 60 ¥ 24 days = $0:02 = 0.000694 days $0:02 0:000694 days = (0:02 ¥ 0:000 694) $/day = 28:8 $/day = $28.80 /day

2 cents/min =

3 Convert a 6 m/s b 3 cents/min c 5g/mL d 0.8 cents/g e 0.75 kg/m2 f 0.4 mL/s g 0.8 cents/m

to to to to to to to

i i i i i

4 A car travels at 60 km/h. a How far will it travel in 2 12 hours?

m/min cents/h g/L cents/kg kg/Ha L/h $/km

ii ii ii ii ii

m/h cents/day kg/L $/kg t/Ha

iii iii

km/h $/day

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b

How long will it take to travel 225 km?

5 Water is flowing into a tank at the rate of 2 L/min.

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a How much water will flow into the tank in 1 hour? b How long will it take for 75 litres to flow into the tank?

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6 A machine glues labels onto jars at the rate of 230 labels per minute. a How many jars will be labelled in 2 12 hours? b How long will it take to label 20 000 jars (to the nearest minute)?

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7 A fertiliser is to be spread at the rate of 0.2 kg/m2 . 25

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a How much fertiliser would be needed for an area of 600 m2 ? b If the fertiliser is sold in 50 kg bags, how many bags are needed for an area of 600 m2 ? c What area could be fertilised with 1 tonne of fertiliser? 8 A patient in hospital is given an antibiotic solution intravenously at the rate of 80 mL/h. a How much antibiotic solution will the patient receive in 6 12 hours? b How often would 600 mL containers of antibiotic solution need to be changed? c If 1 mL of this solution contains 15 drops, calculate the rate at which the patient receives the antibiotic in drops/minute. 9 Anna is paid at the rate of $11.80/h. a How much would she be paid for working 15 hours? b How long would she need to work to earn $472? c Anna wants to save for a trip. If she works 28 hours one week and 31 hours in the next week, how many more hours does she need to work to have $1000? 10 The conversion rate for Australian dollars ($A) into American dollars ($US) is 0.65 $US/$A, (i.e., $A1 = $US0.65). a How many US dollars would I receive for $A2750? b How many Australian dollars would I receive for $US1300? c Convert $US1800 into $A. 11 Ceiling paint covers at the rate of 16.5 m2 /L. a What area could be painted with three 4 litre cans? b How many litres would be needed to paint an area of 300 m2 ? (Answer to nearest litre.) c How many 4 litre cans would you need to buy to paint this area? 12 A builder quotes the cost of building a new home in a certain area as $4750 /m2 . a How much would it cost to build a home with a floor area of 25 m2 ? b What is the approximate size of the new home that could be built with a budget of $140 000? 13 On a trip of 400 km a car uses 30 litres of petrol. 100

a Express the fuel consumption in L/100 km. b Assuming the same rate of fuel consumption, i how much fuel would the car use for a trip of 500 km ii how far could the car travel on a full tank of 45 litres?

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14 A company has to pay tax at the rate of 30 cents in the dollar (or 30 cents/$).

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a Calculate the tax payable on $130 000. b How much profit did the company make if it paid $45 600 in tax?

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15 The average distance of the Earth from the Sun is 1:49 £ 108 kilometres. Assuming the Earth travels in a circular orbit around the Sun,

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a calculate (to 3 sig. figs.) the distance travelled by the Earth in one complete orbit. b It takes the Earth one year to travel this distance. Using the answer from part a, find the average speed at which the Earth travels through space in i km/h ii km/s. (Take 1 year = 365.25 days.)

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CONCENTRATIONS

The most common methods of expressing the concentrations of chemicals and drugs in solutions or mixtures are ² ² ²

weight per unit of weight, volume per unit of volume, weight per unit of volume,

EXAMPLE Convert

a

i

1

280 g/kg to 25 mL/100 mL to 60 mg/mL to

a b c

i i i

280 g/kg 280 g = 1000 g

ii

= 280 ¥ 1000 g/g = 0.28 g/g b

i

c

i

written written written

g/g mL/mL mg/L

weight/weight volume/volume weight/volume

ii ii ii

g/100 g mL/L g/L

or or or

w/w v/v w/v.

iii

L/L

iii

250 mL/L = 250 ¥ 1000 L/L = 0:25 L/L

0.28 g/g 0:28 £ 100 g = 100 g = 28 g/100 g

25 mL 100 mL = 25 ¥ 100 mL/mL = 0.25 mL/mL 60 mg/mL 60 £ 1000 mg = 1000 mL = 60 000 mg/L

ii

0.25 mL/mL 0:25 £ 1000 mL = 1000 mL = 250 mL/L

ii

60 000 mg/L = 60 000 ¥ 1000 g/L = 60 g/L

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EXERCISE 2I 1 Convert a 350 g/kg b 85 mL/L


25

to to

i i

g/g mL/mL


ii ii

g/100 g mL/100 mL

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c d e f g h i j k l m n o

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580 g/L 90 mg/g 450 mL/L 0.5 kg/L 0.6 g/g 0.05 L/L 8 mg/mL 36 g/200 g 6 mL/200 mL 20 g/100 mL 10 mL/50 mL 7 g/20 g 9 g/150 mL

EXAMPLE

to to to to to to to to to to to to to

i i i i i i i i i i i i i

g/mL mg/mg mL/mL g/L mg/g mL/L mg/L g/g mL/mL g/mL mL/mL g/g g/mL

ii ii ii ii ii ii ii ii ii ii ii ii ii

g/200 mL g/g mL/200 mL g/mL mg/mg mL/mL g/L mg/g mL/L g/L mL/L mg/g g/L

iii

g/100 mL

iii

mg/mL

2

Change a 15 mg/g into a percentage weight per unit weight (%w/w) b 8 mL/200 mL into a percentage volume per unit volume (%v/v) a Express both weights in the same units and simplify 15 mg 15 mg/g = or 15 mg/g 1000 mg = 15 ¥ 1000 g/g = 15 ¥ 1000 mg/mg = 0:015 g/g = 0:015 mg/mg = 0:015 £ 100% g/g = 0:015 £ 100% mg/mg = 1:5% w/w = 1:5% w/w b Express both volumes in the same units and simplify

8 mL 200 mL = 8 ¥ 200 mL/mL = 0:04 mL/mL

Now change the decimal to a percentage = 0:04 £ 100% mL/mL = 4% v/v 100

2

a Change to a %w/w i 65 mg/g iv 8 mg/g

ii v

750 mg/g 6:5 mg/100 mg

iii

b Change to a %v/v i 560 mL/L iv 4 mL/100mL

ii v

35 mL/L 45 mL/200 mL

iii

95

485 g/kg

75

6 mL/L

25

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100

95

75

EXAMPLE Change 12% w/w to

a kg/kg 12% =

25

12 100

b g/g

c mg/g

or 0.12

a 12% w/w = 0.12 kg/kg

b

12% w/w = 0.12 g/g

5

0

c 0.12 g/g = 0:12 £ 1000 mg/g = 120 mg/g

3 Convert a 15% b 28% c 16% d 35%

w/w w/w v/v v/v

to to to to

i i i i

kg/kg kg/kg L/L L/L

ii ii ii ii

g/g g/kg mL/mL mL/L

iii iii iii iii

mg/g g/g mL/L mL/mL

mg/g

iv

EXAMPLE A 250g can of insect spray contains 0.5 g of Methbitcom. Express the concentration of Methbitcom in a g/g b g/kg c as a % w/w. 0:5 g 250 g = 0:5 ¥ 250 g/g = 0:002 g/g i.e., there are 0.002 g of Methbitcom in each gram of the spray.

a 0.5 g of Methbitcom per 250 g can

=

0:002 £ 1000 g 1000 g = 2 g/kg

0.002 g/g =

b

i.e., there are 2 g of Methbitcom in each kilogram of the spray. c Expressing the weights in the same units (from a) 0.002 g/g = 0:002 £ 100% g/g = 0.2% w/w i.e., the insect surface spray contains 0.2% (by weight) of Methbitcom.

100

95

75

4 A 500 mg headache tablet contains 300 mg of aspirin. Express the concentration of aspirin in a mg/mg b mg/100 mg c mg/g d as a % w/w 5 A 20 g tube of acne cream contains 1 g benzoyl peroxide. Express this concentration in a g/g b mg/g c as a % w/w

25

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EXAMPLE Paingo, an ointment for pain relief, contains 10 mg/g of Dapzak. a Express this concentration as a % w/w. b How much Dapzak is there in a 50 g tube of this ointment?

25

5

0

a Simplifying to weight per same unit of weight 10 mg 10 mg/g = 1000 mg = (10 ¥ 1000) mg/mg = 0:01 mg/mg = (0:01 £ 100)% w/w = 1% w/w i.e., Paingo contains 1% of Dapzak. b Using the result of a

Amount of Dapzak

or using equivalent rates

= 1% of 50 g 1 £ 50 g = 100 = 0:5 g or 500 mg

50 £ 10 mg 50 g 500 mg = 50 g

10 mg/g =

i.e., there is 500 mg of Dapzak in 50 g of the ointment.

6 A fungicide used in agriculture contains 218 g/kg of the chemical Fungozell. a Express the concentration of Fungozell in this fungicide in i g/g ii as a % w/w b How much Fungozell would there be in a i 100 g container ii 50 kg bag of this fungicide? 7 A fluoride toothpaste contains 8 mg/g of monofluorophosphate. a Express the concentration of monofluorophosphate in i mg/mg ii as a % w/w b What quantity of monofluorophosphate is there in a 200 g tube of this toothpaste? 100

8 In Squeakclean disinfectant the concentration of the main active ingredients, quaternary ammonium compounds, is 15 mL/L.

95

75

a What amount of these compounds would there be in a i 2 L container ii 5 L container? b Express the concentration of these compounds in i mL/mL ii % v/v c What quantity of these compounds is there in i 500 mL ii 750 mL iii 850 mL of disinfectant?



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9 Each 15 mL of the laxative Rolagar contains 4.83 mL of paraffin liquid. a Express the concentration of paraffin liquid in i mL/mL ii % v/v b How much paraffin liquid is there in a i 1L ii 600 mL bottle of Rolagar?

25

5

0

EXAMPLE Paingo also contains 20% w/w of isopropyl alcohol. a How much isopropyl alcohol is there in a 50 g tube? b Express this amount in i g/g ii mg/g. a Amount = 20% of 50 g 20 £ 50 g = 100 = 10 g

b

i

ii

10 g = (10 ¥ 50) g/g 50 g = 0.2 g/g 0.2 g/g = 0:2 £ 1000 mg/g = 200 mg/g

10 A rash cream contains 15% w/w zinc oxide. a What weight of zinc oxide is there in a 50 g tube of this cream? b Express the concentration of zinc oxide in i g/g ii mg/g

g/kg

iii

11 An antiseptic cream contains 0.3% w/w of Zapenic. a How much Zapenic is there in a 75 g tube of cream? b Express the concentration in i g/g ii mg/g iii

g/kg

12 A spray on cleaner contains 2.4% v/v phosphoric acid. a Find the amount of phosphoric acid in a 500 mL container of this cleaner. b Express the concentration of phosphoric acid in i mL/L ii mL/100mL iii mL/mL

EXAMPLE An anti sunburn lotion contains 72 g/L of titanium dioxide. How much titanium dioxide is there in a 125 mL bottle of this lotion?

100

95

72 g 72g/L = 1000 mL = 0:072 g/mL = (125 £ 0:072) g / 125 mL = 9 g / 125 mL

75

25

i.e., there are 9 g of titanium dioxide in a 125 mL bottle of lotion.

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An insecticide contains 550 g/L of the chemical Tetrathon. a Express the concentration of Tetrathon in i g/mL ii mg/mL b How much Tetrathon is there in a i 100 mL ii 5 L container of insecticide? c How much insecticide is needed to supply 27:5 kg of Tetrathon?

13

95

75

25

5

14 A weed killer contains 100g/L of Demonsate. a Express the concentration of Demonsate in i g/mL ii mg/mL b What mass of Demonsate is there in a i 500 mL ii 5 L container? c How much weed killer is needed to supply 2:5 kg of Demonsate?

0

EXAMPLE Express a concentration of 15 % w/v in

a kg/L

b g/L

c mg/mL

Percentage weight per unit volume concentrations are calculated using kg/L. 15% w/v = 15% of 1 kg/L = 0:15 kg/L

a

b

0.15 kg/L = 0:15 £ 1000 g/L = 150 g/L

c

150 g/L 150 000 mg = 1000 mL = 150 000 ¥ 1000 mg/mL = 150 mg/mL

15 Gammaeat contains 10% w/v of Napzene-iodata. a Express this concentration in i kg/L ii mg/mL b How much Napzene-iodata is there in a 15mL bottle of Gammaeat? 16 A solvent used in the manufacture of glue contains 65% w/v of methyl ethyl ketone. a Express this concentration in mg/mL. b Find the amount of methyl ethyl ketone in a i 500 mL ii 1200 mL container of solvent. 100

17 The drug Petragocin comes in bottles of concentration 350 mg/10 mL

95

a How much of the drug would a patient receive if injected with i 5 mL ii 7 mL iii 25 mL iv 12 mL? b How much should be injected if a patient requires i 280 mg ii 525 mg iii 630 mg?

75

25

18 Normal saline solution is a mixture which contains the same amount of salt as human blood. It is often used to prevent shock from loss of fluid, especially for burn victims. Normal saline solution contains 0.9% w/v of salt.



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100

a b

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75

c

Express this concentration in i g/L ii mg/mL Nurse Lee has to make normal saline solution. How much salt will she need to add to i 1 litre of water ii 5 litres of water iii 600 mL of water? To make a normal saline solution, how much water should she add to i 36 g of salt ii 54 g of salt iii 22:5 g of salt?

25

5

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EXAMPLE Express as a % w/v

a 250 g/L

b 30 mg/mL

a Weight/volume concentrations can be expressed as a % w/v by expressing the weight as a percentage of 1 kg per litre of liquid, concentration = 250 g/L = 0:25 kg/L = (0:25 £ 100)% of 1 kg/L = 25% of 1 kg/L Hence, concentration = 25% w/v i.e., in 1 litre of solution there is 25% of 1 kg of the ingredient. i.e.,

b

30 mg/mL = 30 000 mg/L = 30 g/L = 0:03 kg/L = (0:03 £ 100)% of 1 kg/L = 3% of 1 kg/L Hence, concentration = 3% w/v i.e., in 1 litre of solution there is 3% of 1 kg of the ingredient.

19 An insecticide contains 500 g/L of the chemical Jakdol. a Find the concentration of Jakdol in kg/L. b Express the concentration of Jakdol as a % w/v. 20 Each 5 mL of a cough medicine contains 1.3 g of sorbitol. a Convert the concentration of sorbitol to i g/mL b Hence, express the concentration as a % w/v.

ii kg/L. 100

95

75

EXAMPLE Solution A contains 15g of salt per litre and solution B contains 10g of salt per litre. a How much salt is there in 100 mL of each solution? b If 100 mL of each solution are mixed together, what is the amount of salt in the mixture? c Calculate the concentration of salt in the mixture in g/L.

25

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95

15 g/L =

a For solution A,

=

75

¡

15 1000 ¡ 15 1000

¢

g/mL ¢ £ 100 g/100 mL

= 1:5 g/100 mL i.e., 100 mL of solution A contains 1.5 g of salt. 25

10 g/L =

For solution B,

5

= 0

¡

10 1000 ¡ 10 1000

¢

g/ mL ¢ £ 100 g/100 mL

= 1 g/100 mL i.e., 100 mL of solution B contains 1 g of salt. b Total amount of salt in mixture = 2.5 g. c Concentration of salt

=

total amount of salt total volume of mixture

=

2:5 g 200 mL 2:5 200 g/mL

= =

0.0125 g/mL (0:0125 £ 1000) g/L

=

12.5 g/L

=

21 Solution A contains 20 g of salt per litre and solution B contains 12 g of salt per litre. a How much salt is there in 100 mL of each solution? b If 100 mL of each solution are mixed together, what is the amount of salt in the mixture? c Calculate the concentration of salt in the mixture in g/L. 22 Solution A contains 20 g of salt per litre and solution B contains 12 g of salt per litre. a How much salt is there in 150 mL of solution A? b How much salt is there in 250 mL of solution B? c What is the total amount of salt in a mixture of 150 mL of solution A and 250 mL of solution B? d Calculate the concentration of salt in this mixture in g/L. 23 How much salt would be required to make a 100 mL of 30 g/L solution b c 600 mL of 10 g/L solution d

100

200 mL of 25 g/L solution 1500 mL of 40 g/L solution?

95

24 The label on a bottle containing a vitamin supplement for guinea pigs states that the concentration of vitamins in the solution is 300 g/300 mL. a Convert this concentration to g/mL. b The recommended dosage is 10 g/L of their drinking water. If their drink container has a capacity of 500 mL, what quantity of vitamins should be present in the container to get the recommended dose?

75

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56 100

95

75

25

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c How many millilitres of the vitamin solution are needed to supply this quantity of vitamins? 25 The label on a container of weedkiller states that the concentration of its active ingredient, Demonsate, is 100 g/L. a Express the concentration in g/mL. b How much Demonsate is there in 100 mL of the weedkiller? c If 900 mL of pure water is added to 100 mL of the weedkiller, what is the concentration of Demonsate in this mixture? d The recommended concentration to kill weeds is 2 g/L. How many millilitres of the weedkiller would be needed to make a litre of mixture of this strength?

RATIO A ratio is a comparison between quantities of the same kind.

EXAMPLE Tom’s height was 169 cm and Laura’s was 165 cm. Write down the ratio of a Tom’s height to Laura’s height b Laura’s height to Tom’s height. a

169 165

Tom : Laura = 169 : 165 or

b

Laura : Tom = 165 : 169 or

165 169

Note: The order is important as 169 : 165 6= 165 : 169. A ratio can be written using colon notation or as a fraction. A ratio does not have units. Ratios are simplified by multiplying or dividing each term of the ratio by the same number.

EXAMPLE Simplify the ratio a a

24 : 18

b

16 : 12 : 20

24 : 18 = 4 : 3 Calculator:

b

100

Press: 24

16 : 12 : 20 = 4 : 3 : 5

c

1:5 : 2:7

d

1 34 :

2 3

e

85 cm : 1.2 m

95

75

fDividing each term by 6g

18 SHIFT

Display: 4 j 3

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57


1:5 : 2:7 = 15 : 27 =5:9

c

95

1 34 :

d

75


= 12 £ 74 : 12 £ = 21 : 8

2 3

fMultiplying each term by the LCD,

85 cm : 1.2 m = 85 cm : 120 cm = 17 : 24

e

25

2 3

10g

fMultiplying both terms by

12g

fConverting to the same unitsg fDividing each term by 5g

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0

?

EXERCISE 2J 1 The table below shows the number of vehicles that pass the front of a school in one hour:

Type of Vehicle car truck motorcycle bus other

Write down the ratio of the number of a cars to trucks b trucks to cars c motorcycles to cars d trucks to buses e cars to trucks to motorcycles f trucks to motorcycles to buses. 2 Simplify the ratios a 25 : 35 d 12 : 24 : 18 g 1:6 : 1:8 j 0:256 : 0:8

b e h k

27 : 18 36 : 48 : 72 0:56 : 0:32 1 1 2 : 3

c f i l

m

1 12 : 2 12

n

5 8

p s

25 cm : 1.1 m 1 12 h : 40 min

q t

2.2 kg : 850 g 600 mL : 1.5 L : 2.1 L

EXAMPLE

3

Express in the form n : 1 the ratios

:

2 3

a 25 : 10

a Dividing both numbers by 10,

25 : 10 =

Number 54 13 7 4 2

84 : 48 1:6 : 1:9 0:93 : 0:6 1 12 : 13

o

1 5

r

$1:50 : 80 cents

:2

b 16 : 24: 25 10

:

10 10

= 2:5 : 1 This means that the first number is 2.5 times the second. b Dividing both numbers by 24,

16 : 24 = =

This means that the first number is

2 3

16 24 24 : 24 2 3 : 1 or

100

0:6 : 1

95

(or 0:6) times the second.

75

25

3 Express in the form n : 1 a

35 : 10

b

72 : 40

c

359 : 100

d

24 : 60

5

72 : 80

e

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58 100

95

75

25


4 Simplify and express in the form n : 1. Explain the meaning of the answer. a 3.5 m : 70 cm b 1.2 kg : 800 g c 2 12 cups : d $4:68 : $1:80 e 0.04 Ha : 500 m2 5 Express in the form 1 : n a 50 : 87 b e 200 : 154 f i 20 mL : 1 L j

40 : 90 1 cm : 1 m 50 g : 1 kg

c g

60 : 48 1 mm : 1 m

1 2

cup

125 : 8 2 cm : 5 m

d h

5

6

Category

0

No. who develop lung cancer 10 25 20 3 50 15

Males who have never smoked Males who have ever smoked Males who currently smoke Females who have never smoked Females who have ever smoked Females who currently smoke

in in in in in in

760 300 90 471 1150 132

Ratio (1 : n) 1 : 76

a Copy and complete the table to express the risk of developing lung cancer in the form 1 : n. b Which group of people are most likely to develop lung cancer? c Which group of people are least likely to develop lung cancer?

EXAMPLE

4

The ratio of the number of boys to girls in a school is 7 : 6. If there are 354 girls in the school, how many boys are there? boys : girls = 7 : 6 i.e., boys : 354 = 7 : 6 boys 7 or = (in fraction form) 354 6 i.e., number of boys =

7 6

£ 354 = 413

7 The ratio of the number of boys to girls in a school is 9 : 8. If there are 312 girls, how many boys are there? 8 At an electrical store the ratio of profit to sales is 2 : 7. If the annual sales one year were $145 600, what was the annual profit? 9 The ratio of Ben’s net salary to the tax he pays is 10 : 3. Find his net salary if he paid $14 580 in tax for one year.

100

95

75

EXAMPLE

5

25

The ratio of boys to girls in a school is 8 : 7: If there are 264 boys in the school, how many girls are there?

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59


boys : girls = 8 : 7

95

i.e., 264 : girls = 8 : 7

75

i.e.,

girls 7 = 264 8

i.e., the number of girls =

25

or

7 8

8 264 = girls 7

£ 264 = 231

5

0

10 The ratio of boys to girls in a school is 10 : 9. Find the number of girls if there are 470 boys. 11 The ratio of a girl’s height to that of her mother is 4 : 5. What is the mother’s height if the girl is 172 cm tall? 12 A farmer plants lemon trees and orange trees in an orchard in the ratio 2 : 5. a In one orchard he planted 60 orange trees. How many lemon trees did he plant? b In another orchard he planted 18 lemon trees. How many orange trees did he plant?

EXAMPLE

6

An inheritance of $24 000 is to be divided between Sam and Jamie in the ratio 2 : 3. How much should each receive? The money needs to be divided into 2 + 3 = 5 parts. ) Sam receives

2 5

of $24 000 = $9600

and Jamie receives

3 5

of $24 000 = $14 400

13 $30 000 is to be divided between Hannah and Rachel in the ratio 5 : 3. How much should each receive? 14 If $1500 is divided in the ratio 3 : 1, how much is the larger share? 15 Cordial and water are mixed in the ratio 1 : 8 to make a fruit drink. How much cordial and water would there be in a 180 mL glass of fruit drink? 16 A metal solder is made by combining lead and tin in the ratio 2 : 3. How much tin is needed to make 10 kg of solder? 17 Katrina contributes $50 000 and Paul $30 000 to set up a business. In their first year Katrina is paid a director’s fee of $12 000 from the profit of $68 000. The remainder of the profits is then divided between them in the same ratio as their original contributions. How much of the $68 000 does each receive? 18 The weights of Alison, Vincent and Matthew are in the ratio 4 : 3 : 5. If their combined weights total 168 kg, find the weight of each person. 19 An investment fund has investments in property, shares and government bonds in the ratio 5 : 3 : 2. a If the fund has a total of $1.8 million invested, find the amount invested in each of these three areas.



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b If during the next year the fund manager decides to transfer $160 000 from investments in goverment bonds to shares, find the new ratio of investments in this fund.

100

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25

SOLVING PROBLEMS USING THE UNITARY METHOD

The unitary method involves division to find one unit of the quantity first.

5

0

1

EXAMPLE

The cost of 3 kg of fish is $27.87. What is the cost of 5 kg of this fish? 3 kg cost )

1 kg costs

$27:87 ¥ 3

= $9:29

5 kg cost

$9:29 £ 5

= $46:45

)

?

$27.87

EXERCISE 2K 1 The cost of 5 pairs of socks is $18.75. Find the cost of 6 pairs. 2 If 8 kg of potatoes cost $19.92, what is the cost of 5 kg? 3 The cost of 50 m of rope is $63. Find the cost of 20 m of this rope.

EXAMPLE

2

A discount of 17.5% on a refrigerator results in the price decreasing by $152:25: Find the original price of the refrigerator. 17.5% of the original price = $152:25 ) )

1% of the original price = $152:25 ¥ 17:5 = $8:70 100% of the original price = $8:70 £ 100 = $870

4 A discount of 18% on a stereo system results in the price decreasing by $179.82. Find the original price of the stereo. 5 35% of the students at a school travel by bus. If 259 students travel by bus, how many students are there at the school? 6 Linda and Matthew pay a 15% deposit of $54 000 on a new house. What is the total cost of the house? 7 Harry loses 1.2 kg in the first week of his diet. If this was a 1.5% loss in weight, find his weight before he started the diet.



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95

75

25

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61


95

75

25

5

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8 2% of the Australian fur seal population die each year from entanglement in nets or ropes discarded by fishermen. If approximately 600 seals die each year, estimate the size of the seal population.

EXAMPLE

3

A car dealer sells a car for $19240. This represents the cost of the car to him plus a profit of 30%. For what price did the dealer buy the car?

130% of the cost price = $19 240 ) )

1% of the cost price = $19240 ¥ 130 = $148 100% of the cost price = $14 800

9 A car dealer sells a car for $15 000. This represents a profit of 25% on his cost. What was the cost of the car to the dealer? 10 After a 2:5% increase Sarah’s weekly salary was $697. What was her salary before the increase? 11 Ian has a 60% no claim bonus on his car insurance. This means he receives a 60% discount on the full cost of the insurance. a What percentage of the full cost does he pay? b If he pays $512, what is the full cost of the insurance? 12 The sector graphs show the results of an analysis of burglary claims made to an insurance company: If one year 1125 people were on holidays when their home was burgled, how many were a shopping b at work c on the premises? d How many claims were analysed by the insurance company?

Where were you when the burglar came? Shopping On holidays

b

How much of each ingredient would be needed to make this recipe for i 6 people ii 9 people? Penny has 1.5 kg of pork. How many people can she feed using the above recipe, assuming she has enough of all the other ingredients?

At work

31% Other

11% 11%

Visiting

13 The following is a recipe for stir fried pork for 4 people: a

7%

9%

31% On the premises

100

95

600 g pork 3 tbsp peanut oil 8 spring onions 2 12 tsp grated ginger 1 clove garlic 2 tbsp lime juice 1 3 cup chicken stock 200 g sliced beans

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14 A gardener spread 1.5 kg of sulphate of ammonia over 50 m2 of vegetable garden. At this rate of application a what area of garden could be fertilised with 6 kg of sulphate of ammonia? b How much fertiliser would be needed for 800 m2 of garden?

25

SPREADSHEET APPLICATION

5

Enter the quantities for 4 people for the recipe in question 13 into the cells of a spreadsheet and use it to calculate the quantities of each ingredient for 1, 2, 3, ..... people. Print the resulting table with headings.

0

HAVING COMPLETED THIS CHAPTER

You should be able to 2 approximate numbers by rounding off 2 express decimal numbers in scientific notation and vice versa 2 perform calculations with numbers expressed in scientific notation 2 convert fractions and decimals to percentages and vice versa 2 find a percentage of a quantity 2 increase and decrease a quantity by a given percentage 2 determine the percentage increase or decrease in a quantity 2 determine the overall change in a quantity after repeated percentage changes 2 convert between the commonly used metric units for length, mass, capacity and time 2 understand the possible sources of error in measuring 2 determine the limit of reading, absolute error, upper and lower limits and percentage error for a measurement 2 find the maximum possible error when measurements are used in calculations 2 calculate rates 2 convert between units for rates 2 solve problems involving rates 2 perform calculations and solve problems for concentrations expressed as w/w, v/v and w/v 2 simplify and find the ratio of two quantities 2 divide a quantity in a given ratio 2 solve problems using the unitary method.

2

LANGUAGE AND TERMINOLOGY

1 Copy an complete: Scientific notation is also called ........ notation. 2 What sort of numbers is it useful to express in scientific notation? 3 List five possible sources of error in measuring.

100

95

4 What is the meaning of the term “the limit of reading” of a measuring instrument? 75

5 Write down another term for “the greatest possible error” in a measurement. 6 Explain how to determine the percentage error for a measurement. 7 What is the difference between a rate and a ratio? 8 Write down two different meanings of the word “concentration”.

25

5

9 What number is implied by the term “unitary”?

0



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75

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2

DIAGNOSTIC TEST

1 When rounded to 2 significant figures 3950:628 becomes A 3900 B 4000 C 39

D

3950:63

2 Which of the following numbers are written in scientific notation? A 5 £ 10 000 B 50 000 C 5 £ 104

D

50 £ 103

3 7:06 £ 10¡6 A 0:000 070 6

0:000 007 06

C

706 000

D

7 060 000

4 (4 £ 105 ) ¥ (8 £ 10¡3 ) = A 5 £ 108 B

5 £ 107

C

5000

D

5 £ 102

5 5:06 kg = A 0:005 06 g

5060 g

C

0:0506 g

D

506 g

6 Which of the following are not equivalent to 5:3 metres? A 530 cm B 5300 mm C 0:0053 km

D

0:053 km

7 The capacity of a drinking glass would be closest to A 2 mL B 20 mL C 200 mL

D

2L

8 3.3 hours is equivalent to A 3 h, 30 min. B

D

3 h, 18 min

D

2:25

B

B

3 h, 3 min.

C

3 h, 20 min.

9 Which one of the following is not equivalent to 22:5%? 4 45 B 0:225 C 200 A 90 10 17:5% of 21 metres = A 3:675 m

C

367:5 m

D

1:2 m

11 What percentage is $39 of $260 A 0:15% B 15%

C

6:7%

D

670%

12 60 kg increased by 5% is A 65 kg B

C

63 kg

D

3 kg

94%

D

6%

B

36:75 m

5 kg

13 The percentage decrease from 80 kg to 74 kg is A 92:5% B 7:5% C

14 What would be the value of $30000 after 2 years if it increases in value by 20% in the first year and then decreases in value by 20% in the second year? A

$4800

B

$24 000

C

15 The absolute error in the measurement 3:6 litres is A 0:1 L B 0:05 L C

$28 800

D

$30 000

100

95

75

0:5 L

D

3:55 L

16 The mass of a can of soup was measured to be 250 g to the nearest 10 g. The percentage error in this measurement is A 4% B 2% C 0:4% D 0:2%

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17 The length of the side of a square was measured to be 8 cm. The maximum error in stating that the perimeter is 32 cm is A 0:5 cm B 2 cm C 1 cm D 4 cm

95

75

18 A garden hose can fill a 5 litre bucket in 10 seconds. What is the rate of flow in litres per hour? A 180 B 1800 C 30 D 200

25

19 Water leaks from a tap at the rate of 3 drops/min. If 5 drops equal 1 mL, the amount of water wasted in a year is approximately (1 year = 365 days) A 105 L B 7884 L C 1577 L D 315 L

5

0

20 If the current exchange rate is 64 US cents for each $A, how many Australian dollars would you need to exchange to get $US1000? A 640 B 1000 C 1562:5 D 3600 21 20 g/100 mL is equivalent to A 2 g/mL B 0:2 g/L

C

20 kg/L

20% w/v

D

22 The ratio of boys to girls in a school is 5 : 7. If there are 364 girls, then the number of boys is A 260 B 55 C 275 D 77 23 When $56 000 is divided into 2 parts in the ratio 5 : 3 the larger part is A $33 600 B $21 000 C $35 000

D

$28 000

24 If 6 loaves of bread cost $12:84, the cost of 10 loaves is A $21:40 B $2:14 C $128:40

D

$214

25 After receiving a 20% discount, Melissa paid $68 for a pair of jeans. The amount she saved was A $13:60 B $17 C $85 D $51 If you have any difficulty with these questions, refer to the examples and questions in the exercise indicated.

Question Section

?

2A

1 A

2, 3, 4 B

5, 6, 7, 8 C

9-14 D

15, 16 F

17 G

18, 19, 20 H

21 I

22, 23 J

24, 25 K

REVIEW SET

1 Round off 3659:063 to a the nearest 100 d 3 significant figures

b e

the nearest whole number 5 significant figures

c f

2 decimal places 1 significant figure

100

2 State whether or not the following numbers are in standard notation c 2:04 £ 10¡6 : a 6 £ 1000 b 15 £ 107 3 Express in standard notation a 105 000 000

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b

0:000 062

4 Calculate, writing answers in standard notation b a (4:1 £ 108 ) £ (6 £ 105 ) c

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(8 £ 105 )4

d

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(1:96 £ 10¡3 ) ¥ (1:4 £ 107 ) p (8:41 £ 10¡12 )

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5 Convert to a percentage b 1 23 a 58

c

0:065

6 Convert to a i fraction ii decimal a 27% b 145% c 2:4% 7 Find a 43.5% of 22 metres

b

d

7 14 %

1 12 % of $5400

5

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8 What percentage is a $12.60 of $180 c 55 minutes of 2 12 hours? 9

a

Increase 80 kg by 3%.

b

454 g of 2 kg

b

Decrease $290 by 15%.

10

a Find the percentage increase from $4.80 to $5.10. b Find the percentage decrease from 70 kg to 67.2 kg.

11

a What would be the value of $25 000 after 2 years if it increases in value by 8% in the first year and then decreases in value by 5% in the second year? b Calculate the overall percentage change in value over the 2 years.

12 Convert a 5.24 km to m

b

3.06 m to cm

c

15.9 cm to mm

d

2.1 m to mm

13 Convert a 1078 m to km

b

1460 cm to m

c

3295 mm to cm

d

560 mm to cm

14 Convert a 7.08 t to kg

b

5.654 kg to g

c

4.2 g to mg

15 Convert a 19 300 kg to t

b

2085 g to kg

c

600 mg to g

16 Convert a 7.8 kL to L

b

4.26 L to mL

17 Convert a 15 500 L to kL

b

70 mL to L

18 Convert to minutes a 3.5 hours

b

3 h 50 min

19 Convert 258 seconds to a minutes b

100

minutes and seconds

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20 For each of the following measurements, find i the limit of reading ii the greatest possible error iii the lower and upper limits of the true measurement iv the percentage error (to 1 decimal place) a

7.5 metres

b

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280 g to the nearest 10 g

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21 The a b c d e f g

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length and breadth of a rectangle were measured to be 6 cm and 4 cm. Calculate the perimeter using these measurements. Write down the lower and upper limits of the true length and breadth. Find the lower and upper limits of its true perimeter. Hence, what is the maximum error in the answer in a? Calculate the area using the measurements given. Find the lower and upper limits of the true area. Hence, find the maximum error in the answer to e.

22 A truck used 46 litres of diesel on a trip of 250 km. Express the rate of fuel consumption in L/100 km.

0

23 Convert a 12 t/Ha to kg/m2 24 Convert a 280 g/kg to b 25 mL/100 mL to c 15 g/100 mL to

b i i i

9 m/s to km/h

g/g mL/L g/mL

ii ii ii

g/100 g mL/mL kg/L

iii iii iii

%w/w %v/v %w/v

25 A cough mixture contains 10% w/v of codeine. How much codeine is there in a 200 mL bottle of the mixture? 26 Simplify the ratios a 40 : 24 d 45 cm : 1.2 m

b e

1:6 : 3 $1:50 : 80cents

27 Express each of the above ratios in the form

i n:1

c

3 4

: 1 58

ii 1 : n

28 The ratio of boys to girls in a school is 8 : 9. If there are 256 boys, how many girls are there? 29 One day at the school canteen the ratio of bread rolls to sandwiches sold was 3 : 2 and the ratio of sandwiches to pies sold was 1 : 4. If the canteen sold 72 pies on this day, how many bread rolls were sold? 30 Divide $48 000 into 2 parts in the ratio 3 : 5. 31 The cost of 8 pens is $6:32: What is the cost of 5 pens? 32 Penny received a discount of $133.50 on a TV set. This was 15% of the original price. Find the original price of the TV. 33 After a 3% wage increase, Julie’s weekly wage increased to $494.40. What was her wage before the increase?

?

2B

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REVIEW SET

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1 Round off 1472.634 to a the nearest 10

b

2 significant figures

2 Express in scientific notation a 749 000

b

0.000 003

c

2 decimal places

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3 Calculate (1:4 £ 107 ) £ (4:5 £ 108 ).

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4 Complete the table:

Fraction

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1 2

1 3

1 4

1 5

1 6

1 7

1 8

1 9

1 10

Decimal Percentage 25

5 Kieren invests $8000 in a share fund. In the first year the shares increase in value by 12% and in the second year they increase by 15%. Calculate the overall percentage change in the value of the shares.

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0

6 The length of a table was measured to be 154 cm, to the nearest centimetre. a b c d

Write down the limit of reading for this measurement. What is the greatest possible error? Determine the lower and upper limits of the true length. Calculate the percentage error in this measurement.

7 Sue buys 2.8 m of dress material for $99.68. What is the cost per metre of the material? 8 Water flows into a tank at the rate of 8 L/min. a How much water will flow into the tank in 3 hours and 25 minutes? b If the tank has a capacity of 4.24 kilolitres, how long will it take to fill the tank? 9 Simplify the ratios a 36 : 45

b

1 14 : 2

c

1:5 : 2:4

10 If 6 dinner plates cost $29.34, what would be the cost of 8 dinner plates?

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2C

REVIEW SET

1 Round off 0.005 06 to a 2 significant figures

b

2 decimal places

2 Jin’s weekly salary increased from $380 to $391.40. Calculate the percentage increase. 3 Complete a 5.37 m = ...... cm

b

19.6 kg = ...... g

c

0.07 L = ...... mL

4 The weights of 2 bags of potatoes were measured, to the nearest kg, to be 49 kg and 51 kg. a b c d

What is the total weight of the 2 bags using these measurements? Write down the lower and upper limits of the true weight of each bag. Calculate the lower and upper limits of the total weight of the 2 bags. Hence determine the maximum error in the answer to a.

5 Convert a 60 km/h to m/s

b

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50 g/m2 to kg/Ha 25

6 Change a 10 g/kg to g/g

b

40 mL/100mL to mL/L

c

280 g/500 mL to mg/mL

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7 “Shinee” car polish contains 350 mL/L of liquid hydrocarbons. a What quantity of liquid hydrocarbons are present in a 375 mL bottle of polish? b Express this concentration as a % v/v.

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8 $15 000 is divided between 2 people in the ratio 5 : 3. What is the smaller share? 9 The weight of meat decreases when it is cooked. If the ratio of the weights of raw meat to cooked meat is 1:25 : 1, calculate the weight of a 400 g piece of meat after it has been cooked.

25

10 Sean went on a diet and in the first week he lost 1.5 kg. This represented a 2% decrease in his weight. Calculate his weight before he started the diet.

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REVIEW SET 1 Change to a percentage b 0:196 a 17 40

c

3

2 The number of children vaccinated at a clinic doubled from one year to the next. Comment on the statement: “There was a 200% increase in the number of children vaccinated.” 3 Round off 2.0695 to a 1 b

2

4 Convert a 13.65 m to cm

b

c

3

d

3460 kg to t

5 Calculate a (1:08 £ 10¡6 ) ¥ (7:2 £ 10¡5 )

c

b

4 significant figures

276 seconds to minutes and seconds. p 1:96 £ 1020

6 A photocopier is bought for $15 000. If it depreciates by 28% of its value each year, how long will it take for its value to fall below its scrap value of $1200? 7 The a b c

base and perpendicular height of a triangle were measured to be 15.4 cm and 12.5 cm. Find the area of the triangle using these measurements. Calculate the range within which the true area lies. What is the maximum error in using a as the area?

8 Bill the builder purchased $2800 worth of materials from the hardware store. He receives a trade discount of 16% and then a further 5% discount if he pays within 3 days of receiving the account. What is the overall percentage discount if he pays within 3 days? 9 A car uses 32 litres of petrol to travel 250 km. a Calculate the petrol consumption in L/100 km. b At this rate of consumption,how much petrol (to nearest litre) would it use to travel 650 km, and c how far could it travel on 56 L?

100

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10 Each 5 mL of a cough mixture contains 15 mg of Sephazosine. a Convert this concentration to i mg/mL ii kg/L iii a %w/v b How much Sephazosine is there in i a 7.5 mL dose ii a 200 mL bottle of this cough mixture?

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CHAPTER

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Earning money

AREA OF STUDY

This chapter deals with the range of ways individuals earn and manage money. The main mathematical ideas investigated in this chapter are: 8 different methods of calculating income 8 allowances, loadings, bonuses, royalty payments, commission and holiday loading calculations 8 piecework payments 8 government allowances, pensions and health payment calculations 8 charges associated with financial institutions 8 budgeting 8 common household bills.

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EARNING MONEY FM1 (Chapter 3)

A

EARNING AN INCOME

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The earning of money from working is the basis of a person’s income in our society. Without an income a person is unable to satisfy their basic needs. There are many ways of receiving money for work done. Most people work for an employer who pays them for the work done. Earnings based on hours worked are called wages. Earnings that are a fixed amount regardless of the number of hours spent working are called salaries. This section deals with salary and wage payments. A salary is an income that is usually stated as an amount per annum, although the salary may be paid weekly, fortnightly or monthly. People who are paid salaries include nurses, government employees, teachers and other professionals.

per annum means per year.

Wages are based on an hourly rate for a certain number of hours per fortnight. Wages are usually paid on a fortnightly basis. People who are paid wages include factory workers, shop assistants, building workers, mechanics and other workers. In converting from yearly salaries to monthly, fortnightly and weekly amounts the following are used: 1 year = 365 days = 12 months = 52 weeks = 26 fortnights

EXAMPLE

Fortnightly = weekly ´ 2.

1

Jenny’s salary is $31 200 per annum. Express this salary as an amount a monthly b weekly c fortnightly

a

monthly

b

= $31 200 ¥ 12 = $2600

EXAMPLE

weekly = $31 200 ¥ 52 = $600

2

Michiko earns $283 per week. a How much does Michiko earn per year? b How much does Michiko earn per month? a Yearly = $283 £ 52 = $14 716

c

fortnightly = $31 200 ¥ 26 = $1200

100

1 month is not 4 weeks!

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b Monthly = $14 716 ¥ 12 = $1226:33

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EARNING MONEY FM1 (Chapter 3) 100

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EXERCISE 3A 1 Caterina earns $86 924 per annum. Calculate her weekly pay. 2 Express the following annual salaries as amounts i monthly ii weekly a $38 903 b $27 106 c e $5076 f $50 990

iii fortnightly $30 098 d

3 Ashley earns $379 per week. How much does Ashley earn a per year b per month?

$10 987

Remember there are 52 weeks in a year.

4 Calculate the annual salary for each of the following: Name C Cooke D Moussa B Boje T Lambert 5

Weekly wage $423:97 $510:53 $315:27 $285:19

Annual Salary

a David receives an annual salary of $21 398:45: How much does he earn each week? b Debra receives an annual salary of $35 200. How much does she earn each week? c Russell receives an annual salary of $46 837:25: How much does he earn each fortnight? d Kristy receives an annual salary of $78 500. How much does she earn each fortnight? e Iain receives an annual salary of $43 195:27. How much does he earn each fortnight? f Grace receives an annual salary of $18 573. How much does she earn each month?

There are 26 fortnights in a year.

There are 12 months in a year.

6 Find the difference between the following annual salaries

100

a Kylie: $568:30 a week Stacey: $27 321 a year b Roland: $1803 a fortnight Paul: $55 271 a year c Joshua: $2610 a month Aaron: $693:20 a week

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7 Which pay option offers the highest weekly wage? By how much? a A $903:20 a week or B $45 103:20 a year b A $953:29 a fortnight or B $2405 a month c A $1638 a month or B $23 512 a year



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EXAMPLE

3

Monica is paid $8.65 per hour. How much does she earn if she works a 6 hours in one day b 6 hours per day for 5 days?

25

Pay = $8:65 £ 6 = $51:90

a 5

b

Pay = $8:65 £ 6 £ 5 = $259:50

0

8 Aaron works as an assistant in a clothing store and is paid $8:75 an hour for a 35 hour week. Calculate his weekly wage. 9 Madonna works as a receptionist. If she is paid $7:90 an hour, how much does she earn in a 38 hour week? 10 David works at “Pizza Barn” delivering pizzas. Calculate his weekly wage if he earns $11:20 an hour, based on the following hours worked in a given week: a 16 hours b 24 hours c 35 hours d 41 hours 11 Kristie earns $7:43 an hour. Calculate her weekly wage if she works a 18 hours b 35 hours c 40 hours

d

48 hours

12 Rod, Anastasia, Sue and Zoltan work at the Reebok factory. If the hourly rate of pay is $8:40, calculate i

the total number of hours worked in one week by each employee shown

ii

each employee’s weekly wage.

a Day M T W T F

Name: Rodney In Out 7.00 am 4.00 pm 8.00 am 4.00 pm 7.00 am 5.00 pm 11.00 am 4.00 pm 9.30 am 2.30 pm Total:

b Hours

Day M T W T F

..........

Name: Anastasia In Out 7.40 am 5.40 pm 10.20 am 4.20 pm 9.00 am 3.00 pm 8.30 am 3.30 pm 8.35 am 3.35 pm Total:

Hours

.......... 100

c Day M T W T F

Name: In 7.30 am 8.20 am 9.15 am 8.00 am 6.00 am

Sue Out 5.00 pm 4.50 pm 5.15 pm 4.30 pm 6.00 pm Total:

d Hours

..........

Day M T W T F

Name: In 6.00 am 7.20 am 8.00 am 8.20 am 9.30 am

Zoltan Out 4.30 pm 5.50 pm 3.30 pm 2.50 pm 5.30 pm Total:

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Hours

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13 Copy and complete the following table

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Wage Table M

T

Day W

S Smith

7

5

¡

4 12

9

$11:70

D De Souza

11

4

3 12

6 12

7

$10:15

5

A Bentley

6 12

¡

¡

3

9 12

0

C Johnson

4

2 12

7

2

1 12

K Kernell

3

7 12

5

4 12

¡

P Patrick

5 12

9

2 12

¡

6 12

75

Name 25

T

F

Hours Worked

Hourly Rate

Weekly Wage

$248:90 $12:17 $174:80 $16:83

14 James works as a casual at the local supermarket. He is paid $7:05 an hour from Monday to Friday and $10:35 an hour on weekends. During a week of his school holidays he worked from 3:00 pm till 6:00 pm Monday to Friday and 8:30 am till 1:30 pm on Saturday and Sunday. a How many hours did he work on week days? b How many hours did he work on weekends? c Calculate his income for the week.

EXAMPLE

4

Angus works 5 days per week, 8 hours per day. If he were paid $7.80 per hour, how much would he earn in a year?

There are 52 weeks in a year. Pay per day

Pay per week

Annual pay

= $7:80 £ 8

= $62:40 £ 5

= $312 £ 52

= $62:40

= $312

= $16 224

Angus earns $16 224 per year.

100

15 An electrician works a 38-hour week. Find his yearly pay if he earns $28.40 per hour.

95

16 Ikbar works a 9-hour day, six days per week. His hourly rate is $9.27. Find his pay for 1 year. 17 Jim works 26 days per month. He averages 7 hours per day. Find his annual pay if his hourly rate is $10.17. 18 Nato is paid $9.60 per hour. He works 4 hours per day seven days a week. Find his pay for one year.

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EXAMPLE

5

Convert an annual salary of $47 424 to a a weekly salary

b an hourly salary based on 38 hours per week.

a weekly = $47 424 ¥ 52 = $912

b hourly = $912 ¥ 38 = $24

0

19 Indoo receives an annual salary of $63 028. Convert this to a weekly salary and find Indoo’s average hourly rate if she works 52 hours per week.

The numbers of hours worked per week vary according to the job.

20 Guido works 7 hours a day five days per week. Find his hourly rate if his annual income is $38 900: 21 Steve is paid a salary of $78 904 per annum. Find his a weekly pay rate b hourly rate if he works 45 hours per week.

INVESTIGATION 1: TIME PERIODS Armin and Richard were comparing their pay. Richard says that he earns more per week than Armin. Richard earns $23¡520 per annum, and he says that this is $490 per week. Armin earns $480 per week. As the yearly pay for Richard is correct, Richard has made an error in calculating weekly pay . 1 Find Armin’s annual pay. 2 Find Richard’s correct weekly pay. 3 Who really earns more? 4 Richard said that he divided the annual pay by 12 to make it monthly and then by 4 to make it weekly. Find the mistake that Richard made. Explain.

RESEARCH PROJECT Find out exactly how many days there are in a year. Investigate the reason for leap years. Are all years which are divisible by four, leap years? Why not? 100

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75

ALLOWANCES AND LOADINGS

Most employers expect their workers to work a certain number of hours per day and hours worked in excess of these are paid at an overtime rate. The two main overtime rates are time and a half and double time.



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Employees who are required to work under difficult, unpleasant or dangerous conditions are often paid an allowance that is above the normal rate of pay. Some examples include an allowance for cleaning toilets, working in confined spaces, working in hot conditions, dealing with toxic substances, etc. Other allowances that are paid are for uniform, drycleaning of uniform, travelling, meals, tools and so on.

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An annual leave loading is paid to employees taking their annual holidays and is added to their pay.

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RESEARCH PROJECT 1 Investigate the rates paid to various workers as special allowances. 2 Investigate the history of the annual leave loading. Many employers are phasing out or trying to phase out leave loading. Based on your research give your opinion on why this is the case.

EXAMPLE

1

Convert an hourly rate of $10.20 to an overtime rate of a time and a half

b double time.

a Time and a half rate = $10:20 £ 1:5

b Double time = $10:20 £ 2

= $15:30

EXAMPLE

= $20:40

2

Janice works 10 hours on a Sunday. The Sunday rate of pay is $7:80 per hour for for the first 4 hours, then “time and a half” for additional hours worked. Calculate Janice’s wage for Sunday. Number of hours worked is 10: )

4 hours at normal rate and 6 hours at ‘time and a half’. Wage at normal rate

Wage at ‘time and a half’ ) Janice’s wage is

“Time and a half ” means the hourly rate is 1Qw_ times greater.

100

= 4 £ $7:80 = $31:20

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1 12

= 6£ £ $7:80 = $70:20 25

= $31:20 + $70:20 = $101:40

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EXERCISE 3B 1 Convert these normal pay rates to overtime rates at time and a half. a $7:83 b $14:35 c $11:59

d

$18:70

2 Convert these normal pay rates to overtime rates at double time. a $8:35 b $19:58 c $12:65

d

$9:43

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3 Brian is paid $8.42 per hour. Calculate his weekly wage if he works 38 hours at normal time and 6 hours overtime at time and a half. 4 Vikash is paid $10.24 per hour for a 35-hour week. He is paid overtime at double time for all other hours. Calculate his pay in a week where he works 44 hours. 5 Anna earns $6.35 for the first 30 hours in a week. She receives time and a half for the next 5 hours and double time after that. Find her pay for a 42-hour week. 6 Karen earns $8:70 an hour. Calculate her weekly wage if she worked 33 hours at a normal rate and 6 12 hours at time and a half. 7 Alexia works 16 hours at $10:10, 4 hours at ‘time and a half’, and 3 hours at double time. Calculate her weekly wage. 8 Shane is paid $7:10 per hour. His normal working week is 35 hours. He is paid time and a half for the first 5 hours worked beyond the 35 hours and double time for any hours after that. Calculate his weekly wage if he worked 49 hours. 9 Jontey works from 8:00 am to 4:00 pm on a Sunday. He is paid time and a half for the morning work and double time for the afternoon. If the normal hourly rate is $6:90, calculate Jontey’s earnings for Sunday. 10 The table which follows shows the hours worked by employees at Active Sports. Calculate the weekly wage for each if ²

the normal hourly rate is $9:40,

²

overtime of time and a half is paid for hours worked beyond 4 hours from Monday to Friday,

²

time and a half is the hourly rate on Saturdays.

Name

M

Wage Table for Active Sports Day/Hours Total Hours Worked T W T F S Normal Rate Time & a Half

M Goodwyn

8

8 12

8

9

4

6

100

b J Osbourne

9

9

9

7

8

3 12

95

G Dale

10

9 12

9

9 12

10

2

75

4 12

¡

¡

10

9

9

a

c

d E Evans

Weekly Wage

e

M Harkin

7

6

4 12

5

2

7 12

25

f

A Fenech

10

9

8

9

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EXAMPLE

3

Vicki earns $9.45 per hour as a cleaner. If she cleans the toilets she receives an allowance of $ 12.00 per day. Find her weekly pay if she works for 4 hours each weekday and cleans the toilets on Wednesday and Friday.

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5

Normal pay

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= $9:45 £ 4 £ 5 = $189

Total pay = $189 + $24 = $213

Toilet allowance = $12 £ 2 = $24

11 Rose is paid $10.35 per hour welding bins. If it is over 30o C she is paid an extra $6.50 per day. Find her pay for a 35-hour week with 2 days over 30o C. 12 Harry works in a timber yard and is paid $11.40 per hour. He receives a dust allowance of $12.50 per day. Find his pay for a 5-day, 40-hour week. 13 Georgina works in a restaurant and receives a uniform allowance of $6.50 per night. Her hourly rate of pay is $12.85. Find her pay if she works 5 nights for 6 hours per night. 14 Louie works as a plumber’s assistant and is paid $6.80 per hour. He is paid a digging allowance of $8.50 per day if he has to dig trenches and $12.80 per day if he works on open sewers. Find his pay in a week where he works 50 hours, digs on three days and works on an open sewer for two days. 15 Benny is a security guard and is paid $14.25 per hour. If he works on the door he is paid $25 danger money. Find Benny’s wage for a week where he is on the door for 3 nights and works 35 hours. 16 Nabil works in a factory and is paid $12.54 per hour. If he works the furnace he is paid a heat allowance of $2.27 per hour in addition to his normal rate. Find his weekly pay if he works a total of 32 hours spending 8 of them working the furnace. 17 Jenny works in a bar and is paid $13.90 per hour. If she makes cocktails she receives an extra $3.20 per hour. Find her pay in a week where she works 20 normal hours and 10 hours making cocktails.

SPREADSHEET APPLICATION The following spreadsheet calculates an employee’s wage including overtime at time and a half. A 1 2 3 4 5 6 7

Wage Calculator Hours Normal Overtime Normal hours O/T hours Hourly rate

B Mon 8 2 37 6 $7.60

C Tues 8 Normal pay O/T pay Total pay

D Wed 8 3 $281.20 $68.40 $349.60

E

F

Thur 5 -

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C:\...\NSWGM\NSWGM_03\077NG03.CDR Wed Jan 19 09:22:53 2000


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The following view shows the formulas used. 95

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1 2 3 4 5 6 7

A Wage Calculator Hours Normal Overtime Normal hours O/T hours Hourly rate

B Mon 8 2 =SUM(B3:F3) =SUM(B4:F4) 7.6

C

D

Tues 8 Normal pay O/T pay Total pay

Wed 8 3 =B5*B7 =B6*1.5*B7 =D5+D6

E

F

Thur 5 -

Fri 8 1

How would you change cell D 6 in the spreadsheet if the overtime rate was double time instead of time and a half?

C

COMMISSION

Real estate agents and car salespersons are examples of people paid by commission. A commission is a percentage of the total cost of the goods sold paid to the salesperson. The more sold the greater the income.

EXAMPLE

1

Larry sold a car valued at $22 800. His commission for selling the car was 6% of the selling price. Calculate Larry’s commission. Commission = 6% of $22 800 =

6 100

£ $22 800

= $1368 ) Larry earned $1368 from his sale of $ 22 800.

EXAMPLE

2

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David is a real estate salesman. He is paid a commission of 0.6% of the selling price of any house he sells. If in March he sells houses to the value of $468 000, find his income for March.

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Income = 0.6% of $468 000 = 0:6 ¥ 100 £ $468 000

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= $2808

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EXERCISE 3C 1 Find the commission, to the nearest whole dollar, of

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a

25% of $42 000

b

18% of $6 782

c

7% on $840

d

15 12 %

on $132

e

2:7% on $80 000

f

22:5% on $12 500

g

3% on $97 400

h

6:4% on $48 870

i

14% on $150 000

j

17 13 %

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on $287 648

2 Darren sold a house valued at $350 000. Calculate the commission from the sale if his rate of commisson was 4:8%. 3 Josephine sold a car valued at $48 000. Calculate her commission if the rate of commission was 7%. 4 An investment advisor receives 6% commission on money invested. How much does he earn for advising a firm on an $753 000 investment? 5 Karen and Tony both work for “Up Market Realty”. Karen sold a home for $235 000 and Tony sold two units, each valued at $85 000. The commission rate was 8%. a Calculate Karen’s commission. b Calculate Tony’s commission. c Who made more from the sales? By how much?

EXAMPLE A salesperson earns $130 a week plus 13% commission on sales. If her weekly sales were $1780, find a her commission

a

Commission = 13% of $1780 13 £ $1780 = 100 ) commission = $231:40

b her total earnings for the week.

b

Total earnings = base wage + commission = $130 + $231:40 ) total earnings = $361:40

6 Jenny earns $165 per week plus 10 12 % commission on sales. Calculate her total earnings for each week if her sales are as follows: a $124 b $881 c $2 763 d $3 882 e $4 009 f $5 720 7 John earns $110 per week plus 15% commission on sales. Calculate his total earnings for a week if his sales are $2050: 8 In one week, Mark sold goods to the value of $143 000. If his base wage is $145, calculate his weekly wage if his commission rate on sales is 13:7%:



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EXAMPLE

4

An auctioneer sells 40 chairs at $18:75 each. What is the total sale value? Determine his percentage rate of commission if he earns $75 commission.

a b 25

Remember that to change to percentage we multiply by 100%.

5

0

a

Total sale value

b

= 40 £ $18:75

Commission % =

commission earned £ 100% total sale value

=

75 £ 100% 750

= $750

= 10%

9 A car salesperson earned $7 853:20 commission for 1 week. If his total sales were $54 160, calculate the rate of commission as a percentage. 10 Determine the rate of commission if Sue earned $9 348 on sales valued at $76 000. 11 Bart earned $446:20 on sales valued at $4 600. Karla earned $213:18 on sales valued at $2 090: a Who has the better rate of commission? b What is this commission rate? 12 Copy and complete the following table: Name a

E Swift

b

M Keane

c

S Wise

d

G Walsh

e

G Potts

f

L Monroe

Commission rate

Total of Sales

4:4%

$72 000

8:2%

$115 872

$4 634:88

$64 500

$9 481:50

$183 600 $286 472

12 13 %

Commission earned

$16 472:14

100

95

$350 700

75

13 In one week a used car salesman sells three Ford Falcons each priced at $25 360 and earns 4 14 % commission on his total sales. Find his commission for the week. 2 12 %

commission on the 14 A real estate agent sold a house valued at $276 000. If he received 1 first $150 000 and 2 % commission on the remainder, find his total commission.



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SALES AGENCY CAREER

25

We need achievement driven sales people to sell and co-ordinate advertisement placement in our publications. The successful applicants will: (1) Enjoy meeting a diverse range of business people. (2) Be prepared to work hard to enjoy a good lifestyle. (3) Receive a supported introduction program to ensure a flying start.

Having secured this position, Felicity is paid a salary of $1940 per month plus commission of 4 12 % on her sales. During May she made sales totalling $39 423. Calculate her May earnings.

Our ideal agents will be commission orientated and have a year's direct sales or business experience.

5

To apply contact JOCK FLINSTONE on 0411 325 644

PUMA PRINTING PTY. LTD.

0

37 Left Street Georgetown 8034

HHP34566/97 A

16 Madison works in sales. She is given the choice of being paid by 8% commision on sales only, or a base salary of $200 per week plus 2 12 % commission on sales. If Madison sells $3050 worth of goods in a week which method of payment would pay the most? By how much is she better off? 17

WA N T E D

REAL ESTATE SALES CONSULTANTS 2 positions, must be licensed, currently employed/or looking for a start but must have completed full schooling.

Further details, George or Milli (0096) 1542 6666

Jenny responded to this advertisement and was offered a retainer of $640 per month, $140 per month car allowance, plus 1 12 % commission on any houses she sells. If during her first month of work for her new company, she sold four houses valued at $121 000, $162 000, $87 000 and $196 000 respectively, determine her total income for the month.

18 Justine is a car salesperson. She is paid $800 per month plus 12% commission on the value of all the cars she sells. Value is calculated by subtracting the trade in price from the selling price. During April she sells the following cars. Find her income for the month of April.

Model Commodore Falcon Datsun 260Z Excel

Car price $15 990 $11 250 $6250 $13 990

Trade in $5350 $3800 none $7850

19 George is a confecSales value Commission rate tionary salesperson. He Up to $15 000 No commission is paid $360 per week $15 001 to $25 000 Nil plus 4% of sales over $15 000 plus commission on all $25 001 to $40 000 $600 plus 3% of sales over $25 000 sales. His commission $40 001 and over $1 050 plus 1.5% of sales over $40 000 is calculated using this scale. Calculate George’s earnings if he achieved the following sales: a March sales $24 890 b April sales $36 002 c May sales $14 050 d June sales $53 821 20 Rhonda works for Ray Green real estate. She is paid a retainer of $180 per week, a $50 car allowance and 43% of the selling commission of any house sold. Ray Green receives the other 57%. During the week she sells a house that generates a selling commission of $8900. Calculate a Rhonda’s share of the $8900 selling commission b Rhonda’s weekly earnings c Ray Green’s share of the selling commission

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d Rhonda’s earnings if the house earned $6500 commission instead of $8900.

100

21 Reginaldo is paid 6.5% commission on sales in excess of $1500. Find his commission on sales of a $5467 b $4321 c $560 d $9800

95

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25

D

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BONUS AND HOLIDAY LOADING

0

Employees sometimes get extra payments in addition to their normal wages. A bonus is a gift/incentive for employees who have worked hard over a period of time. Employers use a bonus to encourage the employees to work even harder! Holiday loading is an increase in an employee’s pay whilst on holidays. The holiday loading is a fixed percentage of the employees normal pay. It is usually at a rate of 17¡¡¡ %.

1

EXAMPLE

Renee worked extremely hard throughout the year. Her reward was a bonus of 6% of her yearly income. If her yearly income was $58 000, calculate the value of her bonus. Bonus = 6% of $58 000 =

6 100

£ $58 000

58 000

×

6

÷

100

c f

9% of $123 468 15% of $271 096

=

= $3480 Renee’s bonus is $3480:

?

EXERCISE 3D 1 Determine the bonus for each annual salary: a 5% of $53 000 b 8% of $92 500 d 10% of $196 320 e 10% of $253 452

100

2 Zofia earns $590 per week. Calculate her total weekly wage if she is given a bonus of a 5% of her weekly wage b 12% of her weekly wage c 20% of her weekly wage d 14% of her weekly wage.

EXAMPLE

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2

25

Lee is entitled to 4 weeks annual leave. The leave loading is 17 12 %. If his weekly pay is $368, find his pay for his holiday.

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4 weeks pay = $368 £ 4 = $1472

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75

Loading = 17 12 % of $1472 = 17 12 ¥ 100 £ $1472

25

= $257:60

5

) holiday pay = $1472 + $257:60 = $1729:60

0

3 Allie works as a personal assistant and is paid $538 per week. If her 4 weeks holiday pay attracts a loading of 17 12 %, find her pay for her holidays. 4 The holiday loading is 17 12 %. Find the loading on 4 weeks pay for a c

a weekly wage of $611 b a fortnightly wage of $1054 an hourly rate of $8.45 for a 38 hour week.

5 If 17 12 % holiday loading is given on 4 weeks normal pay, find the holiday loading for a b c d e

John, who earns $1700 in 4 weeks Paul, who earns $720 in a fortnight Sid, who earns $430 in a week Jenny, who earns $12:50 an hour (for a 30 hour week) Angie, who earns $15:65 an hour (for a 35 hour week).

6 Raymond earns $96 325 per year. At the end of the year he was given a bonus equal to 13% of his yearly wage. Calculate his bonus. 7 Ximena was given 4 weeks holiday plus 17 12 % holiday loading. If her normal weekly pay is $637:52, calculate a her normal 4 week pay b her holiday loading for 4 weeks c her total holiday pay. 8 Paul earns $14:30 an hour and works a 40 hour week. Calculate his holiday loading if he is given 17 12 % of 4 weeks wages. 9 Slavo earns $35 600 per year. At the end of the year his company made a profit of $129 561. As a reward, Slavo was given a bonus of 1% of the profits. a What was his bonus? b What was his total income for the year? 10 Casey received a 17 12 % holiday loading on four weeks normal wages. She normally works a 35-hour week. Her 4 weeks holiday pay and loading is $1855.56. a b

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wage plus loading = 117 Qw_ %

75

25

Find her normal weekly pay. Find her normal hourly pay rate.

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11 Samantha received a 17 12 % holiday loading on four weeks normal wages. She normally works a 38-hour week. Her 4 weeks holiday pay and loading is $2232.50. a Find her normal weekly pay. b Find her normal hourly pay rate.

100

95

75

PIECEWORK AND ROYALTIES

25

People employed making items may be paid for each item produced. This is piecework. An author will receive a percentage of the selling price of books they have written. This is called a royalty. These are two ways of receiving income.

5

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EXAMPLE Enrique paints wooden fire engines. He is paid $4.38 per fire engine painted. In a week he paints 143 fire engines. How much is he paid? Pay = $4:38 £ 143 = $626:34

?

EXERCISE 3E 1 Austin spray paints wooden toys and is paid $6.90 per toy. Find his pay if he paints 168 toys. 2 Shania assembles skateboards. She is paid $15.00 per board assembled. How much does she earn for assembling 43 skateboards? 3 Ricky waxes surfboards for the surf hire shop. He is paid $6.70 per board waxed. Find his pay for a day in which he waxes 15 boards. 4 Ellen works in a belt factory. Each belt completed earns her $4.20. Find her wage for a week in which she makes 157 belts. 5 A factory worker is paid $3.66 for each garment completed. Calculate his wage if he completes 562 garments. 6 A doctor charges each person $21.90 per consultation. If the doctor sees, on average, 4 people an hour, calculate his earnings for an eight-hour day.

100

EXAMPLE

95

Antonia makes ‘Naughty Dolls’. She is paid $26 for each doll she makes.

75

a How much does Antonia earn if she makes 18 dolls? b If it takes Antonia 5 hours to make a doll, what is her hourly rate? 25

a Income = 18 £ $26 = $468

b Hourly rate = $26 ¥ 5 = $5:20

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H:\...\NSWGM\NSWGM_03\084NG03.CDR Tue Jan 18 10:09:15 2000


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7 Ivan refelts billiard tables and is paid $90 per table completed. If he completes 8 tables in a 35 hour week, find his a weekly pay b hourly rate if each table takes 6 hours.

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25

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8 Allan strings new tennis racquets and is paid $7.50 per racquet strung. In a 40-hour week Allan strings 120 racquets. Find his a weekly pay b hourly rate. 9 Sarah works peeling prawns. She is paid $6.90 per kilogram peeled. In 30 hours she peels 75 kg of prawns. Find her a weekly pay b hourly rate. 10 Nadia is a hairdresser who works at home. She charges $17 for a hair cut, $42 for a cut and colour and $65:20 for a perm. In one week she completes 7 haircuts, 14 cuts and colours and 2 perms. a Find her income for the week. b If Nadia worked a total of 32 hours during this week, calculate her hourly rate of pay. c Calculate her profit for the week, if her expenses were $287:30: 11 A dress-maker charges $19:00 for major alterations and $11:20 for minor alterations. a Calculate her wage, if in one week she completed 15 major alterations and 23 minor alterations. b Determine her hourly rate if she spent a total of 16 hours working on alterations.

EXAMPLE

3

Employees at the Fancy Hanky factory are paid on a piecework basis at the rate of 39 cents per article produced up to 100 items per day and 53 cents per article for each piece produced in excess of 100 items per day. Calculate Romina’s wage if she produces the following

Day Mon Tues Wed Thurs Fri

Number of hankies 84 124 117 93 133

Mon = 84 £ $0:39 = $32:76

Tuesday = 100 £ $0:39 + 24 £ $0:53 = $51:72

Wednesday = 100 £ $0:39 + 17 £ $0:53 = $48:01

Thursday = 93 £ $0:39 = $36:27

100

95

Friday = 100 £ $0:39 + 33 £ $0:53 = $56:49

75

Total income = $32:76 + $51:72 + $48:01 + $36:27 + $56:49 = $225:25

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12 Calculate the weekly wage of the following employees who are paid on a piecework basis at the rate of 17 cents for each piece up to 200 pieces per day and then 25 cents for each piece produced in excess of 200 pieces. a

Monday 230

Tuesday 250

Wednesday 186

Thursday 223

Friday 192

I Kuriakin

Monday 290

Tuesday 210

Wednesday 175

Thursday 238

Friday 177

M Smart

Monday 300

Tuesday 294

Wednesday 256

Thursday 159

Friday 225

N Nine

Monday 220

Tuesday 224

Wednesday 278

Thursday 359

Friday 110

b

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N Solo

c

d

EXAMPLE

4

Martin is paid a royalty of 11.5% on sales of his book. His sales for a year are $327 400. Find his royalty. Royalty = 11:5% of $327 400 = 11:5 ¥ 100 £ $327 400 = $37651 13 Calculate the royalties for these authors Author Royalty a M Golovchenko 8:7% b N Smythe 7:3% c R Ireland 12:9% d P Goodacre 10:5% e L Jones 6:9% f R Spraggon 3:8%

F

Sales $43 590 $367 098 $23 098 $358 900 $21 665 $3 560

INCOMES FROM THE GOVERNMENT

Some people may receive pensions or allowances from the government. These include pensions for old age, disabilities, sole parent, etc. Some allowances include youth, job start etc.

Youth allowance rates Circumstance Fortnightly payment Single, no children - Under 18, at home $146:40 - Under 18, away from home $267:40 - 18 and over, away from home $267:40 - 18 and over, at home $176:00 Single with children $350:20 Partnered, no children $267:40 Partnered with children $293:60

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Chart D shows the personal income test for youth allowance.

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Chart D(a)

Personal Income test for Youth Allowance

75

Income test

For full Allowance (pf)

For part Allowance (pf)

Students unemployed Students unemployed Students unemployed Students unemployed Students unemployed

up to $230 up to $60 up to $230 up to $60 up to $230 up to $60 up to $230 up to $60 up to $230 up to $60

$462.00 $292.00 $504.00 $634.90# $634.90# $672.30# $672.30 $502.30# $753.10# $583.10#

Students

up to $230

Students

up to $230

$561.60 $716.70# $672.30#

Family situation 25

Single, under 18, at home 5

Single, 18 and over, at home 0

Single or partnered, away from home Partnered, with dependants Single, with dependants Long-term unemployed aged 21 or over entering full-time study at home away from home Partnered, no children

* For unemployed, fortnightly income between $60 and $140 reduces fortnightly allowance by 50 cents in the dollar for each dollar in excess of $60 up to $140. Income above $140 reduces payment by 70 cents in the dollar, for each dollar in excess of $140. For students, fortnightly income between $230 and $310 reduces fortnightly allowance by 50 cents in the dollar. Income above $310 reduces payment by 70 cents in the dollar. For students the income test is subject to income bank credit. Partner income which exceeds cut-out point reduces fortnightly allowance by 70 cents in the dollar. # These figures may be higher if you are eligible for Rent Assistance or Pharmaceutical Allowance.

‘pf ’ stands for per fortnight.

EXAMPLE

1

Jenny is a student living at home. She is 17 and entitled to youth allowance.

100

a What is her youth allowance if she has no income? b How much could she earn and still keep full youth allowance? c How much could she earn before she loses her allowance completely?

95

75

a From the table Jenny is entitled to $146.40 . b From chart D Jenny could earn up to $230. c From chart D Jenny must earn less than $462.00 .

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EXERCISE 3F 1 Andrea is a student living away from home. She is 18 and entitled to youth allowance. a What is her youth allowance if she has no income? b How much could she earn and still keep full youth allowance? c How much could she earn before she loses her allowance completely? 2 Goran is a student living at home. He is 16 and entitled to youth allowance. a What is his youth allowance if he has no income? b How much could he earn and still keep full youth allowance? c How much could he earn before he loses his allowance completely? 3 Andres is unemployed living at home. He is 19 and entitled to youth allowance. a What is his youth allowance if he has no income? b How much could he earn and still keep full youth allowance? c How much could he earn before he loses his allowance completely? 4 Sylvana is unemployed living away from home. She is 17 and entitled to youth allowance. a What is her youth allowance if she has no income? b How much could she earn and still keep full youth allowance? c How much could she earn before she loses her allowance completely?

EXAMPLE

2

Marcella is 19, living at home and unemployed. She earns $85 per fortnight, and is entitled to youth allowance. Calculate the amount of youth allowance Marcella will receive. Full youth allowance for Marcella would be $176.00. She loses some as her income exceeds $60. Reduction for income between $60 and $140 is 50 cents in the dollar. Excess = $85 ¡ $60 Reduction = 25 £ $0:50 = $25 = $12:50 Marcella’s allowance = $176:00 ¡ $12:50 = $163:50

5 Ingrid is 17, living at home and unemployed. She earns $105 per fortnight, and is entitled to youth allowance. Calculate the amount of youth allowance Ingrid will receive. 6 George is 19, living away from home and unemployed. He earns $130 per fortnight, and is entitled to youth allowance. Calculate the amount of youth allowance George will receive. 7 Theresa is 18, living at home and a student. She earns $250 per fortnight, and is entitled to youth allowance. Calculate the amount of youth allowance Theresa will receive.

95

The reduction for students begins at $230.

75

8 Venus is 16, living at home and a student. She earns $280 per fortnight, and is entitled to youth allowance. Calculate the amount of youth allowance Venus will receive.



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EXAMPLE

3

Dukan is 18 and has a partner and one dependant. He is a full time student and is entitled to youth allowance. Dukan works part-time earning $350 per fortnight. Calculate his youth allowance. Dukan earns more than $230 pf but less than $672.30 pf and so is entitled to a part allowance. From the youth allowance rate table, full youth allowance in these circumstances is $293.60 . Reduction for income in excess of $310 is $350 ¡ $310 = $40 above $310 Reduction = 40 £ $0:70 = $28 Reduction for income between $230 and $310 Reduction = 80 £ $0:50 = $40 Total reduction = $40 + $28 = $68

$310 ¡ $230 = $80 at the 50 cents in the dollar rate.

Dukan’s allowance = $293:60 ¡ $68 = $225.60 pf

9 Xavier is 17 and lives at home. He is a full time student and entitled to youth allowance. Xavier works part-time earning $360 per fortnight. Calculate his youth allowance. 10 Yoshii is 20, lives away from home and has a partner. He is a full time student and is entitled to youth allowance. He earns $462.00 p/f. Calculate his youth allowance. Use Chart C to answer the next two questions.

Income Test for Pensions – Chart C Income test for pensions Family situation

For full pension (pf)*

For part pension (pf)

Single

up to $102

less than $845.80

Single + 1 child

up to $126

less than $869.80

Couple (combined)

up to $180

less than $1414.40

Additional children

add $24 per child

add $24 per child

100

95

* * #

Income over these amounts reduces the rate of pension payable by 50 cents in the dollar (single), 25 cents in the dollar each (couple). + Pharmaceutical Allowance included. These figures may be higher if Rent Assistance is paid with your pension.

75

25

11 Val and Lionel are entitled to the age pension. They are a couple with a combined income of $453.00 p/f. The full pension is $305.90 each p/f. Calculate the pension paid to Val and Lionel.



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12 Maurice is single with an income of $620 p/f. He is entitled to the age pension. If the full pension is $366.50 p/f calculate the pension Maurice will receive.

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13 Use the Internet to find the latest values for youth allowance and pension. Recalculate some of the questions using these. The address is: http://www.centrelink.gov.au

75

INVESTIGATION 2:

25

5

PENSIONS AND ALLOWANCES

1 Find at least three examples of pensions or allowances. 2 List the criteria for obtaining these pensions or allowances. 3 Find out the amount paid to people on these pensions or allowances.

0

4 Find the amounts that they may earn and still retain the full pension or allowance. 5 Find the reasons for having these pensions or allowances. 6 Where does the money come from to pay for these?

SPREADSHEET APPLICATION Design a spreadsheet that will calculate part pensions and part allowances. Use the latest information from the internet or the tax office.

G

DEDUCTIONS

In order to calculate net pay all expenses are deducted. These include superannuation, tax, health fund, social club, voluntary savings, and others. Net income = gross income ¡ all deductions.

EXAMPLE

1

Won Tae receives a gross weekly wage of $565.43. His weekly deductions are: tax of $128.30, union fees of $6.20, medical insurance of $9.40 and superannuation of $28.20. Calculate his net pay. Total deductions = $128:30 + $6:20 + $9:40 + $28:20 = $172:10

100

Net pay = Gross pay ¡ deductions = $565:43 ¡ $172:10 = $393:33

?

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75

EXERCISE 3G 1 Tony receives a gross weekly wage of $463.90. His weekly deductions are; tax of $95.25, union fees of $4.30, medical insurance of $10.40, and superannuation of $18.45. Calculate his net pay.

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2 Calculate net pay if a gross pay is $380.25 per week, tax is $63.75, superannuation is $13.68, and union fees are $7.30 b gross pay is $411.92 per week, tax is $73.70, superannuation is $12.65, union fees are $3.00, and medical is $15.95 c gross pay is $575.95 per week, tax is $132.25, superannuation is $38.40, medical is $28.20, bank savings $50 and house payment is $150. 3 Silvana receives net pay of $321.99. Her tax was $97.95, union fees $6.90, superannuation $23.85 and house payment $120. Find her gross pay.

0

4 Calculate gross pay if a b c d

net pay is $359.55, tax is $80.10, medical is $14.80, superannuation is $10.50 net pay is $439.01, tax is $116.50, union fee is $3.75, superannuation is $50.35 net pay is $289.47, tax is $95.15, medical is $13.85, superannuation is $45.00 net pay is $185.46, tax is $80.10, medical is $28.60, superannuation is $75.40, and house payment is $200.

MEDICAL INSURANCE Medical insurance costs can be calculated from the following table.

NSW PREMIUMS - SINGLE (The 30% Rebate has been deducted from these premiums)

Direct Debit/Payroll Deduction

Advance Pay

Hospital cover weekly Top Hospital

fortnightly monthly

halfyearly

quarterly

yearly

$13.17

$26.34

$57.05

$171.15 $339.95

$672.70

$11.30 $9.88 $7.03

$22.60 $19.76 $14.06

$48.95 $42.80 $30.45

$146.85 $291.85 $128.40 $255.15 $91.45 $181.75

$577.60 $504.90 $359.65

$11.34

$22.68

$49.10

$147.40 $292.85

$579.55

$9.50 $8.39

$19.00 $16.78

$41.15 $36.30

$123.45 $245.35 $108.95 $216.45

$485.55 $428.35

$7.87 $6.88 $5.83

$15.74 $13.76 $11.66

$34.05 $29.80 $25.25

$102.25 $203.05 $84.45 $177.55 $75.80 $150.40

$401.90 $351.40 $297.70

Top Hospital Excess Level 1 Excess - $150 Level 2 Excess - $250 Level 3 Excess - $500 Intermediate Hospital Intermediate Hospital Excess Level 1 Excess - $150 Level 2 Excess - $250 Basic Hospital Level 1 Excess - $300 Level 2 Excess - $500 Level 3 Excess - $1000

EXAMPLE

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2

75

Jenny is single and takes out top hospital cover. She pays her premium in advance. Find her annual insurance premium.

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5

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5 Harry is single and takes out intermediate hospital cover. Find his premium per a week, if paid by payroll deduction b month, if paid by payroll deduction c quarter, if paid in advance d yearly, if paid in advance.

75

6 Lasallian takes out basic hospital cover for singles. She chooses the $1000 excess. a b c d

25

5

Find her weekly premium. Find the amount she pays for a year making the weekly payment. How much would she save by paying the annual premium in advance? How much does she save per week by having a $1000 excess rather than a $300 excess?

0

7 For each of these rates calculate the saving by paying yearly in advance over paying weekly by payroll deduction a Top cover, $250 excess b Intermediate cover, no excess c Intermediate cover, $250 excess. 8 Paula pays $41.15 per month for medical insurance. What cover does she have? 9 Find a c e

the cover for the following premiums $30.45 per month $13:76 per fortnight $84.45 per quarter

b d

$16.78 per fortnight $19.76 per fortnight

SUPERANNUATION

The employer’s superannuation contribution is not a deduction as it is paid directly to the superannuation fund as an extra payment above the employee’s income. Any superannuation contribution made by the employee is a deduction. It is also a tax deduction.

EXAMPLE Paul earns $36 092 per annum. His employer pays superannuation of 7% on his behalf and Paul contributes 4% himself. a b c d

How much superannuation is paid on Paul’s behalf? How much does Paul contribute? What is Paul’s income less his superannuation contribution? What is the total amount paid into Paul’s superannuation fund? 100

a

c

Employer’s contribution = 7 ¥ 100 £ $36 092 = $2526:44

b

Paul’s income (less super) = $36 092 ¡ $1443:68 = $34 648:32

d

95

Paul’s contribution = 4 ¥ 100 £ $36 092 = $1443:68

75

Total super contribution = $2526:44 + $1443:68 = $3970:12

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10 Arif Arif a b c d

earns $41 922 per annum. His employer pays superannuation of 7% on his behalf and contributes 4% himself. How much superannuation is paid on Arif’s behalf? How much does Arif contribute? What is Arif’s income less his superannuation contribution? What is the total amount paid into Arif’s superannuation fund?

25

11 Repeat question 10 for these people. 5

0

a b c d e f

Name Fielding Louise Bianca Peter Nato Ito

Annual income $32 764 $28 193 $12 943 $41 098 $22 940 $33 901

Employer contribution 7% 7% 7% 7% 7% 7%

Employee contribution 1% 7% 2% 5% 0% 6%

12 Wai earns $549 per week. His employer contributes 7% superannuation and Wai contributes 3%. Calculate Wai’s weekly income and his total weekly superannuation.

EXAMPLE

4

Calculate the annual net pay for an employee whose gross pay is $23 926 pa. He has union fees of $276, tax of $4156.84, employee superannuation contribution is 4% of gross income and basic private (single) health insurance with $300 excess is paid yearly in advance. Calculate deductions: 4 £ 23 926 Superannuation = 100 = $957:04

Health = $297.70 from the table (page 91)

Net income = $23 926 ¡ $4156:84 ¡ $957:04 ¡ $297:70 ¡ $276 = $18 238:42

13 Calculate the annual net pay for Jahmalia whose gross pay is $37 955 pa. Jahmalia has union fees of $421, employee superannuation contribution is 5% of gross income, and intermediate private (single) health insurance with no excess, paid in advance. 14 Theresa earns $38 902 per annum. She pays 36% tax, 6% superannuation and medical of $672.70 per annum. Find a the amount of tax Theresa pays b the amount of superannuation Theresa pays c Theresa’s net annual pay d Theresa’s net weekly pay. 15 A particular superannuation scheme pays its members a retirement benefit of 20% of their average salary over the past five years, times the number of years they have been in the scheme. i.e., retirement payment = 20% of average salary £ number of years. Find the retirement benefit for a person a with an average salary of $46 981, having contributed for 15 years.



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b with an average salary of $39 243 having contributed for 20 years. c with salaries of $38 942, $39 210, $29 881, $33 125, $40 010 for the past 5 years, having contributed for 8 years. 16 Another superannuation fund pays 58% of the final year’s income of an employee as an annual payment. Mikhael retires after earning $45 108 in his last year. He pays tax of 33%. Find his a gross retirement annual income b tax payable c net annual income d net monthly income. 17 On retirement, Allen Jones receives a yearly superannuation payment equivalent to 59% of his final year’s salary. If his final year’s salary was $74 560, how much is his yearly superannuation payment? 18 Carol West’s yearly superannuaton is 68% of her final year’s salary of $138 000: Calculate a her yearly superannuation b her monthly income (before tax) c her monthly income after tax, if the rate of tax is 37%:

H

FINANCIAL INSTITUTIONS

Most banks and financial institutions impose charges to keep your account. The charges vary from institution to institution. There are government charges as well. Some types of charges include: ² ² ²

A flat account keeping fee per month. A fee per withdrawal. Different types of withdrawal may attract different fees. A fee if the balance falls below a preset amount.

The government charges are FID and GDT. FID is Financial Institutions Duty and it is charged at the rate of 0.06% on money deposited into an account. Table 1 GDT charges GDT is Government Debit Tax and it is Withdrawal amount Charge charged per withdrawal from any account $1 to $99.99 30 cents linked to a cheque account. $100 to $499.99 70 cents $500 to $4999.99 $1.50 $5000 to $9999.99 $3.00 Over $10 000 $4.00

EXAMPLE

1

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Calculate the FID on a deposit of $1860.

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FID = 0.06% of $1860 = 0:06 ¥ 100 £ $1860 = $1:116 + $1:11

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EXERCISE 3H 1 Calculate the FID on these deposits. a $2600 b $3890 c $1496

d

$5895

e

$58

f

$25 980

2 In one month Evros deposits $380, $1355, $682 and $1109. Calculate the FID for the month. 25

3 Kelly has her salary of $628 per week deposited into her account. Calculate the FID charge for the year.

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EXAMPLE

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Adrin has a cheque account. Calculate the GDT for a withdrawal of a $4523.31 b $53.33 a $4523.31 lies between $500 and $4999.99 . From the table, GDT = $1.50 . b $53.33 lies between $1 and $499.99 . From the table, GDT = 30 cents.

4 Find the GDT on withdrawals of a $6573 b $4912 c

$39:21

d

5 The following table shows the charges levied by the INT bank. Find the charges if Brendan has a three over the counter withdrawals b five ATM withdrawals c ten telephone transfers d 2 over the counter withdrawals, 3 ATM withdrawals and 6 telephone transfers. 6 The table shows the charges levied by the Republic bank. Find the charges if Clarise has a six EFTPOS transactions b eight Republic ATM withdrawals c ten non-republic ATM transactions d

$375

e

$81.01

f

Activity Withdrawal over the counter ATM withdrawal Telephone fund transfer

Activity EFTPOS transaction or direct debit ATM withdrawal Over the counter withdrawal Non-Republic bank ATM transaction

$23 908 Charge $2.00 50 cents 40 cents

Charge $0:40 $0:60 $2:00 $1:25 100

4 over the counter withdrawals, 2 Republic ATM withdrawals, 5 Non-Republic ATM transactions, eleven EFTPOS transactions.

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INVESTIGATION 3:

ACCOUNT CHARGES

1

1 Obtain the charges for various accounts with different financial institutions.

2

2 Compare each and decide which account would be better for you.

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3 Which account would suit a person who makes six deposits totalling $7580, and ten withdrawals totalling $6985 per month. The balance of their account always exceeds $2500.

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I

BUDGETING

A budget is a financial plan for the future which estimates future expenses and income and calculates a projected surplus or deficit of money. A budget enables a person to plan for holidays or other expensive activities like weddings etc. There are five main steps to follow in setting up a budget. 1 Decide whether the budget will be weekly, fortnightly, monthly or annual. 2 Estimate income from all possible sources. 3 Estimate all expenses. Plan for emergencies. 4 Calculate the surplus, hopefully, or the deficit. 5 Revise the budget if there is a deficit or if you want a greater surplus. Remember fixed expenses like rent, insurance etc cannot be altered, but expenses like food, clothing, and entertainment, can be altered.

EXAMPLE

1

Yick and Gwen want to prepare a budget. They decide on an annual budget. Their income and expenditure information is as follows. Income:

Yick’s net pay is $370 per week Gwen’s net pay is $345 per week Interest of $800 in July

Expenditure:

Home loan repayment Home insurance Council rates Water Food Electricity Telephone Car registration Car green slip Car repayments Car insurance Car running Clothes Entertainment Other expenses

$720 per month $320 $415 pa due September $82 per quarter due March, June, Sept, Dec $95 per week $215 per quarter due Feb, May, Aug, Nov $82 every two months from Feb $165 pa $405 pa $130 per month $199 pa $40 per week $100 per month $65 per week $40 per week

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Use this information to prepare their budget.

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Income:

Yick’s pay Gwen’s pay Interest Total income

$370 £ 52 $345 £ 52

$19 240 $17 940 $800 $37 980

Expenditure:

Home loan repayment Home insurance Council rates Water Food Electricity Telephone Car registration Car green slip Car repayments Car insurance Car running Clothes Entertainment Other expenses Total expenses Annual expected surplus

$720 £ 12 $320 pa $415 pa $82 £ 4 $95 £ 52 $215 £ 4 $82 £ 6 $165 pa $405 pa $130 £ 12 $199 pa $40 £ 52 $100 £ 12 $65 £ 52 $40 £ 52

$8640 $320 $415 $328 $4940 $860 $492 $165 $405 $1560 $199 $2080 $1200 $3380 $2080 $27 064 $10 916

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EXERCISE 3I 1 Complete this annual budget. Income:

Harry’s pay Rose’s pay Interest Total income:

$380 £ 52 $365 £ 52 $500

Expenditure:

Home loan repayment Home insurance Council rates Water Food Electricity Telephone Car registration Car green slip Car repayments Car insurance Car running Clothes Entertainment Other expenses Total expenses: Annual expected surplus:

$845 £ 12 $390 pa $654 pa $96 £ 4 $139 £ 52 $342 £ 4 $109 £ 6 $165 pa $405 pa $270 £ 12 $458 pa $55 £ 52 $120 £ 12 $80 £ 52 $50 £ 52

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2 Gunter has just left school and has a job that clears $190 per week. He pays $35 per week board to his parents, $30 on fares and lunches, $50 on entertainment, $35 per week on clothes and $20 per week on other expenses. Prepare Gunter’s annual budget. 3 Libby has just started a university course and will receive $100 per week from Austudy and about $85 per week for part time work. Her expenses are: rent $40 per week, car running expenses $30 per week, and entertainment of $30 per week. She has to pay car insurance of $150, registration of $160, green slip of $415 and car repayments of $75 per month. a Prepare Libby’s budget. b List some expenses Libby has not included with estimates.

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4 Daniella and Nick are planning a $6000 holiday at the end of the year. Prepare their budget and determine if there is a sufficient surplus. Daniella’s net pay is $290 per week and Nick’s net pay is $285 per week. Their expenses are listed below: Expenditure:

Home loan repayment Home insurance Council rates Water Food Electricity Telephone Car registration Car green slip Car repayments Car insurance Car running Clothes Entertainment Other expenses

$750 per month $255 $320 pa due September $75 per quarter due Feb, May, Aug, Nov $95 per week $160 per quarter due March, June, Sept, Dec $75 every two months from Feb $165 pa $405 pa $120 per month $290 pa $35 per week $80 per month $70 per week $50 per week

If there is insufficient funds, modify their budget to enable the holiday.

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An annual budget is difficult to use. A monthly budget gives an up to date idea of money flow. Here is a sample spreadsheet that displays a monthly budget using the figures from the example.

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MONTHLY BUDGET

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Item

Annual amount

Income: Yick’s wage Gwen’s wage Interest Total income Expenditure: Home loan payment Home insurance Council rates Water Food Electricity Telephone Car Registration Green slip Car Repayments Car insurance Car running Clothes Entertainment Other expenses Total expenses Monthly difference Cumulative difference

0

Jan

Feb

Mar

Apr

May

June

19 240 17 940 800 37 980

1 603 1 495

1 603 1 495

1 603 1 495

1 603 1 495

1 603 1 495

1 603 1 495

3 098

3 098

3 098

3 098

3 098

3 098

8 640 320 415 328 4 940 860 492 165 405 1 560 199 2 080 1 200 3 380 2 080 27 064

720 27 35

720 27 35

720 27 35

720 27 35

412

412 215 82 14 34 130 17 173 100 282 173 2 414 684 1 665

720 27 35 82 412

412

412 215

720 27 35 82 412

14 34 130 17 173 100 282 173 2 117 981 981

1 Complete the budget above for the rest of the year. 2 As bills are paid and income is received the budget can be updated to give an accurate indication of funds. This part spreadsheet shows this for January: As January has been completed, continue the budget for the next five months using the information about actual bills and income in this table.

The monthly difference is the income actual expenses for each month.

Only variable expenses have changed.

The cumulative difference sums these.

14 34 130 17 173 100 282 173 2 199 899 2 564

82 14 34 130 17 173 100 282 173 2 199 899 3 463

Expenditure: Home insurance Council rates Water Food Electricity Telephone Car Registration Green slip Car Repayments Car insurance Car running Clothes Entertainment Other expenses Total expenses Monthly difference Cumulative difference

14 34 130 17 173 100 282 173 2 332 766 4 229

320 415 328 1 140 860 492 165 405 1 560 199 2 080 1 200 3 380 2 080 14 624

82 14 34 130 17 173 100 282 173 2 281 817 5 046

27 35 104

14 34 130 17 160 180 400 350 1 451 1 647 1 647

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Item Expenditure: Water Food Electricity Telephone Car running Clothes Entertainment

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Monthly Budget Jan Feb Mar Apr May June

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160 180 400

88 153 100 240 140 150 170 150 300 350 510

97 97

89 110 286 153 175 180 150 145 200 140 90 390 375 380

Should this budget be modified? If so, make changes to make it more accurate. 4 5 6

Use your own numbers to complete the year for the budget in question 3. Use the figures from question 1 in Exercise 3I to create a monthly budget for Harry and Rose. Use this table of actual expenses to evaluate and manage the budget for Harry and Rose from question 5.

Item Expenditure: Water Food Electricity Telephone Car running Clothes Entertainment

8

Monthly Budget Jan Feb Mar Apr May Jun Jul

600 610 330 110 250 240 100 140 350 360

95 590 585 602 350 115 205 240 195 130 125 165 340 385 380

Aug Sep Oct Nov Dec

100 89 97 620 597 650 600 585 620 750 335 315 135 98 125 101 211 215 214 220 224 199 189 30 115 108 55 68 150 200 396 450 410 320 541 290 350

Does the budget in question 7 need modification? If so, make a new budget for Harry and Rose.

GROUPWORK ACTIVITY Working in pairs, each person should develop a budget. Have the other person make up bills, etc, and modify your budget to suit. Try to make it as real as possible.

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BILLS

There are a number of costs associated with living in a home. These include rates if you are the owner of the property, electricity, telephone, gas and insurance. This section examines some of the accounts sent with respect to these charges.

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EXERCISE 3J 1 Examine the electricity account and answer the following questions.

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ACCOUNT NUMBER:

32519930

AMOUNT DUE:

25

$ 210.55

DATE DUE: 5

For account enquiries, Telephone: 131 002 (NSW) Facsimile: (02) 853 6026 For general information, Telephone: 131 081 For emergency service. Telephone: 131 003

0

Monday, 8 Nov 1999

See over for payment options ELECTRICITY ACCOUNT

MRS JAN MURPHY 15 THE WAY CRESCENT HOMETOWN 2213

Summary of Your Account Description

Amount $183.15 $183.15 CR

BALANCE FROM LAST ACCOUNT LESS PAYMENT(S) RECEIVED TO 08/09/99

$0.00

SUBTOTAL AMOUNT OF THIS ACCOUNT FROM 23/08/99 TO 24/10/99

$210.55

TOTAL AMOUNT DUE

$210.55

Your Electricity Usage 26.2

kWh per day 21.3

19.8

19.2

NOV

DEC

24.8

MAR

MAY

20.9

kWh per day

AUG

SEP

23.9

AUG

SEP

17.2 15.2

JUL

22.6 20.6 17.2

DOMESTIC UNITS

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NOV

DEC

14.0

MAR

MAY

JUL

OFF PEAK UNITS

What is the account number? What is the due date? How much is due? How much was the previous payment? Has the bill increased or decreased? By how much? Express the increase/decrease as a percentage of the bill. How many kilowatts at the domestic rate were used per day for September? How many kilowatts at the off peak rate were used per day for September? Which month used the greatest number of kW per day, at the domestic rate?

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2 Examine the water bill and answer the following questions.

Sydney

WATER Last bill

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BILL ENQUIRIES

$169.10

132 092

SYDNEY WATER CORPORATION LIMITED ACN 063 279 649

Payments

Balance

$169.10

$0.00

This bill

8am to 8pm (Mon-Fri)

Total amount due

$195.20

$195.20 Please pay by

Date of issue 3 November 1999 5

24/11/99

MRS JAN MURPHY 0

15 THE WAY CRESCENT

Account Number

HOMETOWN 2213

6234 1999 Penrith Customer Centre

Account for residential property Charges Water service Sewerage service

1 Oct 99 - 31 Dec 99

Usage charge Water

29 Jul 99 - 2 Nov 99 114 kilolitres at 90.0 cents per kilolitre

Your average daily usage

$ 20.00 72.60

102.60

See over for details Total amount due

$195.20

Litres

2500 2000 1500

1652 1187

1084

1000 litres = 1 kilolitre

1000 500 0

This bill

Last bill

Same time last year

Continued overleaf

a What is the account number? b What is the due date? c How much is due? d How much was the previous payment? e Has the bill increased or decreased? By how much? 100

f Express the increase/decrease as a percentage of the bill. g How many kilolitres were used for the billing month?

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h What was the sewerage charge?

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i What was the water usage charge? j Was the water usage greater than or less than the previous year? By how much?

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k What is the telephone number for bill enquiries?

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3 Examine this telephone bill and answer the following questions.

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Telstra Bill 75

Telstra Corporation Limited ACN 051 775 556

Account Number 856 7947 300

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Opening balance

Date of issue

BILL ENQUIRIES

T 667 122 251-9

23 / 08 / 95

132200

Balance

Total of this bill

Bill number

We received

Total amount payable

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$149.45

$149.45

$0.00

$98.46

$98.46

0

Payment due by

27 Sep 99 MRS JAN MURPHY 15 THE WAY CRESCENT HOMETOWN 2213

Item

Account Summary

Your Reference

02 5361 3604

$

98

Call Charges

to 02 Sep

81.65

99

Service and equipment

to 07 Sep

16.81 Total of this bill

$98.46

$160 Calling Patterns Compared With Last Bill Local Calls Calls to Mobiles Information Calls

$120

down by

$17.00

$80

down by

$2.65

$40

down by

$1.50

$0

Same time Last Year Total Bill

Sep-98 Mar-99 Jun-99 Sep-99

a What is the account number? b What is the bill number? c When is the due date? 100

d How much is the payment due? e How does this bill compare to the last bill?

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f How does the bill compare with the same time last year?

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g How much was the last payment? h How much were the call charges?

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i What was the service and equipment charge?

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4 Examine this rates notice and answer the following questions.

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PENRITH CITY COUNCIL 601 High Street, Penrith 209 Queen Street, St Marys Rate Enquiries: (02) 4732 7676 Other Enquiries: (02) 4732 7777

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CHEQUES/MONEY ORDERS send payments to: Agency Payments Locked bag 636 PENRITH N.S.W. 2751 CORRESPONDENCE:

PO Box 60 PENRITH N.S.W. 2751

INSTALMENT NOTICE CREDIT CARD PAYMENTS PHONE: l31991 (7 DAYS/24H) Operator requires Company Number 190 and Notice Number

BILLER CODE No: 10090 REF 287442359

Notice Number 31498382 MRS JAN MURPHY 15 THE WAY CRESCENT HOMETOWN 2213

Account Number 194213 Payment Due 30/11/99

PLEASE DEDUCT PAYMENTS MADE ON OR AFTER 15/10/99 INSTALMENT DUE 30/l l/99

$164.70

TOTAL

$164.70

INTEREST CHARGES ACCRUE ON A DAILY BASIS AT 9.5% P/A ON OVERDUE RATES & CHARGES

Alan Travers General Manager Methods of Payment and other Information see the reverse side of this Notice.

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What is the notice number? What is the account number? When is the payment due? How much is the payment? Where is correspondence addressed? Where would the payment be posted? What is the telephone number for rate enquires? What is the telephone number for credit card payments? When can you ring? How much interest is charged on overdue rates?


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1 Here is a list of terms used in this section. Write a description of each term in a sentence.

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annual leave, annual leave loading, commission, deductions, double time, gross pay, net pay, overtime, piecework, royalty, time and a half 25

2 List five other words or terms from this chapter and write their meanings. 5

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You should be able to 2 2 2 2 2 2 2 2 2 2 2 2

convert between annual, weekly, fortnightly and monthly incomes calculate incomes that include special allowances or loadings including holiday loading calculate pay when overtime is paid calculate incomes based on commissions, piecework and royalties understand and calculate bonus payments use charts and tables to calculate pensions, government allowances and health care payments calculate net pay understand the difference between gross pay and net pay calculate superannuation contributions and payments calculate charges incurred using financial institutions prepare and manage a budget understand and draw information from common household bills.

?

DIAGNOSTIC TEST 1 Katrina receives an annual salary of $19 370. Her weekly salary is A $1616.17 B $403.54 C $372.50

D

$745

2 Zoltan is paid $8.95 per hour. His earnings for a 26 hour week are A $232.70 B $8.95 C $465.40

D

$349.05

3 Andre works 9 hours per day, five days per week. He is paid $8.43 per hour. His annual pay is A $75.84 B $379.35 C $18 208.80 D $19 726:70 4 Mal receives a salary of $63 428 per annum. He works 6 days per week, 8 hours per day. His hourly rate is A $25.41 B $27.52 C $27.53 D $50.82 5 Karen works 8 hours at normal time and 2 hours at time and a half. Her day’s pay at $7.90 per hour is A $79.00 B $86.90 C $118.50 D $11.85

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6 Paula is paid $7.85 per hour normal rate. She works 30 normal hours, 6 hours at time-and-a half and 3 hours at double time. Her pay is A $235.50 B $306.15 C $341.48 D $353.25

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7 Dommo sold a car for $28 950. He receives 6% commission. His commission is A $1737 B $173 700 C $28 950 D $482 500 25

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8 Marek is paid a weekly retainer of $400 and 13% commission on sales in excess of $5000. His pay in a week with sales of $8350 is A $1085:50 B $435.50 C $835:50 D $1685:50 9 Alexia is paid $362 commission on sales of $4525. Her percentage rate of commission is A 0.08% B 8% C 12.5% D 3:62% 10 Zanthia receives a bonus for Christmas of 6.2% of her salary. If her salary is $27 695, then the bonus is A $1717:09 B $276:95 C $33.02 D $6:20 11 Paulette is paid $456 per week. She receives a holiday loading of 17 12 % on her four weeks holiday. Her total holiday pay is A $79.80 B $319.20 C $775.20 D $2143:20 12 Sue picks mushrooms and is paid $2.15 per box. Her pay for picking 53 boxes is A $24:65 B $53 C $4.05 D $113.95 13 Charles is paid a royalty of 8% on total sales. His royalty for sales of $356 990 is A $28 559:20 B $3569:90 C $44 623:75 D $35 699 14 The single under 18 living at home rate is $146.40. It reduces by 50 cents for every dollar in excess of $230 p/f. Sharon is 17, living at home and earning $250 pf. Her allowance is A $146.40 B $376.40 C $136.40 D $10 15 Bernadette receives gross pay of $35 866, has tax deductions of $6874, medical of $425 and superannuation of $1240. Her net pay is A $28 992 B $27 327 C $44 405 D $390 657 16 Trina has an annual salary of $41 908. Her final superannuation payment is 63% of her salary. Her payment is A $68 310:04 B $2 640 204 C $15 505:96 D $26 402:04 17 The FID rate is 0.06%. The FID on an account with deposits of $36.80, $456.90 and $123.76 is closest to A $617.46 B $37.04 C $37.05 D 6 cents 18 In budgeting for telephone charges, the bill is A fixed B changeable

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partly fixed and changeable

D

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Check your answers to the diagnostic test. Check the relevant section from the table to review any questions that you answered incorrectly. Question Section

1, 2, 3, 4, A

5, 6 B

7, 8, 9 C

10, 11 D

12, 13 E

14 F

15, 16 G

17 H

18 I

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REVIEW SET

1 Lucy earns $72 493 per annum. Calculate her weekly pay. 2 An electrician works a 42-hour week. Find his yearly pay if he earns $26.50 per hour.

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3 Indera receives an annual salary of $57 801. Convert this to a weekly salary and find Indera’s average hourly rate is she works 48 hours per week. 4 Brian is paid $7.63 per hour. Calculate his weekly wage if he works 35 hours at normal time and 7 hours overtime at time and a half. 5 Jack is a security guard and is paid $13.25 per hour. If he works on the door he is paid $15 danger money. Find Jack’s wage for a week where he is on the door for 3 nights and works 30 hours. 6 A real estate agent is paid 1.2% commission on any sales made. Find his commission on a house that sells for $276 500. 7 Georgina works as a personal assistant and is paid $496 per week. If her 4 weeks holiday pay attracts a loading of 17 12 %, find her pay for her holidays. 8 Mark waxes surfboards for the surf hire shop. He is paid $7.40 per board waxed. Find his pay for a day in which he waxes 11 boards. 9 Martin is paid a royalty of 10.8% on sales of his book. His sales for a year are $63 580. Find his royalty. 10 Lorena is unemployed living away from home. She is 16 and entitled to youth allowance. Use the charts in Section F to answer these questions. a What is her youth allowance if she has no income? b How much could she earn and still keep full youth allowance? c How much could she earn before she loses her allowance completely? 11 Joan and Don are entitled to the age pension. They are a couple with a combined income of $459.00 p/f. The full pension is $305.90 each p/f. Use the table in Section F to calculate the pension paid to Joan and Don. 12 Calculate net pay if gross pay is $390.25 per week, tax is $73.75, superannuation is $23.68, and union fees are $6.30. 13 Calculate gross pay if net pay is $163.21, tax is $82.10, medical is $27.60, superannuation is $55.40, and house payment is $210. 14 Harry is single and takes out basic hospital cover with a $500 excess. Use the table in Section G to find his weekly premium. 15 Ronnie earns $432 per week. His employer contributes 7% and Ronnie contributes 3% to Ronnie’s superannuation fund. Calculate Ronnie’s weekly income and his total weekly superannuation contribution. 16 Penny has her salary of $537 per week deposited into her account. Use the table in Section H to calculate the FID charge for the year. 17 Armin has a cheque account. Use the table in Section H to calculate the GDT for a withdrawal of $365.29.

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3B

REVIEW SET

1 Jenny earns $62 475 per annum. Calculate her weekly pay. 2 Akbar works a 7-hour day, six days per week. His hourly rate is $11.28. Find his pay for 1 year.

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3 Richard is paid a salary of $57 932 per annum. a Find his weekly pay rate. b Find his hourly rate if he works 35 hours per week.

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4 Visnoo is paid $11.95 per hour for a 30-hour week. He is paid overtime at double time for all other hours. Calculate his pay in a week where he works 38 hours. 5 Hewey works as a plumber’s assistant and is paid $5.30 per hour. He is paid a digging allowance of $11.50 per day if he has to dig trenches and $15.80 per day if he works on open sewers. Find his pay in a week where he works 42 hours, digs on two days and works on an open sewer for three days. 6 Robert sells cars valued at $35 675. Find his pay if his commission is 4.5%. 7 Tabitha received a 17 12 % holiday loading on four weeks normal wages. She normally works a 32-hour week. Her 4 weeks holiday pay and loading is $2587.90. a Find her normal weekly pay. b Find her normal hourly pay rate. 8 Lynne works in a belt factory. Each belt completed earns her $2.20. Find her wage for a week in which she makes 394 belts. 9 Levi is paid a royalty of 12.7% on sales of his book. His sales for a year are $628 149. Find his royalty. 10 Mitchell is a student living at home. He is 17 and entitled to youth allowance. Use the charts in Section F to answer these questions. a What is his youth allowance if he has no income? b How much could he earn and still keep full youth allowance? c How much could he earn before he loses his allowance completely? 11 Wade is single with an income of $610 pf. He is entitled to the age pension. If the full pension is $366.50 pf, calculate the pension Wade will receive. Use the charts in Section F. 12 Calculate net pay if gross pay is $635.28 per week, tax is $128.54, superannuation is $27.60, medical is $26.20, bank savings is $40, and house payment is $175. 13 Calculate gross pay if net pay is $372.55, tax is $89.10, medical is $34.80, and superannuation is $15.50 . 14 Brent is single and takes out top hospital cover. Use the table in Section G to find his weekly premium.

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15 Kang earns $728 per week. His employer contributes 7% and Kang contributes 3% to Kang’s superannuation fund. Calculate Kang’s weekly income and his total weekly superannuation. 25

16 Sally has her salary of $762 per week deposited into her account. Use the table in Section H to calculate the FID charge for the year.

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REVIEW SET

1 Erica earns $29 881 per annum. Calculate her weekly pay. 2 Jenny works 28 days per month. She averages 8 hours per day. Find her annual pay if she is paid $10.52 per hour. 3 Ronaldo works 9 hours a day five days per week. Find his hourly rate if his annual income is $26 031. 4 Mona earns $9.28 for the first 35 hours in a week. She receives time and a half for the next 8 hours and double time after that. Find her pay for a 52-hour week. 5 Regina works in a restaurant and receives a uniform allowance of $8.50 per night. Her hourly rate of pay is $15.23. Find her pay if she works 5 nights for 4 hours per night. 6 The commission paid to a jewellery salesperson is 5.8% of the selling price of any item. If Robyn sells a ring for $5250, how much is Robyn’s commission? 7 Ling works as a personal assistant and is paid $553 per week. If her 4 weeks holiday pay attracts a loading of 17 12 %, find her pay for her holidays. 8 Michelle assembles skateboards. She is paid $17.00 per board assembled. How much does she earn for assembling 28 skateboards? 9 Steven is paid a royalty of 9.8% on sales of his book. His sales for a year are $265 800, find his royalty. 10 Fabrice is unemployed living at home. He is 17 and entitled to youth allowance. Use the charts in Section F to answer these questions. a What is his youth allowance if he has no income? b How much could he earn and still keep full youth allowance? c How much could he earn before he loses his allowance completely? 11 Maurice is single with an income of $625 pf. He is entitled to the age pension. If the full pension is $366.50 pf, calculate the pension Maurice will receive. Use the charts in Section F. 12 Calculate net pay if gross pay is $456.31 per week, tax is $93.47, superannuation is $15.89, union fees are $7.00, and medical is $23.50. 13 Calculate gross pay if net pay is $486.21, tax is $108.65, union fee is $2.78, and superannuation is $39.27 .

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14 Larry is single and takes out basic hospital cover with the maximum excess. Use the table in Section G to find his weekly premium. 15 Jai earns $480 per week. His employer contributes 7% and Jai contributes 5% to Jai’s superannuation fund. Calculate Jai’s weekly income and his total weekly superannuation contribution. 16 Mary has her salary of $539 per week deposited into her account. Use the table in Section H to calculate the FID charge for the year.

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REVIEW SET

1 Pauline earns $42 531 per annum. Calculate her weekly pay. 2 Fielding is paid $8.37 per hour. He works 5 hours per day six days a week. Find his pay for one year. 3 Jim is paid a salary of $62 408 per annum. a Find his weekly pay rate. b Find his hourly rate if he works 38 hours per week. 4 James is paid $8.22 per hour. Calculate his weekly wage if he works 35 hours at normal time and 5 hours overtime at time and a half. 5 Ross works in a timber yard and is paid $15.40 per hour. He receives a dust allowance of $2.50 per day. Find his pay for a 5-day, 38-hour week. 6 Art salespeople are paid 3.2% of the selling price of any works they sell. Find the commission on a painting selling for $2340. 7 Endora received a 17 21 % holiday loading on four weeks normal wages. She normally works a 36-hour week. Her 4 weeks holiday pay and loading is $2185.70 . a Find her normal weekly pay. b Find her normal hourly pay rate. 8 A factory worker is paid $4.35 for each garment completed. Calculate his wage if he completes 429 garments. 9 Wade is paid a royalty of 8.3% on sales of his book. His sales for a year are $149 000. Find his royalty. 10 Agneska is a student living away from home. She is 19 and entitled to youth allowance. Use the charts in Section F to answer these questions. a What is her youth allowance if she has no income? b How much could she earn and still keep full youth allowance? c How much could she earn before she loses her allowance completely? 11 Carol and John are entitled to the age pension. They are a couple with a combined income of $428.00 pf. The full pension is $305.90 each pf. Calculate the pension paid to Carol and John. Use the charts in Section F. 12 Sam receives a gross weekly wage of $511.03. His weekly deductions are: tax of $103.23, union fees of $5.20, medical insurance of $11.80, and superannuation of $19.50. Calculate his net pay. 13 Calculate gross pay if net pay is $305.61, tax is $93.12, medical is $15.50, and superannuation is $38.00. 14 Perry is single and takes out intermediate hospital cover with the minimum payable excess. Use the table in Section G to find his weekly premium.

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15 Tsui earns $512 per week. His employer contributes 7% and Tsui contributes 6% to Tsui’s superannuation fund. Calculate Tsui’s weekly income and his total weekly superannuation. 16 Jenny has her salary of $754 per week deposited into her account. Use the table in Section H to calculate the FID charge for the year. 17 Arif has a cheque account. Use the table in Section H to calculate the GDT for a withdrawal of $5 268.33 .

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CHAPTER

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Algebraic modelling

AREA OF STUDY

This chapter deals with basic algebraic skills. The main mathematical ideas investigated in this chapter are: 8 working with number patterns 8 simplifying algebraic expressions 8 substituting into formulas and evaluation 8 solving linear equations.

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ALGEBRAIC MODELLING AM1 (Chapter 4)

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LINEAR NUMBER PATTERNS

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For centuries there has always existed a fascination for numbers and patterns. 2, 4, 6, 8, ...... is an example of a number pattern or number sequence, where the dots ...... show that the pattern continues indefinitely.

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Each of the numbers in the pattern is called a term.

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In the case of 2, 4, 6, 8, ...... the first term is 2, the second term is 4, and so on. Number patterns are formed in many ways. Some of them are formed by adding, subtracting, multiplying or dividing by a fixed number.

EXAMPLE

1

Describe each of these patterns in words and write the next three terms of the number patterns. a d

5, 7, 9, 11, ..... 64, 32, 16, 8, .....

b

32, 29, 26, 23, .....,

c

1 1 9, 3,

1, 3, .....,

a This pattern is formed by starting with 5 and adding 2 to each number to get the next number. The next three terms are 13, 15, 17. b This pattern is formed by starting with 32 and subtracting 3 from the previous term. The next three terms are 20, 17, 14. c This pattern is formed by starting with terms are 9, 27, 81.

1 9

and multiplying each term by 3. The next three

d This pattern is formed by starting with 64 and dividing the terms by 2. The next three terms are 4, 2, 1.

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EXERCISE 4A 1 These number patterns were formed by adding or subtracting. Describe each pattern in words, copy the pattern and write the next three terms. a 5, 8, 11, 14, ..... b 2, 6, 10, 14, ..... c 30, 26, 22, 18, ..... d ¡6, ¡1, 4, 9, ..... e 200, 175, 150, 125, ..... f ¡13, ¡10, ¡7, ¡4, ..... g 8, 3, ¡2, ¡7, ..... h 50, 42, 34, 26, ..... i 14, 25, 36, 47, ..... 1 1 l 13 , 1, 1 23 , 2 13 , ..... j ¡50, ¡45, ¡40, ¡35, ..... k 5, 4 2 , 4, 3 2 , .....

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2 These number patterns were formed words, copy the pattern and write the a 1, 2, 4, 8, ..... d 50 000, 10 000, 2000, 400, ..... 1 1 g 25 , 5 , 1, 5, .....

by multiplying or dividing. Describe each pattern in next three terms. b 14 , 1, 4, 16, ..... c 243, 81, 27, 9, ..... e 128, 64, 32, 16,..... f 1, 6, 36, 216, ..... 1 1 h 2401, 343, 49, 7, ..... i 100 , 10 , 1, 10, .....

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3 These number patterns were formed by adding, subtracting, multiplying or dividing. Describe each pattern in words, copy the pattern and write the next three terms. a 4, 7, 10, ..... b 13 , 1, 3, ..... c 40, 20, 10, ..... d 11, 7, 3, ..... e ¡3, ¡14, ¡25, ..... f ¡1, 3, ¡9, ..... g 36, 6, 1, ..... h 32, 16, 8, ..... i 71, 63, 55, ..... j ¡11, ¡6, ¡1, ..... k 5, 1, 0.2, ..... l 625, 250, 100, .....

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EXAMPLE For a b c

the number pattern 2, 5, 8, 11, ...... write the next three terms of the number pattern write a rule in words to describe the pattern find the 10th and 50th terms. 2, 5, 8, 11, 14, 17, 20, ...... Add three to get the next term. To find the 10th term add 9 lots of 3 to the first term 2, i.e., 2 + 9 £ 3 = 29. To find the 50th term add 49 lots of 3 to the first term 2, i.e., 2 + 49 £ 3 = 149.

a b c

4 For each number pattern

a d g

i ii iii

3, 5, 7, ...... 18, 17, 16, ...... ¡6, ¡3, 0, ......

write the next three terms write a rule in words to describe the pattern find the 10th and 50th terms. b e h

8, 10, 12, ...... 200, 190, 180, ...... 16, 14, 12, ......

c f i

45, 50, 55, ...... 7, 12, 17, ...... 7, 14, 21, ......

EXAMPLE Write the next four numbers in the pattern using the first term and the rule given. a 5, ‘subtract 3’ b 18 , ‘times 4’ a

b

5, 5 ¡ 3 = 2, 2 ¡ 3 = ¡1, ¡1 ¡ 3 = ¡4, The pattern is 5, 2, ¡1, ¡4, ¡7, ...... 1 8,

1 8

£ 4 = 12 ,

The pattern is

1 2 1 1 8, 2,

£ 4 = 2,

2 £ 4 = 8,

¡4 ¡ 3 = ¡7, ...... 100

8 £ 4 = 32, ......

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2, 8, 32, ......

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5 Write the next four numbers in the pattern using the first term and the rule given: a 5, ‘add 2’ b 6, ‘subtract 2’ c 5, ‘multiply by 2’ d ¡15, ‘add 6’ e 12, ‘subtract 5’ f 32, ‘divide by 2’ 1 1 g 5000, ‘divide by 5’ h 20 , ‘multiply by 10’ i ‘multiply by 3’ 3,



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EXAMPLE Using the starting number and the rule given, find the 10th term of these number patterns: 1 16 ,

a

starting with 5, “add 4”

b

starting with

a

The pattern is 5, 9, 13, 17, ......

b

The pattern is

The 10th term would be 5 plus 9 lots of 4. i.e., 5 + 9 £ 4 = 41

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“multiply by 2”

1 1 1 1 16 , 8 , 4 , 2 , 1, 2, ...... 1 would be 16 times 2,

The 10th term 9 times, i.e., 1 (2 £ 2 £ 2 £ :::::: £ 2) 16 £ | {z } 9 times 1 £ 29 = 16 = 32

6 Using the starting number and the rule given, find the 10th term: a starting with 4 “add 2” b starting with 6 “add 4” c starting with 17 “subtract 3” d starting with 11 “subtract 5” 1 81

e

starting with 5 “multiply by 2”

f

starting with

g

starting with 5 “multiply by 2”

h

starting with 262 144 “divide by 4”.

“multiply by 9”

EXAMPLE Consider the matchstick pattern: a Draw the next two diagrams. b From the diagrams write the number pattern for the number of matchsticks. c Find the number of matchsticks in the tenth diagram. d Write a rule, in words, for the number of matchsticks needed to make each diagram. e If there are 99 matchsticks, how many triangles would be in the largest diagram possible?

a b

By counting, the pattern is: 3, 5, 7, 9, 11, ......

c

Extending the number pattern gives: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, ......

d

1 triangle = 1 + 2 matches 2 triangles = 1 + 2 + 2 matches = 1 + (2 £ 2) matches 3 triangles = 1 + (3 £ 2) matches 4 triangles = 1 + (4 £ 2) matches

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Rule is: For T triangles we need 1 + (T £ 2) matches. e

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For 99 matches, then T £ 2 = 98: i.e., 49 triangles can be made with 99 matches.

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7 Copy these matchstick patterns, draw the next two, and write down the number pattern for the number of matchsticks in each diagram. a b

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8 For each of the matchstick patterns in question 7, write a rule in words for finding the number of matchsticks to make each shape. 9 Find the number of matches needed for the 30th diagram in each of the matchstick patterns in question 7. 10 If you had 40 matches, find the largest of each diagram in question 7 that you could make. How many matches would be left over?

B

SUBSTITUTION

When the pronumeral x is replaced by a number in an expression like 3x¡5 the answer is another number.

For example, if x = 2, then

3x ¡ 5 = 3(2) ¡ 5 =6¡5 =1

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When substituting, always place the substituted value within brackets.

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and we say that 2 was substituted for x, and the value of the expression is 1 when x = 2.



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EXAMPLE

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If a = 5, b = ¡2 and c = 3 find the value of

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2a ¡ c = 2(5) ¡ (3) = 10 ¡ 3 =7

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a 2a ¡ c

p a ¡ 2b

b

p a ¡ 2b p = (5) ¡ 2(¡2) p = 5+4 p = 9 =3

c c ¡ 4b2

c ¡ 4b2 = (3) ¡ 4(¡2)2 = 3¡4£4 = 3 ¡ 16 = ¡13

c

EXERCISE 4B 1 For a = 2, b = 5, find the value of a

a+b

b

b¡a

c

2a + 3b

d

a ¡ 2b

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b2

f

2b2

g

(2b)2

h

b3

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4a + b

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4(a + b)

k

(a + b)2

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a2 + b2

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ab 20

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5(2a + 3b)

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8ab 4

p

a2 b 10

c

2p + 5q

d

3p ¡ 2q

g

(p ¡ q)2

h

p2 ¡ q2

k

8 ¡ (3p ¡ 4q)

l

(p + q)(p ¡ q)

2a + c p b ¡ 3c

d

5¡c¡b

h

5 ¡ (a ¡ b)

2 For p = 2 and q = ¡5, find the value of a

p+q

b

e

1 3 (p

+ q)

f

i

¡5(3p ¡ 2q)

j

p¡q p+q 2 4 ¡ (p ¡ q)

3 For a = ¡3, b = 3 and c = ¡2, find the value of a e

a+b+c p a + 4b

EXAMPLE

b

a¡b¡c

c

f

3a + 2c

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2

Copy and complete the following table of values for 4x ¡ 3:

3

x 4x ¡ 3

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When x = 3, 4x ¡ 3 = 4(3) ¡ 3 = 12 ¡ 3 =9

When x = 2, 4x ¡ 3 = 4(2) ¡ 3 = 8¡3 =5

When x = 1, 4x ¡ 3 = 4(1) ¡ 3 =4¡3 =1

When x = 0, 4x ¡ 3 = 4(0) ¡ 3 =0¡3 = ¡3

When x = ¡1 4x ¡ 3 = 4(¡1) ¡ 3 = ¡4 ¡ 3 = ¡7

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Thus

x 4x ¡ 3

3 9

2 5

1 1

0 ¡3

¡1 ¡7

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4 Copy and complete the following tables of values: b a x 0 1 2 3 4 3x + 1

x 5 ¡ 2x

¡2

d

a 7 ¡ 4a

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x x +3

¡1

0

e

x 3 ¡ x2

¡2

¡1

0

1

2

f

p 4p + p2

g

y y+2 y

¡3

¡1

1

3

5

h

q 3q ¡ 2q2

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2

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0

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EXAMPLE

¡3

¡1

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Substituting into formulae gives number patterns.

3

a By substitutions, show that the expression 3n + 2 for n = 1, 2, 3, .... generates the number pattern 5, 8, 11, 14, 17, ..... b Describe this pattern in words. c Find the 200th term. a If n = 1, If n = 2, If n = 3, 3n + 2 3n + 2 3n + 2 = 3(1) + 2 = 3(2) + 2 = 3(3) + 2 =5 =8 = 11 ) the expression 3n + 2, where n = 1, 2, 3, ..... generates the pattern 5, 8, 11, 14, 17, ......

If n = 4 3n + 2 = 3(4) + 2 = 14 etc.

b To get the next term, add 3 to the previous term. c Substitute n = 200 into 3n + 2 i.e., 3n + 2 = 3(200) + 2 = 602

EXAMPLE

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Find the first 4 terms of the pattern described by n ¡ 2, for n = 1, 2, 3, ......

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If n = 1, n¡2 = 1¡2 = ¡1

If n = 2, n¡2 =2¡2 =0

If n = 3, n¡2 =3¡2 =1

If n = 4 n¡2 = 4¡2 = 2 etc.

) the pattern is ¡1, 0, 1, 2, .....

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5 By substituting n = 1, sequence it describes: a n¡3 e 3n i 4n m 7 ¡ 4n q n2 64 u 2n

2, 3 and 4 into each expression, find the first 4 terms of the number b f j n r v

7¡n 3n + 3 4n + 2 17 ¡ 4n 2n 243 3n

c g k o s

12 + n 3(n + 1) 4(n + 2) ¡3 ¡ 4n 5n

d h l p t

5¡n 3n + 6 4n ¡ 7 25 ¡ 4n 2n2 ¡ 3

w

n(n + 1)

x

n(n + 1)(n + 2)

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a

Show that the number pattern 18, 17, 16, 15, ...... can be expressed as 19 ¡ n, for n = 1, 2, 3, ...... and hence find the 50th term.

b

Show that the number pattern 8, 12, 16, 20, ...... can be expressed as 4(n + 1), for n = 1, 2, 3, ...... and hence find the 105th term.

c

Show that the number pattern 1, 3, 5, 7, ...... can be expressed as 2n ¡ 1, for n = 1, 2, 3, ...... and hence find the 300th term.

d

Show that the number pattern 3, 0, ¡3, ¡6, ...... can be expressed as 6 ¡ 3n, for n = 1, 2, 3, ...... and hence find the 60th term.

e

Show that the number pattern 1, 4, 9, 16, ...... can be expressed as n2 , for n = 1, 2, 3, ...... and hence find the 100th term.

7 Using a = 2 and b = ¡5, and showing all working, check whether these pairs of expressions are the same. a c e g i

4(a + b) and 4a + 4b 2(3a + b) and 6a + 3b 3a ¡ 5(a + b) and ¡2a ¡ b (3a ¡ 2b)(4a + 7b) and 12a2 ¡14 ab (a + b)3 and a3 + b3

b d f h j

4(a + b) and 4a + b 3a ¡ 5(a + b) and 2a ¡ 5b 3a ¡ 5(a + b) and ¡2a ¡ 5b (3a ¡ 2b)(4a + 7b) and 12a2 ¡ 8ab ¡ 14b2 (3a ¡ 2b)(4a + 7b) and 12a2 + 13ab ¡ 14b2

8 Using a = 3 and b = ¡4, evaluate c 4a + 4b d ¡10 + 2a: a 4(a + b) b a2 + b ¡ 9 e What do you notice about the value of all these expressions? f Does this necessarily mean that all these expressions are equal? g Test using other values of a and b to find which, if any, of the expressions are always equal. 100

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Find v given that u = 50, a = 10, t = 6 and v = u + at v = u + at v = 50 + 10 £ 6 v = 110

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a Find v given that u = 5, a = 10, t = 7 and v = u + at: b Find s given that u = ¡3, t = 5, a = 10 and s = ut + 12 at2 . 360 c Find d given that n = 12 and d = 180 ¡ n 2Rn d Find a given that R = 8, n = 24 and a = n+1

10 The volume of a sphere, V , is found using the formula V = 43 ¼r3 , where r is the radius. Find the volume of a sphere with radius a 6 cm b 8.5 cm c 1.3 metres 11 A formula for calculating the bend allowance, B, in sheet metal is 2¼(R + T2 ) £ A where B is the bend allowance B= 360 T is the thickness in mm A is the number of degrees in the angle of bend and R is the radius of curvature in mm. Find B when: a T = 3, A = 5, R = 12 b T = 15, A = 50, R = 60 12 The body mass index, B, is a measure used to determine if weight is within an acceptable range. The range is 21 < B < 25: m The body mass index formula is B = 2 where m = mass in kg, h = height in metres. h a Find B for a person weight 65 kg and height 1.65 m. b Find B for a person weight 100 kg and height 1.95 m. c Find B for a person weight 50 kg and height 1.5 m. d Find B for your weight and height. 13 A sick child requires medication. The adult dose of the medication is 20 mL. The child is 16 months old and has mass of 7.9 kg. mA where D = infant dosage a Find the dosage using Fried’s rule. D = 150 m = age of infant in months and A = adult dosage. yA b Find the dosage using Young’s rule: D = where y = age of child in years and (y + 12) A = adult dosage. c Find the dosage using Clark’s rule: D =

kA where k = mass of child in kg and 70 A = adult dosage.

d How much would you give the child? Explain.

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14 Repeat question 13 for a child 20 months old weighing 12 kg, with an adult dose of medication of 30 mL.

ADDING AND SUBTRACTING TERMS


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Algebraic expressions can be simplified by collecting like terms. Like terms are the only terms that can be added or subtracted.


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EXAMPLE

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Simplify, where possible, by collecting like terms a 3a + 4a b 11b ¡ b c 2ab + 3ab

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3a + 4a = 7a

d

3x2 + 2x

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11b ¡ b = 11b ¡ 1b = 10b

b

d 3x2 + 2x 2ab + 3ab = 5ab

c

is in simplest form as x2 and x are unlike terms

EXERCISE 4C 1 Simplify, where possible, by collecting like terms a 4a + 5a b 3x + 2x c e 2x + x f x + 3x g j 7x + 3 k i a2 + a2 o m 17x ¡ x n x2 + x 2 2 q 3b ¡ b r 2ba + 3ab s u xy + 2yx v 3p ¡ 2p w

5x ¡ 2x 3x ¡ 2x x2 + 10x2 7b + b 11n ¡ 11n 15 ¡ 8

d h l p t x

b+b 3x ¡ x 17x ¡ 7 7b ¡ b 3ab + ba 7bca ¡ 5abc

EXAMPLE Simplify, where possible, by collecting like terms a

6d + 3d + 5

b

4x + 5 + 2

c

3x ¡ x2 + 4x

a

6d + 3d + 5 = 9d + 5

b

4x + 5 + 2 = 4x + 5 + 2 = 4x + 7

c

3x ¡ x2 + 4x = 3x + 4x ¡ x2 = 7x ¡ x2

2 Simplify, by collecting like terms a 6x ¡ 3x + 7x c 9a2 + 13a2 ¡ 17a2 e 4x2 y + 13x2 y ¡ 8x2 y g 11xy ¡ 5xy ¡ 6xy i 4p2 q + 3p2 q ¡ 5p2 q ¡ p2 q

b d f h

7y ¡ 2y ¡ 3y 7ab ¡ 2ab + 11ab 7q + q + 4q ¡ 10q 6ab ¡ 3ab ¡ 2ab + 7ab

3 Simplify, where possible, by collecting like terms a 5x + 4x ¡ 2 b m+7+4 d 2y + y + 3 e p + 3p ¡ 5 g 6t + 4 ¡ 3t h 16n ¡ 16 + 5n j 7k + k ¡ 8 k 4a2 ¡ a2 ¡ 7a 2 2 m 2m ¡ m + 5n n 3x + 2y + 5x



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x+6+x 7 + 4x ¡ x p2 + p + 4p2 5cd + 2dc ¡ 2 3a + 2b ¡ a

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r u x

6a + 3a ¡ ab x ¡ x2 + 2x 5x2 ¡ 3x2 + 6x3

c

x ¡ 5 ¡ 2x ¡ 1

c

x ¡ 5 ¡ 2x ¡ 1 = x ¡ 2x ¡ 5 ¡ 1 = ¡x ¡ 6

4 Simplify, where possible, by collecting like terms a 4x ¡ 7x b ¡4x ¡ 7x d 6d ¡ d e ¡6d ¡ d g 4n ¡ 11n h ¡4n + 11n j 3a + 2 ¡ 6a k 5¡d¡8 m ¡3g ¡ (¡g) n 5a ¡ (¡a) ¡ 2a

c f i l o

¡4x + 7x ¡6d + d ¡4n ¡ 11n x ¡ (¡2x) 4ac ¡ 5ca

5 Simplify, where possible, by collecting like terms a a + 3 + 2a + 7 b 5 + 2a + 3 + 4a d 2a + 3b + 3a + b e 3a2 + a + a2 + 2a g 3 + 6y ¡ 1 + 2y h n2 + n ¡ n + n2 2 2 j x + 2x ¡ x + 5x k 3x ¡ 5x ¡ 3 + 2x m 7 ¡ 5x ¡ 7x + 3 n 3p + 7p ¡ 8 + 4p p 8 ¡ 7x ¡ 5 + 3 q x ¡ 8 ¡ 7x ¡ 5

c f i l o r

3a + 2 + a + 4 ab + b2 + 2ab + 2b2 18c + 5 ¡ 4 ¡ 11c 7 ¡ 5p + 3p ¡ 12 3x + 7x ¡ 2 ¡ x x2 + x + 2 ¡ 5x

100

p s v

95

75

25

5

0

123

2ab + b2 + 2b2 n + n + 2n2 5mn ¡ 8m ¡ 4mn

q t w

6x + 4x ¡ 10 5 + 2a ¡ 1 a2 b + 2ab2 + 4a2 b

EXAMPLE Simplify, where possible, by collecting like terms a 4x ¡ 7x ¡ 5 +x b 5 ¡ 5a ¡ a + 7 a

4x ¡ 7x + x ¡ 5 = ¡2x ¡ 5 fas 4 ¡ 7 + 1 = ¡2g

b

5 + 7 ¡ 5a ¡ a = 12 ¡ 6a

6 Simplify, where possible, by collecting like terms a ¡8l ¡ 4 + 3l ¡ 6 b x2 + 2x ¡ 5x ¡ x2 c 3x ¡ 2y ¡ (¡x) + y d ab + b ¡ 2ab ¡ 3b e ¡x ¡ 5 ¡ 2x ¡ 3 f 5t ¡ (¡t) + 6 ¡ t g 3a ¡ 2 ¡ a + 3 ¡ a h a2 b + a2 b ¡ 3a2 b + 7b i 5d ¡ c + d ¡ 2c + 2 j 4p5 + 5p4 ¡ p5 ¡ 6p4 k 8m2 ¡ 5n2 + 3n2 ¡ 4m2 l 6s2 t + 5s2 ¡ 8s2 t ¡ 9s2 100

95

MULTIPLICATION AND DIVISION OF ALGEBRAIC TERMS

75

Algebraic terms can be multiplied or divided to form single expressions.

25

If there are algebraic fractions then they may be cancelled in the same way as fractions, using normal algebraic rules.

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EXAMPLE

95

Simplify a ¡3 £ 4x d (¡4x) £ (¡2x2 )

75

b e

5x £ 2y 2m2 £ 3n2

c f

2x £ (¡x2 ) (¡5x)2

25

5

a

¡3 £ 4x = ¡3 £ 4 £ x = ¡12x

b

5x £ 2y =5£2£x£y = 10xy

c

2x £ (¡x2 ) = 2 £ ¡1 £ x £ x2 = ¡2x3

d

(¡4x) £ (¡2x2 ) = ¡4 £ ¡2 £ x £ x2 = 8x3

e

2m2 £ 3n2 = 2 £ 3 £ m2 £ n2 = 6m2 n2

f

(¡5x)2 = ¡5 £ x £ ¡5 £ x = 25x2

b

4x £ 7

c

¡2 £ 8x

e

3x £ x

f

3x £ 7x

0

?

EXERCISE 4D 1 Simplify a

4 £ 3x

d

¡5 £ 4x 2

g

¡3x £ 2x

h

(¡2x)(¡x)

i

(¡3x)3

j

(¡x2 )(¡2x)

k

3d £ (¡2d)

l

2a2 £ 3a2

m

a3 £ (¡3a)

n

(¡gh) £ (hr)

o

xyz 2 £ xy2

p

pq £ pr2

q

7m2 £ 5n

r

¡3p £ 9q

s

1 3

of 15a

t

3 4

u

1 5x

v

(¡5p) £ 7pq

w

(¡6m2 ) £ (¡3mn)

x

4x2 £ 5xy

of 12x2

£ 20y 2

2 Simplify a

3 £ 4x £ 7

b

9 £ 5y £ 2x

c

3x £ 4 £ 6

d

5y £ 3y £ 3

e

7x £ 4y £ 5z

f

(¡6x) £ 2y £ (¡4x)

g

8a £ (¡5b) £ (¡3a)

h

6mn £ (¡3np) £ 2m

i

(¡3xy) £ (¡4xz) £ (¡5yz)

j

2ab £ (¡3bc) £ 7ac

k

2

(3x) £ (¡2xy)

l

2

100

2

6pr £ (¡2p)

95

3 Multiply to simplify

75

a

(2x)2

b

(¡2x)2

c

a £ 2a2

d

a £ (2a)2

e

¡8 £ b2

f

(¡8b)2

g

(¡x)2

h

(¡x)3

i m

(¡3x) £ 10 3

(¡2n)

j n

¡ab £ (¡2a) 2

a £ (¡a)

k o

2

(¡ab) 2

2

x £ (¡3x )

25

2

l

¡3x £ (¡2x )

p

(¡4x2 )2

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EXAMPLE a ¡10x ¥ 5

Simplify

¡10x ¥ 5 ¡10x = 5 1 ¡5 £ 2 £ x = 5 = ¡2x 1

a 25

5

0

4 Simplify ¡3a a a

c (¡14a) ¥ 21a2

b 20pq ¥ 4p b

20pq ¥ 4p 20pq = 4p 1

(¡14a) ¥ 21a2

c

¡14a 21a2 1 1 ¡2 £ 7 £ a = 3£7£a£a =

1

5£4£p£q 4£p = 5q 1 1 =

¡2 = 3a

1

1

b

¡3a 3

c

¡3a ¡6

d

8a 4a

e

8a ¡a

f

8a ¡4

g

a2 a

h

¡a2 2a

i

(¡a)2 a

j

(¡a)2 a2

k

¡15a 50

l

¡15a ¡5

m

¡27x2 3x

n

a2 b ¡a

o

¡6ab ¡3b

5 Write a e i m q

in fraction form and simplify 8x ¥ 4 b 20y ¥ 5 15pq ¥ 5p f 6ab ¥ 3b 5ab ¥ 3a j ¡14pq ¥ 12q 3p ¥ 15p n 8r ¥ (¡24) 2 16x ¥ 16x r ¡5p ¥ 10pq

c g k o s

35p ¥ (¡7) ¡8st ¥ 4s ¡20ab ¥ 15b ¡30ab ¥ 6a 7m ¥ 14m2

15r ¥ 3 12mn ¥ 3n 5xyz ¥ 10xy 5xy ¥ 10yz 11ab ¥ 11ab

d h l p t

EXAMPLE Simplify

a

a

m 3 £ m 6

m 3 £ m 6 1

1

b

b

3 £ m2 m 3 £ m2 m

c

3 a2 £ a 2 3 a2 £ a 2

c

3 1 £m£m = m

=

1 = 2

3m = 1

3a = 2

2

1

4 n2 £ n 2

2

a

100

4 n2 £ n 2

d

3 a2 £ a 2

3 m = £ m 6 1

d

4 n2 £ n 2 1 1 2n = 1

95

n

=

1

= 2n

= 3m

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6 Simplify the following products a

a a2 £ 2 3

b

a2 3 £ 2 a

c

a3 £a 2

d

a2 £

e

3 £ a2 a

f

3 £ a2 a2

g

3£

a3 3

h

¡6 £

i

c £ c3 5

j

c 1 £ 2 5 c

k

a b2 £ x x

l

b a2 £ 2 b a

m

a ( )2 b

n

m3 £

o

m2 £

p

1 m3 £ m2 2

75

25

5

0

3 m2

2 m3

a2 3 a2 2

7 Simplify a

3 a2 £ 2 a

b

2 a2 £ a 4

c

1 1 £ b 3

d

2 1 £ n n

e

5 c £ 5 c2

f

c c2 £ 5 7

g

c2 c £ 3 5

h

a b £ x x

i

a c £ b d

j

m3 £

2 m

k

m m2 £ n n

l

3 g2 £ g 4

8 Simplify a

pq 6 £ 3 p

b

x3 12 £ 4 xy

c

pq a2 q £ a p

d

15x 8y 3 £ 2 2y 3x

e

ab 12 £ 6 ac

f

9x3 5y 3 £ 2 yz 6x

g

3mn 7g 3 £ 14g 2 6m

h

7st 14s £ 2 3r 9t

i

25xy 12z 2 £ 32 5y3

j

d3 ab £ 2 d a b

k

xyz 6x2 £ 2 4a xy

l

21mn 15p2 £ 5np 7n2

E

EXPANSIONS

Grouping symbols are used to write algebraic terms more simply. These terms need to be expanded back out to individual terms to further simplify.

EXAMPLE

1

Expand and simplify a

100

95

a 3(x + 4)

3(x + 4) =3£x + 3£4 = 3x + 12

b

b 4(2a + 7) 4(2a + 7) = 4 £ 2a + 4 £ 7 = 8a + 28

c 5x(3 ¡ 2x) c

75

5x(3 ¡ 2x) = 5x £ 3 ¡ 5x £ 2x = 15x ¡ 10x2

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127

EXERCISE 4E 1 Expand and simplify

75

25

a

3(x + 2)

b

5(y ¡ 2)

c

2(a + 2)

d

5(b ¡ 3)

e

6(m + 4)

f

9(ab + c)

g

2(2a + 1)

h

4(1 ¡ 2c)

i

7(2a + 7)

j

7(1 ¡ 3d)

k

5(2e + 9)

l

a(a + 4)

m

x(4 ¡ x)

n

3x(x + 4)

o

3x(2x ¡ 1)

p

4(2x + 3y ¡ 1)

q

7(6x ¡ 4y ¡ 2)

r

7x(2x ¡ 5y ¡ 4)

5

0

EXAMPLE a ¡3(x + 5)

Expand

b

¡3(x + 5)

a

¡5(4 ¡ 2y) ¡5(4 ¡ 2y)

b

= ¡3 £ x + (¡3) £ 5

= ¡5 £ 4 ¡ 5 £ (¡2y)

= ¡3x ¡ 15

= ¡20 + 10y = 10y ¡ 20

2 Expand and simplify a

¡3(x + 3)

b

¡4(5 ¡ x)

c

¡4(2x + 1)

d

¡2(4 ¡ x)

e

¡3(2p + 3)

f

¡6(a + 3)

g

¡5(m + 2n)

h

¡x(x + y)

i

¡a(3b + 3d)

j

¡b(7 ¡ 2b)

k

¡y(x + 3)

l

¡a(3a ¡ 5)

m

¡6(¡x + y ¡ 2z)

n

¡4(a ¡ b ¡ 1)

o

¡x(x + 3 + y)

EXAMPLE a ¡(x ¡ 3)

Expand

¡(x ¡ 3)

a

b ¡(6 + y) ¡(6 + y)

b

= ¡x + 3

c ¡3y(5y ¡ x) c

¡3y(5y ¡ x)

100

= ¡15y2 + 3xy

= ¡6 ¡ y

95

75

3 Expand a

¡(x + 3)

b

¡(5 ¡ x)

c

¡(x + 8)

d

¡(7p ¡ 7)

e

¡(2x + 4)

f

¡(5 ¡ 3x)

g

¡(8y + 4x)

h

¡(7q ¡ 8p)

i

¡(5a + 7b)

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4 Expand a ¡3a(2 ¡ b) d ¡5m(p + m) g ¡3a(a + b ¡ 1)

¡4y(x + 3) ¡x2 (x + 7) ¡5x(x ¡ t + 2)

b e h

c f i

¡4x(4 ¡ x) ¡4c(3cd ¡ t) ¡7q(q + 3r ¡ 2)

EXAMPLE

5

Expand and simplify 0

a 5 ¡ (3 ¡ 4x)

b 2 + 3(x ¡ 5)

5 ¡ (3 ¡ 4x)

a

b

2 + 3(x ¡ 5)

= 5 ¡ 3 + 4x

= 2 + 3x ¡ 15

= 2 + 4x

= 3x ¡ 13

= 4x + 2

5 Expand and simplify a 6 ¡ (3x + 7) d 6x ¡ (4 ¡ 4x) g 3x ¡ (2x ¡ 7) j 3(8 ¡ x) ¡ 11x

b e h k

7 ¡ 2(1 ¡ 2x) 5x ¡ 5(2x + 1) 5 ¡ 4(x ¡ 2) 7(x ¡ 3) + 1

c f i l

15 ¡ (5x + 8) 6 ¡ 5(1 ¡ 2x) 8 ¡ 6(3x ¡ 2) 9(x ¡ 2) ¡ 9x

EXAMPLE Expand and simplify

a 4(x + 6) + 3(5 ¡ x)

4(x + 6) + 3(5 ¡ x)

a

b x(3 ¡ x) ¡ 3(4 ¡ 3x) b

x(3 ¡ x) ¡ 3(4 ¡ 3x)

= 4x + 24 + 15 ¡ 3x

= 3x ¡ x2 ¡ 12 + 9x

= 4x ¡ 3x + 24 + 15

= 12x ¡ x2 ¡ 12

= x + 39


4(x + 3) + 3(x ¡ 1)

b

4(y + 1) + 2(y + 3)

c

7(p + 1) ¡ 3(p ¡ 2)

100

d

3(1 ¡ x) ¡ 3(x ¡ 1)

e

d(d + 1) + d(d ¡ 1)

f

d(d + 4) ¡ d(d ¡ 4)

95

g

n(n + 7) + n(2n + 1)

h

n(n + 2) ¡ n(2n + 1)

i

7(2 + 3x) + 3(x + 5)

75

j

x(x + 6) ¡ 3x(2 ¡ x)

k

a(b + c) ¡ b(c + a)

l

3x(x + 1) + 3(x + 2)

m 5x(x + 3) ¡ 5(5 + x)

n

¡(x ¡ 3) ¡ 2(3 ¡ x)

o

2x(1 ¡ x) + 2(x + 6)

p

¡2x(3 ¡ x) + 3(x ¡ 4)

q

¡4(x + 3) ¡ 5x(x + 1)

r

¡x ¡ 5 ¡ 7(x + 1)

s

5(x ¡ 4) ¡ (8x + 10)

t

b(b + 5) ¡ (6b ¡ 3)

u

¡(7 ¡ 3x) ¡ (5x + 2)

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EXAMPLE

6

Expand and simplify a

25

5

129

a 2x2 (7 ¡ x) + x(x ¡ 1)

b 3x2 (x ¡ 5) ¡ x(3x ¡ 4)

2x2 (7 ¡ x) + x(x ¡ 1) = 14x2 ¡ 2x3 + x2 ¡ x = 15x2 ¡ 2x3 ¡ x

b

3x2 (x ¡ 5) ¡ x(3x ¡ 4) = 3x3 ¡ 15x2 ¡ 3x2 + 4x = 3x3 ¡ 18x2 + 4x

0


3x2 (5 ¡ x) + x(x ¡ 2)

b

6x2 (7 ¡ 2x) + x(2x ¡ 5)

c

3x2 (3x + 7) + x(4 ¡ 2x)

d

2x(3x ¡ 8) ¡ 4x2 (2x ¡ 3)

e

5x(7 ¡ 4x) ¡ 5x2 (4 ¡ 3x)

f

2x2 (3x ¡ 8) ¡ 3x2 (6 ¡ 3x)

F

ONE STEP EQUATIONS An algebraic equation has two algebraic expressions that are equal to each other.

The simplest are called one step equations as it takes one step to solve them. To solve an equation you can

²

add the same number to both sides of the equation

²

subtract the same number from both sides of the equation

²

multiply each side of the equation by the same number.

²

divide each side of the equation by the same number.

The aim of solving an equation is to find the value of the pronumeral. This is done by using one or more of the operations described.

EXAMPLE

1

Solve these equations

a x + 9 = 17

100

b p ¡ 12 = 30

95

a

p ¡ 12 = 30 p ¡ 12 + 12 = 30 + 12 p = 42

x + 9 = 17 x + 9 ¡ 9 = 17 ¡ 9 x=8

b

Check: If x = 8, x + 9 = 8 + 9 = 17 X

Check: If p = 42, p ¡ 12 = 42 ¡ 12 = 30 X

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EXERCISE 4F 1 Solve a e i m q u y

these equations p + 8 = 16 a + 16 = 31 f ¡ 28 = 36 d + 18 = 40 x + 30 = 38 f ¡ 12 = 20 T + 12 = 42

EXAMPLE

d¡3= 7 q ¡ 16 = 30 p + 18 = 40 f ¡ 5 = 21 y ¡ 7 = 33 p ¡ 16 = 42 s + 32 = 51

c g k o s w

2

a 3x = 18

3x = 18

42 7x = 7 7 x=6

Check: If x = 6, 3x = 3 £ 6 = 18 X

these equations 5m = 25 9m = 63 3a = 15 2q = 42 13L = 39 11y = 132 10r = ¡140

b f j n r v z

6p = 30 20x = 180 8N = 56 4m = 40 12z = 72 8d = 72 9c = ¡72

a

x =5 2

x =5 2 x 2£ =5£2 2 x = 10

Check: If x = 10,

Check: If x = 6, 7x = 7 £ 6 = 42 X

c g k o s w

7m = 63 6p = 72 14z = 56 8w = 168 4T = 80 2p = ¡16

d h l p t x

18b = 90 5g = 60 12d = 48 7c = 63 15a = 90 5q = ¡25

The inverse operation of division is multiplication.

3

Solve these equations a

L + 14 = 19 x ¡ 3 = 26 d ¡ 15 = 41 x + 21 = 48 z ¡ 8 = 63 r + 48 = 49

7x = 42

b

x=6

EXAMPLE

d h l p t x

b 7x = 42

3x 18 = 3 3

2 Solve a e i m q u y

d ¡ 6 = 10 c + 9 = 19 y ¡ 4 = 17 g ¡ 11 = 21 p ¡ 24 = 58 L + 15 = 61

The inverse operation of multiplication is division.


b f j n r v z

b b

10 x = =5 X 2 2

p =6 8

100

95

p =6 8 p 8£ = 6£8 8 p = 48

Check:

If p = 48,

75

25

p 48 = =6 X 8 8

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3 Solve these equations x a =2 7

75

25

b

x =2 4

c

p = 11 q

d

m = 14 5

e

p =2 8

f

d =7 10

g

y = 12 3

h

q =3 7

i

d =5 6

j

x =6 4

k

d =7 2

l

c =5 22

5

m 0

q u

m =8 8 y =7 7 a =4 6

n r v

p =5 12 x =4 17 s =3 2

o s w

f =5 14 q =6 5

p t

t =9 15

d = 12 3 e =8 19 d = 15 18

x

Think about what you are doing and remember to balance the equation.

4 Solve these equations a

a =9 4

b

3d = 18

c

x ¡ 9 = 24

d

5m = 105

e

d + 12 = ¡30

f

p + 9 = 20

g

x = 20 5

h

y = ¡11 7

i

y + 4 = ¡15

j

2n = ¡96

k

8 + b = 20

l

q ¡ 9 = ¡7

EXAMPLE

4

Check, by substituting the solution into the equation, to see if the answer is correct. p a x + 15 = 27 b = 17 c 5m = ¡45 9 [x = 12] [m = ¡8] [p = 142] Substitute the value of the pronumeral into the equation. p = 17 a x + 15 = 27 b 9 12 + 15 = 27 142 9 = 17 27 = 27 15 79 = 17 These are equal ) x = 12 is correct.

100

c

These are not equal ) p = 142 is not correct.

95

5m = ¡45 5 £ ¡8 = ¡45 ¡40 = ¡45

75

25

These are not equal ) m = ¡8 is not correct.

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5 By substituting into the equation, determine whether or not the solution is correct

100

x + 18 = 9 ) x = ¡9

a

95

3x = 18 x=6

c

p ¡ 12 = 3 ) p = ¡7

a ¡ 32 = ¡11

f

41 ¡ d = ¡22

b )

75

d )

25

e

)

m + 82 = 97

g

5

d = 64 4 d = 16

)

0

h

m = 15

t = 104 13 ) t=8

j

G

a = 21

x = 143 11 ) x = 13 5p = 85

k )

)

9m = ¡54

i ) l

p = 13

d = 71

m = ¡6

q ¡ 52 = 108 )

q = 160

TWO STEP EQUATIONS

The techniques needed to solve two step equations are the same as for one step equations except two techniques need to be applied. Always add or subtract first to get the pronumeral on one side of the equation and the numbers on the other side. Then multiply or divide to find the answer.

EXAMPLE

1


5x + 7 = 37 5x + 7 ¡ 7 = 37 ¡ 7 5x = 30 5x 30 = 5 5 x=6

b

8x ¡ 9 = 79 8x ¡ 9 + 9 = 79 + 9 8x = 88 8x 88 = 8 8 x = 11

a 5x + 7 = 37

b 8x ¡ 9 = 79

Check: fsubtract

7 from both sidesg

fdivide both sides by 5g

5x + 7 = 37 (5 £ 6) + 7 = 37 30 + 7 = 37 37 = 37 ) correct X

Check:

fadd

9 to both sidesg

fdivide both sides by 8g

8x ¡ 9 = 79 (8 £ 11) ¡ 9 = 79 88 ¡ 9 = 79 79 = 79 ) correct X

100

95

75

?

EXERCISE 4G 1 Solve the following two-step equations a 3x ¡ 9 = 3 b 4p + 8 = 24 d 6x + 7 = 61 e 4p + 12 = 52



25

c f

6x ¡ 2 = 82 5y ¡ 6 = 44

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g j m p s v

95

75

25

2x ¡ 32 = 16 13d + 9 = 126 3p ¡ 6 = 54 7p + 33 = 117 4q ¡ 21 = 115 7T ¡ 8 = 27

5

0

EXAMPLE

h k n q t w

11p ¡ 10 = 34 5x + 15 = 65 9a + 12 = 57 6r ¡ 14 = 94 8s ¡ 12 = 252 11s + 16 = 115

i l o r u x

7m + 8 = 50 10c ¡ 7 = 153 9m + 12 = 120 6q + 16 = 100 10r ¡ 30 = 100 4m + 9 = 73

2

Check, by substituting the solution into the equation, to see if the answer is correct a 3x ¡ 9 = 36 [x = 11] b 49 ¡ 5x = 29 [x = 4] a

If 3x ¡ 9 = 36 then (3 £ 11) ¡ 9 = 36 33 ¡ 9 = 36 24 = 36 as these are not equal ) x = 11 is not correct

b

If 49 ¡ 5x = 29 then 49 ¡ (5 £ 4) = 29 49 ¡ 20 = 29 29 = 29 as these are equal ) x = 4 is correct.

2 By substituting into the equation, determine whether or not the solution stated is correct a

4x + 8 = 60 [x = 13]

b

5x ¡ 9 = 11 [x = 3]

c

3x + 4 = 13 [x = 3]

d

9 ¡ 2p = ¡15 [p = 12]

e

10 + 5m = ¡20 [m = ¡6]

f

22 + 3c = 46 [c = 7]

g

6d ¡ 5 = 31 [d = 9]

h

11y ¡ 72 = ¡193 [y = ¡11]

i

14q + 18 = 116 [q = 7]

EXAMPLE

3

a 9 ¡ 7x = 5

Solve

a )

9 ¡ 7x = 5 9 ¡ 7x ¡ 9 = 5 ¡ 9 ) ¡7x = ¡4 ¡7x ¡4 ) = ¡7 ¡7 ) x = 47

b 17 = 8 ¡ 4x

b f¡9g f¥ ¡ 7g

)

17 = 8 ¡ 4x 17 ¡ 8 = 8 ¡ 8 ¡ 4x ) 9 = ¡4x 9 ¡4x ) = ¡4 ¡4

100

f¡8g

95

f¥ ¡ 4g

75

¡ 94 = x

) )

x = ¡ 94

)

x = ¡2 14

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75

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3 Solve a d g j m p s v

for x 4x + 7 = ¡1 3x ¡ 4 = ¡6 8x ¡ 7 = ¡11 5 = 3x + 7 6 ¡ x = ¡5 5 ¡ 4x = ¡7 11 = 3 ¡ 2x 6 = ¡1 ¡ 7x

0

EXAMPLE Solve

b e h k n q t w

4x ¡ 1 = ¡9 5x + 8 = 2 3x + 6 = ¡7 6x ¡ 7 = ¡1 3 ¡ 4x = 15 3 ¡ 7x = ¡2 15 ¡ 2x = ¡1 ¡15 = 3 ¡ 6x

3x + 6 = 0 4x ¡ 9 = 1 6 + 7x = ¡2 ¡1 = 2x + 6 3 ¡ 2x = 7 17 ¡ 2x = ¡1 8 = 3 ¡ 2x 11 = ¡4 ¡ 3x

c f i l o r u x

4

m ¡ 5 = ¡2 3 m ¡ 5 = ¡2 3 )

m ¡ 5 + 5 = ¡2 + 5 3 m =3 ) 3 1 m £3 = 3£3 ) 1 3 )

fadding 5 to both sidesg

fmultiplying both sides by 3g

m=9

4 Solve for x x +3=8 a 2 x d + 3 = ¡4 6

b e

x ¡1=4 3 x ¡2=4 7

c f

x + 2 = ¡3 5 x ¡ 6 = ¡1 10

5 Check the given solution by substitution and say whether or not it is correct a

2x + 8 = 15

[x = 7]

b

7 ¡ 5x = 9

[x = ¡ 25 ]

c

¡15 = 6 ¡ 7x

[x = 3]

d

x ¡3=6 5

[x = 95 ]

EXAMPLE

5

100

95

If y = 5x ¡ 3 find x when y = ¡18: y = 5x ¡ 3 ¡18 = 5x ¡ 3 ¡15 = 5x ¡3 = x i.e., x = ¡3

75

fsubstitute the value for y g

3 to both sidesg fdivide by 5g fadd

25

5

0



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6 a c e

95

75

If y = 3x ¡ 5, find x when y = ¡14: If y = 7 ¡ 5x, find x when y = 8: If y = 5 ¡ 7x, find x when y = ¡5:

H

25

b d f

135

If y = 4x + 2, find x when y = 11: If y = 4 ¡ 3x, find x when y = ¡3: If y = 3x ¡ 5, find x when y = 0:

FURTHER EQUATIONS

5

When solving equations with the pronumerals on both sides it may be necessary to not only add and subtract numbers from both sides, but, to add and subtract pronumerals from both sides. The first step is to add or subtract the pronumerals to move them to one side. It does not matter which side. Next add or subtract to move the numbers to the other side of the equation. Then solve as normal by multiplying or dividing.

0

EXAMPLE Solve

1

a 5x + 2 = 3x ¡ 5

a )

5x + 2 = 3x ¡ 5 5x + 2 ¡ 3x = 3x ¡ 5 ¡ 3x ) 2x + 2 = ¡5 ) 2x = ¡7 ) x = ¡ 72 )

b )

?

b 15 ¡ 2x = 11 + x

x=

fsubtracting 3x from both sidesg fsubtracting 2 from both sidesg fdividing both sides by 2g

¡3 12

15 ¡ 2x = 11 + x 15 ¡ 2x + 2x = 11 + x + 2x ) 15 = 11 + 3x ) 15 ¡ 11 = 11 + 3x ¡ 11 ) 4 = 3x ) x = 43 = 1 13

fadding

2x to both sidesg

fsubtracting 11 from both sidesg fdividing both sides by 3g

EXERCISE 4H 1 Solve the following equations with a 6x + 2 = 2x + 18 d 4x ¡ 4 = 5x ¡ 1 g 3x ¡ 12 = x + 6 2 Solve a d g j m


integer solutions b 3x + 7 = 12 ¡ 2x e 3 ¡ x = 2x + 9 h 5x ¡ 9 = 11 + 7x

5 + x = 9 ¡ 3x 9 ¡ 2x = 3 ¡ 5x 3x ¡ 5 = 10 ¡ 2x

c f i

100

95

75

6x + 7 = 4x ¡ 2 2x ¡ 3 = 5x + 7 2x + 5 = 9 ¡ 3x 5a + 3 = a ¡ 3 11a ¡ 7 = 5a + 15

b e h k n


5x + 3 = 2x + 7 3 + x = 17 + 5x 4x ¡ 5 = 5x + 9 4 ¡ 3s = 2s + 15 3y ¡ 5 = ¡14 ¡ 5y

5 + 2x = 11 ¡ 3x 15 ¡ 3x = 2 ¡ 5x 5 ¡ 7x = 3x + 7 9x ¡ 4 = 3 + 5x 7p = 15 ¡ 4p

c f i l o

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5

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136


100

95

75

25

5

0

EXAMPLE

2

By substituting check the solutions to the following equations a 2x ¡ 5 = 10 ¡ 3x; [x = 3] b 5x + 2 = 2x ¡ 7; a

2x ¡ 5 = 10 ¡ 3x 2(3) ¡ 5 = 10 ¡ 3(3) 1=1 ) x = 3 is the solution

b

[x = 2]

5x + 2 = 2x ¡ 7 5(2) + 2 = 2(2) ¡ 7 12 6= ¡3 ) x = 2 is not the solution

3 By substituting check the solutions of the following equations a

3x + 9 = 4 ¡ 2x

[x = ¡1]

b

9a + 2 = 7a ¡ 4

[a = ¡3]

c

7a ¡ 5 = 3 ¡ a

[a = 2]

d

15 ¡ 2x = 6 + x

[x = 3]

e

2x ¡ 3 = 7 ¡ 4x

[x = 53 ]

f

5x ¡ 7 = 3 + x

[x = 3 12 ]

EXAMPLE

3

a 5(x + 1) ¡ 2(x ¡ 2) = 7

Solve a

b 3(x + 1) = 5x + 3(2x ¡ 1)

5(x + 1) ¡ 2(x ¡ 2) = 7 ) 5x + 5 ¡ 2x + 4 = 7 ) 3x + 9 = 7 ) 3x + 9 ¡ 9 = 7 ¡ 9 ) 3x = ¡2 ) x = ¡ 23

b

)

fexpanding the bracketsg fcollecting ‘like’ termsg fsubtracting 9 from both sidesg fdividing both sides by 3g

3(x + 1) = 5x + 3(2x ¡ 1) ) 3x + 3 = 5x + 6x ¡ 3 ) 3x + 3 = 11x ¡ 3 3x ¡ 3x + 3 = 11x ¡ 3x ¡ 3 ) 3 = 8x ¡ 3 ) 6 = 8x 8x 6 = ) 8 8 )

3 4

=x

i.e., x =

fexpanding the bracketsg fcollecting ‘like’ termsg fsubtracting 3x from both sidesg fadding

3 to both sidesg

fdividing both sides by 8g 100

3 4

95

75

4 Solve for x given that all answers are integers a 3(x + 1) ¡ 2(x ¡ 4) = 14 c 4(x ¡ 5) + 5(x + 1) = 3 e 4(x ¡ 2) = 3x + 4(x ¡ 2) g 6 ¡ x = 2 ¡ 3(x + 2)



b d f h

2(x ¡ 5) + 3(x + 2) = ¡14 2(x ¡ 1) = 3(x + 5) ¡ 17 2(x ¡ 1) = 4(2x + 1) ¡ 7x 7 ¡ 2(x + 5) = 2(2x ¡ 1) ¡ 5x

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95

75

25

5 Solve for x a 2(x + 1) ¡ 1 = 9 c 3(x + 2) ¡ 7 = 12 e 4(2x ¡ 1) + 3 = 0 g 3 ¡ 2(x + 1) = ¡6 i 5x ¡ 4(4 ¡ x) = x + 3 k 2(x ¡ 1) = 8 ¡ (3 ¡ x)

137

5(1 ¡ 3x) = ¡7 2(x + 1) + 3(x ¡ 1) = 9 11 ¡ 2(x ¡ 1) = 10 7 ¡ (2 ¡ x) = 5x 3 ¡ x = 7 ¡ 2(x + 1) x + 7(4 ¡ x) = 7x + 3(x ¡ 1)

b d f h j l

5

4

0

EXAMPLE

If y = 3 ¡ 5(x + 4), find x when y = ¡32: y = 3 ¡ 5(x + 4) ¡32 = 3 ¡ 5(x + 4)

fsubstituting y

¡32 = 3 ¡ 5x ¡ 20 ¡32 = ¡17 ¡ 5x ¡15 = ¡5x 3=x ) x=3

6

y y y y y y y y

If If If If If If If If

a b c d e f g h

5

Solve for x

x 2 = 3 5

a

5

15 £ 1

) )

17 to both sidesg fdividing by ¡5g fadd

4 x = 7 3

b

x 2 = 3 5

a )

fcollecting like termsg

= 7 ¡ 3(x + 2), find x when y = ¡7: = 5 ¡ 4(x ¡ 3), find x when y = 42: = 4 ¡ 5(2x ¡ 5), find x when y = 16: = 14 ¡ 3(2x ¡ 8), find x when y = 9: = 3x ¡ 2(5x + 1), find x when y = ¡8: = 4x ¡ 3(5 ¡ 2x), find x when y = 12: = 3(2x ¡ 1) ¡ 4(x + 2), find x when y = ¡7: = 4(1 ¡ 3x) ¡ 2(1 ¡ x), find x when y = 3:

EXAMPLE

x 2 = £ 15 3 5

)

3

21 £

1

5x = 6 x=

6 5

1

) )

To solve equations involving algebraic fractions multiply both sides by the lowest common denominator.

c

4 x = 7 3

b 3

= ¡32g

x =4 2

4 x = £ 21 7 3

95

x =4 2

c 7

)

1

2£ 1

1

12 = 7x x=

100

)

75

x = 4£2 2 25

x=8

12 7

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6

100

95

75

25

EXAMPLE Solve for x

a

4x + 3 = ¡2 5

b

¡2 1

b

4x + 3 = 5 ¶ µ 1 4x + 3 = 5£ 5

a

5

0

1

)

3£

¡2 £5 1

) 4x + 3 = ¡2 £ 5 ) 4x + 3 = ¡10 4x + 3 ¡ 3 = ¡10 ¡ 3 ) 4x = ¡13 ) x = ¡ 13 4

7 Solve for x x =7 a 3

1 3 (2x ¡

1) = ¡4

1 3 (2x 1 3 (2x

¡ 1) = ¡4 ¡ 1) = ¡4 £ 3

2x ¡ 1 = ¡12 2x ¡ 1 + 1 = ¡12 + 1 2x = ¡11 x = ¡ 11 2

fsubtracting from both sidesg fdividing both sides by 4g

b

2x = ¡6 5

c

x + 1 = ¡4 2

d

x ¡ 2 = ¡3 4

e

x¡1 =8 2

f

x+5 = ¡5 3

g

2x + 7 = ¡1 3

h

1 2 (3x

i

1 + 2x =7 7

j

1 ¡ 2x =0 2

k

1 5 (4 ¡

l

1 4 (5

EXAMPLE Solve for x

) ) )

3x) = ¡2

7 a

¡ 2x) = ¡1

The lowest common denominator is 35.

2 3x + 1 = 3 5

3x + 1 2 = 3 5 5 3 2 3x + 1 = £ 15 15 £ 31 51

a

+ 1) = ¡3

b

3x ¡ 1 2x = 5 7 3x ¡ 1 2x = 5 7 7 5 2x 3x ¡ 1 = 35 £ 35 £ 51 71

b

5(3x + 1) = 2 £ 3 ) 15x + 5 = 6 15x + 5 ¡ 5 = 6 ¡ 5 ) 15x = 1 1 x = 15

100

95

)

)

7(3x ¡ 1) = 5 £ 2x ) 21x ¡ 7 = 10x 21x ¡ 10x ¡ 7 = 10x ¡ 10x ) 11x ¡ 7 = 0 ) 11x = 7 7 ) x = 11

75

25

5

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8 Solve for x

95

a

1 4x + 1 = 3 2

b

3x + 1 2 = 5 3

c

7x ¡ 2 2 =¡ 4 5

d

2x + 1 3 = 5 4

e

3x + 1 3 = 4 5

f

3x + 2 x¡1 = 5 3

g

x+2 1¡x = 5 6

h

x+1 x = 2 7

i

2x ¡ 1 3x = 4 5

75

25

5

0

Extension:

When either the LHS and RHS of a fractional equation has more than one term we solve it by multiplying both sides of the equation by the lowest common denominator (LCD).

EXAMPLE Solve for x

8 a

2x x ¡ =5 3 2

b

2x x ¡ = 5 has LCD of 6 3 2 µ ¶ ³x´ 2x ) 6 ¡6 = 6£5 3 2

a

)

b

x 3x ¡3= 5 8 3x x ¡3= has LCD of 40 5 8 µ ¶ ³x´ 3x ¡ 40 £ 3 = 40 ) 40 5 8 )

2(2x) ¡ 3(x) = 30 )

4x ¡ 3x = 30 )

9 Solve for x x x + =2 a 2 5

x = 30

8x ¡ 120 = 15x )

¡120 = 15x ¡ 8x

)

¡120 = 7x )

x=

¡120 7

b

x 2x 5 ¡ = 2 3 6

c

3x x ¡ = 11 2 8

d

x x + =0 2 4

e

x 5x + =0 3 6

f

x 3x + = 1 12 3 4

g

2x x ¡ = ¡2 5 2

h

x 7x ¡2= 3 12

i

2x 3x ¡ =2 3 7

100

95

75

SPREADSHEET APPLICATION Use a spreadsheet program to solve the equation 3x ¡ 7 = 9 ¡ x given that the solution is an integer.

25

The spreadsheet needs to have three columns labelled x, 3x ¡ 7, and 9 ¡ x.

5

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140 100

95

75

25


In cell A1 enter x, in cell B1 enter 3x ¡ 7, in cell C1 enter 9 ¡ x, in cell A2 enter 0, in cell B2 enter =A2 ¤ 3 ¡ 7, in cell C2 enter = 9¡A2, Use the fill down command to find the values for each side of the equation. The answer is the value of x that gives the same value in each column. It can be seen that x = 4 is the solution.

5

0

x 0 1 2 3 4 5 6

3x ¡ 7 ¡7 ¡4 ¡1 2 5 8 11

9¡x 9 8 7 6 5 4 3

x = 4 gives the same value for both expressions

What to do: 1 Change the spreadsheet to solve these equations with positive integer solutions. a 5x ¡ 3 = 53 ¡ 2x b 3x + 5 = 35 ¡ 2x c 19 ¡ 2x = 7x ¡ 44 d 6x + 11 = 41 ¡ 4x e 3x ¡ 17 = 33 ¡ 7x f 9x + 15 = 79 ¡ 7x 2 Change the spreadsheet to solve these equations with negative integer solutions. a 4x ¡ 3 = 13 + 6x b 7x ¡ 3 = ¡25 + 5x c 8 ¡ 7x = 2 ¡ 8x d 3 ¡ 5x = 39 ¡ 3x 3 Explain how to modify the spreadsheet to solve equations that do not have integer solutions. 4 Solve: a 3x ¡ 7 = 6 ¡ 9x

b

5x + 23 = 2x ¡ 8

INVESTIGATION 1:

c

8 ¡ 7x = 4x + 59

GRAPHICS CALCULATORS

Most graphics calculators have an equation solver function. Find out how to use this function and use it to solve some of the equations in the previous exercises. What other types of equations can your graphics calculator be used to solve?

I

SUBSTITUTION AND EVALUATION OF FORMULAE A formula is an equation which connects two or more variables.

It is normal for one of the variables to be expressed in terms of the other(s). 100

The subject of the formula is the variable which is written in terms of the other variables. After finding the subject, it may be evaluated directly using substitution.

95

75

FORMULA SUBSTITUTION

If a formula contains two or more variables and we know the value of all but one of them, we can use the formula to find the value of the unknown variable. 25

The Method:

² ²

Write down the formula. State the values of the known variables.

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² ²

95

75

EXAMPLE

Substitute into the formula to form a one variable equation. Solve the equation for the unknown variable.

1

Find a given that u = 50, v = 110, t = 6 and v = u + at.

25

5

v = u + at 110 = 50 + a £ 6 110 = 50 + 6a 110 ¡ 50 = 50 + 6a ¡ 50 60 = 6a a = 10

0

?

fsubstituting the values into the formulag fsubtracting 50 from both sidesg

EXERCISE 4I Find a given that v = u + at and u = 20, v = 150 and t = 5. Find u given that v = u + at and a = 20, v = 135 and t = 5.

1

a b

2

a Find s given that v = 20, a = 10, u = 10 and v 2 = u2 + 2as. b Find a given that u = 5, t = 5, s = 100 and s = ut + 12 at2 . c Find n given that d = 162 and d = 180 ¡ d Find R given that a = 6, n = 24 and a =

360 . n 2Rn . n+1

3 The formula to convert temperature measurements from degrees Celsius, C, to degrees Fahrenheit, F , is F = 95 C + 32. Find C when a F = 120 degrees b F = 60 degrees c F = 212 degrees 4 The formula for finding the circumference (perimeter) C, of a circle of diameter d, is C = ¼d where ¼ is the constant with value approximately 3:14159:::::: Find a the circumference of a circle of diameter 7:8 cm b the diameter of a circle with circumference 300 cm.

d

5 Another formula for calculating the circumference C, of a circle of radius r, is C = 2¼r. Find a the circumference of a circle of radius 9:4 cm b the radius of a circle of circumference 150 metres. 6 When a car travels a distance d kilometres in time t hours, the average speed s kmph, for the d journey is given by the formula s = : Find t a the average speed of a car which travels 300 km in 2 12 hours b the distance travelled by a car in 3 14 hours if its average speed is 90 kmph c the time taken for a car to travel 865 km at an average speed of 100 kmph.



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95

75

25

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142 100

95

75

25

5

0


7 To find the area of a triangle with sides a, b and c units long we find s, its semi-perimeter using the formula p a+b+c s= , and then use A = s(s ¡ a)(s ¡ b)(s ¡ c): 2 A triangle has sides of length 4 cm, 5 cm and 6 cm. Find its semi-perimeter and hence its area.

a

c

b

8 The volume of a sphere, V , is found using the formula V = 43 ¼r3 , where r is the radius. Find the radius of a sphere with volume a 20 cm3 b 140 cm3 c 0.8 m3 T A )£ 2 360 A is the number of degrees in the angle of bend R is the radius of curvature in mm.

9 A formula for calculating the bend allowance, B, in sheet metal is B = 2¼(R + where B is the bend allowance T is the thickness in mm and Find A when a T = 3, B = 5, R = 12 b T = 15, B = 50, R = 60

m 10 The body mass index B = 2 where m = mass in kg, i.e., h weight, and h = height in metres is a measure used to determine if weight is within an acceptable range. The range is 21 < B < 25.

For people on earth, mass and weight are the same.

a Find the weight for a person height 1.65 m and B = 22. b Find the weight for a person height 1.85 m and B = 19. c Find the weight for a person height 1.45 m and B = 28.

EXAMPLE

2

The time T seconds, for a pendulum of length L metres, to swing back and forth once is given r by the formula: L T = 2¼ where g + 10 m/s2 g a Find the time for a pendulum length 3 metres to swing back and forth once. b Find the length of a pendulum with T = 5. r T = 2¼

a )

T = 2¼

q

L g

fsubstitute the values into the formulag

3 10

f2 decimal placesg

100

95

+ 3:44 seconds

75

r T = 2¼

b

r )

5 = 2¼

L g

fsubstitute the values into the formulag 25

L 10

fdivide by

2¼ g

5

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r

100

5 2¼

=

¢ 5 2 2¼

=

)

95

75

)

25

5

¡

L 10

L 10

143

fsquare both sidesg

fmultiply by

¡

10g

¢ 5 2 2¼

)

L = 10 £

)

L + 6:33 metres

0

point of support

11

The time taken for one complete p swing of a simple pendulum is given by T = 15 l where l is the length of the pendulum (in cm) and T is the time (called the period) in seconds. Find

l

pendulum

a

the time for one complete swing of the pendulum if its length is 60 cm

b

the length of a pendulum if it is to have a period of exactly one second.

p The formula D + 3:56 h gives the approximate distance (D km) to the horizon which can be seen by a person with eye level h metres above the level of the sea. Find a the distance of the horizon when a person’s eye level is 15 m above sea level b how far above sea level a person’s eye must be if the person wishes to see the horizon at a distance of 40 km.

12

h

D

Earth

EXAMPLE When a stone is dropped down a well the total distance fallen, D metres, is given by the formula D = 12 gt2 where t is the time of fall (in seconds) and g is the gravitational constant of 9:8 ms¡1 . Find a the distance fallen after 6 seconds 1 b the time (to the nearest 100 th second) taken for the stone to fall 150 metres.

D 100

95

75

D = 12 gt2

a )

D=

1 2

where g = 9:8 and t = 6:

£ 9:8 £ 62

metres

) D = 176:4 metres ) the stone has fallen a distance of 176:4 metres.

Calculator: 0:5

×

9:8

×

6 x2

25

=

5

0

H:\...\NSWGM\NSWGM_04\143NG04.CDR Thu Jan 20 12:05:10 2000


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144 100


D = 12 gt2

b

1 2

where D = 150, g = 9:8

£ 9:8 £ t2

95

)

150 =

75

)

150 = 4:9 £ t2

)

150 = t2 4:9 r

150 ) t= fas t > 0g 4:9 ) t + 5:5328:::::: ) time taken is 5:53 seconds.

25

5

0

13 D

Calculator:

( 150

÷

4:9 )

=

When a cricket ball is dropped from the top of a building the total distance fallen is given by the formula D = 12 gt2 where D is the distance in metres and t is the time taken in seconds. Find, given that g + 9:8, a b

the total distance fallen in the first 4 seconds of fall the time taken to fall 100 m.

14 A circle’s area A, is given by A = ¼r 2 where r is its radius length. Find a the area of a circle of radius 6:8 cm b the radius of a circular swimming pool which must have an area of 250 m2 . r

15 A cylinder of radius r, and height h, has volume given by V = ¼r2 h. Find a the volume of a cylindrical tin can of radius 10 cm and height 17:5 cm b the height of a cylinder of radius 5 cm given that its volume is 80 cm3 c the radius of copper wire of volume 100 cm3 and length 1000 m.

h

16 The formula for calculating the total surface area of a sphere of radius r is given by A = 4¼r2 . Find a b

the total surface area of a sphere of radius 8:3 cm the radius of a spherical balloon which is to have a surface area of 1 m2 . [Answer in cm.]

r

100


95

75

1 Write a definition in a sentence for each of these terms from this chapter: linear, substitution, formula, like terms, simplify, grouping symbols, expand, equation, variables, pronumeral.

25

2 Based on this chapter write a paragraph describing how algebra is used in mathematics.

5

3 Research the difference between the terms mass and weight.

0



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95

You should be able to: ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤

75

25

5

0

?

4

extend and interpret linear patterns make substitutions for pronumerals and evaluate expressions substitute into formula to extend number patterns add and subtract like terms simplify algebraic terms using multiplication and division expand and simplify expressions with grouping symbols solve equations solve equations resulting from substitution into formulae.

DIAGNOSTIC TEST

1 The 10th term in the sequence ¡7, ¡4, ¡1, 2, ..... is A 17 B 20 C 23

D

26

2 The 10th term in the sequence 40, 34, 28, 22, ..... is A 4 B ¡2 C ¡8

D

¡14

3 Using the rule “start at ¡6 and add 4” the 10th term would be A 28 B 30 C 36

D

40

4 The number of matches needed for the fifth diagram in this pattern is A 60 B 5

13

D

16

50 78

D

8

6 The value of s = ut + 12 at2 with u = 0, t = 5, a = 10 is A 125 B 130 C 2500

D

2505

7 Simplified 7ab ¡ 5ab ¡ 4ab equals A 6ab B ¡2ab

C

16ab

D

¡16ab

8 Simplified x2 + 2x equals B 3x3 A 2x3

C

4x

D

x2 + 2x

5 For a = 7 and b = ¡3 the value of A

64

B

a2 ¡ 5b 8

3

9 Simplified ¡8x + 7y ¡ 4x ¡ 9y equals A ¡14xy B ¡12x ¡ 2y 10 Simplified 8x £ (¡4x) £ 5 equals A 4x + 5 B ¡20x 11 Simplified A

¡2a

¡6a2 4a

C is C

100

C

¡8x ¡ 6y

D

¡x ¡ 13y

C

160x2

D

¡160x2

75

equals B

95

25

2a

C

¡

3a 2

¡3 2a

D

5

0



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100

12 Simplified 95

A 75

4a3 15 £ 2 5 2a

6a5

equals B

6a

13 The expansion of ¡3(8 ¡ 5x) is equal to A 15x ¡ 24 B ¡24 ¡ 15x 25

C

12a

D

6a2

C

¡24 ¡ 5x

D

24 + 15x

D

¡6 + x

D

6x ¡ 11

14 The expansion and simplification of 8 ¡ (x ¡ 14) is a A 22 ¡ x B 22 + x C ¡6 ¡ x

5

15 The expansion and simplification of 5(2x ¡ 3) ¡ 4(x ¡ 1) is

0

A

6x ¡ 19

B

14x ¡ 11

C

14x + 11

16 The expansion and simplification of 5x2 (7 ¡ 3x) + 4(2x2 ¡ 7) equals A

15x3 +43x2 ¡28

B

43x2 ¡15x3 ¡28

17 The solution to p ¡ 16 = 42 is A p = 26 B p = 36

C

28x3 ¡ 28

D

28x2 ¡ 28

C

p = 48

D

p = 58

C

x=

11 35

D

x=

C

x+5=8

D

5x + 2 = ¡13

C

4x = 16

D

x+5= 9

D

¡x = 18 ¡ 2

D

1 ¡ 2x = 5

18 The solution to 11x ¡ 10 = 25 is A

x = 25

B

2 x = 3 11

19 x = ¡3 is the solution to the equation A 3x ¡ 7 = 2 B 4x ¡ 3 = 15 20 x = 4 is not a solution of which equation x = 16 A x¡4= 0 B 4

21 A correct line in the solution of 6x ¡ 2 = 5x + 18 is A ¡x = 18 + 2 B x = 18 ¡ 2 C x = 18 + 2 22 x = 2 is a solution of 4x + 1 =1 A 3

B

3x + 2 =2 4

C

5x ¡ 1 = ¡1 3

15 11

1 2

23 A cone has volume V = 13 ¼r2 h. The cone with h = 10 and r = 3 has V closest to A 31 B 94 C 314 D 30 Question 1, 2, 3, 4, 5, 6 7, 8, 9 10, 11, 12 13, 14, 15, 16 17, 20 18, 19, 21 22 23 Section A B C D E F G H I

?

4A

100

95

REVIEW SET

75

1 For the number pattern 7, 10, 13, 16, ...... a write the next 4 terms c find the 50th term

b

write a rule describing the pattern 25

2 Using n = 1, 2, 3, ...... find the first 5 terms of the number sequence described by the rule T = 5n + 3.

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3 If x = 4 and y = 6 find the value of 3x ¡ 5y

95

4 Copy and complete this table

25

5

0

¡2

x 3x ¡ 7

75

5 For the number pattern 90, 85, 80, 75, ...... a write the next 3 terms

¡1

0

1

2

5

find the 40th term

b

6 The volume of a sphere, V , is found using the formula V = 43 ¼r3 , where r is the radius. Find the volume of a sphere with radius a 7 cm b 3:5 cm c 0:6 metres 7 Simplify a 2p + 3p

b

8m ¡ m

c f

4a + 5b ¡ 8a a a £ 3 5

d

2a + 3b + 4a ¡ 8b

e

(¡4x)2

g

6L3 2L2

h

a3 6 £ 2 12 a

b

¡3(x + 5)

c

¡(5 ¡ x)

e

8(2x ¡ 5) ¡ 4x

f

6(p + 5)¡3(2p + 7)

8 Expand and simplify if possible a 3(x ¡ 9) d g

5 ¡ (2x + 8) 2

5x (7 ¡ 2x) + 6x(2x ¡5)

2

h

2x (3x + 7) + x(4 ¡ 2x)

9 Solve the following equations a

d ¡ 8 = 36

b

x + 9 = 20

c

d

11x = 55

e

4x ¡ 7 = 39

f

x = 72 8 3x ¡ 12 = 60

g

10 + 8n = 72

h

12 ¡ 6c = 85

i

3(2x + 1) = 4

j

3x ¡ 2 = x + 6

k

2(3 ¡ x) = 3(x + 5)

l

4 ¡ 3(4 ¡ x) = 8

m

3(x ¡2) + 5(x + 1) = 15

n

1 (3x + 1) = 4 3

o

(3x ¡ 1) 4x = 3 5

10 The formula to convert temperature measurements from degrees Celsius, C, to degrees Fahrenheit, F , is F = 95 C + 32. Find C when a

F = 200 degrees

b

F = 20 degrees

c

F = 55 degrees 100

11

a Find u given that v = 150, a = 10, t = 8 and v = u + at:

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b Find a given that u = 12, t = 5, s = 140 and s = ut + 12 at2 . c Find n given that d = 170 and d = 180 ¡ d Find R given that a = 8, n = 36 and a =

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360 . n

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2Rn . n+1

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4B

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REVIEW SET

1 For the number pattern 2, 4, 6, 8, ...... a write the next 4 terms c find the 30th term

write a rule describing the pattern

b

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2 Using n = 1, 2, 3, ...... find the first 5 terms of the number sequence described by the rule T = 3n ¡ 5:

0

3 If x = ¡2 and y = 3 find the value of 5x ¡ 7y. 4 Copy and complete this table

¡2

x 9 ¡ 5x


¡1

b

0

1

2

5

find the 50th term

6 The surface area of a sphere, A, is found using the formula A = 4¼r2 , where r is the radius. Find the surface area of a sphere with radius a 4 cm b 7:5 cm c 0.8 metres 7 Simplify a 9p + 7p

b

22m ¡ m

c f

5a + 3b ¡ 8a a a £ 4 3

d

7a + 9b + 5a ¡ 8b

e

(¡7x)2

g

9L3 6L2

h

5 a3 £ 15 a2

b e h

¡5(x + 5) c 5(2x ¡ 5) ¡ 7x f 2 3x (3x + 4) + 3x(9 ¡ 2x)

¡(8 ¡ x) 3(p + 5)¡8(2p + 5)

x =5 3 4x ¡ 12 = 15 7(2x + 5) = 4 12 ¡ 3(5 ¡ x) = 9

8 Expand and simplify if possible a 7(x ¡ 9) d 9 ¡ (2x + 4) g 7x2 (7 ¡ 5x) + 4x(2x ¡ 5) 9 Solve the following equations a

d ¡ 9 = 23

b

x + 4 = 65

c

d g j

11x = 121 10 + 3n = 32 4x ¡ 2 = x + 9

e h k

3x ¡ 7 = 30 12 ¡ 9c = 43 3(3 ¡ 7x) = 4(x + 5)

f i l

m

4(x ¡ 2) + 2(x + 1) = 33

n

(3x + 1) 1 = 5 2

o

(5x ¡ 1) 7x = 3 5

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10 The formula to convert temperature measurements from degrees Celsius, C, to degrees Fahrenheit, F , is F = 95 C + 32. Find C when a F = 120 degrees b F = 10 degrees c F = 0 degrees 11

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a Find u given that v = 120, a = 10, t = 7 and v = u + at. b Find a given that u = 15, t = 5, s = 130 and s = ut + 12 at2 .

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c Find n given that d = 120 and d = 180 ¡ 360 n

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d Find R given that a = 11, n = 30 and a = 2Rn n+1

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4C

REVIEW SET

1 For the number pattern ¡15, ¡11, ¡7, ¡3, ...... a write the next 4 terms b c find the 50th term


2 Using n = 1, 2, 3, ...... find the first 5 terms of the number sequence described by the rule T = 20 ¡ 4n: 3 If x = ¡5 and y = 8 find the value of 4x2 ¡ 6y: 4 Copy and complete this table

x 6x2 ¡ 5

¡2


b

¡1

0

1

2

5

find the 10th term

6 The surface area of a cylinder, A, is found using the formula A = 2¼r(r + h) where r is the radius and h is the height. Find the surface area of a cylinder with radius 15 cm and height 10 cm. 7 Simplify a 12p + 7p

b

(¡11x)2

f

e

22m ¡ 7m a2 a £ 4 7

8 Expand and simplify if possible a 7(2x ¡ 5) d 13 ¡ (3x + 4) g 9x2 (3 ¡ 4x) + 8x(3x ¡ 5)

c g

6a + 7b ¡ 11a 24L3 6L2

d h

7a + 12b + 6a¡8b 12 a3 £ 2 15 a

b e h

¡5(3x + 5) c 6(5x ¡ 5) ¡ 12x f 2 6x (3x+5)+ 3x(11¡ 2x)

¡(12 ¡ x) 4(p + 5)¡ 8(3p + 5)

x =2 3 7x ¡ 12 = 19 2(4x + 5) = 12 18¡ 5(6¡ x) = 22 (8x ¡ 1) 7x = 4 3


d ¡ 9 = 11

b

x + 4 = 14

c

d g j

11x = 77 14 + 3n = 23 6x ¡ 4 = 3x + 8

e h k

f i l

m

2(x¡6) + 2(4x + 1) = 24

n

3x ¡ 8 = 12 23 ¡ 4c = 11 4(3 ¡ 7x) = 4(3x + 5) (4x + 1) 1 = 3 7

o

10 The time T seconds, for a pendulum of length L metres, to swing back and forth once is r L give by the formula: T = 2¼ where g + 10 m/s2 . g a Find the time for a pendulum length 2.5 metres to swing back and forth once. b Find the length of a pendulum with T = 8.

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a Find u given that v = 50, a = 10, t = 3 and v = u + at. b Find a given that u = ¡5, t = 5, s = 50 and s = ut + 12 at2 . 360 c Find n given that d = 150 and d = 180 ¡ . n 2Rn : d Find R given that a = 2:5, n = 12 and a = n+1

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4D

REVIEW SET

1 For the number pattern 5, ¡1, ¡7, ¡13, ...... a write the next 4 terms b c find the 120th term


2 Using n = 1, 2, 3, ... find the first 5 terms of the number sequence described by the rule T = 15 ¡ 3n. 3 If x = 12 and y = ¡4 find the value of 6x2 ¡ 11y: 4 Copy and complete this table

x 4x2 ¡ 1

¡2

5 For the number pattern 13 , 1, 3, 9, ...... a write the next 3 terms

b

¡1

0

1

2

5

find the 10th term

6 The volume of a cylinder, V , is found using the formula V = ¼r2 h where r is the radius and h is the height. Find the volume of a cylinder with radius 12 cm and height 25 cm. 7 Simplify a ¡5p + 7p

b

3m ¡ 12m

c f

2a + 7b ¡ 15a a3 a2 £ 3 7

d

8a + 16b + 3a ¡ 9b

e

(¡3x)2

g

30L3 6L2

h

24 a3 £ 2 18 a

b

¡8(4x + 7)

c

¡(11 ¡ 4x)

f

6(p + 2)¡ 4(3p + 9)

8 Expand and simplify if possible a 4(2x ¡ 9) d

21 ¡ (5x + 8)

e

4(2x ¡ 8) ¡ 15x

g

9x2 (3¡5x) + 5x(4x¡ 5)

h

¡7x2 (3x + 3) + 5x(14¡6x) 100


d ¡ 5 = 11

b

x + 7 = 14

c

d g j

7x = 77 11 + 7n = 53 5x ¡ 3 = 3x + 1

e h k

3x ¡ 4 = 92 27 ¡ 9c = 31 2(3 ¡ 4x) = 6(3x + 2)

f i l

n

5 (4x + 3) = 5 7

m 2(x¡4)¡2(4x+1) =¡8

o

x =2 6 8x ¡ 12 = 11 4(3x + 5) = 13 2 ¡ 5(7 ¡ x) = 11

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(7x ¡ 5) 2x = 2 7

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10 A formula for calculating the bend allowance, B , in sheet metal is A where B is the bend allowance, T is the thickness in mm, B = 2¼(R + T2 ) £ 360 A is the number of degrees in the angle of bend, and R is the radius of curvature in mm. Find A when T = 1.5, B = 9.4, R = 4.

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11

a Find u given that v = 100, a = ¡5, t = 10 and v = u + at. b Find a given that u = ¡24, t = 15, s = 110 and s = ut + 12 at2 .

5

360 : n 2Rn . d Find R given that a = 3:9, n = 18 and a = n+1 c Find n given that d = 144 and d = 180 ¡

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CHAPTER

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Data collection and sampling AREA OF STUDY

This chapter deals with the planning of data collection and the sampling process. The main mathematical ideas investigated in this chapter are: 8 the different ways of presenting questions 8 the classification of data as categorical or quantitative 8 identification of the target population 8 bias in sampling 8 random numbers 8 the random sample 8 the capture-recapture technique 8 the stratified random sample 8 the systematic sample 8 the suitability of sample types for a given situation 8 the effect of sample size in estimating the nature of population.

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DATA COLLECTION AND SAMPLING (Chapter 5)

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INTRODUCTION

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Statistics refers to the science of collecting facts and information. The information is organised into a form which can easily be analysed and from which conclusions and predictions can be accurately made.

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The facts or pieces of information collected are called data. Data may be collected by counting, measuring and asking questions.

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Statistics are used by governments and businesses to make informed decisions on the provision of services such as health, transport and commerce. Statistics are used by manufacturers for quality control and in medical research to test new drugs. Market research surveys are used to gauge consumer preferences of goods and services. Information collected over many years is used to make weather forecasts. The process of statistical inquiry includes the following steps: ² ² ² ² ² ²

posing questions collecting data organising data summarising and displaying data analysing data and drawing conclusions writing a report.

In this chapter we will investigate the first two of these steps.

A

QUESTIONING

Questioning can be by interview, either in person or by phone, or by questionnaire. The usefulness of the responses to the questions asked will depend on the relevance, clarity and structure of the questions. Questions should use simple language, be unambiguous so that there can be no doubt about their meaning, and they should be free from bias, that is, free from unfair influence. The main types of questions used in questionnaires are: ²

Free-response or open-ended questions For example, “What TV programs do you like to watch?” The person answers the question in his/her own way.

²

Yes or No questions For example, “Do you like to watch the news?” The person answers yes or no to the question.

²

True or False questions These are similar to yes or no questions.

²

Tick the box type questions For example, “Which of the following do you like to watch?” News, Sport, Drama, Comedy, Soapies

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DATA COLLECTION AND SAMPLING (Chapter 5) 100

²

Response Scales For example, “I like to watch the news.” Circle a number to indicate your opinion of the above statement: 1 Strongly disagree 2 Disagree 3 Don’t know 4 Agree 5 Strongly agree

²

Continuum Scale For example “I like to watch the news.” Place a cross on the line to indicate your opinion of this statement:

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EXAMPLE

strongly agree

1

Write a question, of each of the types above, which could be used to investigate your school uniform. Discuss the advantages/disadvantages of each. Free-Response: “What do you like/dislike about the present school uniform?” This question would possibly collect a lot of information but because there would be many different responses it might be difficult to organise and interpret the responses. Yes or No: “Would you like to change the present school uniform?” The responses to this question will indicate whether there is a desire to change the uniform but will not indicate what change, if any, is wanted (this of course could be done by further questions). True or False: “The present school uniform is satisfactory.” Comment is as for the Yes and No questions. Tick a box: “Which of the following items of school uniform would you like to be changed?” ¤ Boys ¤ Shirt ¤ Trousers ¤ Shoes ¤ Pullover ¤ Girls ¤ Blouse ¤ Skirt ¤ Shoes ¤ Pullover The responses to this question will give information about what particular changes are wanted. Response Scales: “Circle the response which most closely agrees with your opinion of the present school uniform.” 1 Strongly like 2 Like 3 No opinion 4 Dislike 5 Strongly dislike The responses to this question will give a clearer indication of the depth of feeling concerning a change but will not indicate what change, if any, is wanted. Note: When using response scales some thought should be given to the number of choices. An even number of choices may force an opinion one way or another, in particular circumstances, but sometimes it may be desirable to allow a neutral choice by having an odd number of choices.

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When writing questionaires, the questions should be simple and straightforward so that there is only one interpretation possible. They should show respect for the privacy of individuals, should not involve any calculations and should avoid any bias. For example, “Do you agree that ....?”. In general, the simpler the question and the more precise the response required, the more accurate the data will be. It is a good idea to trial your questions on a few people to check that they are achieving the purpose for which you are designing them.

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EXERCISE 5A 1 List some of the advantages and disadvantages of a personal interviews b phone interviews c questionaires for collecting information from people. You could consider time, cost, the response rate (the number of people prepared to respond to the questions), the truthfulness of answers etc. d Discuss your answers with the class. 2

i ii

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Write a short questionnaire, using each of the different types of questions discussed above, for each of the following investigations. Trial your questions on a small group and if necessary modify them.

a What change, if any, would students like to your school uniform? (Consider how many students want a change, boys/girls, juniors/seniors, what items, ....?) b What are the most popular radio stations and why? (Consider types of programs, types of music, boys/girls, when they listen, how often, ....?) c Are students satisfied with the food available at the school canteen? (Consider who uses the canteen, when, how often, how much is spent, types of food, ....?) d What are the smoking and/or drinking habits of the senior students at your school? (Consider how many students of each gender, when, how often, the cost, ....?)

RESEARCH PROJECT Have you ever taken part in market research? Explain what market research is and how market research companies use statistics. Write a short report.

B

CLASSIFICATION OF DATA

When collecting data, we are interested in a particular property or characteristic of a group of people or objects. The particular characteristic that we are interested in is called the variable. A categorical variable is one which describes a characteristic. It can be divided into categories. For example, colour, quality, gender are categorical variables. A quantitative variable is one which has a numerical value. Further, quantitative variables can be either discrete or continuous.

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A discrete variable takes exact numerical values. It is often the result of counting.

75

For example, the number of children in a family, marks in a test, shoe size. A continuous variable takes numerical values within a certain range. It is usually the result of measuring. For example, if the weight of a student is given as 48 kg the exact weight could be anywhere between 47:5 and 48:5 kg. Weight is a continuous variable.



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EXAMPLE

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Classify the following variables as categorical, quantitative discrete or quantitative continuous. a the number of heads when 3 coins are tossed b the brand of toothpaste used by the students in a class c the heights of a group of 15 year old children

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a The values of the varaibles are obtained by counting the number of heads. The result can only be one of the exact values 0, 1, 2 or 3. It is quantitative discrete data. b The variable describes the brands of toothpaste. It is categorical data. c This is numerical data obtained by measuring. The results can take any value between certain limits determined by the degree of accuracy of the measuring device. It is quantitative continuous data.

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EXERCISE 5B 1 For each of the following investigations, classify the variable as categorical, discrete or continuous: a the number of goals scored each week by a netball team b the heights of the members of a football team c the most popular radio station d the number of children in an Australian family e the number of loaves of bread bought each week by a family f the pets owned by a class of students g the quality rating of different brands of washing machines h the number of leaves on the stems of plants i the amount of sunshine in a day j the number of people who die from cancer each year k the amount of rainfall in each month of the year l the countries of origin of immigrants m the most popular colours of cars n the number of pets owned by the students of a class o the gender of school principals p the time spent doing homework q the sports played by students in high schools r the stopping distances of cars s the number of cars passing through an intersection t the marks scored in a class test u the items sold at the school canteen v the number of matches in a box w the pulse rates of a group of athletes x the reasons people use taxis y the fuel consumption of different cars 2 Give 3 of your own examples of a categorical data b discrete data Discuss your answers with the class.

c

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continuous data

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C:\...\NSWGM\NSWGM_05\157NG05.CDR Fri Jan 21 09:19:39 2000


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RESEARCH PROJECT

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1 What statistics does the World Health Organisation (WHO) collect and how is it used? (The Internet address is www.who.int) 2 The United Nations collects statistics. Research what data is collected and how it is used. (The Internet address is www.un.org)

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IDENTIFYING POPULATIONS BEING INVESTIGATED

One of the first decisions to be made when collecting data is from whom, or what, the information is to be collected. There are two ways in which this can be done. These are: a census or a sample. A census is a method which involves collecting data about every individual in the whole population. The individuals may be people or objects. A census is detailed and accurate but is expensive, time consuming and often impractical. A sample is a method which involves collecting data about a part of the population only. It is cheaper and quicker than a census but is not as detailed or as accurate. Conclusions drawn from samples always involve some error.

EXAMPLE

1

Would a census or sample be used to investigate a the length of time an electric light globe will last b the causes of car accidents in NSW c the number of people who use Britex toothpaste? a Sample. It is obviously impractical to test every light globe produced - there would be none left for sale! b Census. An accurate analysis of all accidents would be required. c Sample. It would be very time consuming to interview the whole population to find out who uses, or does not use, Britex toothpaste.

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EXERCISE 5C

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1 State whether a census or a sample would be used for each of the investigations in question 1 of Exercise 5B. Discuss any problems with the class. 2 Give 3 examples of data which would be collected by using a a census b sample

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RESEARCH PROJECT

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The Australian Bureau of Statistics (ABS) is responsible for the Australian census. 1 How often is the Australian census taken? What information is collected in this census? 2 What information is made available? 3 How is the information used by governments and businesses? 4 Research the role and information collected by the ABS. (The ABS produces Yearbooks and many other free publications. They can be found on the Internet at www.abs.gov.au)

D

BIAS IN SAMPLING

The most common way of collecting information is by using a sample. The purpose of a sample is to provide an estimate of a particular characteristic of the whole population. Therefore the challenge in selecting a sample is to make it as free from prejudice as possible and large enough to be representative of the whole population. A biased sample is one in which the data has been unfairly influenced by the collection process and is not truly representative of the whole population.

EXAMPLE

1

Suggest the possible bias in each of the following samples: a a phone survey during the day b a survey of people on a train station c a survey of a football crowd d 10 people are tested with a new drug developed to cure the common cold a The sample would be biased towards people who are at home during the day, i.e., it does not include people who go to work. b The sample would be biased towards people who catch the train, i.e., it does not include people who use other forms of transport or work at home. c The sample would be biased towards people who attend football matches. For example, there would probably be more males than females at football matches. d The sample is not big enough to make a valid conclusion.

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EXERCISE 5D

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1 Explain and discuss any possible bias in the following samples: a a phone survey on a Saturday night b a survey of the people at a bus stop c a survey of the people in a supermarket carpark d a survey of people at the beach e a survey of people in your street

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f businesses selected from the Yellow Pages phone directory g people selected from the electoral roll 2 Comment on any possible bias in the following situations: a Year 7 students are interviewed about changes to the school uniform. b Motorists stopped in peak hour are interviewed about traffic problems. c Real estate agents are interviewed about the prices of houses. d Politicians are interviewed about the state of the country’s economy. e People are asked to phone in to register their vote on an issue. f An opinion poll is conducted by posting a questionnaire to people. g A manufacturing company tests a sample of its products every Monday morning. h A survey of 20 people indicated that 80% of people watch the Channel 9 News. i A company claims that 4 out of 5 dentists recommend their brand of toothbrush. j An advertisement claims that “Dog breeders recommend Buddy dog food.”

EXAMPLE

2

Sometimes people use biased samples to enhance their claims for their products or to support a particular point of view. For example, if you wanted the local council to upgrade its swimming pools, what sample of people would you choose to survey? A sample of people at the swimming pool. You would expect the people who use the pool to be biased very favourably towards this proposal.

3 Describe the sample you would choose if you wanted to gather support for the following: a Stop smoking in public places. b Improve the local bus survice. c Increase the number of books in the public library. d Improve the facilities in the school’s senior common room. e Stop logging in National Parks. f Increase unemployment benefits. g Reduce tariffs on farm products. h Reduce bank fees. 4 Describe samples with the opposite bias to those in question 3.

EXAMPLE

3

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Sometimes we want a sample which is biased towards a certain group. For example, what would be a suitable sample to survey if we wanted to:

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a investigate the most popular movies b collect information about the cleanliness of trains?

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a People who go to the movies, such as the people in the foyer of a movie theatre. b People who travel on trains.

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5 What would be a suitable sample to survey if we wanted to. a investigate the suitability of bus timetables on a certain route b collect information about the effects of road tax on company profits c gather data about the effects on a town of closing its only bank d investigate the success of a new treatment for ticks and fleas on dogs e find the most popular brand of dishwasher f investigate bank charges?

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RESEARCH PROJECT

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Investigate the aims and purpose of “Choice” magazine. How does this publication use statistics? Give some examples. How do they avoid bias in their results? (Choice magazine is probably available in your shool library or can be found at www.choice.com.au on the Internet.)

E

TABLES OF RANDOM NUMBERS

Much of the work involving sampling depends on the concept of a random number. A random number is one which is chosen in such a way that all possible numbers have an equal chance of being chosen. One way to do this involves writing all the possible numbers on pieces of paper and drawing one piece from a hat. For example, if we wanted a random (whole) number less than 21 then we could write the numbers 1 to 20 on pieces of paper, place them in a hat, mix them and then choose one. If the number of possibilities is large then it is more convenient to use lists of random numbers generated from prepared tables, computers or calculators. Below is an excerpt from a table of random numbers: 48047 08695 90070 10132 58547 85566 40942 80553 27328 64584

EXAMPLE

45381 58112 98873 27359 01331 81574 42373 58331 85758 20776

33232 96070 89846 13017 62538 71965 38710 62724 45342 86792

35178 91910 50953 41045 79181 20977 39916 74004 98884 42340

46971 18868 92529 13817 33071 48005 08187 09344 36034 83522

85879 52251 68249 65603 63766 83418 00133 91315 79836 62139

31458 99827 54949 87615 73613 58738 16288 25791 94902 14038

22016 32581 83829 55691 24470 98771 64277 40296 80442 88433

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Use the table above to select a random sample of five a 1 digit numbers b 2 digit numbers c 3 digit numbers

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i Start at any digit in the table, say the 7 in row 5 and column 16, as shown. ii Now select the next five numbers to the right, left, down or up. Moving to the right generates the random numbers 7, 9, 1, 8, 1, 3 to the left generates the random numbers 7, 8, 3, 5, 2 down the column generates the random numbers 7, 2, 3, 7, 9, 4 up the column generates the random numbers 7, 4, 5, 9, 3 Any of these sets of numbers would be satisfactory.

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The crossed off numbers have already been selected.

b

i Start at any digit, say the 6 in row 5 and column 11. ii Select pairs of digits by moving to the right, left, down or up. Moving to the right generates 62, 53, 87, 91, 81 to the left generates 62, 31, 13, 70, 54 down generates 62, 71, 38, 62, 45, 86 up generates 62, 13, 89, 96, 33

c

i Start at any digit, say the 8 in row 6 and column 22. ii Select groups of 3 digits by moving to the right, left, down or up. Moving to the right generates 800, 583, 418, 587, 389 to the left generates 800, 774, 209, 965, 471 down generates 800, 818, 934, 603, 352 up generates 800, 307, 381, 252, 886

Notes:

1 If you run out of digits in a row (or column), continue by moving down (or up) to the next row, or across (left or right) to the next column. 2 If, for example, a 3 digit number such as 005 is generated, it is read as the number 5.

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EXERCISE 5E 1 By starting in the table given with the 2 in row 6 and column 16, select 5 random one digit numbers by moving a to the right b to the left c down the column d up the column 2 Use the method in question 1 to select 5 random two digit numbers. 3 Use the method in question 1 to select 5 random three digit numbers. 4 Using the table of random numbers given, select 20 one digit random numbers. Check the randomness of these numbers by a counting the numbers of odd and even numbers. Are they about the same? b How many of the numbers are smaller than 5, 5 or bigger? c Combine your results with the rest of the class and repeat a and b. 5 Choose 20 two digit numbers from the table of random numbers given. Check the randomness of these numbers by a counting the number of odd and even numbers. Are they about the same? b How many of the numbers are smaller than 50, 50 or bigger? c Combine your results with the rest of the class and repeat a and b. d Discuss your results.

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6 Choose 20 three digit numbers from the table of random numbers given. Check the randomness of these numbers by a counting the number of odd and even numbers. Are they about the same? b How many of the numbers are smaller than 500, 500 or bigger? c Combine your results with the rest of the class and repeat a and b. d Discuss your results.

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EXAMPLE

2

Use the table to find 10 random numbers which are less than 60. Choose 2 digit numbers, as shown in Example 1, rejecting any that are bigger than 60. Suppose we start at the 2 in row 3, column 24 and move across the row to the right. The first number is 29, the next is 68 but this is bigger than 60 so we reject it, the next number is 24 etc. Our final list will be 29, 24, 49, 1, 32, 27, 35, 30, 17, 41.

7 Starting at the 2 in row 4, column 6 and moving down the column, find 10 random numbers which are less than a 80 b 35 8 Starting at the 1 in row 6, column 12 and moving down the column, find 10 random numbers which are less than a 700 b 280 (Use 3 digit random numbers.)

F

CALCULATOR GENERATION OF RANDOM NUMBERS

Most modern scientific calculators have a random number generator. On Casio calculators the function key is labelled “ RAN¡# ” and it generates random numbers from 0:000 up to 0:999. On a Casio Calculator, press SHIFT RAN¡# . A number such as 0:364 will appear.

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To obtain a one digit random number, ignore the decimal point and the first two digits. This gives the random number 4.

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To obtain a two digit random number, ignore the decimal point and the first digit. This gives the random number 64. To obtain a three digit random number, ignore the decimal point. This gives the random number 364 .

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Check the instructions in your calculator manual for obtaining random numbers.

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EXERCISE 5F

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Repeat questions 4, 5, 6, 7, 8 of Exercise 4E using a calculator instead of the table to generate the random numbers.

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GENERATING RANDOM NUMBERS ON A COMPUTER

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Random numbers can be generated using a spreadsheet. The following instructions apply to Microsoft EXCEL. Format cells to 0 decimal places ² Type = RAND ( ) * 100 Select Fill down (or Fill across) This produces a list, or row, of 2 digit random numbers. ²

To produce a list of 3 digit random numbers Type = RAND ( ) * 1000 Select Fill down

²

To produce a list of random numbers from a to b Type = RAND ( ) * (b ¡ a) + a Select Fill down

An example of this is shown below:

A B C D E F G 1 RANDOM NUMBERS between any two numbers 2 2 digit 3 digit start = 0 finish = 100 3 4 40 646 73 5 78 685 14 6 96 418 66 7 10 197 81 8 98 254 50 9 82 783 35 10 03 876 31 11 97 489 58

The formula in cell A4 is = RAND ( ) * 100. The formula in cell B4 is = RAND ( ) * 1000. The formula in cell E4 is = RAND ( ) * ($G$3¡ $D$3) + $D$3. They are then filled down. Try changing the values in D3 and G3.

EXERCISE 5G Use a spreadsheet to answer the questions following. 1 Produce a list of a 20 random 2 digit numbers c 20 random 1 digit numbers.

b

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20 random 3 digit numbers

75

2 Using Fill down and Fill across make rows and tables of random numbers. 3 Produce a list of random numbers a from 0 to 50 b d from 50 to 100 e


25

10 to 50 less than 60 (i.e., 0 to 60)


c f

0 to 100 less than 250

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H

THE RANDOM SAMPLE

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In a random sample, each member of the population has an equal chance of being selected. In choosing random samples it is usual to assign numbers to each member of the population and then numbers are selected at random. The members of the population whose numbers are selected form the sample. If the population is small then the random numbers could be chosen by, for example, drawing them from a hat. However, if the population is large then this method is not practical. For these cases we use random numbers generated by tables, calculators or computers.

EXAMPLE

1

a Explain how a random sample of ten students could be chosen from a class of 30. b Which students would be selected if we use the following list of random numbers? 59147 79223 43889 43702 15558 75042 59875 53705 32924 26711 34092 18732 13930 67314 23133 05916 03882 29477 a Assign the numbers 1 to 30 to each member of the class. Write the numbers on cards, shuffle, spread out and then select any 10 cards. The students whose numbers are selected form the sample. or 10 random numbers not bigger than 30 could be selected from a table or by using a calculator. b Choosing 2 digit numbers less than 31 from the list given (starting at 59 and moving to the right), the students whose numbers are 14, 23, 2, 15, 5, 11, 9, 21, 13, 6 would be selected.

EXAMPLE

2

100

95

Explain how a random sample of 80 students could be selected from a school of 950 students. Assign each of the numbers 1 to 950 to the students of the school. In this case it would be very time consuming to write numbers on cards, etc., so it would be simpler to select 80 random numbers from 1 to 950 using a table or calculator. The students whose numbers are selected form the sample.

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EXERCISE 5H

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1 A factory superviser wishes to interview a sample of workers concerning the use of safety equipment. a Explain two methods that she could use to choose a random sample of five workers from the 25 in her factory. b Which workers would be selected if she used the following list of random numbers to do this? 49246 36704 13499 48926 61279 36535 32337 15455 26345 95085 05701 30998 95907 47062 44630 49743 60358 26022 2 A supermarket manager wants a sample of his shop assistants to complete a questionnaire on rosters. a Explain how ten shop assistants could be selected at random from the 70 employed by the supermarket. b Which shop assistants would be selected if he used the list of random numbers given in question 1b to do this? 3 Explain how a random sample of 100 students could be selected from a school of 760.

EXAMPLE

3

Use the list of random numbers from question 1 to select, at random, 6 squares from the grid drawn below.

5 4 3 2 1 1

2

3

4

5

1

2

3

4

5

5

Number the columns and rows as shown. Select six ordered pairs from the list of random numbers in Exercise 5H, question 1. If we start at the first pair of digits and move to the right, suitable ordered pairs would be (2, 4), (1, 3), (3, 5), (3, 2), (3, 3), (5, 4). These squares are shaded alongside:

4 3 2 1

100

4 a

b

Use the list of random numbers given in 1 to select, at random, ten squares from the grid drawn alongside. Copy the diagram and shade the squares selected.

8 7 6 5 4 3 2 1

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5

30 students are seated in a classroom in 5 rows of 6 desks as shown in the diagram given.

6

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4

a

3

80553 58331 62724 74004 09344 91315 25791 40296 27328 85758 45342 98884 36034 79836 94902 80442

2

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1

5

Select six students (desks) at random from the room by choosing random ordered pairs from the following list of random numbers.

1

2

3

4

5

b

Copy the diagram and shade the desks selected.

0

6 Repeat question 5 if the desks are numbered as shown:

5 4 3 2 1 0 0

7 A gardener spreads flower seeds over a large plot of ground. To estimate the number of seeds which germinate he divides the plot of land into eighty 1 m by 1 m squares as shown in the diagram alongside. He selects ten squares at random and counts the number of seedlings in each square. By averaging the number of seedlings in each square he can estimate the total number of seedlings in the garden.

1

2

4

3

7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9

a Use the list of random numbers in question 5 to select, at random, ten squares. b If the number of seedlings in each square selected is 21, 17, 18, 13, 27, 9, 22, 10, 18, 25, calculate i the average number of seedlings per square ii an estimate of the total number of seedlings in the plot. 8

a Divide the photograph of Candychocs into a grid of squares. b Select five squares at random. c Count the number of Candychocs in each square. d Calculate the average number of Candychocs in each square. e Estimate the total number of Candychocs in the

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INVESTIGATION 1:

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4 5 6 7

1 Assign a number to each student in your class. 2 Write your age, in years and months, next to your number on the board. 3 Choose a random sample of five students from your class, using random numbers. Calculate the average age of the five students in your sample. Compare the average of your sample with the averages of the other students ’ averages. Calculate the average age of the class using the information on the board. Compare the sample averages with the actual class average. Discuss and comment on any findings.

RESEARCH PROJECT Find out how geographers use grid sampling to determine percentages of land types.

I

THE “CAPTURE–RECAPTURE” METHOD

The “capture-recapture” technique is often used by biologists to collect data about the size of various animal populations A random sample is captured, “tagged” and then released. Some time later another random sample is captured and the number of “tagged” animals in this sample is used to predict the total population, based on the assumption that the proportion of tagged animals is the same in the total population as it is in the sample.

EXAMPLE

1

A biologist catches a random sample of 120 fish from a lake. Each one is tagged and released back into the lake. One month later another random sample of 80 fish is caught and it is found that 6 of them have tags. Estimate how many fish there are in the lake.

If we let T represent the total number of fish in the lake, then the fraction of tagged fish in 120 6 the whole lake is . The fraction of tagged fish in the sample = . T 80 We assume that the fraction of tagged fish in the sample is the same as the fraction of tagged fish in the whole lake. 120 6 Hence = T 80

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80 T = 120 6

i.e.,


100

80 6

)

T =

)

T = 1600


25

£ 120 i.e., there are approximately 1600 fish in the lake.

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100

or

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Population Sample

75

Size T 80

No. Tagged 120 6

Using proportion, T : 120 = 80 : 6 T 80 i.e., = 120 6

25

80 6

)

T =

)

T = 1600

£ 120

This method depends for its accuracy on both the first and second samples being random (i.e., all fish have an equal chance of being caught) and that, when released, the tagged fish mix thoroughly with all the others.

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EXERCISE 5I 1 A biologist catches 100 fish in a dam, tags them and releases them back into the dam. One week later he catches 90 fish of which 12 have tags. Estimate how many fish there are in the dam. 2 A Park Ranger catches and tags 50 rabbits and then releases them. A few days later he catches another 50 rabbits in the same area and finds that 4 of them are tagged. Estimate how many rabbits there are in this area. 3 A scientist at a trout farm takes 200 trout from a pond, tags them and releases them back into the pond. A week later he catches 100 trout and finds that 5 of them are tagged. Estimate the number of trout in the pond. 4 A researcher investigating the kangaroo population in an area catches and tags 40 kangaroos before releasing them back into the same area. A week later he catches another 40 kangaroos and finds that 8 of them are tagged. Determine the number of kangaroos in this area.

PRACTICAL ACTIVITIES You can simulate the capture/recapture technique as follows: Place a large (unknown) number of toothpicks in a bowl. Take out a small number (say 10) from the bowl and mark (tag) each of these with a texta pen. Put the marked (tagged) toothpicks pack in the bowl and mix them all thoroughly. Now take a handful of the toothpicks from the bowl and determine what fraction of this sample are marked. Use this fraction to estimate the number of toothpicks in the bowl. a Repeat the simulation by marking 20 toothpicks. b Check your estimates by actually counting the number of toothpicks in the bowl. 100

J

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STRATIFIED RANDOM SAMPLE

75

When we select a random sample of 10 students from a mixed class of 30 then it is possible that we would get, for example, 10 boys. If it is important for our purpose that we have a proportional mix of boys and girls then simple random sampling may not produce a satisfactory sample. When we select a random sample of 80 students from a school of 950 then it is possible that we would get 80 students from the same Year group. If it is important that we have a proportional

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mix of students from each Year group, then again simple random sampling may not produce a satisfactory sample. In these cases we can use a stratified random sample.

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In this method the population is divided up into subgroups (strata), based on different characteristics such as age, gender, income etc. The stratified random sample is then made up of random samples from each subgroup. The size of each random sample is proportional to the relative size of the subgroup.

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EXAMPLE

0

1

Of the 130 students in Year 7, 70 are boys and 60 are girls. If we were to select a stratified random sample on gender, what proportion of our sample should be boys and what proportion should be girls?

The number of students to be selected from each subgroup is proportional to the relative size of each subgroup.

?

70 out of the 130 students are boys, hence made up of boys.

70 130

or

7 13

or 53:8% of the sample should be

60 out of the 130 students are girls, hence made up of girls.

60 130

or

6 13

or 46:2% of the sample should be

EXERCISE 5J 1 Calculate the proportion of boys and girls that should be chosen in a stratified random sample, based on gender, of a 140 students consisting of 75 boys and 65 girls b 180 students consisting of 76 boys and 104 girls c 870 students consisting of 420 boys and 450 girls. 2 The numbers in each year at a school are 7 122

Year Number

8 131

9 135

10 120

11 94

12 81

Total 683

In a stratified random sample, based on year groups, what proportion of students should be chosen from each year? 100

3 Repeat question 2 for the following school numbers: 95

a

b

Year Number

7 156

Year Number

7 86

8 184

9 196

10 199

9 98

10 78

8 88

11 175 11 61

12 162 12 48

Total 1072

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EXAMPLE

2

A school of 950 students is made up as follows: Year Number of students

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7 130

8 165

9 180

10 190

11 150

12 135

If a stratified random sample is to be taken based on these subgroups (school years) a what proportion of students should be chosen from each year? b How many students should be chosen from each year if we want a stratified random sample of 80 students? c Explain how to obtain this sample. The number of students to be selected from each subgroup is proportional to the relative size of each subgroup. i.e.,

a

Year

Proportion

+ 165 950 £ 80 + 13:9 + 14

c We select a random sample of

or 13:7%

10

8

165 950

or 17:4%

11

150 950

or 15:8%

9

180 950

or 18:9%

12

135 950

or 14:2%

and total =

Year 10

11 14 15 16 13 11 80

Proportion

7

b No. of students to be selected from Year 9 Year 7 + 130 950 £ 80 + 10:9 + 11 Year 8

Year

130 950

190 950

or 20%

+ 180 950 £ 80 + 15:2 + 15

Year 11

+ 150 950 £ 80 + 12:6 + 13

= 190 950 £ 80 = 16

Year 12

+ 135 950 £ 80 + 11:4 + 11

students from Year 7 students from Year 8 students from Year 9 students from Year 10 students from Year 11 students from Year 12 students from whole school.

4 A school of 859 students is made up as follows Year Number of students

7 125

8 142

9 175

10 160

11 134

12 123

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If a stratified random sample is taken based on these subgroups (school years) 75

a what proportion of students should be chosen from each year? b How many students should be chosen from each year if we want a stratified random sample of 60 students? c Explain how to obtain this sample.

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5 A school of 439 students is made up as follows 7 96

Year Number of students

95

8 84

9 78

10 65

11 56

12 60

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If a stratified random sample is to be taken based on these subgroups (school years) 25

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a what proportion of students should be chosen from each year? b How many students should be chosen from each year if we want a stratified random sample of 40 students? c Explain how to obtain this sample. 6 In Year 11 at Greengate High School there are 69 girls and 51 boys. In a stratified random sample based on gender a what proportion of girls and boys should be in our sample? b How many girls and boys should we select for a stratified random sample of 40 students? c Explain how to obtain this sample. 7 Of the 500 plants in a nursery, 300 are exotic plants and 200 are Australian native plants. If we were to select a stratified random sample based on these 2 subgroups a what proportion of each subgroup should be in our sample? b How many of each type of plant should we select for a stratified random sample of 60 plants? c Explain how to obtain this sample. 8 The ages of the people who live in a small country town are shown in the following table: Age No. of people

0-9 180

10 - 19 835

20 - 29 260

30 - 39 350

40 - 49 280

50 - 59 250

60 - 69 200

70 120

It is proposed to do a dental check by choosing a stratified random sample of the population based on these age groups. a What proportion of each age group should be included in the sample? b What number of people in each age group should be selected for a sample of 40? c Explain how to obtain this sample. 9 A manufacturing company produces the following number of electric light globes each week. Globe Number

40 watts 2500

60 watts 4800

80 watts 3400

100 watts 6300

If a stratified random sample based on these globe types is taken each week for a quality check a what proportion of each type of globe should be selected for the sample? b What number of each type of globe should be checked for a sample of 50 globes? c Explain how to obtain this sample.

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10 Of the 150 students in Year 11, 87 travel to school by bus, 23 ride bicycles, 30 travel by private car and 10 walk. In order to find the average time of travel to school for Year 11 students it is decided to take a stratified random sample based on these subgroups (methods of travel).

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a What proportion of each subgroup should be in the sample? b How many of each subgroup should there be in a sample of 30 students?

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11 What subgroups could be used in a stratified random sample to investigate

100

a the number of people with private health care cover b the political party people intend to vote for at the next election?

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SYSTEMATIC SAMPLE

5

Again each member of the population is assigned a number. To obtain a systematic sample the first number is selected at random and then we select every nth number, that is, we select numbers at regular intervals. The value of n, i.e., the size of the regular interval, depends on the size of the sample wanted and the size of the population.

0

EXAMPLE

The interval shows how often you select items.

1

For a population of 240 items, at what intervals should we select items to obtain a systematic sample of a 20 b 40 c 60? Divide the size of the population by the size of the sample wanted. a 240 ¥ 20 = 12 Hence, we should select every 12th item.

?

b 240 ¥ 40 = 6

Hence, we should select every 6th item.

c 240 ¥ 60 = 4

Hence, we should select every 4th item.

EXERCISE 5K 1 A machine produces 300 items each day. At what intervals should we select items to obtain a systematic sample of a 25 b 50 c 10 d 15 items? 2 An electoral roll has 1200 names on it. How often should we select names to obtain a systematic sample of a 100 b 200 c 150 d 40 names?

EXAMPLE

2

Explain how to obtain a systematic sample of from a year containing 150 students.

100

a 10

b 15

95

c 25 students

75

Assign the numbers 1 to 150 to each student. a Divide the size of the population by the size of the sample wanted. 150 ¥ 10 = 15: Hence, we start at any number in our list and then select every 15th number.

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For example: i

If we started at the student with the number 7, then we would select the numbers 7, 22, 37, 52, 67, 82, 97, 112, 127, 142 to form our sample.

ii

If we started at the student with the number 24 we would select the numbers 24, 39, 54, 69, 84, 99, 114, 129, 144, (159 =) 9.

iii

If we started at the student with the number 56, then we would select the numbers 56, 71, 86, 101, 116, 131, 146, (161 =) 11, 26, 41.

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5

Note how in ii and iii we go back to the start of the list. 0

b 150 ¥ 15 = 10. Start at any number and then select every 10th number. c 150 ¥ 25 = 6. Start at any number and then select every 6th number.

3

a Explain how to obtain a systematic sample of 10 students from a year containing 200 students. b What students would be selected if we started at the student with the number i 10 ii 15 iii 30 iv 45 v 113?

4

a Explain how to obtain a systematic sample of 15 names from a roll containing 180 names. b What names would be selected if we started at the name with the number i 5 ii 10 iii 25 iv 50 v 120?

EXAMPLE

3

Explain how to select a systematic sample of 20 names from a list of a 146

b 156 names

a Number the names from 1 to 146. 146 ¥ 20 = 7:3 Hence, start at any number and then select every 7th number until we have a sample of 20 names. b Number the names from 1 to 156. 156 ¥ 20 = 7:8 Hence, start at any number and then select every 8th number until we have a sample of 20 names. 100

5 Explain how to obtain a systematic sample of 10 names from a list of a 62 b 87 c 124 d 239

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185 names

75

6 A machine produces 200 numbered engine parts in a day. a How often should parts be selected for a quality check if a systematic sample of i 5 parts ii 12 parts is required each day? b Choose a starting number at random and write down the numbers of the engine parts you would select for each of the samples in a.

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SUITABILITY OF SAMPLE TYPES

EXERCISE 5L What sampling methods would be suitable for the following?

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1 A machine produces 5000 bolts in a day. A sample of 50 bolts/day is required for a quality control check. 2 There are 600 people on a town’s electoral roll. A sample of 40 people is required to complete a questionnaire. 3 A sample of people is wanted to investigate average weekly expenditure on food. 4 A farmer wishes to determine the number of sheep on his property. 5 A sample of the people entering a football game is wanted. 6 The school principal wants a sample of teachers to estimate how much homework is set each night. 7 A shoe company employs 5 machinists. A weekly quality control check of their work is required. 8 A sample of Year 11 students is required to estimate the number of left-handed people in the group. 9 A grid sample is required by a geographer to estimate the proportions of various types of land use (for example, grazing, crops, forest, buildings, etc.). 10 A sample to estimate the average weekly income of senior students is wanted.

M

THE EFFECT OF THE SAMPLE SIZE

A good sample is one which has almost exactly the same characteristics as the total population. We say that a good sample is “representative” of the population. In this section we will investigate what effect the size of a sample has on how well it represents the whole population.

EXERCISE 5M 1 The results when 34664 14624 52415 33663 43356 64132 26563 21264

a normal die is rolled 200 times are 31362 55142 63611 42553 55244 65132 63514 62453 21634 46323 55651 26435 23245 61224 32616 51326

recorded below: 63144 45213 12646 35236 53142 25145 21621 16231

56443 24546 26513 21652

54346 13251 42214 31643

The average (mean) of these scores is 3:5 (rounded to 1 decimal place).

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95

a What is the average of the first 5 scores? Is this sample representative of all the scores? b What is the average of the last 5 scores? Is this a good sample? c Choose two other groups of 5 scores and calculate the average of each. Are they good samples? d In general, do you think that samples of 5 are big enough to be representative of the average of the whole population?

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e What is the average of i the first 10 scores ii the last 10 scores? f Choose two other groups of 10 scores and calculate the average of each. g i By selecting 3 digit numbers from the table of random numbers given below, select 10 numbers not bigger than 200 (start at the first digit). 48047 08695 90070 10132

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5

45381 58112 98873 27359

33232 96070 89846 13017

35178 91910 50953 41045

46971 18868 92529 13817

85879 52251 68249 65603

31458 99827 54949 87615

22016 32581 83829 55691

ii Use these random numbers to select a sample of 10 scores from the results of our experiment with the die. (For example, the first 3 digit number 6 200 is 133. Hence find the 133rd result in the table of results above. The 133rd result was a 1, etc.)

0

iii Calculate the average of this sample. h

i Obtain a systematic sample of 10 scores (start at the 4th number, say). ii What is the average of this sample?

i Compare the averages of these samples of 10 with the population average. Are samples of 10 big enough to be representative of the population? Discuss and comment. j

i Using your calculator, find 40 random numbers not bigger than 200. ii Use these random numbers to select a random sample of 40 scores from the results of our experiment. iii Calculate the average of these 40 scores.

k

i Obtain a systematic sample of 40 scores from the 200 scores. ii What is the average of this sample?

l Compare the averages of these samples of 40 with the population average. Are samples of 40 big enough to be representative of the population? Discuss and comment. m How well do the averages of our samples compare with the average of the whole population? Comment on the effect of the sample size. 2 The 200 students in year 11 and 12 of a high school were asked whether (y) or had ever smoked a cigarette. The replies, as they were received, were nnnny nnnyn ynnnn yynyy ynyny ynnyn nyynn yynyn ynnyn nnyyy yyyyy nnnyy nnnnn nnyny yynny nynnn ynyyn nnyny ynnnn yyyyn yynnn nynyn nynnn yynny nyynn yynyn ynynn ynnyy nyyny ynynn nyynn nnnyy ynyyn yyyny ynnyy nnyny

not (n) they yynyy ynyyy nyyyn ynnnn

Is this a census or a sample? Is the data categorical or quantitative? Calculate the proportion of all students who said they had smoked. What proportion of the first i 5 ii 10 students said they had smoked? Are these samples representative of all Year 11 and 12 students? e Use a set of random numbers to select a random sample of i 5 ii 10 iii 20 iv 40 students from the 200 students above and calculate the proportion of each sample who said that they had smoked. f Obtain a systematic sample of i 5 ii 10 iii 20 iv 40 students from these 200 students and calculate the proportion of each sample who said that they had smoked. When arranged within the categories male and female, the replies are:

a b c d

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Male

95

Female

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nnyny yyynn nnyny nnyyy nnyny nnyny

nnnny nnynn ynnyn yynyy ynnyy nn

ynyyy nnyny yynyn nnnyy nyyny

nyynn nyynn ynynn nynnn ynynn

yynyn nnyny nyyyn nnyny nyynn

ynnyn nnynn yyyny yynny nnnyy

yynyy nyynn ynn nynnn ynyyn

177

ynyyy ynyyn ynyyn ynnyy

g Select a stratified random sample, based on gender, of i 10 ii 20 iii 40 students and calculate the proportion of each sample who said they had smoked. h Comment on whether or not the samples in e, f and g are representative of the whole population. What effect does the size of the sample have on estimating the responses of the total population? 3 State whether you think the following samples would be satisfactory in estimating the nature of the whole population: a The proportion of boys and girls in your class to estimate the proportion of boys and girls in Year 11 in your school. b The proportion of boys and girls in your class to estimate the proportion of boys and girls in Year 11 across the state.

RESEARCH PROJECT Investigate the contribution made to statistics by John Graunt Carl Friedrich Gauss Sir Ronald Fisher W Edwards Deming

Florence Nightingale John Tukey


You should be able to: 2 design different types of questions to collect information 2 be familiar with the advantages/disadvantages of different methods of collecting information from people 2 classify data as categorical, discrete or continuous 2 identify whether a census or sample is appropriate for data collection 2 recognise that the purpose of a sample is to estimate a particular characteristic of the whole population 2 understand the concept of bias in a sample 2 determine random numbers from a table, calculator or computer 2 describe and use random, stratified and systematic samples to collect data 2 describe and use the capture-recapture technique to estimate population size 2 determine the appropriate sample type for a given situation 2 recognise the effect of sample size in estimating a characteristic of a population.

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5

100

95


Below is a list of key words used in this chapter. Explain the meaning of each. bias, continuum, questionnaire, categorical, quantitative, discrete, continuous, population, census, sample, random number, survey, random sample, strata, stratified, proportion, systematic sample, representative

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5

DIAGNOSTIC TEST

1 Which of the following is not an example of a quantitative continuous variable? A weights of students B wages of factory workers C daily temperature D time to get to work 2 Which of the following is an example of data that could not be collected by a census? A the life of a light globe B the number of vehicle accidents caused by speed C the number of tries scored by a rugby team D the age of school teachers in NSW 3 Explain any possible bias in the following samples a a phone survey on a Friday night b a survey of the audience at the opera 4 Comment on any possible bias in the following situations a Year 10 are interviewed about the school uniform b A factory tests a sample of its products every Friday afternoon 5 Write down the steps to find one digit random numbers on your calculator. 6 Write down the instructions to generate two digit random numbers using a spreadsheet. 7 A biologist catches a random sample of 100 fish from a lake, tags them and releases them back into the lake. A month later he catches another random sample of 80 fish and finds that 5 of them are tagged. The number of fish in the lake is approximately A 160 B 400 C 500 D 1600 8 Of the 150 students in Year 11, 70 are boys and 80 are girls. In a stratified random sample based on gender, the proportion of boys who should be chosen is 7 8 B 15 C 15 D 87 A 78 9 For the information in 4, the number of girls who should be chosen for a sample of 60 students is A 53 B 28 C 32 D 69 10 A machine produces 6000 items in a week. To obtain a systematic sample of 200 we could start with the 50th item and then select item numbers A 80, 110, 140, 170, ...... B 200, 400, 600, 800, ...... C 110, 170, 230, 300, ...... D 250, 450, 650, 850, ...... 11 To obtain the average time it takes for Year 11 students to get to school, the most suitable sample type would be a A random sample B stratified random sample based on mode of travel C systematic sample D the capture-recapture technique

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12 To estimate the number of boys and girls in Year 11 in NSW a suitable sample would be the number of boys and girls in A your class B the mathematics classes at your school C Year 11 at your school D Year 11 in a selection of schools in different regions

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If you have any difficulty with these questions, refer to the examples and questions in the exercise indicated. Question 1 2 3, 4 5 6 7 8, 9 10 11 12 Section B C D F G I J K L M

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5A

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REVIEW SET

1 State one advantage and one disadvantage of using personal interviews to collect data. 2 Classify the following data as categorical, discrete or continuous a the number of cloudy days in a month b the cost of petrol in country areas c the dress sizes available in a store 3 State whether a census or sample would be used for each of the investigations in 2. 4 Comment on any possible bias in the following situations a a phone survey on Saturday night b a questionnaire is sent to a sample of people chosen from the electoral roll 5 Explain how a random sample of 12 students could be chosen from a class of 30. 08695 90070 10132

6

58112 98873 27359

96070 89846 13017

91910 50953 41045

18868 92529 13817

52251 68249 65603

99827 54949 87615

32581 83829 55691

a Use the table of random numbers given to find a random selection of 5 squares from the grid below. 5 4 3 2

100

1

95

1

2

3

4

5 75

b Copy the grid and shade the squares selected. 7 A park ranger catches and tags 50 rabbits and then releases them. A month later he catches another random sample of 40 rabbits, in the same area, and finds that 8 of them are tagged. Estimate how many rabbits there are in this area.

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100

8 A school of students is made up as follows: 95

Year Number of students

75

7 85

8 82

9 89

10 96

11 75

12 73

If a stratified random sample based on year groups is to be taken a what proportion of each year should be selected? b How many from each year should be selected for a sample of 100 students?

25

5

9 Explain how to obtain a systematic sample of 20 students from a roll of 300.

0

5B

?

REVIEW SET

1 State one advantage and one disadvantage of using phone interviews to collect data. 2 Write down 2 examples of categorical data. 3 Would a census or sample be used to investigate a the number of people who use Freshspray deodorant b the consumption of alcohol by senior students in your school? 4 Describe the sample you would choose if you wanted to gather support for a improved netball facilities at the local park b more lines for senior students at the school canteen. 5 Describe samples with the opposite bias to those in 4. 08695 90070 10132

6

58112 98873 27359

96070 89846 13017

91910 50953 41045

18868 92529 13817

52251 68249 65603

99827 54949 87615

32581 83829 55691

Choose 5 random 3 digit numbers less than 200 from the table above. Start at the first digit and move to the right. 7 A footwear factory employs 4 machinists who produce the following numbers of pairs of shoes in a week: Machinist Pairs of shoes

Vince 56

Sophie 43

Talia 51

Mohammed 50

The manager wants to carry out a quality check by using a stratified random sample of the shoes produced by each machinist. a What proportion of shoes from each machinist should be checked? b How many shoes produced by each machinist should be selected for a sample of 20 shoes? 8 What sampling method would be suitable for a survey of the number of people who own mobile phones?

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REVIEW SET

75

1 State one advantage and one disadvantage of collecting information by posting a questionnaire.

25

2 Classify the following data as categorical, discrete or continuous a the eye colours of a group of students b the amount of sunshine in a day c the number of children in a family

5

3 Give an example of data that could be collected using a a census b sample

0

4 Describe a suitable sample to survey if you wanted to collect data about a the punctuality of the local bus service b the cleanliness of the council swimming pool. 5

a Use your calculator to find 10 random 2 digit numbers. b How many of the numbers are odd, even? c How many numbers are less than 50, 50 or bigger?

6 A scientist at a trout farm takes 120 trout from a pond, tags them and releases them back into the pond. A week later he catches 100 trout and finds that 5 of them are tagged. Estimate the number of trout in the pond. 7 A machine produces 300 numbered engine parts in a day. a How often should parts be selected for a quality check if a systematic sample of i 20 ii 12 parts is required each day? b Choose a starting number at random and write down the numbers of the engine parts you would select for each of the samples in a. 8 What method of sampling would be suitable for a cattle grower to estimate the number of cattle on his property.

?

5D

REVIEW SET

1 A TV station invites viewers to phone in their response to a question. What are the advantages and disadvantages of this method of data collection? 2 Write down two examples of discrete data. 3 State the advantages and disadvantages of using a census to collect information. 4 Write down the instructions to generate a list of 20 random 1 digit numbers using a spreadsheet.

100

5 Comment on any possible bias in the following samples a car tyre distributors are interviewed about the best value tyre to buy b a systematic sample of the people from 5 pages of the telephone directory

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6 A factory has 120 employees. A sample of 25 employees is to be chosen to complete a questionnaire. a Explain how a random sample using a table of random numbers could be used to select the sample.

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b Explain how to obtain a random sample without using random numbers. c Explain how to obtain a systematic sample of employees. 7 Explain the meaning of the term “bias”. 8 Write down the purpose and characteristics of a good sample.

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CUMULATIVE REVIEW (CHAPTERS 2 - 5) 100

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2–5

CUMULATIVE REVIEW 1

25

5

a Round off 4263:091 to i the nearest 100 ii iii 2 decimal places iv v 5 significant figures vi b State whether or not the following numbers are in i 6 £ 10 000 ii 15 £ 108 c Express in standard notation i 105 000

0

183

ii

the nearest whole number 3 significant figures 1 significant figure standard notation iii 2:04 £ 10¡6

0:000 062 7

d Calculate, writing answers in standard notation i (4:1 £ 108 ) £ (7 £ 105 ) ii (8 £ 105 )3 e Convert to a percentage i 38

ii

1 13

f Convert 43% to a i fraction

ii

decimal

g Find i 43:5% of 15 metres

ii

1 12 % of $7650

h What percentage is i $12:60 of $240 ii iii 35 minutes of 2 12 hours?

454 g of 3 kg

iii

0:037

i Increase 80 kg by 5%. j Find the percentage decrease from 70 kg to 65:8 kg. k Convert i 6:73 km to m ii 1750 cm to m iii 4:3 g to mg iv 47 000 kg to t v 6:35 L to mL vi 90 mL to L vii to minutes 4 hours 17 min l For i ii iii iv 2

the measurement 6:8 metres, find the limit of reading the greatest possible error the lower and upper limits of the true measurement the percentage error (to 1 d.p.)

a Pauline earns $36 028 per annum. Calculate her weekly pay. b Fielding is paid $6:58 per hour. He works 5 hours per day six days a week. Find his pay for one year. c Louise is paid $11:40 per hour. Calculate her weekly wage if she works 40 hours at normal time and 8 hours overtime at time and a half.

100

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d Hunter works in a timber yard and is paid $12:40 per hour. He receives a dust allowance of $2:50 per day. Find his pay for a 5-day, 37 hour week. e Art salespeople are paid 6:2% of the selling price of any works they sell. Find the commission on a painting selling for $6250.

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f Endora received a 17 12 % holiday loading on four weeks normal wages. She normally works a 32-hour week. Her 4 weeks holiday pay and loading is $2185:70. i Find her normal weekly pay. ii Find her normal hourly pay rate.

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75

g A factory worker is paid $5:35 for each garment completed. Calculate his wage if he completes 324 garments. h Micky earns $484 per week. His employer contributes 7% and Micky contributes 3% to Micky’s superannuation fund. Calculate Micky’s weekly income and his total weekly superannuation.

25

5

0

CUMULATIVE REVIEW (CHAPTERS 2 - 5)

3

a For the number pattern ¡16, ¡12, ¡8, ¡4, ....... i write the next 4 terms ii write a rule describing the pattern iii find the 50th term b Using n = 1, 2, 3, ...... find the first five terms of the number sequence described by the rule T = 20 ¡ 6n. c If x = ¡5 and y = 8 find the value of 4x2 ¡ 6y. d For the number pattern 1024, 512, 256, ....... i write the next three terms ii

find the tenth term

e The surface area of a cylinder, A, is found using the formula A = 2¼r(r + h) where r is the radius and h is the height. Find the surface area of a cylinder with radius 12 cm and height 8 cm. 4 Simplify a 11p + 7p

b

21m ¡ 7m

c

2a + 7b ¡ 11a

f

a2 a £ 6 7

d

13a + 12b + 6a ¡ 8b

e

(¡10x)2

g

30L3 6L2

h

12 a3 £ 2 18 a

5 Expand and simplify if possible a 9(2x ¡ 5) c ¡(15 ¡ x) e 4(5x ¡ 5) ¡ 12x g 8x2 (3 ¡ 4x) + 8x(3x ¡ 5) 6 Solve the following equations a d ¡ 9 = 12 x =6 c 3 e 3x ¡ 8 = 19 g 14 + 3n = 29 i 2(4x + 5) = 18 k 5(3 ¡ 7x) = 4(3x + 5) m

2(x ¡ 6) + 5(4x + 1) = 24

o

(8x ¡ 1) 7x = 4 5



b d f h

¡8(3x + 5) 19 ¡ (3x + 9) 3(p + 5) ¡ 8(3p + 5) 7x2 (3x + 3) + 3x(11 ¡ 2x)

b

x + 4 = 12

d

11x = 55

100

f h j l

7x ¡ 12 = 16 23 ¡ 4c = 14 8x ¡ 4 = 3x + 8 15 ¡ 5(6 ¡ x) = 22

95

(4x + 1) 1 = 5 7

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7

a Name the main types of interviews and give an advantage and disadvantage of each. b Classify the following data as categorical, discrete or continuous. i number of rainy days in a month ii the cost of LPG in country areas iii shoe sizes in a department store c Explain how a random sample of ten students from a form of 120 may be chosen. d The owner of a trout farm catches and tags 30 fish. A week later she catches 200 fish and 8 are tagged. Estimate the number of fish. e Explain the statistical meaning of the term “bias”.

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CHAPTER

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6

Application of area and volume AREA OF STUDY

This chapter deals with area and volume. The main mathematical ideas investigated in this chapter are: 8 calculation of areas 8 classifying prisms and pyramids 8 identifications of nets of solids 8 the sketching of 3D solids 8 the calculation of surface area and volume 8 units of capacity. 100

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APPLICATION OF AREA AND VOLUME (Chapter 6)

A

100

95

AREAS INVOLVING TRIANGLES AND RECTANGLES

75

This section revises the areas of triangles and rectangles, and composite shapes involving rectangles and triangles.

25

The area of a closed figure is the number of square units it contains.

5

The most frequently used units of measurement of area are 0

² ² I

square millimetres (mm2 ) square metres (m2 )

² ²

square centimetres (cm2 ) hectares (ha)

RECTANGLE

i.e.,

breadth

Area = length £ breadth Area = lb

length

I

TRIANGLE

height

base

base

base

Area = 12 (base £ height) i.e.,

EXAMPLE

Area = 12 bh

1 100

Find the area of the rectangle 3.4 cm

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75

6.8 cm

Area = length £ breadth = 3:4 £ 6:8 cm2 = 23:12 cm2

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APPLICATION OF AREA AND VOLUME (Chapter 6) 100

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?

EXERCISE 6A 1 Find the area of each rectangle 9 cm 12 m a b

d

c

7 cm

14 cm

14 m

25

5 cm

5

e

0

f

g 10.3 cm

12.9 cm

7 cm

A square is also a rectangle.

1.3 cm

4.7 cm

3.7 cm

2 Find the area of each rectangle, given Length Breadth a 12 cm 3 cm b 18 cm 5 cm c 16 mm 8 mm d 4m 2m e 5 cm 2 cm

EXAMPLE

8 cm

f g h i j

Length 6:2 cm 15:9 m 11:3 cm 4:3 mm 7:7 cm

Breadth 4 cm 6:2 m 5:8 cm 1:7 mm 0:8 cm

2

Find the areas of the following triangles a b

c

2.5 cm

3m

8m 5m

6 cm 12 m

a

Area = 12 base £ height =

1 2

£ 12 £ 5 m2

= 30 m2

b


1 2

c

£ 8 £ 3 m2


= 12 m2

1 2

£ 2:5 £ 6 cm2

= 7:5 cm2

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95

3 Find the area of each triangle a b

75

c

d

6 cm 2 cm

4 cm

C:\...\NSWGM\NSWGM_06\189NG06.CDR Fri Mar 10 15:17:41 2000


3 cm

25

5 km

6 cm

5

7 km

8 cm

0

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9m

e

95

f

g 5 cm

4m

75

50 m

2 cm

8 cm

32 m

25

5

0

h

EXAMPLE

5 cm

3

Find these shaded areas a

b

4 cm 5m 6 cm

2m 3m

8 cm

a Area = Arectangle + Atriangle = 8 £ 6 + 12 £ 8 £ 4 cm2 = 64 cm2

b

3m

Area = Alarge rectangle ¡ Asquare = 5 £ 8 ¡ 2 £ 2 m2 = 36 m2

4 Find these shaded areas a

b

4m

6 cm

c 5m

10 m

8 cm 10 cm

4m

8m

d

e

2m

3m

2m

f 8 cm

5m

2m

4m

5 cm

12 cm

9 cm

2 cm 20 cm

6m

100

10 cm

95

g

h

i

25 cm

3m 20 cm

11 m

75

6m 25

5m

5

8 cm

50 cm 7m



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4

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EXAMPLE

Q

R 3 cm

Find the area of quadrilateral PQRS

B 14 cm 5 cm

A

P

25

5

S

Area = Area of triangle A + Area of triangle B

0

=

1 2

£ 14 £ 3 +

1 2

£ 14 £ 5

2

= 56 cm 5 Find these areas a

b

c 10 m

6 cm

5 cm

S

R

Q

P

12 m

5 cm

2 cm PQ = 8 cm

d

RS = 13 cm

e

22 m 15 m

20 m

TU = 28 m

f

26 m

10 m 16 m

18 m

EXAMPLE

U

T

8m

23 m

25 m

12 m

5

Find the area of the field PQRST.

63 m 51 m

18 m 48 m

R 16 m

S

24 m

X 17 m 32 m

Y T

17 m

10 m

Q

Z 15 m

P

Area ¢RQP = 12 £ RP £ YQ = 12 £ 58 £ 32 = 928 m2

Area ¢PTZ = 12 £ PZ £ TZ = 12 £ 15 £ 17 = 127:5 m2

100

95

Area ¢SRX = 12 £ RX £ SX = 12 £ 16 £ 24 = 192 m2

Area TSXZ = Arectangle + Atriangle = 17 £ 27 + 12 £ 7 £ 27 = 553:5 m2

75

25

2

Total Area = (928 + 127:5 + 553:5 + 192) m = 1801 m2

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0

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6 Find the areas of these fields, measurements in metres. J a b

75

D

c

E

10 K

29

40 E

28

55

8

25

A 5

Q

27 P

32

R

20

15

15

D

25

I

8

0

35

39

L

20

45

F 14

B

C

25

25

80

C B

110

70

H

A

7 A rectangular lawn is surrounded by a concrete path 1 m wide. If the lawn is 18 m by 30 m, find the total area of concrete. 8 A rectangular lawn measuring 8 m by 5 m is planted within a rectangular garden plot measuring 12 m by 7 m. Find the area not planted with lawn. 9 A concrete driveway is 23 m long and 3 m wide. Determine the total cost of pebblecreting the driveway if the material costs $15 per square metre and the cost of hired labour is $4 per square metre. 10 A farmer wishes to spread 200 kg of superphosphate per hectare. What weight of superphosphate is required to fertilise a 600 m by 400 m paddock? Give your answer in tonnes. 11 A rectangle is 12 cm by 8 cm. If the length of the rectangle is increased by 3 cm, how must the breadth be varied so that the area remains the same? 12 The dimensions of a room are length 3500 mm, width 4200 mm, and height 2200 mm. If the combined area of the door and windows is 5:5 m2 , calculate the amount of paint (to 2 d.p.) required to paint the room. One litre of paint covers 12 m2 , and the walls and ceiling need two coats of paint.

INVESTIGATION 1:

4200 mm

2200 mm 3500 mm

MODELLING TASK 100

We wish to find the area of a field using a traverse survey. Surveying involves the measurement of distances and angles in order to determine the positions of boundaries and the major features of areas of land.

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THE TRAVERSE SURVEY (OFFSET METHOD)

In this method the surveyor walks along a straight line, often a diagonal of the area to be surveyed, measuring distances along this line and distances to major features when they are at right angles to this line. The measurements are recorded in the surveyor’s notebook or fieldbook.

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To survey the field ABCDE: 95

75

25

5

E

1 Choose any suitable diagonal. Use AD, say.

29

2 Measure the lengths of the perpendicular offsets from the corners B, C and E to the diagonal AD, i.e., measure EP, BQ and CR.

A

Q

27

32

R

P 20 39

35

3 Measure the lengths AP, PQ, QR, and RD along the diagonal.

D

15

B

C

0

4 Record the measurements in a fieldbook as follows (assuming the measurements on the diagram):

E29

D 94 79 47 27 A

The distances along the diagonal from A to the foot of each perpendicular offset (i.e., the distances AP, AQ, AR, AD) are shown between the vertical lines.

39C 35B

The numbers outside the vertical lines give the lengths of the offsets and indicate whether they are on the right or left when traversing the diagonal AD.

5 Here is an example of drawing a sketch. Draw a sketch of the field PQRST given the following fieldbook entries:

S24 T17

R 58 42 25 15 P

32Q

R

In words: 15 metres from P, along the diagonal PR, there is a perpendicular offset on the left of length 17 metres. 25 metres from P (i.e., a further 10 metres along the diagonal PR) is an offset on the right of length 32 metres. 42 metres from P (i.e., a further 17 metres along PR) is an offset on the left of length 24 metres. It is then a further 16 metres to R.

16 S

24 17 32 T

Q

17 10 15 P

6 The surveyor has a very precise instrument called a theodolite with which he/she can accurately find the perpendicular offsets. You are not likely to have this instrument, however, there are other methods, of varying degrees of accuracy, which you could use: Make a simple cross with 2 pieces of timber at right angles to each other (2 T-Squares are quite suitable). B

P

d

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95

Point one arm of the cross along the diagonal AB. Walk along the diagonal until the other cross member is in line with a vertex (P ) and then measure the distance (d) from the diagonal to the vertex.

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What to do: 1 Mark out a field with 5 corners, on the playground (or on a sports field, etc) using witches hats (or chairs etc).

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2 Make a traverse survey (offset method) of your field using a fieldbook to record all your measurements. 3 Calculate the area of the field.

25

4 Write a short report detailing exactly what you did and how you did it. Include all measurements and calculations. Comment on the accuracy of your method and your results.

5

0

SOLIDS, NETS AND CROSS-SECTIONS The net of a solid is a two dimensional shape that may be folded or shaped to form a solid.

?

EXERCISE 6B 1 Match the nets with the solids a b

A

B

2 Draw nets for the following solids a b

d

e

c

d

C

D

c

f

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h

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EXAMPLE

1

When a solid is “sliced” a cross-section is formed. Sketch the cross-section for each of the following solids a b c

25

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The cross-section is a square.

a

b

A cube is bounded by plane surfaces only.

The cross-section is a circle.

c

A cylinder is bounded by plane and curved surfaces.

The cross-section is a circle.

A sphere is bounded by a curved surface only.

The cross-section is viewed from a direction at right-angles to the plane.

3 Sketch the cross-sections when the following solids are “sliced”. a b c

d

e

f

100

95

PRISMS

75

A right prism is a solid with two ends that are parallel to each other and of the same shape and size. All other sides must be rectangular. 25

If the solid is placed on one of the ends that end is called the base. A prism is named according to the cross-section parallel to the end. This must be the same shape and size as the end.



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100

EXAMPLE

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75

2

Draw a diagram and vertical cross-section of a a triangular prism a

25

a hexagonal prism

b b

5

0

Triangular cross-section

Hexagonal cross-section

PYRAMIDS

A pyramid is a solid with a polygon for a base and triangular faces which come from the base to meet at a point. Pyramids are named according to the shape of their base.

EXAMPLE

3

Draw a diagram of a a square-based pyramid

b

a

a triangular-based pyramid

b

4 Draw a diagram of a a cube d sphere g triangular-based pyramid

b e h

cylinder hexagonal prism rectangular-based pyramid

c f

rectangular prism cone

100

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5

a b c d e


Which of the solids in question 4 are prisms? Sketch their cross-sections. Which of the solids in question 4 are pyramids? Draw their bases. Which of the solids in question 4 are neither prisms nor pyramids?


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6 Name the solid which best resembles a a basketball b a broom handle d a six-sided die e a cornflakes packet g a four-sided die h the top secton of a funnel

c f

197

a tennis ball container a carrot

7 Sketch each of the solids from question 6. 25

5

8 The diagram shows a large cube with a smaller cube removed from one corner. a Sketch the diagram.

0

b Draw another sketch with cubes removed from two corners. c Draw another sketch with cubes removed from four corners. 9 Match each net in the first column with the correct solid and the correct name. Net

Solid

Name

a

(A)

(1) pentagonal-based pyramid

b

(B)

(2) triangular prism

c

(C)

(3) tetrahedron

d

(D)

(4) cylinder 100

95

75

e

(E)

(5) square-based pyramid 25

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100

INVESTIGATION 2:

OCTAHEDRA

95

Equipment needed: graph paper, scissors. 75

25

An octahedron has 8 equilateral triangular faces.

5

0

Ask your teacher to enlarge the graph paper by using a photocopier, or construct some of your own using a 60o set square. What to do: 1 Use scissors to cut out the shape.

Can this shape be folded about its inner lines to form an octahedron?

2 Experiment with

and

to try to find a suitable octahedron ‘net’. 3 Try to find as many different octahedron ‘nets’ as you can.

C

DRAWING 3D OBJECTS

Solids are often drawn with different views. Sometimes they are drawn as plans with elevations from the front, back and sides. Other times they are drawn as isometric diagrams. In an isometric drawing the lines going back from the front face are inclined at 30o . Distances are preserved and the easiest way of drawing them is by using isometric graph paper.

?

100

EXERCISE 6C

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75

1 Use centicubes or other blocks to make the following solids a

b

c

d

e

f 25

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EXAMPLE

199

1

Draw the following solids as isometric projections. Use the dark line as the starting place. a b

25

5

A different starting edge gives a different view of the diagram.

0

a

b

Note:

² ²

The edge chosen is the one appearing closest to us. . are drawn along lines on the graph paper. All other lines

2 Draw isometric projections of each of the solids in question 1, using the darkly drawn edge as the starting point. 3 Redraw the solids in question 1 using a different starting line. 4 Copy these oblique projections and then draw them as isometric projections on isometric grid paper. Use the darkly drawn edge as the starting line for your drawing. a

b

c

d

e

f

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OBLIQUE PROJECTIONS

When drawing an object using an oblique projection, the lines going back from the front face are inclined at 45o and these lengths are halved.

25

5

An oblique projection of a cube appears in Example 2 which follows. 0



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25

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EXAMPLE

2

Draw a cube using an oblique projection. Steps for drawing: 1 Draw the front face. 2 Draw lines back from the front face at 45o : 3 Make these lines half the normal length. 4 Complete the cube. 5 Draw dotted lines for the hidden edges.

5 Redraw these isometric projections as oblique projections. a b

d

e

INVESTIGATION 3:

c

f

PERSPECTIVE DRAWINGS

Isometric drawings can be criticized as being non-realistic. Perspective drawings are an attempt to make objects look as we see them and, in fact, as a photograph would capture them. A sense of depth is created by making objects smaller the more distant they are from the spectator , and parallel lines are made to converge (meet at a point) as they recede into the distance. Perspective drawings have one or two vanishing points VP1 and VP2 . These vanishing points lie on the horizon and lines to the vanishing points seem to vanish.

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One-point Perspective With distance from the spectator i the edges of the road get closer together and meet at VP ii telegraph poles decrease in size iii spaces between poles also get smaller .

vanishing point (VP)

5

0

VP1

Two-point Perspective Consider a 2 £ 2 £ 2 cube: Notice the use of two vanishing points.

VP2

In two-point perspective drawings the only true measurements are those on the common vertical line. All other measurements are affected by the perspective. Technical drawers use measuring lines or grids to draw the correct perspective lengths. What to do: 1 Drawing a cube using one-point perspective. a Copy the diagram accurately. VP b Join each point (as shown for point A) to the vanishing point using dotted lines. D

A

B

c Draw all the edges along the dotted lines. Make the edges of the cube half the length of each dotted line. d Draw the back face of the cube by joining the ends of the edges drawn in c. (These lines must be horizontal or vertical.)

2 cm

C

2 Draw these solids using the vanishing point given a b VP VP

c

VP

100

95

d

VP

e

VP

f

VP

75

25

5

0



C

M

Y

K


202 100

95

75

3


a Copy the diagram accurately. b Use B as the vanishing point to draw a 3-dimensional staircase. c Repeat b using A as the vanishing point.

A

B

25

5

0

Here is an arrow drawn in perspective. In this drawing a measuring line with equal intervals of 1:4 cm between each mark is used to form a grid. The intervals on the measuring line are 1:4 cm as this is the approximate length of the diagonal of a 1 cm square. Lines are drawn from the measuring line to each of the vanishing points. A grid is formed. The arrow is then drawn on the grid.

} } 1.4 cm

4 Copy the arrow from the example by first drawing a measuring line with 1:4 cm spacings and then forming a grid. 5 Copy and complete this perspective drawing of a chessboard.

100

95

75

25

5

0



C

M

Y

K



Extension

95

6 Copy and complete the perspective drawing of this chessboard cube.

203

75

25

5

0

7 Draw an object of your own choice using perspective.

D

SURFACE AREA The surface area of any shape refers to the sum of the areas of each face.

TOP

SID

E B OT TO M

BACK

SID

This box has 6 faces: · 2 sides · front and back · top and bottom.

E

FRONT

?

EXERCISE 6D 1 State the number of faces for each solid a

b

c

100

d

e

95

f

75

25

5

0



C

M

Y

K


204


100

95

75

EXAMPLE

1

Calculate the total surface area of this rectangular prism The rectangular prism has 6 faces.

25

Area of 2 sides = (2 £ 32 cm2 ) Area of front + Area of back = (2 £ 24 cm2 ) Area of top + Area of bottom = (2 £ 56 cm2 )

top 5

50 cm2

2

0

cm 32 e sid

24 cm2 front

) total surface area = (2 £ 32) + (2 £ 24) + (2 £ 56) cm2 = 64 + 48 + 112 cm2 = 224 cm2

2 Calculate the surface area of each solid, using the area given a

b

c

Sloping face: 234 cm2

Side: 180 cm2

Back: A= 60 cm2

Top: 96 cm2

20 cm2

Side: 50 cm2

Front: 70 cm2

d

Base: 170 cm2

e

f 6 cm

2

2

m

A

=

0c 20

cm

2

cm 24 A= Base: A = 40 cm2

A = 85 cm

2

Base: A = 160 cm2

A = 48 cm2

EXAMPLE

A

8 =2

A= 36 cm2

18.2 cm

2

Calculate the surface area of each solid. a

100

95

75

b 25

4 cm 5 cm 4 cm



5

7 cm

0

C

M

Y

K



205

100

95

Surface area = 6£ area of one face =6£s£s = 6 £ 4 £ 4 cm2 = 96 cm2

a

75

b

25

5

0

Area of front and back = 2£ area of front = 2 £ 7 £ 4 cm2 = 56 cm2

(Rectangle)

Area of sides = 2£ area of one side = 2 £ 5 £ 4 cm2 = 40 cm2

(Rectangle)

Area of top and bottom = 2£ area of top =2£7£5 = 70 cm2

(Rectangle)

) surface area = 56 + 40 + 70 cm2 = 166 cm2

3 Calculate the surface area of the cubes drawn below a

b

c

9 cm

5 cm 6 cm

d

e

f

12 cm

15 cm 3m

4 Calculate the surface area of a cube with sides of the following lengths a 4:8 cm b 0:7 m c 17 mm d 9:03 cm e 0:31 m f 14:4 cm 100

5 Calculate the surface area of these rectangular prisms a

b

c

95

75

15 cm

4 cm

7 cm

25 cm

3 cm

10 cm

8 cm

5

8 cm

4 cm



25

0

C

M

Y

K


206 100


d

e

f 30 m

2m

95

6 cm

8 cm

75

45 mm 60 mm

12 cm

25

6 Calculate the surface area of the following rectangular prisms, given the dimensions stated

5

0

130 m

17 mm

a b c d e

Length 9 cm 11 cm 8m 12 mm 15:3 m

Breadth 12 cm 20 cm 9m 15 mm 6:4 m

Height 7 cm 5 cm 4m 6 mm 4:3 m

Drawing a diagram can help.

7 Find the surface area of the solids formed by the following nets a b c (6 squares)

5 cm

6 cm 10 cm

4 cm

EXAMPLE

6 cm

3

5 cm

Calculate the surface area of the triangular prism. 4 cm 15 cm 6 cm

Step 1:

Total surface area of 2 triangles (front and back) = 2£ area of one triangle = 2 £ 12 (B £ H) =2£

1 2

£ 6 £ 4 cm2

100

95

Back/Front

= 24 cm2 Step 2:

Surface area of 2 rectangles (identical) = 2£ surface area of one rectangle =2£L£B = 2 £ 15 £ 5 cm2 = 150 cm2

75

25

5

Two sides

0



C

M

Y

K


207


Step 3:

95

75

Surface area of 1 rectangle (base) =L£B = 15 £ 6 cm2 = 90 cm2

Base

) total surface area = 24 + 150 + 90 cm2 = 264 cm2

25

5

0

8 Calculate the surface area of the following triangular prisms a

b

c 17 cm 10 cm

8 cm

13 m

15 cm

12 m 25 cm 4 cm

12 cm

18 m

5m

d

e

16 cm

f 5 cm

41 cm

13 cm

9 cm

12 cm

4 cm

15 cm 13.6 cm

80 cm

8 cm

6 cm

5 cm

g

h

i 15 cm

17 cm

9 cm

4.8 cm 24.4 cm

8 cm

20 cm 15 cm

21.6 cm

7.8 cm 24 cm

28 cm

EXAMPLE

4

100

95

Calculate the surface area of the composite prism

75

10 cm 8 cm 25

6 cm

5

12 cm



7 cm 0

C

M

Y

K


208


100

95

75

Step 1: [front and back ) find composite area £ 2] Area2 = L £ B Area1 = 12 B £ h = 12 £ 6 cm2 = 12 £ 12 £ 8 cm2 = 72 cm2 = 48 cm2

25

) total area = 48 + 72 cm2 = 120 cm2

5

) area of front/back = 2 £ 120 = 240 cm2

1

Triangle

2

Rectangle

0

Step 2:

[sides]

Area = 2 £ L £ B = 2 £ 7 £ 6 cm2 = 84 cm2

Step 3: [roof]

Step 4: [base]

Area = 2 £ L £ B = 2 £ 10 £ 7 cm2 = 140 cm2

Area = L £ B = 12 £ 7 cm2 = 84 cm2

) total surface area = (front/back) + (sides) + (roof) + (base) = 240 + 84 + 140 + 84 cm2 = 548 cm2

9 Calculate the total surface area of the following composite shapes a

17 cm

b

c

5.7 cm

8 cm 8 cm

28 cm

15 cm

18.2 cm

7 cm 9 cm

12 cm

6.2 cm

4 cm

32.9 cm

5 cm 3 cm

d

e 12.9 cm

f

38 cm

17.8 cm

50 cm

9.2 cm

33 cm 12.3 cm

26 cm

15 cm 50 cm

6.4 cm

18.1 cm

EXAMPLE

17 cm

8 cm

29 cm

14 cm

5

100

Find the surface area of these pyramids a

95

75

b 11.2 cm

10.1 cm 8 cm

25

3 cm

6 cm

5

10 cm



0

C

M

Y

K


209


a There are 4 equal sized triangles and one square.

95

) surface area = 4£ area of ¢ + area of 2 4 £ ( 12 £ 6 £ 8) + 6 £ 6 cm2 = 132 cm2

75

25

b There are 2 pairs of equal sized triangles and one rectangle. ) surface area = 2 £ ( 12 £ 10 £ 10:1) + 2 £ ( 12 £ 3 £ 11:2) + (10 £ 3) cm2 = 164:6 cm2

5

0

10 Find the surface area of these pyramids a b

c

9 cm

7 cm

10 cm

d

3.6 cm

e 10.8 cm

10.2 cm

f 8.6 cm

7.1 cm

4 cm

21.4 cm

20.9 cm

11.8 cm

7.5 cm

8 cm

11

4.8 cm

9 cm

15.3 cm

12.3 cm

a A gift box measures 10 cm by 12 cm by 20 cm. Calculate its total surface area. b Gift wrapping paper costs $3:50 a sheet. If each sheet covers 290 cm2 , calculate the total cost of wrapping the gift box. (Remember: You must purchase complete sheets.)

12

Belinda's room Tania's room 2.2 m 3.1 m 5.6 m

a b c d

2m

6.3 m

100

2.3 m

95

Calculate the wall area of Belinda’s room. Calculate the wall area of Tania’s room. Who has the greater wall area? A 4 litre tin of paint covers approximately 10 square metres and costs $34:95. Calculate i the number of 4 litre tins of paint needed to paint Tania’s room ii the cost of purchasing paint for Tania’s room.

75

25

5

0



C

M

Y

K


210 100


13 Which cereal box has the greater surface area?

95

75

32 cm

16 cm

25

18 cm

5

0

25 cm

12 cm

7 cm

The interior walls and floor of an inground swimming pool are to be repainted. Calculate

14 2.3 m

b how many cans of paint are needed if one can covers 70 m2 c the cost of repainting, if each can costs $82:50:

20 m 9m

E

a the total surface area of the four walls and the floor of the pool

CALCULATING THE VOLUME OF A SOLID The volume of a solid is the amount of space it occupies.

When a solid has a uniform cross-section (face), the volume of the solid can be determined using Volume = area of face ´ height

i.e.,

EXAMPLE

Volume = the area of the face of a shape multiplied by the height.

1

The “height” is perpendicular to the front face or base.

Find the volume of this triangular prism with front face having area 12 cm2 and height (depth) 13 cm.

100

95

A =12 cm2

13 cm

75

V = area £ height = 12 cm2 £ 13 cm = 156 cm3

25

5

0



C

M

Y

K


211


95

75

?

EXERCISE 6E 1 Calculate the volume of the following solids given the area of the face of the shape and the height a c b

25

A =15 cm2

5

A =36 cm2 5 cm

0

15 cm

3.2 cm

d

e

A =21 cm2

f

2

A =45.3 cm

18 cm

A =25 cm2

A = 9.2 cm2

5 cm 1.9 cm

EXAMPLE

2

Find the volumes of the following solids a b 10 cm

4 cm

7.5 cm

a

The “end” of a cylinder is a circle.

10 cm

6 cm

Volume = length £ breadth £ height = 7:5 cm £ 6 cm £ 4 cm = 180 cm3

b

Volume = area of end £ height = ¼r2 £ h = ¼ £ 5 £ 5 £ 10 + 785:4 cm3

2 Find the volume of the following solids a b

100

c

95

75

14 cm

2m 9m

7m 25

5

7 cm

6 cm 0



C

M

Y

K


212 100


d

e

f 1 cm

95

75

6 cm

12 cm

12 cm

30 cm

18 cm

8 cm 25

5

0

3 Determine correct to the nearest cm3 , the volume of these cylinders a b c 4.2 cm

9.6 cm

18 cm 10.3 cm 20 cm 2.3 cm

4 Consider the following cylinders. Which has the greatest volume? 20 cm

B A

32 cm

15 cm 4 cm

EXAMPLE

20 cm

C

10 cm

3

Find the volume of a sphere with diameter 12 cm. The volume of a sphere radius r units is given by V = 43 ¼r3

100

V = 43 ¼r3

Radius = diameter ¥ 2 = 12 ¥ 2

=

= 6 cm

= 904:8 cm3

4 3

r

95

£ ¼ £ 63

75

25

5 Find the volume of a sphere with a radius 10 cm b


5

diameter 18 cm


c

radius 8:3 cm

C

M

0

Y

K



213

INVESTIGATION 4: VOLUMES OF SOLIDS AND PYRAMIDS

95

75

25

5

0

Equipment needed: cardboard, scissors. What to do: 1 Construct and cut out the nets of the open cube and the open square pyramid using the information on the diagrams. The cube and the pyramid have the same base and height. Use sticky tape to make the objects. Fill the pyramid with sand and see how many pyramids of sand it takes to fill the cube. Construct others of your own and test. 2 Construct and cut out the nets of the open cylinder and the open cone using the information on the diagrams. The cylinder and the cone have the same circular base and height. Use sticky tape to make the objects. Fill the cone with sand and see how many cones of sand it takes to fill the cylinder. Construct others of your own and test.

open cube

25 mm

open pyramid with square base

30.6 mm

25 mm

94 mm open cylinder

20 mm

open cone 216° 20 mm

15 mm

F

FURTHER VOLUME

PYRAMIDS AND CONES:

100

h

h

h

95

75

square-based pyramid

triangular-based pyramid

From the investigation the volume of a pyramid is given by:

cone

Volume = 13 (area of base £ height)

25

i.e., V = 13 Ah

5

0



C

M

Y

K


214


100

EXAMPLE

95

1

Find the volumes of the following solids a

75

b

8 cm 6 cm

12 cm 25

5

10 cm

0

Volume

a

?

b

Volume =

1 3 1 3

£ area of base £ height

= 13 £ area of base £ height

=

= 13 £ 10 £ 10 £ 12 cm3 = 400 cm3

= 96¼ cm3 + 301:6 cm3

£ (¼ £ 62 ) £ 8 cm3

EXERCISE 6F 1 Find the volume of the following solids a b

c

12 cm

10 cm

10 cm

3 cm 8 cm

10 cm

8 cm

d

e

f 8 cm

12 cm

7 cm

6 cm 5 cm

4 cm

5 cm

4 cm

10 cm 9 cm

3 cm 5 cm

5 cm

100

95

15 cm

6 cm

c

12 cm

8 cm

2 Find the volume of the following composite solids (to 1 d.p.) a b

75

12 cm

3 Find the volume of the following a a 10 m by 8 m by 5 m rectangular prism b a triangular prism with triangular base 4 cm and height 5 cm and prism height of 6 cm



C

M

Y

K

25

5

0


215


95

75

b a triangular prism with triangular base 4 cm and triangular height 5 cm and prism height of 6 cm c a cylinder of base radius 7 cm and height 10 cm d a sphere of radius 4 cm e a cone with base radius 8 cm and height 12 cm f a square based pyramid of height 12 cm and base edges 6 cm.

25

5

4 Find the volume of the following a b

0

c

area 20 cm² 5 cm

5 cm

20 cm

8 cm

10 cm

d

e

f

15 cm

3m 1m

8 cm

15 cm

3m 1m

12 cm

20 cm

5 Find the total volume of a piece of steel which has length 6 m and cross-section as shown

1 cm 8 cm

6 cm

6 A rectangular swimming pool 6 m by 5 m by 2 m deep, costs $0:50 per cubic metre per month to maintain. What is the cost of maintaining the pool over a year? 7 Concrete costs $150 per cubic metre. What will it cost to concrete a driveway 20 m long and 3 m wide to a depth of 12 cm?

G

CAPACITY

100

95

75

The capacity of a container is the amount of fluid that it may contain. The units of capacity are ² millilitres (mL)

²

litres (L)

²

kilolitres (kL)

²

25

megalitres (ML) 5

Capacity and volume are related by these conversions. 0



C

M

Y

K


216


100

1 millilitre = 1 cm3 1 litre = 1000 mL = 1000 cm3

95

75

1 kL = 1000 L = 1 m3

Note: 1 ML = 1000 kL = 1 000 000 L 25

EXAMPLE

5

0

1

Find the capacity in kilolitres of a cylindrical rain water tank of height 3 m and diameter 4 m. Volume = area of base £ height = ¼ £ 2 £ 2 £ 3 m3 = 37:6991 m3

3m

) capacity = 37:6991 kL fas 1 kL = 1 m3 g + 37:70 kL

?

4m

EXERCISE 6G 1 Find the capacity in kL of a cylindrical tank diameter 6 m and height 5 m. 2 Find the capacity of this swimming pool in kL.

1.5 m

6m 10 m

3 Find the capacity of a dam, in ML, with cross sectional area 5000 m2 and average depth of 8 m. 4 Find the capacity, in mL, of a drink cup in the shape of a cone with perpendicular height of 15 cm and diameter of 10 cm. 10 m

5 A circular swimming pool has a diameter of 10 m and a depth of 2:3 m. Calculate a its volume to the nearest m3 b the number of litres of water required to fill the pool.

2.3 m

6 A water tank is 3 m in diameter and has a height of 6:2 m. Calculate the maximum capacity of the water tank, to the nearest litre. 7

a Calculate the capacity of each water tank to the nearest litre. b Which tank has the greater capacity? By how many litres?

100

95

75

3.8 m

B A

25

2.1 m

5

1.5 m

3m



C

M

0

Y

K


217


95

75

25

8 A circular swimming pool is 1:8 metres in depth and has a radius of 1:2 metres. If the pool is exactly half-full, how many litres of water are contained in the pool? (Answer correct to the nearest litre.) 9 A large container is 3:1 m high and has a radius of 0:8 m. It contains a poisonous substance which consists of 16 hydrochloric acid. a Calculate the capacity of the container to the nearest litre. b Determine the amount of hydrochloric acid, in litres.

5

10 A rectangular tank has internal dimensions 4 m by 4 m by 2 m. Find its capacity in kilolitres.

0

11 A cylindrical tank has base area 8 m2 and height 3 m. Find its capacity in kL. 12 A lake has average depth 2 m and surface area 35 ha. Find its capacity in kL. 13 A building is in the shape of a pentagonal prism with a pentagonal pyramid on top. The base area is 140 m2 and the prism and pyramid both have perpendicular height 8 m. Find the volume of air contained in kL.

SPREADSHEET APPLICATION This spreadsheet examines how the area of a rectangle of fixed perimeter changes as the length and breadth vary. The information is entered as appears below.

Per. (cm) 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

Len. (cm) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

Bre. (cm) 10.0 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

A 1 Per. (cm) 2 20 3 20 4 20 Area (cm2) 0.00 4.75 9.00 12.75 16.00 18.75 21.00 22.75 24.00 24.75 25.00 24.75 24.00 22.75 21.00 18.75 16.00 12.75 9.00 4.75 0.00

B

C

Len. (cm) 0 =B2+0.5 =B3+0.5

D

Bre. (cm) 10 =C2-0.5 =C3-0.5

Area (cm2) =B2*C2 =B3*C3 =B4*C4

Area of rectangle with fixed perimeter 25.0

20.0

Area & Lengths

Using the fill down command the following spreadsheet and graph from the spreadsheet are produced:

15.0

100

10.0

95

5.0

75

0.0

25

0

2

4

6

Len. (cm)

8 10 12 14 16 18 20 22 Bre. (cm)

Area (cm2)

5

0



C

M

Y

K


218 100

95

75

25


Questions: 1 2 3 4 5

Describe how the area changes as the length and breadth change. When is the area a maximum? Describe the rectangle with maximum area. Try this spreadsheet using other perimeters and increments. Try some of your own rectangles.

5

0

SPREADSHEET APPLICATION – MAXIMUM/MINIMUM You are provided with enough material to make a container with surface area of 660 cm2. You are instructed to make the ‘best’ container possible. You must explain why this is the ‘best’ container. In the following spreadsheet we are assuming that the container is a square prism. This may not be the ‘best’ container but it is a starting point. The information is entered into the spreadsheet as shown below.

A B C 1 L (cm) B (cm) H (cm) 1 =(660-2*A2^2)/(4*A2) 2 1 3 =A2+1 =B2+1 =(660-2*A3^2)/(4*A3)

D

E

Vol. (cm 3) =A2*B2*C2 =A3*B3*C3

S.A. 660 660

The fill down command is used to give the following spreadsheet, and hence a scatter plot graph showing this information. A

B

C

D

E

L (cm) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

B (cm) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

H (cm) 164.50 81.50 53.50 39.25 30.50 24.50 20.07 16.63 13.83 11.50 9.50 7.75 6.19 4.79 3.50 2.31 1.21 0.17

Vol. (cm3) 164.50 326.00 481.50 628.00 762.50 882.00 983.50 1064.00 1120.50 1150.00 1149.50 1116.00 1046.50 938.00 787.50 592.00 348.50 54.00

S.A. 660 660 660 660 660 660 660 660 660 660 660 660 660 660 660 660 660 660

F

G

H

I

Volume of Square Prism (fixed surface area) 1200 1000 Volume (cm3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

800 600 400 200 0 0

2

4

6

100

8 10 12 14 16 18 Length (cm)

95

Vol. (cm3) 75

From this information the container that would hold the most has a maximum volume of 1150 3 cm with dimensions 10 cm £ 10 cm £ 11:5 cm. There may be a slightly larger volume if smaller increments are used for the changes in lengths. It may be that maximum volume is the ‘best’ container or there may be some criteria.

25

5

0



C

M

Y

K



219

100

95

75

What to do: 1 Modify the spreadsheet to check if there is a greater volume. [Use 0:1 increments.] 2 Investigate other shaped containers. For example: rectangular prism, cylinder, etc.

25

SPREADSHEET APPLICATION – THE ‘BEST’ CONTAINER

5

0

You are asked to design a container that will hold 1000 mL. It needs to be the “best” container. What shape container will you choose and what will its dimensions be? You must use a spreadsheet to explain your choice. It can be similar to the one in Investigation 5. Here are some examples.

a

b

c h

x cm x cm Square prism.

(x + 2) cm

r Cylinder

Rectangular prism with length 2 cm more than the width.

Modify your spreadsheets for other solid shapes. For example: square pyramid, rectangular prism with length 2:5 cm longer than the width, rectangular pyramid, sphere, etc.

6


Go through the chapter and write the definition of any new terms. If there are no terms that are new, write the definition of ten mathematical words from the chapter.

100


95

You should be able to: 2 calculate the areas of triangles, rectangles and composite figures 2 draw and recognise solid figures and nets of solids 2 calculate the surface area of right prisms, and square and rectangular pyramids 2 calculate the volume of right prisms, cylinders, pyramids, cones and spheres 2 calculate capacity.

75

25

5

0



C

M

Y

K


220 100

95

75

?


6

DIAGNOSTIC TEST

1 The area of this triangle is A 7 cm2 B 14 cm2

C 1:75 cm2

3.5 cm

D 2 cm2 4 cm

25

5

0

2 The shaded area of this figure is A 32 m2 B 2 C 26 m D

23 m2 60 m2

4m 3m

3 The area of this figure is A 231 m2 C 115:5 m2

B D

168 m2 32 m2

5m 5m 6m

7m

8m 6m

4 A solid with the same cross section throughout its length is a A prism B pyramid C cone 5 A square-based pyramid has sides of A 4 squares and 1 triangle C 4 rectangles and 1 triangle

B D

D

4 triangles and 1 square 6 squares

6 The surface area of this solid is A 300 cm2 B 21 cm2 C 140 cm2 D 280 cm2

5 cm 6 cm

10 cm

7 The surface area of this square pyramid is A 120 cm2 B 1440 cm2 2 C 384 cm D 624 cm2

sphere

10 cm

12 cm

8 The volume of this cylinder is closest to A 565 cm2 B 266 cm2 C 267 cm2 D 259 cm2

5 cm

12 cm

9 The volume of a sphere diameter 15 cm is closest to A 14 137 cm3 B 14 138 cm3 C 1767 cm3

100

95

D

3

1768 cm

10 The volume of this square pyramid is closest to B 260 cm3 A 259 cm3 C 266 cm3 D 267 cm3

75

8 cm 25

5

10 cm 0



C

M

Y

K



95

75

25

221

11 The volume of a cone radius 5 cm and height 8 cm is closest to A 629 cm3 B 628 cm3 C 210 cm3

D

209 cm3

12 The capacity in kL, of a cubic tank side length 8 m is closest to A 64 kL B 512 kL C 640 kL

D

512 000 kL

If you have any difficulty with these questions, refer to the examples and questions in the exercise indicated. Question 1, 2, 3 4, 5 6, 7 8, 9 10, 11 12 Section A B D E F G

5

0

?

6A

REVIEW SET

1 Find the area of these figures a

b 7 cm

3m 10 m

6 cm

C

c

d 20 cm

12 m B

A 15 m

30 cm 50 cm

AB = 30 m D

2 Draw a net of a square pyramid. 3 Draw and name a prism with a rectangular cross-section. 4 Which solid has this net?

100

5 Draw an isometric projection of this solid

95

75

25

5

0



C

M

Y

K


222


100

6 Calculate the surface area of these solids a

95

b

75

9.2 cm 13 cm 12 cm

25

5 cm

5

11.6 cm

4 cm

7 Calculate the volume of these solids a

0

b

20 cm

4 cm 8 cm

6 cm 10 cm

8 Find the capacity in mL of an ice cream cone radius 7 cm and perpendicular height 14 cm.

?

6B

REVIEW SET


b 3.9 cm

4 cm

6 cm

1.6 cm

c

d 12 m 10 m

8m

10 m

4m

6m

7m

6m 8m

6m

2 Draw a net of a rectangular prism. 3 Draw and name a prism with a circular cross-section.

100

4 Which solid has this net?

95

75

r 25

5

0



C

M

Y

K



223


95

75

25


5

b 12 cm

25.8 cm

0

7.6 cm 8 cm


b

4.1 m

6 cm

5.2 m 15 cm 8 cm

8 Find the capacity in ML, of a dam with cross-sectional area of 6000 m2 and an average depth of 10 m.

?

6C

REVIEW SET


b

3.2 m

8.2 cm

1.9 m

c

d 20 m 100

5 cm 35 m

30 m

95

5 cm 75

12 cm

15 m 10 m

2 Draw a net of a cylinder.

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6.3 m

4 cm 10 cm

7.8 m

15.3 m

7 Calculate the volume of these solids a 6.9 cm

6 cm

b 11 cm

9 cm

8 Find the capacity in kL of a cylindrical tank diameter 8 m and height 4 m.

?

6D

REVIEW SET

1 Find the area of these figures a 21.8 mm

b 2.6 m

15.3 mm

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4.9 m

c

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d B

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50 m

12 cm 8 cm

40 m

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10 cm A

AB = 100 m

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b 15.3 m 12.1 m

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CHAPTER

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Displaying data

AREA OF STUDY

This chapter is about using tables and graphs to display data. The main mathematical ideas investigated in this chapter are: 8 frequency distribution tables 8 cumulative frequency 8 relative frequency 8 grouped data, class, class centre 8 stem and leaf plots 8 sector graphs 8 divided bar graphs 8 dot plots 8 column and bar graphs 8 line graphs 8 radar charts 8 misleading graphs 8 frequency histograms and polygons 8 cumulative frequency graphs.

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DISPLAYING DATA (Chapter 7)

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In Chapter 5 we discussed the planning and management of data collection. We will now look at the recording and displaying of the data that has been collected. Raw data, when collected by observation or experiment, is usually in random order. It is difficult to gain information from the data when it is in this form.

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One of the aims of statistics is to rearrange and condense the raw information into a form which is more easily read so that patterns and characteristics may be identified, conclusions drawn and inferences may be made.

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FREQUENCY DISTRIBUTION TABLES

A frequency table is a table which displays the frequency (number of times it occurs) for each of the categories of the data.

EXAMPLE

1

The method by which the students of a class travelled to school on a particular day is recorded below, using the code walk (W), cycle (C), bus (B), train (T) and motor car (M) WCBTC BBBWB BBCBT CMCBT MMTMM MTCCB Rearrange this information into a frequency distribution table using a tally column. Method of travel Walk Cycle Bus Car Train

Tally jj © jj © jjjj © © © © jjjj jjjj © j © jjjj © © jjjj Total

Frequency 2 7 10 6 5 30

Note: ² Every 5th tally mark is placed through the 4 preceding tally marks. (This makes counting easier.) ² The frequency is the total of the tally marks, that is, the number of times a particular mode of travel is used. ² Always check that the total of the frequency column is the same as the number of observations recorded.

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EXERCISE 7A

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1 The colours of cars passing the front of a school in a 30 minute period are recorded below using the code: white (W), blue (B), grey (G), red (R), other (O) BWROW BRGWO BGRWW GBRWO GBRWG BRRGW BRGOW BWGRB WWBRG WBRWB

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a Rearrange this information into a frequency distribution table using a tally column. 0



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DISPLAYING DATA (Chapter 7) 100

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b How many cars passed the front of the school in this time period? c What was the most popular colour in this survey? d Calculate the percentage of each colour. 2 The minimum daily temperatures were recorded for a month as follows (in o C) 22 21 20 22 23 22 21 23 19 21 21 23 21 20 22 21 20 20 22 20 23 19 20 19 20 18 19 17 18 22 a Organise this information into a frequency distribution table using a tally column. b On how many days was the temperature recorded? c On how many days did the lowest minimum temperature occur? d How many days had a minimum temperature of 20o C? e What month of the year could this be? Discuss. 3 The eye colours of a group of students are recorded below using the code: blue (B), brown (b), green (G), grey (g) BbgBb bGbBg bBgbG bbBgB BgbBG BbgbB a Organise this data into a frequency distribution table using a tally column. b How many students were observed? c What is the most common colour in this sample? 4 The marks out of 10 for a mental arithmetic test were 9 10 4 9 7 9 3 2 9 5 8 7 1 8 3 6 2 10 0 8 7 4 6 1 6 5 7 7 5 3 3 5 8 2 6 7 a Rearrange the data into a frequency distribution table using a tally column. b How many students sat for the test? c How many scored full marks? d How many scored zero? e What percentage of students scored 5 or more?

5 9

9 8

INVESTIGATION 1: TRAVELLING TO SCHOOL Investigate the modes of travel of the students in your class. Present the data in a frequency distribution table and write a short report on your findings. W ould the results for your class be typical of the whole school? Comment.

B

CUMULATIVE AND RELATIVE FREQUENCY

A convenient way of answering many of the questions in the above exercise is to add a cumulative frequency column and a relative frequency column.

The cumulative frequency of each score is found by adding the frequencies of all the scores up to and including that particular score. The relative frequency of each score is the fraction of times that the score occurs.

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For the purpose of comparison, it is often convenient to express this fraction as a percentage (sometimes called the percentage relative frequency).

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EXAMPLE

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Copy the frequency distribution table and i add a cumulative frequency column ii add a relative frequency column How many packets contained i 40 or less ii less than 38 Candychocs? What fraction of packets contained i 37 ii 40 Candychocs? What percentage of packets contained i 38 ii 39 Candychocs?

Number of Candychocs 36 37 38 39 40 41 42

Frequency 1 5 8 13 7 4 2

a Number of Candychocs

b

Frequency

Cumulative Frequency

36

1

1

37

5

(1 + 5 =)

6

38

8

(6 + 8 =)

14

39

13

(14 + 13 =)

27

40

7

(27 + 7 =)

34

41

4

(34 + 4 =)

38

42

2

(38 + 2 =)

40

Total

40

Number of packets containing 40 or less Candychocs = cumulative frequency of the score 40 = 34

ii

Number of packets containing less than 38 Candychocs = number of packets containing 36 or 37 Candychocs = cumulative frequency of 37 =6 5 40

or

i

d

i 20%

1 8

ii

1 40 5 40 8 40 13 40 7 40 4 40 2 40

=

2:5%

=

12:5%

=

20%

=

32:5%

=

17:5%

=

10%

=

5%

1

i

c

Relative Frequency

100%

7 40

ii 32:5% 100

?

EXERCISE 7B 1

The numbers of children in the families of a class of students is summarised in the given frequency table. a Copy the frequency distribution table and i add a cumulative frequency column ii add a relative frequency column.

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Number of children 1 2 3 4 5

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Frequency 3 9 7 4 2

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b c d

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How many families had What fraction of families had What percentage of families had

3 or less 3 2

i i i

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less than 5 children? 4 children? 5 children?

ii ii ii

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0

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c d

Copy the frequency distribution table and i add a cumulative frequency column ii add a relative frequency column. In how many games did she score i 7 or less ii less than 10 goals?

In what fraction of games did she score i 8 In what percentage of games did she score i 6

frequency 4 3 7 5 0 3 2 1

ii 5 goals? ii 11 goals?

3 The contents of 60 boxes of matches were counted and the results are shown below 51, 50, 48, 48,

50, 48, 50, 53,

50, 49, 49, 52,

48, 51, 50, 49,

49, 50, 51, 50,

50, 53, 52, 51,

49, 48, 50, 53,

53, 49, 49, 48,

50, 51, 48, 50,

48, 50, 52, 51,

51, 52, 50, 49,

49, 49, 51, 50,

50, 50, 49, 53,

51, 52, 50, 48:

S

49, 52, 51, 50,

HE

5

goals/match 5 6 7 8 9 10 11 12

MA TC

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The number of goals scored by the goal shooter in a netball team for a season is shown in the frequency table

a Organise this data into a frequency distribution table and add a cumulative frequency column and a relative frequency column. b How many matchboxes contained i exactly 50 ii less than 50 iii 50 or less iv more than 50 matches? c What percentage of matchboxes contained i 49 ii 50 iii 51 iv 50 or less v 50 or more matches?

C

GROUPED FREQUENCY DISTRIBUTION TABLES

Consider the following marks (out of a possible 100) of 50 students in a test 70 68 22 74 65

31 57 45 51 55

49 45 53 48 51

54 56 29 69 42

55 54 61 75 94

74 85 20 61 64

50 44 13 40 38

34 66 52 41 3

60 56 41 33 51

68 60 50 62 79

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If we arranged this data into a frequency distribution table, we get Score Frequency

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3 13 20 22 29 31 33 34 38 40 41 42 44 45 48 49 50 51 52 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 1 2 3 1

25

Score 53 54 55 56 57 60 61 62 64 65 66 68 69 70 74 75 79 85 94 Frequency 1 2 2 2 1 2 2 1 1 1 1 2 1 1 2 1 1 1 1



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You can see that we are unable to obtain very much information from the rearrangement of the raw data into this form, apart from getting a rank order of the scores. It was also very time consuming to prepare.

Class 0-9 10 - 19 20 - 29 30 - 39 40 - 49 50 - 59 60 - 69 70 - 79 80 - 89 90 - 99

In cases such as these we can condense the data into a less complex form by grouping it into classes, or class intervals, and finding the frequency of each class, again using a tally column. For example:

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Note that each class has the same size as each includes 10 possible scores. For example, the class 0 - 9 includes the scores 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. We could say the scores are grouped in tens.

Tally j j jjj jjjj © jjjj © jjjj © © © jjjj © jjjj jjjj © © © j © jjjj jjjj © © jjjj j j Total

Frequency 1 1 3 4 9 14 11 5 1 1 50

From this distribution we can now easily determine that, for instance, 28% (14 out of 50) of the scores lie between 50 and 59, 36% (18 out of 50) scored less than 50, etc. We could also condense the data further by grouping as follows: Again each class interval has the same size as each includes 20 possible scores. We could say that the scores are grouped in twenties.

Class 0 - 19 20 - 39 40 - 59 60 - 79 80 - 99

Tally jj © jj © jjjj © © © © © © © jjj © jjjj jjjj jjjj jjjj © © © © © j © jjjj jjjj jjjj jj Total

Frequency 2 7 23 16 2 50

Hence, if the data we are collecting has a large range of values then we can reduce the complexity of the distribution and put it into a more useable form by grouping it into classes, sometimes called class intervals. In order to make interpretation of the table as easy as possible, when grouping data it is desirable that: ²

the classes should not overlap. For example, if we had formed classes 0 - 10, 10 - 20, 20 - 30, .... then we would not know in which class to put the score 10, or 20 etc.

²

there should be no gaps between class intervals. For example, if we were organising the above data into classes 0 - 4, 5 - 9, 10 - 14, etc. then, even though there would be no scores in the class interval 5 - 9, this class would be included.

²

each class interval should be the same size. (When there are extremely low or high scores it may be convenient to make the bottom and top class intervals open, for example, less than 300, 300 - 319, 320 - 329, ......, 390 - 399, 400 or more. However any further statistical analysis then becomes very difficult.)

The disadvantage of using a grouped frequency distribution is that some details are lost. For example, in the first distribution above there are 9 scores in the 40 - 49 class but we cannot determine from the table whether the scores are all at one end of the interval or evenly spread through it.

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EXERCISE 7C 1 The marks scored by 30 students in a test are shown below 28 51

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35 32

8 57

61 24

45 61

68 44

47 34

23 42

52 60

44 39

22 47

36 32

20 76

44 53

54 62

a Organise this data into a frequency distribution table, using classes i 0 - 4, 5 - 9, 10 - 14, 15 - 19, .... ii 0 - 9, 10 - 19, 20 - 29, 30 - 39, ...... iii 0 - 19, 20 - 39, 40 - 59, ...... iv 0 - 49, 50 - 99 b Which of the above groupings gives the clearest “picture” of the results of the test? Discuss any advantages/disadvantages of each grouping. c Explain why the use of the following groupings of scores would be unsatisfactory: i 0 - 5, 5 - 10, 10 - 15, ...... ii 0 - 9, 20 - 29, 30 - 39, 40 - 49, ...... iii 0 - 4, 5 - 14, 15 - 29, 30 - 34, ...... 2 A random sample of light globes was tested and the time (£ 100 hours), to the nearest 100 hours, each one lasted is recorded below 10 12 10

9 9 10

13 7 9

5 11 11

2 15 13

11 10 12

8 11 5

12 10 11

4 4 8

12 8 16

7 8

13 11

8 12

14 14

10 13

a Organise this data into a frequency distribution table using class intervals i 0 - 4, 5 - 9, 10 - 14, 15 - 19 ii 1 - 5, 6 - 10, 11 - 15, 16 - 20 b Would the following class intervals be suitable to use: i 0 - 3, 3 - 6, 6 - 9, ...... ii 0 - 3, 4 - 7, 8 - 11, ...... iii 0 - 9, 10 - 19, 20 - 29, ...... ? c How many light globes were tested? d Using both tables from a where necessary, find how many light globes had a recorded life of i less than 1000 hours ii more than 1000 hours iii 1000 hours? 3 A group of Year 11 students decided to have a competition to find out who could hit a golf ball the furthest. Below is recorded, to the nearest metre, the best shot of each of the students 244 220 232 195 242 200 229 205 213 224 216 250 227 186 207 236 225 212 227 230 209 235 234 251 195 206 221 190 198 210 222 235 230 209 254 193 203 228 191 242 204 218 212 218 239 223 252 215 220 198 207 205 226 235 202 196 188 224 245 230 a Copy and complete the following table:

Distance 180 - 189 190 - 199 200 - 209 210 - 219 220 - 229 230 - 239 240 - 249 250 - 259

Tally

Frequency

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b How many golfers entered the competition? c Add a cumulative frequency column and a relative frequency column. d How many golfers had the length of their best shot recorded as i less than 200 m ii less than 230 m? e What percentage of golfers had the length of their best shot recorded as i 250 m or more ii 220 m or more but less than 230 m iii less than 200 m?

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4 A marriage counsellor asked his clients to keep a record of the number of arguments they had in a one week period. The results collected were

Number of arguments 0-2 3-5 6-8 9 - 11 12 - 14

a How many different responses are included in each class interval? b How many clients responded to the counsellor’s request? c What percentage of these clients had less than 3 arguments in the week? d What percentage had more than 8 arguments? e Can you determine how many clients had 6 arguments? f Can you determine the lowest and highest number of arguments?

Frequency 8 18 10 3 1

5 A nurseryman takes a random sample of seedlings Height (mm) Frequency and measures their height to the nearest millime300 - 324 3 tre. The results are recorded alongside 325 - 349 18 a How many different heights, to the nearest 350 - 374 47 millimetre, are possible in each class interval? 375 - 399 32 400 - 424 14 b How many seedlings are there in this sample? 425 449 6 c Can you determine how many plants were 350 mm high? d Can you determine the smallest and largest heights gained by the plants in this sample? e Copy the table and add a cumulative frequency column and a relative frequency column. f How many seedlings had a recorded height which was less than i 350 mm ii 400 mm? g What percentage of seedlings had grown to 425 mm or more? h Any seedlings which have a recorded height less than 325 mm are discarded by the nursery. What percentage of seedlings will be discarded? 100

D

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STEM AND LEAF PLOTS

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The main disadvantage of grouped distribution tables is that some information is lost. For example, it is not possible to determine the lowest and highest scores from the table.

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Stem and leaf plots are another way of displaying information. They are used to group and rank data to show the range and distribution of the data.

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The stem is the first digit (or digits) and the leaf is the final digit of a number. 95

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A stem may have any number of digits but a leaf has exactly one.

EXAMPLE

1

The results in a mathematics class test are given below 43, 45, 46, 22, 65, 65, 23, 53, 45, 26, 46, 61, 51, 57, 55, 55, 66, 57, 42, 41, 63, 70, 57, 65, 48, 23, 67, 62, 70, 46 a Draw a stem and leaf plot to represent this data. b What are the lowest and highest scores? c How many students scored i 46 ii 50 iii 70 iv a mark in the sixties? a In this stem and leaf plot, the tens digit forms the stem and the units digit forms the leaf. This means that for the mark 45, the stem is the 4 and the leaf is the 5. Stem Leaf 2 2363 This row represents the numbers 22, 23, 26 and 23. 3 There were no scores in the thirties. 4 356562186 5 3175577 6 55163572 7 00 The leaves are now put into ascending numerical order.

Stem 2 3 4 5 6 7

Leaf 2336 1 1 1 0

23556 668 355777 2355567 0

b Lowest score = 22, highest score = 70. c

i ii iii iv

Number Number Number Number

of of of of

students who students who students who students who

scored scored scored scored

46 =3 50 =0 70 =2 a mark in the sixties = 8 100

With a stem and leaf plot ² ² ² ²

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all of the data is used and displayed the largest and smallest measurements can be found the clustering of data can be more easily seen the length of the leaf row indicates the number of scores belonging to that stem.

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EXERCISE 7D 1

a Draw a stem and leaf plot, using the stems 3, 4, 5 and 6, for the scores

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b c d e f

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32, 45, 37, 65, 48, 57, 33, 42, 59, 62, 47, 51, 50, 47, 36, 68, 60, 50, 63, 47 What are the lowest and highest scores? How many times does the score i 50 ii 40 occur? Which scores occurs the most often? How many scores are in the sixties? How many scores are less than 50?

0

2

a Draw a stem and leaf plot, using the stems 5, 6, 7, 8 and 9, for the scores 63, 54, 87, 50, 61, 74, 77, 81, 50, 69, 55, 56, 68, 75, 87, 51, 60, 85, 62, 54, 78, 54, 69, 68, 50, 60, 57, 65, 85, 74, 90, 58, 67, 81, 90, 54, 58, 63, 78, 54 b What are the lowest and highest scores? c How many times does the score i 50 ii 60 iii 70 iv 80 occur? d Which score occurs the most often? e How many scores are in the fifties? f How many scores are i less than 60 ii 80 or more?

3

a Draw a stem and leaf plot, using the stems 12, 13, 14 and 15, for the scores 132, 154, 124, 156, 125, 145, 130, 141, 126, 151, 140, 139, 128, 145, 126, 146, 128, 130, 140, 158, 154, 142, 149, 145, 150, 126, 130, 148, 158, 145 b c d e

What are the lowest and highest scores? How many times does the score i 150 Which score occurs the most often? How many scores are i less than 130

ii 158

iii 120 occur?

ii 150 or more?

4 Draw stem and leaf plots for the scores a 16, 24, 13, 8, 22, 4, 5, 26, 14, 10, 2, 20, 11, 23, 8, 7, 24, 8, 12, 9 b 48, 62, 74, 43, 60, 79, 47, 40, 63, 71, 40, 62, 68, 47, 49, 61, 48, 42, 71, 65 c 263, 258, 281, 265, 274, 270, 283, 270, 254, 265, 280, 274, 256, 280, 251, 263, 280, 278, 259, 260 5 The following stem and leaf plot shows the time spent (hours) watching TV, by a group of students during one week a b c d e f

Stem 0 1 2 3

Leaf 356 022 245 011

8 3 5 4

9 5559 578 6

How many students were surveyed? What was the least and greatest number of hours of TV watched? How many students watched less than 10 hours of TV a week? How many students watched more than 30 hours of TV a week? Draw a grouped frequency table to represent this data, using class intervals of i 1 - 5, 6 - 10, 11 - 15, etc. ii 1 - 9, 10 - 19, 20 - 29, etc. Comment on the advantages and disadvantages of the stem and leaf plot compared with the grouped frequency tables.

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EXAMPLE

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2

The number of people who live in each house in a particular street is recorded below 4, 2, 3, 4, 5, 2, 4, 3, 2, 5, 1, 4, 5, 5, 3, 7, 2, 3, 4, 4 Draw a stem and leaf plot for this information.

5

Stem 0

0

Leaf 12222333344444455557

This stem and leaf plot gives no more information that an ordered list of numbers. In order to show the information in a more meaningful way the stem is split as follows: The stem 0 will contain the leaves 0, 1, 2, 3, 4. The stem 0¤ will contain the leaves 5, 6, 7, 8, 9. The stem and leaf plot will then become Stem 0 0¤

Leaf 122223333444444 55557

It can be seen from the second plot that most households have 4 or less people.

6 For a group of Year 11 students, the number of hours per week spent doing homework is shown below 31, 30, 28, 32, 29, 32, 32, 29, 32, 31, 29, 27, 29, 31, 27, 32, 32, 31, 29, 28, 29, 33, 31, 27, 27, 26, 27, 29, 18, 32, 30, 30, 28, 31, 28, 28, 27, 28, 33, 31 a b c d

E

How many students were surveyed? Draw a stem and leaf plot for this data using stems 1, 2 and 3. Draw another stem and leaf plot for the data using stems 1, 1¤ , 2, 2¤ , 3, 3¤ . What information is shown in the second stem and leaf plot that is not evident in the first?

SECTOR GRAPHS

The aim of using graphs to display data is to present the information in a way which is visually attractive. Although there is some loss of detail, compared with a table, it is often easier to see trends and relationships. Often both a table and a graph are used to convey the maximum amount of information.

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In a sector graph (sometimes called a pie chart), each category is represented by a sector of a circle. 25

The area of each sector is proportional to the size of each category and hence each sector angle is proportional to the size of each category.

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EXAMPLE

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1 The sector graph shows the method of travel to school by a group of 60 students.

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Car

Measure the sector angle for each category. What fraction of the students travel by i bus ii train iii car iv walk v bicycle? How many students travel by i bus ii train iii car iv walk v bicycle?

a b

Walk 25

Train Bicycle

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c Bus

a Measure each of the sector angles with a protractor

b Hence, the fraction of students who travel by i bus = = iii car = =

180 360 1 2

ii train =

48 360 2 15

iv walk =

=

=

60 360 1 6

42 360 7 60

v

Category bus train car walk bicycle Total

bicycle = =

Sector angle 180o 60o 48o 42o 30o 360o

30 360 1 12

c Hence, the number of students who travel by

?

i

bus =

180 360

£ 60

= 30

ii

train =

60 360

£ 60

= 10

iii

car =

48 360

£ 60

=8

iv

walk =

42 360

£ 60

=7

v

bicycle =

30 360

£ 60

=5

Total

= 60

EXERCISE 7E 1

Eye colour of class

a

Measure the sector angle for each category.

b

What fraction of students have i blue ii brown iii grey iv green eyes?

Grey Brown Green

c

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If there are 24 students in the class, find the number of students who have i blue ii brown iii grey iv green eyes

Blue



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Measure the sector angle for each category. If the store sold 36 dishwashers, how many of each brand did it sell?

2 a b

Brands of dishwashers sold Westinghouse

Hoover

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25

Bosch

5

Dishlex

ASEA

0

3

What was the most common method of entry? Measure the sector angle for each category. What percentage of illegal entries were through i a forced door ii an open door? If there were 10 500 illegal entries in 1999, how many were through i a forced window ii an open window?

a b c

How burglars got into homes in 1999 Open window Open door

d Other

Force window

Force door

EXAMPLE

2

In a survey, 50 salesmen were asked the country of manufacture of their car. The results were a Write down the fraction of cars built in each country. b For a sector graph, calculate the size of the sector angle for each country. c Draw a sector graph to illustrate this information. a The fraction of cars manufactured in Australia = Germany =

17 50 3 50

Japan = Other =

19 50 5 50

¡ =

1 10

¢

Korea=

6 50

Country Australia Japan Korea Germany Other

¡

=

3 25


¢ 100

b Hence the sector angle for 17 50

95

19 50

£ 360 Australia = = 122:4o + 122o

Japan = £ 360 = 136:8o + 137o

3 Germany = 50 £ 360 = 21:6o + 22o

5 Other = 50 £ 360 o = 36

6 50

Korea = £ 360 = 43:2o + 43o

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c

Country of manufacture of cars

Korea

95

Germany 75

Japan Other 25

5

0

Australia

4

5

Meat beef lamb chicken pork

Sales (kg) 160 120 90 80

The table shows the quantities of meats sold by a butcher. a Write down the fraction of each type of meat sold. b For a sector graph, calculate the size of the sector angle for each category of meat. c Draw a sector graph to illustrate this information.

The table shows the percentage of the workforce in each of the industry categories given.

Industry Agriculture Manufacturing Construction Hospitality Finance Public Administration

For a sector graph, calculate the size of the sector angle for each category. Draw a sector graph to illustrate this information.

a b

6 Channel 9 7 10 ABC SBS

TV audience (%) Sydney Nationally 36 33:6 26:4 29:1 21:3 20:4 13:2 13:8 3:1 3:2

% of workforce 5 26 12 35 16 6

The table shows the percentage of the TV audience gained by each of the major channels, both for Sydney and nationally. Draw a sector graph to illustrate the set of results a for Sydney b nationally

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95

Many different types of graphs can be made using the chart option (or equivalent) of a spreadsheet package. For example, we could use a spreadsheet to draw a sector graph for the data in Example 2 of the previous exercise. We enter the data in the first two columns of a spreadsheet as shown.

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Highlight the table. Select the Chart Option. Choose Pie Chart

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Other Germany Australia Korea

25

Japan

5

0

Investigate the options available such as the labelling, including the table, etc. Use a spreadsheet to draw pie charts for the data in the previous questions 4, 5 and 6.

Sector graphs are useful, especially for categorical data, to illustrate how one thing is divided into parts. They show the sizes of each category relative to each other and relative to the whole group. The disadvantages are that there is quite a lot of actual measurement of angles required, quite a lot of calculations needed and the answers in most cases, are approximations.

F

DIVIDED BAR GRAPHS

Divided bar graphs are also useful for illustrating how a whole thing is divided into parts. They have all the advantages of sector graphs, apart from maybe being less attractive visually. The calculations and measurements required are easier.

EXAMPLE

1

In a survey, 50 salesmen were asked the country of manufacture of their car. The results were Country Frequency a b a

Australia 17

Japan 19

Korea 6

Germany 3

Other 5

Calculate the percentage of cars manufactured in each country. Hence, draw a divided bar graph to illustrate this data. 100

Country Australia

Frequency 17

Japan

19

Korea

6

Germany

3

Other Total

5 50

Percentage of total 17 50 £ 100% = 34% 19 50 6 50 3 50 5 50

95

£ 100% = 38%

75

£ 100% = 12% £ 100% = 6%

25

£ 100% = 10%

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100

b Step 1: 95

Step 1:

75

25

5

0

Draw a rectangular bar of convenient length, i.e., one that can easily be divided into percentages. ( 100 mm is a convenient length.) Divide the bar into parts in the proportion of the percentages found above, that is, length of part representing = 34 mm Australia = 34% of 100 mm = 38% of 100 mm = 38 mm Japan = 12% of 100 mm = 12 mm Korea = 6 mm Germany = 6% of 100 mm = 10% of 100 mm = 10 mm Other Hence, the divided bar graph is Country of manufacture of cars a ali str Au

?

n pa Ja

rea Ko

y an rm ther Ge O

EXERCISE 7F 1

Meat beef lamb chicken pork

Sales (kg) 160 120 90 80

The table shows the quantities of meats sold by a butcher. a

Calculate the percentage of each type of meat sold.

b

Draw a divided bar graph, of length 100 mm, to illustrate this information.

2 The table shows the percentage of the workforce in each of the industry categories given. Draw a divided bar graph, of length 100 mm, to illustrate this information.

3 Channel 9 7 10 ABC SBS

TV audience (%) Sydney Nationally 36 33:6 26:4 29:1 21:3 20:4 13:2 13:8 3:1 3:2

Industry Agriculture Manufacturing Construction Hospitality Finance Public Administration

% of workforce 5 26 12 35 16 6

The table shows the percentage of the TV audience gained by each of the major channels, both for Sydney and nationally. Draw a divided bar graph, of length 200 mm, to illustrate the set of results a for Sydney b nationally

100

95

75

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100

G

95

DOT PLOTS

75

A dot plot is a column or row of dots. The number of dots is the frequency of the variable. 25

EXAMPLE

5

0

1

Brand Dishlex Hoover Westinghouse Bosch ASEA

The number of each brand of dishwasher sold by an electrical store in one week is shown in the table Draw a dot plot to illustrate this data. The use of 2 mm grid paper makes this easier.


Dishlex Hoover Westinghouse

D is hl ex H W o es ov tin e gh r ou s Bo e sc h AS EA

Bosch

EXAMPLE

ASEA

2

The colours of cars passing the front of a school in a 30 minute period are recorded below using the code: white ( W), blue (B), grey (G), red (R), other (O) BWROW BRGWO BGRWW GBRWO GBRWG BRRGW BRGOW BWGRB WWBRG WBRWB Draw a dot plot from the information given. Working through the data, as for a tally column in a frequency table, we get

Car colours

White Blue Grey Red Other 100

?

95

EXERCISE 7G 1 A survey of the number of children in 25 families is summarised alongside: Draw a dot plot for this information.

Number of Children 1 2 3 4 5

Frequency (No. of families) 3 9 7 4 2

75

25

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244 100

95

75

25

2


Outcome 2 heads 1 head 0 heads

Draw a dot pot for this data.

3 The eye colours of a group of students are recorded below using the code: blue (B), brown (b), green (G), grey (g)

5

0

The results when two coins are tossed 25 times are:

Frequency 8 11 6

BbgBb bGbBg bBgbG bbBgB BgbBG BbgbB Draw a dot plot for this data. 4 The method by which the students of a class travelled to school on a particular day is recorded below, using the code: walk (W), cycle (C), bus (B), train (T) and motor car (M) WCBTC BBBWB BBCBT CMCBT MMTMM MTCCB Draw a dot plot for this data. 5 In a survey, 60 salesmen were asked the country of manufacture of their car. The results were

Country Australia Japan Korea Germany Other

Draw a dot plot.


6 The marks out of 10 for a mental arithmetic test were 9 0

10 8

4 7

9 4

7 6

9 1

3 6

2 5

9 7

5 7

8 5

7 3

1 3

8 5

3 8

6 2

2 6

10 7

5 9

9 8

Draw a dot plot. 7 Was the dot plot a convenient and time efficient method to use for 5 and 6? Comment. 8 Can you draw a dot plot for the information given below? Comment.

Channel 9 7 10 ABC SBS

TV audience (%) Sydney 36 26:4 21:3 13:2 3:1 100

95

A dot plot is a convenient and efficient method for illustrating a small set of data consisting of a small number of categories/possible outcomes. It also has the advantage that it can be used directly for unsorted data. For large sets of data, the dot plot is very time consuming to prepare, and conversely, completed dot plots are difficult to read. It can only be used when the outcomes can be counted.

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H

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75

COLUMN AND BAR GRAPHS

Instead of a column or row of dots it is often easier to draw a plain column, or row.

1

Brand Dishlex Hoover Westinghouse Bosch ASEA

5

?

10 8 6 4 2

AS EA

Bo sc h

r ve oo H

gh ou se

0

D is hl ex

Notes: ² The rectangular columns are the same width and are evenly spaced. ² The vertical axis shows the scale. ² The number of dishwashers is shown by the height of each column.

Number of dishwashers (frequency)

The number of each brand of dishwasher sold by an electrical store in one week is shown in the table Draw a vertical column graph to illustrate this information.

0


W es tin

EXAMPLE

25

EXERCISE 7H 1 The table shows the average percentage of protein in certain foods Make a vertical column graph to illustrate this information.

2

Month Jan/Feb Mar/Apr May/Jun Jul/Aug Sept/Oct Nov/Dec

Sales ($ million) 6:4 7 2:7 3:6 2:9 5:1

Food Beef Rice Eggs Fish Milk

% Protein 70 80 95 70 85

The sales of a real estate company, for the months given, are shown in the table. Draw a column graph to illustrate this data.

3 The ages of the vehicle occupants killed on NSW roads in one year were Represent this information using a column graph.

100

Age Group 0-9 10 - 19 20 - 29 30 - 39 40 - 49 50 - 59 60 - 69 70+

Number Killed 42 110 175 65 80 55 40 60

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When the rectangles are drawn horizontally the graph is usually called a bar graph.

95

75

EXAMPLE

2

City Sydney Melbourne Canberra Brisbane Perth Adelaide Hobart

The average house price for the capital cities of Australia is shown alongside: Illustrate this information on a horizontal bar graph.

25

5

0

Ave. Price ($000’s) 290 240 210 210 190 170 140

Sydney

City

Melbourne Canberra Brisbane Perth Adelaide Hobart

Average price ($000’s) 0

4

Sport AFL Soccer Cricket Rugby League Rugby Union Netball Hockey Indoor Cricket Basketball

50

100

Number of injuries/year 270 160 99 88 58 40 32 30 23

150

200

300

The table alongside shows the number of injuries in each of the 8 main sports played in Australia. Illustrate this information on a horizontal bar graph.

5 The number of milligrams of cholesterol per 100 g of various foods is shown in the table alongside: Draw a horizontal bar graph to illustrate this data.

6

250

Food Lobster Beef Chicken Duck Prawns

Cholesterol (mg/100g) 70 80 105 110 145

100

95

75

Student Reiko James Wendy Hassan Vincent

English 75 65 89 54 62

Maths 84 76 65 78 90

Science 68 84 60 75 88

The English, Mathematics and Science exam marks for 5 students are shown in the table: Draw a column graph to show these results by grouping the subjects together for each student. Following is the start of the graph for you.

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100

English

100

95

Exam mark

Maths

75

25

Science 50

......

...... Student

0

5

Reiko

James

7 The number of absences each day of Lifesaving Week for 2 roll classes is shown in the table Draw a column graph grouping the classes together for each day, for comparison purposes. This has been started below: Day Mon Tue Wed Thurs Fri

No. of absences 11R - 1 11R - 2 3 2 3 3 5 6 6 4 2 3

No. of absences

0

......

6 5 4 3 2 1 0

11R–1 11R–2

.... Mon

....

....

....

Days of week

SPREADSHEET APPLICATION The vertical column graph below was made using the chart option of a spreadsheet. Type the information from Example 1 into the spreadsheet as for a table, highlight the table, select CHART and choose column graph. Investigate all the options.

The horizontal bar graph below, representing the data in Example 2, was also made using a spreadsheet and the CHART option. 100

95

75

25

Use a spreadsheet to draw the graphs in Exercise 7H. Print some of them and paste them in your book.

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I

95

LINE GRAPHS

75

EXAMPLE 25

1

The table alongside shows the variation in price of Minaus shares for one week in 1999. Draw a line graph to show this information.

5

0

Day Price (cents)

Mon 45

Tue 48

Wed 55

Thu 52

Fri 46

Price of Minaus shares Price (cents)

The graph is drawn by plotting the points corresponding to the information given in the table and the joining these points.

60 40 20

Day of week 0

Mon

Tue

Wed

Thu

Fri

Line graphs are useful for showing upward and downward trends in a quantity. They are also useful for finding “in between” values when the quantity on the horizontal axis is continuous.

?

EXERCISE 7I 1 The table shows the temperature (o C) variation on a summer’s day in Sydney, from 7 am to 7 pm Time Temperature

7 13

8 14

9 16

10 17

11 19

12 22

1 25

2 28

3 27

4 27

5 24

6 22

7 18

Draw a line graph to illustrate this data. 2 The monthly sales figures for a computer firm are Month Sales ($000’s)

J 54

F 36

M 30

A 30

M 28

J 34

J 25

A 26

S 31

O 38

N 44

D 56

Draw a line graph for this information. 3 The percentage of each age group of the male population who are overweight is shown below Age Group % Overweight

10 - 19 25

20 - 29 28

30 - 39 39

40 - 49 42

50 - 59 55

100

60 - 69 50

95

Illustrate this information on a line graph.

75

4 The weight of a baby at various ages is shown below Age (months) Weight (kg)

0 3:2

3 5:1

6 7:0

9 8:8

12 10:0

15 10:7

18 11:2

21 11:8

24 12:5

25

5

Illustrate this information on a line graph. 0



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5 The price of CHP shares over a fortnight varied as shown.

95

Day Price ($)

75

M 8:25

T 8:27

W 8:30

T 8:00

F 8:06

M 8:06

T 8:14

W 7:99

T 8:08

F 8:04

Draw a line graph to show this variation in prices. 25


5

0

The line graph alongside, representing the data in Example 1, was made using a spreadsheet. The price for each day is typed in as for a table. Highlight the table, select CHART and choose line graph.

Use a spreadsheet to draw some of the line graphs in Exercise 7I.

J

RADAR CHARTS

A radar chart is a type of line graph, drawn on a spider web grid, which is used for comparing quantities which vary in a cyclic manner.

EXAMPLE

1

John and Lauren own two sports stores in Caringbah and Hurstville. The monthly sales for each store are shown on the radar chart alongside. a

b

c d

What were the monthly sales at the Hurstville store in i March ii August? Determine the monthly sales at the Caringbah store in i January ii July. In which months did each store have their highest sales? In which month(s) were the sales at Caringbah higher than at Hurstville?

Sports store sales Jan $80000 Dec

Feb $60000

Nov

Mar

$40000 $20000

Oct

100

Apr

0

95

75

Sep

May

25

Aug

Jun 5

Jul Hurstville

Caringbah 0



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250


100

a i $50 000 ii $52 000 b i $55 000 ii $30 000 c Both Hurstville and Caringbah stores had their highest sales in December. d The sales at Caringbah were higher than those at Hurstville for the months of March and April.

95

75

25

5

0

?

EXERCISE 7J Shop sales

1 The monthly sales figures for Kerri’s dress shops at Burwood and Parramatta are shown on the radar chart alongside. a b

c

What were the sales at Burwood for the months of i April ii October? What were the sales at the Parramatta shop for the month of i May ii September? In which months did each store have its highest sales?

Jan 40000 Feb

Dec

30000 Nov

Mar

20000 10000

Oct

Apr

0

Sep

d e

In which months did each store record its lowest sales? In which months were the sales at the Burwood shop greater than the sales at the Parramatta shop?

EXAMPLE

May

Aug

Jun Jul

Burwood

Parramatta

2

Draw a radar chart to display the monthly rainfall figures shown in the table below. Month Ja Fe Ma Ap Ma Ju Jy Au Se Oc No De Rainfall (mm) 68 74 55 50 42 48 56 38 30 45 54 46 Monthly rainfall (mm)

Radar charts are most easily drawn using spider web grid. Step1: Indicate the 12 months of the year, at equal intervals of 30o , around the outermost circle. Step2: Choose a suitable scale for the radii of the circles. Step3: Plot points and join them as shown.

Jan 80 Dec

Feb

60 Nov

Mar

40

100

20

95

0

Oct

Apr 75

Sep

May 25

Aug

Jun 5

Jul 0



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2 Draw a radar chart to display the monthly rainfall figures shown in the table below Month Rainfall (mm)

95

75

J 84

F 65

M 50

A 46

M 38

J 52

J 40

A 35

S 30

O 42

N 45

D 35

J 34

J 25

A 26

S 31

O 38

N 44

D 56

3 The monthly sales figures for a computer firm are Month Sales ($000’s)

25

5

J 54

F 63

M 30

A 30

M 28

Draw a radar chart for this information. 0

4 The average number of cloudy days in Brisbane for each month of the year is shown in the table Month J F M No. of cloudy days 9:8 10:0 9:1

A 6:7

M 7:9

J 6:1

J 5:6

A 3:9

S 3:8

O 6:8

N 7:4

D 8:4

Illustrate this information on a radar chart. 5 The table shows the temperature (o C) variation on a summer’s day in Sydney, from 12 midnight to 11 pm Time 12 1 2 3 4 5 6 7 8 9 10 11 Temperature 13 12 12 11 11 12 13 13 14 16 18 21 Time Temperature

12 22

1 24

2 26

3 28

4 27

5 26

6 26

7 24

8 21

9 20

10 17

11 15

Draw a radar chart to illustrate this data.

SPREADSHEET APPLICATION The radar chart below, representing the data in Example 2, was made using a spreadsheet. The rainfall for each month is typed in as for a table. Highlight the table, select CHART and choose radar chart.

100

95

75

25

Use a spreadsheet to draw radar charts for some of the questions in Exercise 7J. Investigate the options.

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95


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MISLEADING GRAPHS

75

EXAMPLE 25

1

The table shows the profits of a company over the five year period 1995 to 1999.

5

Year Profit ($ millions)

0

1995 12:3

1996 12:9

1997 13:2

1998 13:8

1999 14:6

Shown below are five ways of presenting this information graphically. What features, if any, are misleading:

Profit – Graph B

10 5 0 1994 1995 1996 1997 1998 1999 2000 Year

Profit (millions)

Profit (millions)

Profit – Graph A 15

15 14 13 12 1994 1995 1996 1997 1998 1999 2000 Year

Profit – Graph D Profit (millions)

Profit (millions)

Profit – Graph C 15 10 5 0 1994

1995

1996

1997

1998

1999

15 10 5 0

2000

95 97 99

Year

Year

Profit – Graph E 16 15 14 13 Profit (millions)

Graph B has exaggerated the increase in profit by not starting the scale on the vertical axis at zero and by enlarging this scale. Graph C has the opposite effect – diminishing the rate of increase by enlarging the horizontal scale. Graph D, by using a smaller scale on the horizontal axis, gives a different impression again. Graph E has an irregular scale on the vertical axis. Graphs A, C and D are fair, although each gives a different impression. Graphs B and E are misleading.

100

12 10

95

75

5

25

0 1994 1995 1996 1997 1998 1999 2000 Year

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EXAMPLE

95

2

Following are shown three ways of presenting the information in the table graphically. Comment on any misleading features about the graphs.

75

Brand X Y

No. of TV’s sold 4000 8000

25

Number sold

5

Number sold

Number sold

0

8000

8000

8000

6000

6000

6000

4000

4000

4000

2000

2000

2000

0

X

0

Y

0

Y

X

Brand

Y

X

Brand

Brand

The column graph A correctly shows that the sales of brand X are double those of brand Y (as the table does). The width of each column is the same and the height of the second column is twice that of the first. In graph B, both dimensions, height and width, of the picture of a TV have been doubled creating the impression that the sales of brand Y are ( 2 £ 2 =) 4 times those of brand X. In graph C, the 3 dimensions have been doubled creating the impression that the sales of brand Y are (2 £ 2 £ 2 =) 8 times those of brand X.

The main causes of graphs being misleading are: ² ² ² ²

?

the the the the

scale on the vertical axis does not start at zero scale on the vertical axis is irregular scale on the vertical axis is missing use of area or volume to create a false impression.

EXERCISE 7K 1 Describe the misleading or poor features of the following graphs 100

b

Tonnes (’000)

Steel production 25 24 23 22 21 20

Year 92

93

94

95

96

Steel production Tonnes (’000)

a

95

75

40 20 25

10

Year

0 97

98

5

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100

c

d

Vitamin content in dog food

Fish sold at markets

Vitamin

75

Cases (100’s)

95

Vitamin

A

25

D

E

A

D

E

30 20 10

Brand

Week 0

X 5

e

Y

1

f

Milk production

20 ('000s L)

Exports Tonnes (millions)

0

10 A

2

B

Year

40 30 20 10

Year

0

0

1994

Factory

g

1995

1996

Train fares

Dollars

3 2 1 Year

0 90

95

96

GRAPH A

2

GRAPH B 3000 Sales of Nu-Choc Chocolates

Sales of Nu-Choc Chocolates

2600 2500 2400 2300 2200 2100

2000 1500 1000 500

Year

Year 0

2000 90

a b c

2500

91

92

93

94

95

96

90

91

92

93

94

95

96

Which graph gives the impression of rapidly increasing sales? Have sales in fact rapidly increased over this 7 year period? According to graph A, the sales for 1993 appear to be double those of 1992? Is this true?

3 The frequency column graph shows the median house price in each capital city in 1992 and the estimated median house price in 1994. a In which city is the mid-1992 median house price the greatest? b Find Melbourne’s mid-1992 median house price. c What is the estimated percentage median price increase for Adelaide houses from mid-1992 to mid-1994?



MEDIAN HOUSE PRICES 100

280

95

230 75

180 130 25

80 5

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d Looking at the Hobart column, the estimated median house price at mid-1994 will more than double its mid-1992 price. Is this statement true or false? Why?

95

e In one sentence, state why the graph can be misleading. 75

f Redraw the graph accurately. 25

The sector graph alongside indicates factors relating to the delays people have in servicing their motor vehicles.

4

5

10%

0

a

Discuss any points about this sector graph which make it misleading.

b

Draw a column graph representing these figures accurately.

c

Redraw the sector graph labelling the missing sector as “unknown”.

22% 28%

5

12%

a Is the given graph misleading? b Why do you think it was drawn this way? c If the graph were redrawn with an accurate scale would the effect be the same? Explain.

% 18 16 14 12 Jan 95

10 0

6 Operating costs per property

Jan 92 Jan 89

a Is the given graph misleading? b Draw an accurate line graph representing this information. c How would the Water Board collect these figures? d The operating cost for 87=88 is about $465. Is this $465 per person, per house, or something else? 100

7

a Is the given graph misleading? Explain.

95

b What type of graph is this closest to?

75

c Draw an accurate column graph using this information. d Contrast your graph in c with the original graph.

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75

Many other types of graphs, not used so far, are available on a spreadsheet. Try other possibilities using the data from the previous graphing exercises. Investigate the options. Try to make some misleading graphs. Print any that you find interesting. Some examples are shown below.

5

Method of travelling to school walk bicycle bus car train

0

COLUMN GRAPH

10

Number

8 Number

25

4 11 8 6 3

6 4 2 0 walk

bicycle

bus

train

car

Method

10

8

8

6

6 4

LINE GRAPH

10

Number

Number

3-D COLUMN GRAPH

4 2

2

0

0 walk

bicycle

bus Method

car

walk

train

bicycle

3-D PIE CHART train

bus

train

car

Method

train

PIE CHART walk

walk

car

car

bicycle

bicycle

bus bus

L

100

FREQUENCY HISTOGRAMS AND POLYGONS

95

A convenient method of representing a frequency distribution graphically is by means of a frequency histogram. A histogram is basically a column (or bar) graph in which the values of the variable are placed on the horizontal axis and the frequency of the variable on the vertical axis. A frequency polygon is a line graph with the first and last points joined to the horizontal axis to form a polygon.

75

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EXAMPLE

1

Temperature 17 18 19 20 21 22 23

The temperature, in o C, on each day in November was recorded and the results summarised in a frequency table, as shown

25

Frequency 1 2 4 7 6 6 4

The columns are centered about the scores they represent. They are the same width and are joined.

a November temperatures

10

5

0 17

18

19 20 21 22 Temperature (°C)

23

The area of each column represents the frequency of each score. Hence the total area of the histogram represents the total number of scores.

The frequency polygon can be drawn from the histogram by joining the midpoints of the columns

November temperatures

10 Frequency

b

Frequency polygon

5

0 17

18


23

November temperatures

or separately, by plotting points

10 Frequency

0

Draw a a frequency histogram b a frequency polygon of the distribution

Frequency

5

257

5

100

95

0 17

18


75

23

Note that since the area under the polygon should be equal to the area of the histogram then the first and last points should be joined to the points on the horizontal axis where the next score would be found.

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EXERCISE 7L 1 Draw a frequency histogram for each of the following distributions

75

a

25

b

Number of Snacks Frequency Score Frequency

5

0 5

10 7

1 7

2 9

11 11

3 7

12 17

4 6

13 0

5 4

6 2

14 5

0

2 Draw a frequency polygon for each of the following distributions a

Number of goals 0 1 2 3 4

3 On the same diagram, draw a frequency histogram and a frequency polygon for the data given.

EXAMPLE

b

Salary 15 000 20 000 25 000 30 000 35 000 40 000

Mark Frequency

14 4

Frequency 8 9 6 3 1

Frequency 8 6 7 3 0 1 15 6

16 9

17 0

18 2

19 3

20 1

2

Draw a grouped frequency histogram and polygon for the data given Height Frequency

155-159 4

160-164 7

165-169 10

170-174 5

175-179 2

For grouped data, we need to find the midpoint of each class interval, called the class centre, as shown. 10

Class Centre

Frequency

9 8

155-159

(155+159) 2

= 157

4

160-164

(160+164) 2

= 162

7

165-169

(165+169) 2

= 167

10

170-174

(170+174) 2

= 172

5

(175+179) 2

= 177

175-179

7

Frequency

Height

6 100

5 4

95

3

2

75

2 1

The class centres are then placed on the horizontal axis and the histogram and polygon completed.

0

25

157

162

167

172

177 5

Height

0



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75

25

5

0

4 Find the class centre for each class interval and hence draw a grouped frequency histogram and polygon for a b Mass (kg) Frequency Score Frequency 60 - 62 2 41 - 50 3 63 - 65 5 51 - 60 0 66 - 68 7 61 - 70 10 69 - 71 4 71 - 80 8 72 - 74 2 81 - 90 7 91 - 100 2 c

M

No. of patients/day 10 - 19 20 - 29 30 - 39 40 - 49 50 - 59 60 - 69

Frequency 13 15 16 10 4 2

CUMULATIVE FREQUENCY GRAPHS

EXAMPLE

1

Alongside is the information regardTemperature 17 18 19 20 21 22 23 ing the temperature, in °C, on each Frequency 1 2 4 7 6 6 4 day in November, given in a previous example. Add a cumulative frequency column to the table and hence draw a cumulative frequency histogram and a cumulative frequency polygon.

30

Temp. 17 18 19 20 21 22 23

Cumu. Freq. freq. 1 1 2 3 4 7 7 14 6 20 6 26 4 30

25 Cumulative frequency

100

259

20 15 100

10

95

5

75

0 17

18

19

20 21 Temperature

22

23

First draw a histogram, using the cumulative frequency of each score. Next join the top righthand corners of each column by straight lines to form the cumulative frequency polygon.

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260 100


Note:

²

We do not join the last corner to the horizontal axis, so it is not strictly a polygon.

²

Again, we join the upper right hand vertices, not the midpoints of each column, when drawing a cumulative frequency polygon.

²

To draw a cumulative frequency polygon without first drawing the histogram requires a careful consideration of the upper and lower boundaries of each class interval.

95

75

25

5

A cumulative frequency polygon is called an ogive .

0

?

EXERCISE 7M 1 For each of the frequency tables given in 1 of Exercise 7L a add a cumulative frequency column b draw a cumulative frequency histogram and a cumulative frequency polygon. 2 For each of the frequency tables given in 2 of Exercise 7L a add a cumulative frequency column b draw a cumulative frequency histogram and a cumulative frequency polygon.

EXAMPLE

2

Given the information in the table below, add a cumulative frequency column and hence draw a cumulative frequency histogram and cumulative frequency polygon for this grouped data. Height Frequency

155 - 159 4

160 - 164 7

165 - 169 10

170 - 174 5

175 - 179 2

30

155 160 165 170 175

-

159 164 169 174 179

Freq. 4 7 10 5 2

Cu. Freq. 4 11 21 26 28

Using the class centres the cumulative frequency histogram is drawn. To draw the cumulative frequency polygon (the ogive) we then join the top right-hand corners of each column of the histogram


Height

Cl. Cent. 157 162 167 172 177

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3 For each part of 4 in Exercise 7L

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a add a cumulative frequency column to the table b hence draw a cumulative frequency histogram and the ogive, on the same diagram.

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The following is a list of key words used in this chapter. Write a sentence to describe each one. frequency distribution table, cumulative frequency, relative frequency, stem and leaf plot, sector graph, column graph, dot plot, bar graph, divided bar graph, line graph, radar graph, histogram, polygon, ogive

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You should be able to 2 construct frequency distribution tables 2 understand and use cumulative and relative frequency 2 group data into classes 2 construct stem and leaf plots 2 create sector graphs 2 create divided bar graphs 2 create dot plots 2 create column and bar graphs 2 create line graphs 2 create radar charts 2 understand how graphs may be misleading 2 construct frequency histograms and polygons 2 construct cumulative frequency histograms and ogives 2 link types of data with appropriate displays 2 describe the strengths and weaknesses of various forms of display.

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7

DIAGNOSTIC TEST 100

1 The marks out of 10 for the spelling test were 5, 3, 9, 8, 6, 4, 5, 6, 6, 5, 7, 8, 10, 6, 7, 5, 6, 8, 6, 7

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a Organise this information into a frequency distribution table. b How many students sat for the test? 2 For the data in question 1 the cumulative frequency and relative frequency of the score 7 are A 3 and 15% B 15 and 3% C 15 and 15% D 12 and 15% respectively

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3 The distance to travel to school for a group of students is shown below 0:6, 3:4, 0:8, 2:8, 0:9, 1:1, 1:9, 2:4, 3:5, 3:6, 0:8, 1:4, 2:7, 0:6, 1:3, 3:9, 2:1, 2:3, 3:0, 1:6, 1:5, 3:2, 2:1, 1:8, 0:7, 2:3, 3:6, 3:5, 1:6, 1:4, 0:4, 2:8, 3:8, 2:9, 0:8, 1:7, 3:7, 2:5, 1:9, 2:0:

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a Organise this data into a frequency table using classes 0 - 0:4, 0:5 - 0:9, 1:0 - 1:4, 1:5 - 1:9, .... b How many students were surveyed?

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4 Organise the data in question 3 into an ordered stem and leaf plot using stems 0, 1, 2 and 3. 5 The following stem and leaf plot shows the time, in hours, spent studying by a group of Year 11 students during one week The highest and lowest scores are A 7 and 9 B 0 and 9 C 2 and 9 D 2 and 39 respectively

0 1 2 3

2 0 1 0

3 2 1 2

3 2 2 2

677 588899 34456678 9

6 Using the data in 5 the number of students who spent less than 20 hours studying was A 9 B 19 C 15 D 21 7

8

Method of travel Walk Cycle Bus Car Train


To construct a sector graph to display the given information, the angle for the sector representing “Bus” would be A 12o B 30o C 40o D 144o

If a divided bar chart of length 200 mm is used to display this information, the length of the part representing Ladies’ wear would be A 45 mm B 90 mm C 100 mm D 180 mm

Department Men’s wear Ladies’ wear Children’s wear Babywear

% of total sales 25 45 20 10

9 Which of the following statements is not true? A A dot plot is convenient for illustrating small sets of data. B A dot plot can be used directly with unsorted data. C A dot plot is very time consuming for large sets of data. D A dot plot is convenient for displaying large sets of data. 10 The sales figures for a real estate company for one year are shown in the table alongside

a b c d

Months Jan/Feb Mar/Apr May/Jun Jul/Aug Sep/Oct Nov/Dec

Sales (£ $1 million) 2:4 3:1 1:8 1:4 2:8 3:6

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Draw a column graph to illustrate this information. Display this data with a horizontal bar graph. Draw a line graph to illustrate this data. Draw a radar chart to display this data.

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11 Which of the following statements is incorrect? The main causes of graphs being misleading are A the scale on the vertical axis does not start at zero B the scale on the horizontal axis does not start at zero C the scale on the vertical axis is irregular D the use of area or volume to create a false impression.

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Using the frequency histogram alongside, the total number of matches played was

12 5

Number of matches

6 0

A C

5 4

17 6

B 5 D 18

3 2 1 0 0

1

2 3 4 Number of goals

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13 To draw a frequency polygon using the histogram in question 12, the first and last column midpoints should be joined to A any two points on the horizontal axis B the points on the horizontal axis where the next scores would be C any two points on the vertical axis D the bottom left and right corners of the histogram. 14 Draw a cumulative frequency histogram and the ogive for the data in 3. If you have any difficulty with these questions, refer to the examples and questions in the exercise indicated. Question Section

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7A

1 A

2 B

3 C

4, 5, 6 D

7 E

8 F

9 G

10 H, I, J

11 K

12, 13 L

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REVIEW SET

1 The results of a survey of the hair colour of a group of people is shown below using the code: black (B), brown (b), blonde (w), red (r), grey (g) Bbbwg bBBrw wbgBw BBbbw wwwbB grBbw a b c d

Organise the information into a frequency distribution table. How many people were surveyed? What was the most common hair colour in this group of people? Add a relative frequency column to the table above and hence find the relative frequency of people with i black ii blonde hair

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100

e Display the data above using a i sector graph iv horizontal bar graph

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divided bar graph

ii

iii

dot plot

2 The table shows the variation in the price of AMB shares during one week. Day Price ($)

25

M 6:80

T 6:95

W 7:45

T 6:50

F 6:55

5

Illustrate this information using a line graph. 0

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7B

REVIEW SET

The marks scored in a mathematics test by a class are 11, 12, 11, 14, 18, 15, 13, 14, 19, 18, 17, 15, 15, 12, 18, 17, 13, 10, 9, 14, 16, 16, 13, 14, 11, 14, 18, 10, 14, 15 1 Organise this data into a frequency distribution table. 2 Add a cumulative frequency and relative frequency column. 3 How many students scored

a less than 15

b 15 or less?

a 14

4 What percentage of students scored

b more than 18?

5 Draw a frequency histogram and polygon for this data. 6 Draw the ogive for this data.

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REVIEW SET

1 The numbers of each brand of sound system sold by an electrical store one week are shown alongside

Brand Sony National Pioneer Aiwa Phillips

Number 15 8 12 10 5

a How many sound systems were sold this week? b Draw a divided bar chart to display this information. c Draw a vertical column graph.

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2 Draw a radar chart to illustrate the monthly rainfall shown below 75

Month Rainfall (mm)

J 54

F 68

M 52

A 48

M 35

J 46

J 38

A 32

S 30

O 36

N 44

D 46 25

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3 Describe the misleading or poor features of the following graphs 95

Wheat production

Pollution level

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Tonnes × 1000

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70 60 50

40 35 30 25

0

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97

98

95

99

96

Year

97 98 Year

99

Profit (millions)

Wine consumption

13 12 10

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1989

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Year

1

a b

c d

e

REVIEW SET

Copy the cumulative frequency histogram and draw the ogive. How many students scored i less than 12 ii 12 or less iii more than 12 ? Construct a frequency distribution table for this data. What percentage of students scored i 10 ii 15¡? Draw a frequency histogram and polygon.

Mark in test

60 Cumulative frequency (number of students)

7D

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40

30 100

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12 13 Mark

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CHAPTER

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Investing money

AREA OF STUDY

This chapter deals with the mathematics of investments made over a period of time. The main mathematical ideas investigated in this chapter are: 8 the calculation of simple interest over various time periods 8 the use of formulae in interest calculations 8 drawing graphs to compare interest rates 8 the mathematics of shares 8 the calculation of the value of investments 8 the calculation of the price of goods following inflation.

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INVESTING MONEY FM2 (Chapter 8)

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SIMPLE INTEREST CALCULATIONS

Simple interest is the name given to the type of interest that is calculated on the full amount borrowed or invested over the full term of the loan. The amount borrowed is called the principal. The simple interest rate as a percentage is often called the flat rate of interest. The term of the loan is usually expressed in years, but may be given as any time period.

25

The formula for calculating simple interest is given by:

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I = P rn where

EXAMPLE

I P r n

is is is is

the the the the

amount of interest in dollars principal, the amount borrowed percentage interest rate per time period, expressed as a decimal number of time periods.

1

Calculate the simple interest on $5000 at a flat rate of 8% pa over 4 years. I = P rn,

)

where P = $5000 r = 8 ¥ 100 = 0:08 n=4

I = $5000 £ 0:08 £ 4 = $1600

) the interest is $1600.

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EXERCISE 8A 1 Calculate the simple interest on $300 at a flat rate of 6% pa over 3 years. 2 Calculate the simple interest on $750 at a flat rate of 9% pa over 2 years. 3 Calculate the simple interest on $1500 at a flat rate of 11% pa over 5 years. 4 Calculate the amount of simple interest earned on the following investments:

a b c d e f g h i

Principal

Interest rate pa

Years

$2680 $5990 $21 000 $55 000 $12 500 $3650 $11 220 $8100 $44 656

8% 7% 12% 3% 9% 3% 5% 10% 11%

5 8 6 2 4 6 9 8 7

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EXAMPLE

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2

Calculate the simple interest on $2000 at a flat rate of 9.62% pa over 3 years. I = P rn,

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5

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0

where P = $2000 r = 9:62 ¥ 100 = 0:0962 n=3

I = $2000 £ 0:0962 £ 3 = $577:20

) the interest is $577.20 5 Calculate the amount of simple interest earned on the following investments:

a b c d e f g h i

EXAMPLE

Principal

Interest rate pa

Years

$1268 $5360 $33 000 $11 000 $68 500 $4225 $3690 $4100 $27 333

9:62% 5:36% 6:35% 7:28% 14:5% 11:96% 2:36% 3:77% 1:254%

3 6 4 7 8 2 1 6 5

‘sig. figs.’ is short for significant figures.

3

An investment pays 5% interest pa. a Express this as a decimal. b Find the six monthly interest rate as a percentage and a decimal. c Find the quarterly interest rate as a percentage and a decimal. d Find the monthly interest rate as a percentage and a decimal. a Rate = 5% = 0:05 as a decimal b Six months is half a year,

fas 5% = 5 ¥ 100g ) rate = 5% ¥ 2 = 2:5% = 0:025 as a decimal.

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c A quarter of a year is three months, ) rate = 5% ¥ 4 = 1:25% = 0:0125 as a decimal.

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d There are 12 months in a year, ) rate = 5% ¥ 12 = 0:146 666 6:::::: = 0:1467% (to 4 dec pl) = 0:001 467 (to 4 sig. figs.) as a decimal.

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6 An investment pays 8% interest pa. a b c d

Express this as a decimal. Find the six monthly interest rate as a percentage and a decimal. Find the quarterly interest rate as a percentage and a decimal. Find the monthly interest rate as a percentage and a decimal.

7 An investment pays 3% interest pa. a b c d


8 Copy and complete this table of percentage interest rates annual rate

quarterly rate

monthly rate

9% 10% 15% 4:5%

a b c d e f g h i

EXAMPLE

six monthly rate

3:8% 2:6% 1:3% 0:065% 1:95%

4

Calculate the simple interest on $3000 at a flat rate of 8.53% over 16 months. I = P rn,

)

where P = 3000 r = 8:53 ¥ 12 ¥ 100 = 0:007 108 333 3 per month n = 16 months

I = 3000 £ 0:007 108 333 3 £ 16 = $341:20

) the interest is $341:20

The units for r and n must be the same; both in years or both in months.

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9 Calculate the amount of simple interest earned on the following investments:

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a b c d e f g h i

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EXAMPLE

Principal $3000 $6000 $45 500 $7500 $2360 $5630 $562 $380 $12 600

Interest rate 4% 7% 9% 8:2% 4:6% 8:5% 3:52% 7:95% 8:34%

Months 16 18 20 28 30 14 15 16 18

5

If $3500 is invested for 3 years the amount of interest earned is $735. Calculate the annual simple percentage interest rate. I = P rn ) 735 = 3500 £ r £ 3 ) 735 = 10500 £ r ) 10735 500 = r ) r = 0:07 ) percentage rate = 0:07 £ 100% = 7%

10 If $4000 is invested for 4 years the amount of interest earned is $1280. Calculate the annual simple percentage interest rate. 11 If $6200 is invested for 5 years the amount of interest earned is $1240. Calculate the annual simple percentage interest rate. 12 If $190 is invested for 10 years the amount of interest earned is $123.50. Calculate the annual simple percentage interest rate. 100

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SIMPLE INTEREST GRAPHS

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A mathematical model for simple interest may be developed using a graph that compares the interest and the rate for a fixed amount of money over a number of years. This section is best attempted using a graphics calculator or if this is not available then a spreadsheet will do. This enables the variables to be changed and the new graph seen without having to calculate and plot points.

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EXAMPLE

1

Draw a graph showing the amount of interest earned over a period of 10 years if $1000 is invested at 8% pa.

a

Find the interest after 8 12 years.

b

Find the time to earn interest of $550:

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Use the formula that calculates simple interest and substitute the values. I = P rn = 1000 £ 0:08 £ n = 80n Use a table to find some values. Interest at 8% pa

n I

1 80

2 160

3 240

4 320

5 400

10 800

Use these values to draw the graph. a From the graph the interest after 8 12 years is about $680. b The time taken to earn interest of $550 is a little less than 7 years.

Interest

5

800 700 600 500 400 300 200 100 0

2

4

6 Years

8

10

Use TRACE on the graphics calculator to find the values.

EXERCISE 8B 1 An amount of $1000 is invested for 10 years at 5% pa simple interest. a Substitute the values of P = $1000 and I = 0:05 into the formula I = P rn. b Use this formula to complete years 1 2 4 5 10 this table of values. Interest c Use the values to draw a graph showing the amount of interest earned over a period of 10 years. Put years on the horizontal axis and interest on the vertical axis. d Find the interest after 7 12 years. e Find the time to earn interest of $180: 100

2 An amount of $1000 is invested for 10 years at 6% pa simple interest.

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a Use this formula to complete this table of values. years Interest

1

2

4

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b Use the values to draw a graph showing the amount of interest earned over a period of 10 years. Put years on the horizontal axis and interest on the vertical axis. c Find the interest after 9 12 years. d Find the time to earn interest of $250:



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3 On the same set of axes draw the graphs to show A $1000 invested at 9% pa over 10 years B $1000 invested at 11% pa over 10 years C $1000 invested at 8.5% pa over 10 years. a b c d

How How How How

much has each investment earned over 10 years? much is each investment worth after 4 12 years? much more than the others has the best investment earned after 10 years? long does each investment take to earn $500 of interest?

4 On the same set of axes draw the graphs to show A $2000 invested at 8% pa over 10 years B $2000 invested at 12% pa over 10 years C $2000 invested at 10.5% pa over 10 years. a b c d

How How How How

much has each investment earned over 10 years? much is each investment worth after 6 12 years? much more than the others has the best investment earned? long does each investment take to earn $1000 of interest?

RESEARCH PROJECT 1 Find some examples of current annual interest rates for investments. List them. 2 Draw graphs to show the investment at simple interest of $ 100 over 10 years at varying interest rates. 3 Find some examples of current annual interest rates for loans. List them. 4 Draw graphs to show the loan at simple interest of $100 over 10 years at these rates.

SPREADSHEET APPLICATION This spreadsheet draws the simple interest graphs.

A 1 principal 2 percentage rate 3 year 4 1 5 2 6 3 7 4 8 5 9 6 10 7 11 8 12 9 13 10

B 1000 8 interest 80 160 240 320 400 480 560 640 720 800

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Simple interest 8%

The formula in cell B4 is = $B$1*$B$2/100*A4.

1000

Interest

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Use fill down to complete the spreadsheet.

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800 600 400 200 0

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Draw some other graphs.

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6 Years

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COMPOUND INTEREST

When money is invested at compound interest, the interest earned at the end of each time period is added to the principal. This increases the principal that is used to calculate the interest for the next time period. With compound interest you earn interest on the interest.

EXAMPLE

Interest on interest. That’s interesting!

1

$2000 is invested for 3 years at 7% pa interest compounded annually. Find a the amount the $2000 will grow to after three years b the amount of interest earned. a

Time

Working

Start of 1st year

Total balance

Interest at the end of year 7 100

$2000

Start of 2nd year

$2000 + $140

$2140

Start of 3rd year

$2140 + $149:80

$2289:80

Start of 4th year

$2289:80 + $160:29

$2450:09

£ 2000 = $140

7 100 £ $2140 = $149:80 7 100 £ $2289:80 = $160:29

The amount is $2450:09 b The amount of interest earned = $2450:09 ¡ $2000 = $450:09 :

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EXERCISE 8C 1

100

a Copy and complete the table to determine the final value of $2400 invested at 6% pa compound interest for 3 years. Time Start of first year Start of second year Start of third year Start of fourth year

balance + interest $2400 + $144 $2544 + $152:64

Total balance $2400 $2544

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Interest $144 $152:64

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b Calculate the total interest earned.



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a Copy and complete the table to determine the final value of $3600 invested at 5% pa compound interest for 3 years.

2

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Start Start Start Start

25

Time of first year of second year of third year of fourth year

balance + interest $3600 + $180 $3780+

Total balance $3600 $3780

Interest $180 $189

b Calculate the total interest earned.

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a Copy and complete the table to determine the final value of $980 invested at 3% pa compound interest for 4 years.

3


Time of first year of second year of third year of fourth year

balance + interest

Total balance

Interest

b Calculate the total interest earned. 4 Adele decided to invest her savings of $7890 for 5 years at 8.9% pa compound interest. a Copy and complete the table.

Start Start Start Start Start Start

Time of first year of second year of third year of fourth year of fifth year of sixth year

balance + interest

Total balance

Interest

b If Adele intended to buy a car valued at $12 990 when her investment matured, would she have enough to buy the car? Explain. c By how much is she over/under the value of the car? USING THE COMPOUND INTEREST FORMULA

The compound interest formula is: A = P (1 + r)n

EXAMPLE a

where,

A, P, r, n,

the the the the

amount is the final balance or future value principal is the initial value invested or present value interest rate per compounding period as a decimal number of time periods.

2

Use the compound interest formula to calculate the amount of a fixed term investment of $5000 over 5 years at 6.5% pa interest compounding yearly.


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a A = P (1 + r)n

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where

P = $5000 r = 0:065, as the compounding period is annual, the interest rate is the annual rate n = 5, as the compounding period is annual the number of time periods is the same as the number of years.

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) ) )

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A = 5000 £ (1 + 0:065)5 A = 5000 £ (1:065)5 A = $6850:43

b Interest = $6850:43 ¡ $5000 = $1850:43

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a Use the compound interest formula to calculate the amount of a fixed term investment of $4000 over 6 years at 7.5% pa interest compounding yearly. b Find the total interest earned.

6


EXAMPLE

3

Use the compound interest formula to calculate the amount of a fixed term investment of $3500 over 7 years at 6.2% pa interest compounding quarterly. There are 4 quarters in a year. Therefore there are 7 £ 4 = 28 time periods. Quarterly interest rate, r = 0:062 ¥ 4, i.e., the annual rate divided by 4. = 0:0155 ) ) )

A = P (1 + r)n A = 3500 £ (1 + 0:0155)28 A = 3500 £ (1:0155)28 A = $5384:01

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7 Use the compound interest formula to calculate the amount of a fixed term investment of $1200 over 5 years at 8.4% pa interest compounding quarterly. 8 Use the compound interest formula to calculate the amount of a fixed term investment of $950 over 3 years at 4.1% pa interest compounding quarterly.

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9 Calculate the amount using the values in this table.

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a b c d e f g h i

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Principal $700 $450 $12 000 $650 $8540 $2205 $25 000 $675 $3600

rate pa 6% 8% 5% 4% 6:5% 3:8% 2:9% 4:8% 12:5%

years 3 5 2 8 6 4 2 5 3

compounding period quarterly monthly six monthly monthly quarterly monthly six monthly quarterly monthly

10 Which is the better return over a year and by how much? A $5000 is invested at 6% pa compounding yearly, or B 5.9% pa compounded quarterly, or C 5.85% pa compounding monthly.

EXAMPLE

4

Calculate the amount that must be invested at 6% pa interest compounding annually to have $5000 at the end of 4 years. A = 5000,

)

r = 0:06, n=4 n A = P (1 + r) ) 5000 = P (1 + 0:06)4 5000 =P (1 + 0:06)4 )

P = $3960:47

$3960.47 must be invested at 6% compounding interest over 4 years to have $5000.

11 Calculate the amount that must be invested at 7% pa interest compounding annually to have $6000 at the end of 5 years. 12 Calculate the amount that must be invested at 5% pa interest compounding annually to have $1600 at the end of 7 years. 13 Calculate the amount that must be invested at 11.2% pa interest compounding annually to have $10 000 at the end of 6 years.

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EXAMPLE

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Calculate the amount that must be invested at 7.5% pa interest compounding quarterly to have $1700 at the end of 3 years.

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100

Quarterly interest rate = 0:075 ¥ 4 = 0:01875

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Number of quarters = 3 £ 4 = 12 A = 1700,

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r = 0:01875,

n = 12 n

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)

A = P (1 + r) ) 1700 = P (1 + 0:01875)12 ) 1700 = P (1:01875)12 1700 =P (1:018 75)12 ) P = $1360:31 i.e., $1360.31 must be invested.

14 Calculate the amount that must be invested at 8.5% pa interest compounding quarterly to have $2300 at the end of 7 years. 15 Calculate the amount that must be invested at 4% pa interest compounding quarterly to have $1540 at the end of 8 years. 16 Calculate the amount that must be invested at 10.2% pa interest compounding quarterly to have $10 000 at the end of 4 years. 17 Calculate the amount that must be invested at 9% pa interest compounding monthly to have $3000 at the end of 3 years. 18 Calculate the amount that must be invested at 4.5% pa interest compounding monthly to have $950 at the end of 8 years. 19 Calculate the amount that must be invested at 7.2% pa interest compounding six monthly to have $2000 at the end of 10 years. 20 Paul wants to know how much he needs to invest at 4.95% to have $2500 in three years time. Interest compounds monthly. 21 A company needs $20 000 to replace their computer system in 4 years time. How much needs to be invested at 4.95% pa interest compounding quarterly to have this amount available?

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COMPOUND INTEREST GRAPHS

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EXAMPLE

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1

Draw a graph showing the amount the investment grows into over a period of 10 years if $1000 is invested at 8% pa interest compounding annually.

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a Find the value of the investment after 8 12 years. b Find the time for the investment to be worth $1600.

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The values in the table are calculated using the method at the beginning of the exercise or the compound interest formula. Plot these values.

0

a After 8¡12 years the invest ment is worth about $1900: b The investment is worth $1600 after about 6¡¡14 years.

year 1 2 3 4

amount $1; 080:00 $1; 166:40 $1; 259:71 $1; 360:49

year 5 6 7 8

amount $1; 469:33 $1; 586:87 $1; 713:82 $1; 850:93

year 9 10

amount $1; 999:00 $2; 158:92

Compound interest

For year 0 the amount is the original investment.

Amount of investment

$2,000.00

$1,500.00

$1,000.00

$500.00

Years $0.00 0

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2

4

6

8

10

EXERCISE 8D 1 Draw a graph showing the amount an investment grows into over a period of 10 years if $1000 is invested at 7% pa interest compounding annually. a b c

Copy and complete this table using the compound interest formula or a spreadsheet. 6 12

Find the value of the investment after years Find the time for the investment to be worth $1500.

year 1 2 3 4 5 6 7 8 9 10

amount 100

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year 1 2 3 4 5 6 7 8 9 10

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Draw a graph showing the amount of interest earned over a period of 10 years if $2000 is invested at 9.3% pa interest compounding annually. a

Copy and complete this table using the compound interest formula or a spreadsheet.

b c d

Find the amount the investment grows into after 4 12 years. Find the interest earned after 4 12 years. Find the time for the investment to be worth $4000.

3 Draw graphs for these investments of $500 over 10 years. a 6.5% pa compounding annually b 6.2% pa compounding quarterly c 5.9% pa compounding monthly. Which is the best investment?

SPREADSHEET APPLICATION This is the spreadsheet used to input the values into the graph for the example.

The formula in cell B2 is = 1000*(1.08)ˆA2. This is the compound interest formula with P = 1000, r = 0:08, and the value of n given by the number in the first column. Use fill down for the other values. Edit this spreadsheet to graph other values.

E

A B 1 Year Amount 1 $1,080.00 2 2 $1,166.40 3 3 $1,259.71 4 4 $1,360.49 5 5 $1,469.33 6 6 $1,586.87 7 6 $1,600.00 8 7 $1,713.82 9 8 $1,850.93 10 9 $1,900.00 11 9 $1,999.00 12 13 10 $2,158.92

SHARES A share is a part ownership in a company.

One method a company may choose to use to raise capital is to sell shares to investors. The shares in the company when first issued, have a face or par value . The shares are then listed on the stockmarket and the price fluctuates. The price of the share being traded on the stock exchange is called the market price. Any profits made by the company are divided amongst the shareholders. This return to the shareholders is called a dividend. The dividend may be quoted as an amount per share or as a percentage of the face value of the share. The theoretical return on the money invested is called the yield and is calculated according to the following formula:



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Percentage yield =

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amount of dividend £ 100% market price

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A stockbroker carries out the buying and selling of shares on behalf of an investor. The fee charged by stockbrokers is called brokerage and is usually a percentage of the value of the transaction or a fixed fee according to the amount of the transaction.

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Stamp duty and brokerage are charged when buying or selling.

The tax on buying and selling shares is called stamp duty. Stamp duty is currently 15 cents per $100 or part thereof.

EXAMPLE

1

Calculate the total cost to purchase 700 David Jones $1.00 shares with a market price of $ 1.49. Brokerage is 2.5% and stamp duty is 15 cents per $100 or part thereof.

Cost of shares = 700 £ $1:49 = $1043 Brokerage = 2.5% of $1043 = 2:5 ¥ 100 £ $1043 = $26:075 = $26:08

Round the cost of shares up to the next hundred and use this value.

Stamp duty = 1100 ¥ 100 £ $0:15 = $1:65 Total cost = $1043 + $26:08 + $1:65 = $1070:73

?

EXERCISE 8E 1 Calculate the total cost to purchase 600 Coles Myer $1.00 shares with a market price of $7.97. Brokerage is 2.5% and stamp duty is 15 cents per $100 or part thereof. 2 Calculate the total cost to purchase 600 Fosters shares with a market price of $4.65. Brokerage is 2.5% and stamp duty is 15 cents per $100 or part thereof. 3 Calculate the total cost to purchase 300 AMP shares with a market price of $15.05. Brokerage is 2.5% and stamp duty is 15 cents per $100 or part thereof. 4 Calculate the total cost to purchase 450 OPSM shares with a market price of $2.85. Brokerage is 2.5% and stamp duty is 15 cents per $100 or part thereof. 5 Calculate the total cost to purchase 300 MYOB shares with a market price of $9.70. Brokerage is 2.5% and stamp duty is 15 cents per $100 or part thereof.

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Calculate the amount obtained when 1200 David Jones shares are sold at $1.36. Brokerage is $29 fixed and stamp duty is 15 cents per $100 or part thereof. Selling income = 1200 £ $1:36 = $1632

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) stamp duty = 1700 ¥ 100 £ $0:15 = $2:55

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) amount obtained = $1632 ¡ $2:55 ¡ $29 = $1600:45 6 Calculate the amount obtained when 800 Qantas shares are sold at $4.85: Brokerage is $29 fixed and stamp duty is 15 cents per $100 or part thereof. 7 Calculate the amount obtained when 600 Suncorp shares are sold at $2.46. Brokerage is $29 fixed and stamp duty is 15 cents per $100 or part thereof. 8 Calculate the amount obtained when 1000 Prime Life shares are sold at $1.70. Brokerage is $27.50 fixed and stamp duty is 15 cents per $100 or part thereof. 9 Calculate the amount obtained when 400 Wattyl shares are sold at $4.40. Brokerage is $32 fixed and stamp duty is 15 cents per $100 or part thereof. 10 Calculate the amount obtained when 300 ANZ bank shares are sold at $11.54. Brokerage is $45 fixed and stamp duty is 15 cents per $100 or part thereof. 11 Calculate the amount obtained when these shares are sold with brokerage of 2.5% and stamp duty of 15 cents per $100 dollars.

EXAMPLE

a b c d e f g

Share Foodland Meridian GPT Medica CC Amatil National Bank Mirvac Group

Number 800 3000 1500 5000 500 300 900

Selling price $9:44 $0:24 $2:53 $0:66 $5:28 $23:25 $3:16

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A dividend of 34 cents per share is paid on Coles Myer shares with a market value of $8.02. Find the percentage yield.

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amount of dividend Percentage yield = £ 100% market price 34 £ 100 = 802 = 4:24%

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12 A dividend of 6.32 cents per share is paid on Prime Ind shares with a market value of $0.73. Find the percentage yield. 13 A dividend of 25 cents per share is paid on Stockland shares with a market value of $3.30. Find the percentage yield. 14 A dividend of 4.65 cents per share is paid on Rebel shares with a market value of $0.91. Find the percentage yield. 15 A dividend of 10 cents per share is paid on E A Coffee shares with a market value of $3.10. Find the percentage yield.

0

EXAMPLE

4

Calculate the amount of dividend on 1500 shares with a face value of 50 cents if the dividend is 13%. Dividend per share = 13% of 50 cents

Total dividend = 1500 £ 6:5 cents

= 6.5 cents

= 9750 cents = $97.50

16 Calculate the amount of dividend on 1200 shares with a face value of 50 cents if the dividend is 11%: 17 Calculate the amount of dividend on 2000 shares with a face value of 50 cents if the dividend is 5%: 18 Calculate the amount of dividend on 600 shares with a face value of $1.00 if the dividend is 7%: 19 Calculate the amount of dividend on 3500 shares with a face value of $1.00 if the dividend is 4.6%: 20

a Find the total cost of buying 500 CBA shares at $19.45 each. Brokerage is 2.5% and stamp duty is 15 cents per $100 or part thereof. b If the shares are sold 12 months later for $26.95, calculate the amount obtained from the sale. Yield % is calculated using the market price.

c Determine the gain from the investment. d Find the percentage gain made on the original investment.

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21 Copy and complete the following table.

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Face value $1:00 $1:00 $0:50 $0:50 $1:00

Market price $1:80 $1:50 $1:24 $0:86

Dividend per share in cents 9 4 10

Yield % 12

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Dividend % is calculated using the face value.

22 An investor buys the following portfolio of shares from his broker. The end of year dividends are included. Shares 100 Amcor 50 BHP 300 Borel 500 PMG 2000 RBC

Face value $ 1:00 1:00 1:00 0:50 0:50

Market price $ 9:80 19:90 3:60 1:10 0:28

Dividend per share in cents 22 18 12 12 8

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a Find the market value of this portfolio of shares. b Calculate the total cost of purchasing the shares. Brokerage is 2:2% and stamp duty is the usual rate. c Calculate the amount of the total dividend for the year. d Using your answers from a and c, determine the percentage yield for this portfolio. e The shares are then sold with the table Shares Market price $ showing the selling price. Calculate the 100 Amcor 10:20 net profit or loss. Remember brokerage 50 BHP 21:30 and stamp duty. 300 Borel 2:90 500 PMG 1:25 2000 RBC 0:36

5

The following graph shows the performance of a share over a period of time. Use this to estimate the value of the share in the year 2001.

BHP share performance $20.00

Share price

EXAMPLE

$15.00 $10.00 $5.00 $0.00 1989 1991 1993 1995 1997 1999 Year

From the graph an estimate of the value in a year’s time would be about $22:00.

This is just a guess. It is not possible to predict share prices.

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23 The following graphs show the performance of some individual shares over a period of time. Use this to estimate the value of each share in one year’s time. a b Commonwealth Bank Coles Myer $30.00

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$10.00

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Share price

Share price

$8.00 $20.00 $10.00

$6.00 $4.00 $2.00

5

$0.00 1990

0

Year 1992

c

1994

1996

1998

2000

Year 1992

1994

d

AGL

$15.00

$0.00 1990

1996

1998

2000

AMP

$25.00

Share price

Share price

$20.00 $10.00 $5.00 $0.00 1990

Year 1992

1994

1996

1998

2000

$15.00 $10.00 $5.00 $0.00 Jan-98

Date Jul-98

Feb-99

Aug-99

Mar-00

RESEARCH PROJECT 1 Visit the ASX site and play the trading game. www.asx.com.au 2 Choose your own portfolio of shares from five companies. Check the share prices and calculate the profit or loss each month. Don’t forget the costs of buying and selling. The money site will give a summary of share prices each day www.money.ninemsn.com.au 3 A site which gives share history and graphs and charts is www.money.betainvestor.com.au. Use this site to view share histories and make predictions. Make a portfolio of shares and check your predictions.

F

INVESTMENTS

The value of investments may be calculated using prepared tables of values. The following table shows the value of $1.00 invested at various interest rates over various time periods.

Periods 1 2 3 4 5 6 7 8

Interest rate 1% 5% 1:010 1:050 1:020 1:103 1:030 1:158 1:041 1:216 1:051 1:276 1:062 1:340 1:072 1:407 1:083 1:477

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per period 10% 15% 1:100 1:150 1:210 1:323 1:331 1:521 1:461 1:750 1:611 2:011 1:772 2:313 1:949 2:660 2:144 3:059

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EXAMPLE

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Use the table to find the future value of a $5000 invested at 10% compounded annually for 3 years b $380 invested at 1% per quarter compounded quarterly for 5 quarters.

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a Amount = $5000 £ 1:331 = $6655 to the nearest dollar.

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b Amount = $380 £ 1:051 = $399:38 Be careful with the time periods

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EXERCISE 8F 1 Use the table from the example to find the future value of a b c d e f g h i j k l

$8000 invested at 5% compounded annually for 3 years $5000 invested at 10% compounded annually for 7 years $12 000 invested at 10% compounded annually for 6 years $750 invested at 15% compounded annually for 2 years $1120 invested at 1% per quarter compounded quarterly for 6 quarters $10 500 invested at 5% per quarter compounded quarterly for 2 quarters $1140 invested at 10% per quarter compounded quarterly for 5 quarters $8000 invested at 1% per quarter compounded quarterly for 2 years $750 invested at 15% per quarter compounded quarterly for 1 year $1550 invested at 1% per month compounded monthly for 7 months $78 420 invested at 1% per month compounded monthly for 4 months $100 000 invested at 5% per month compounded monthly for 5 months.

2 Use this table of compounded values to answer the following questions. Find the future value of a $5000 invested at 9% compounded annually for 4 years b

$3500 invested at 6% compounded annually for 8 years

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Periods 1 2 3 4 5 6 7 8

Interest 3% 1:030 1:061 1:093 1:126 1:159 1:194 1:230 1:267

rate per 6% 1:060 1:124 1:191 1:262 1:338 1:419 1:504 1:594

period 9% 1:090 1:188 1:295 1:412 1:539 1:671 1:828 1:993

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$1300 invested at 3% per quarter compounded quarterly for 7 quarters $6800 invested at 3% per quarter compounded quarterly for 3 quarters $2350 invested at 6% per quarter compounded quarterly for 2 years $20 000 invested at 3% per quarter compounded quarterly for 8 quarters $700 invested at 3% per quarter compounded quarterly for 5 quarters $890 invested at 6% per month compounded monthly for 3 months


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$8431 invested at 9% per month compounded monthly for 5 months $200 000 invested at 3% per month compounded monthly for 7 months.

3 Use this table of compounded values to answer the following questions. Find the future value of a $6000 invested at 4% compounded annually for 5 years

Periods 1 2 3 4 5 6 7 8

Interest rate 2% 4% 1:020 1:040 1:040 1:082 1:061 1:125 1:082 1:170 1:104 1:217 1:126 1:265 1:149 1:316 1:172 1:369

per period 6% 8% 1:060 1:080 1:124 1:166 1:191 1:260 1:262 1:360 1:338 1:469 1:419 1:587 1:504 1:714 1:594 1:851

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$1500 invested at 8% per quarter compounded quarterly for 8 quarters $2900 invested at 4% per quarter compounded quarterly for 5 quarters $2200 invested at 2% per quarter compounded quarterly for 2 years $15 000 invested at 6% per quarter compounded quarterly for 7 quarters $900 invested at 8% per quarter compounded quarterly for 6 quarters $1150 invested at 2% per month compounded monthly for 5 months $5280 invested at 6% per month compounded monthly for 7 months $260 000 invested at 4% per month compounded monthly for 3 months.

0

4 Use the compound interest formula to calculate the values from question 1 to the nearest cent. Explain the differences in answers.

G

INFLATION AND APPRECIATED VALUE

Inflation is the rise in cost of consumer goods and services and is usually expressed as a percentage rate. The inflation rate reflects the increase in cost of items. We can calculate appreciated values on a year by year basis, or if the rate of inflation is constant over the time period, we could use the compound interest formula. µ price after n years = original price £

This is,

1+

inflation rate 100

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¶n

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EXAMPLE

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The price of petrol increases at the same rate as inflation. The inflation rate over 3 years is 4% pa. If petrol is 68.5 cents per litre now, how much will it be in 3 years time?

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Price after 1 year = 68:5 £ 1:04 = 71.24 cents per litre.

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Price after 2 years = 71:24 £ 1:04 = 74:0896 cents per litre = 74:09 cents per litre.

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This method is used if the rate is not constant.

Price after 3 years = 74:0896 £ 1:04 = 77:053184 = 77.1 cents per litre.

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or using the formula, Price after n years = original price £

¶n µ inflation rate 1+ 100

4 3 petrol price = 68:5 £ (1 + 100 ) = 77.1 cents per litre ( 1 decimal place)

EXAMPLE

2

A new Nissan costs $19990 now. The price increases at the same rate as inflation. If the rate of inflation is 4.8%, find the expected cost of the Nissan in 4 years time.

µ ¶n inflation rate Price after n years = original price £ 1 + 100 ¡ ¢ 4:8 4 ) price of Nissan = 19990 £ 1 + 100 + $24 100 (to the nearest $100)

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EXERCISE 8G 1 A loaf of bread costs $2:05 today. The price increases at the same rate as inflation. The rate of inflation is 5%. Find the expected price of a loaf of bread after a 1 year b 2 years c 3 years d 4 years e 5 years 2 What price would you expect to pay for each of the following items after the number of years shown? The rate of inflation remains constant. 100

Item Microwave oven Camera Cordless phone Office chair VCR Jar of coffee Icecream 4 litres Potatoes 4 kg

present price $499 $850 $253 $140 $385 $4:52 $3:84 $3:88

inflation rate % 3:2 1:8 2:8 3:5 2:3 3:8 4:3 2:9

number of years 3 4 2 3 4 2 3 2

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3 In three successive years the annual rate of inflation was 6.4%, 5.6%, and 4.8%. How much would you expect to pay after these 3 years, for a dozen eggs which cost $2.55 at the beginning of this period? 4 In three successive years the annual rate of inflation was 2.8%, 3.5%, and 4.1%. How much would you expect to pay after these 3 years, for a carton of beer which cost $19.99 at the beginning of this period? 5 In four successive years the annual rate of inflation was 4.6%, 6.5%, 4.9% and 3.8%. How much would you expect to pay after these 4 years, for a dozen roses which cost $13.50 at the beginning of this period? 6 A stamp collection is purchased for $2500. It increases in value by 4.5% per year. Find the expected value of the stamp collection after a 10 years b 20 years c 50 years 7 A signed Don Bradman cricket bat is purchased in 2000 for $3500. It is expected to increase in value by 3% per annum. Find the expected value of the cricket bat after a 5 years b 10 years c 20 years 8 A Samurai sword is purchased for $3 800. It is expected to increase in value by 6% per year. After 10 years it is sold for $8600. By how much did this exceed the expected value? 9 A signed photograph of Peter Brock is purchased for $900. It is expected to increase in value by 4% per year. After 6 years it is sold for $1500. By how much did this exceed or not exceed the expected value? 10 The last signed Balmain jumper is purchased for $50 000. It is expected to increase in value by 9% per year. After 10 years it is sold for $200 000. By how much did this exceed or not exceed the expected value? 11 A signed 1999 World Cup Rugby jumper is purchased for $2000. It is expected to increase in value by 8% per year. After 5 years it is sold for $2500. By how much did this exceed or not exceed the expected value?

EXAMPLE

3

Three years ago a loaf of bread cost $2.12. It now costs $2.36. What is the average annual rate of inflation of the cost of bread over the three years? It is possible to use the compound interest formula since the rate of inflation is constant. Use A = 2:36 as the final amount, P = 2:12 as the principal amount, and n = 3.

)

100

r n ) A = P (1 + 100 ³ r ´3 ) 2:36 = 2:12 1 + 100 2:36 ³ r ´3 ) = 1+ 2:12 100 ³ r ´3 1:113 207 547 = 1 + 100

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p r 3 1:113 207 547 = 1 + 100 r ) 1:036 395 171 = 1 + 100 r ) 0:036 395 171 = and so r + 3:6% 100 ) the average annual inflation rate is 3.6%.

Use the key for years exceeding 3.

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12 Three years ago the price of rump steak was $8.99/kg. It now costs $9.69/kg. Calculate the average yearly inflation rate which would produce this rise. 13 In three years the price of a TV rose from $ 729 to $899. What was the annual rate of inflation (assumed to be constant)? 14 For the Samurai sword in question 8 calculate the annual rate of increase. (Assume constant inflation.) 15 For the collectables in questions 9, 10, 11 calculate the actual annual rate of increase.

8


1 Here is a list of terms used in this section. Write a description of each term in a sentence. compounding period, dividend, simple interest, compound interest, dividend yield, future value, present value, shares, flat rate, annual, brokerage, stamp duty, inflation. 2 What is the difference between simple and compound interest? Write a paragraph. 3 Write a paragraph about the advantages and disadvantages of using graphs as mathematical models for interest calculations.


You should be able to: ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤


calculate simple interest over different time periods, altering the rate where necessary draw simple interest graphs using pen and paper and technology calculate compound interest amounts using tables and the compound interest formula, including time periods other than years calculate the amount needed to be invested at a particular rate to have a specific amount available after a fixed period of time draw and interpret compound interest graphs complete calculations associated with the purchase and sale of shares understand and calculate dividend and yield analyse share performance and extrapolate calculate investments using prepared tables calculate the effect of inflation on the cost of goods calculate the value of collectables based on expected increases in value over time calculate the average rate of inflation.


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8

DIAGNOSTIC TEST

1 The simple interest on $650 at 8% pa over 4 years is A $52 B $208 C $758

D

$20 800

2 The simple interest on $1580 at 7.2% pa over 15 months is A $1706.40 B $3286:40 C $113:76

D

$142:20

3 The annual simple interest rate when $561 interest is earned on $2200 over 3 years is A $561 B 8:5% C 25:5% D 5:61% Use the graph to answer questions 4 and 5. Interest at 8% pa 900 800 700 600

Interest

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500 400 300 200 100

Years 0 0

2

4

6

4 From the graph the interest after 7 years is about A $550 B $860 C

8

$500

10

D

5 From the graph the number of years required to earn $700 interest is about A 8:8 B 8 C 9 D

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$600 8.4

6 The amount when $3550 is invested at 5.8% pa compounded annually for 4 years is A $4373:60 B $823:60 C $898:06 D $4448:06 7 The interest earned when $680 is invested for 5 years at 4.9% pa interest compounding annually is A $863:75 B $183:75 C $166.60 D $846:60 8 The amount when $1630 is invested at 4.3% pa compounded quarterly for 7 quarters is A $2188:66 B $558.66 C $1756:68 D $126:68

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9 The amount that must be invested at 6% pa interest compounding annually to have $4500 at the end of 5 years is A $6022:02 B $1522:02 C $5850 D $3362:66 10 The amount that must be invested at 4.8% pa interest compounding monthly to have $4500 at the end of 3 years is A $3897:61 B $3909:57 C $832.16 D $3899:84

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Use this graph to answer the next two questions. 95

Compound interest

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Amount of investment

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$2,500 $2,000 $1,500 $1,000 $500 $0 0

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2

3

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5 Years

6

7

11 From the graph the value of the investment after 5.5 years is about A $1000 B $1520 C $1500

8

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D

10

$1600

12 From the graph the number of years taken for the investment to be worth $1700 is A 6:8 B 5:8 C 6:4 D 5:7 13 Calculate the total cost to purchase 400 AMP shares with a market price of $15.05. Brokerage is 2.5% and stamp duty is 15 cents per $100 or part thereof. The answer is A $6020 B $6170:50 C $6179:65 D $6179:76 14 Calculate the amount obtained when 500 Qantas shares are sold at $4.85: Brokerage is $29 fixed and stamp duty is 15 cents per $100 or part thereof. The answer is A $2425 B $2395:96 C $2392:25 D $2457:75 15 A dividend of 24 cents per share is paid on Stockland shares with a market value of $3.25 and a face value of $1.00. The percentage yield is A 7:38% B 13:54% C 24% D 3:25% 16 The amount of dividend on 1100 shares with a face value of 50 cents and a market value of 93 cents, if the dividend is 8% is A $81:84 B 4 cents C 43 cents D $44 Use the table to answer the next two questions. Interest rate per period Periods

1%

5%

10%

15%

4 5 6 7

1:041 1:051 1:062 1:072

1:216 1:276 1:340 1:407

1:461 1:611 1:772 1:949

1:750 2:011 2:313 2:660

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17 The future value of $3 500 invested at 5% pa interest compounding annually for 5 years is A $4256 B $4466 C $4690 D $4924:50

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18 The future value of $1300 invested at 10% pa interest compounding quarterly for 6 quarters is A $1380:60 B $1742 C $3006:90 D $2303:60

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19 A box of cereal costs $3.78 today. The price increases at the same rate as inflation. The rate of inflation for the next three years is predicted to be constant at 6% pa. The expected cost of the cereal in 3 years time is A $4:07 B $4:25 C $4:50 D $4:77

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20 A signed Australia Pakistan cricket bat is purchased after Australia’s win in the 1999 series for $600. It is expected to increase in value by 3% pa. Its expected value after 10 years is A $806.35 B $780 C $618 D $3600

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Question Section

8A

?

1, 2, 3 A

4, 5 B

6, 7, 8, 9, 10 C

11, 12 D

13, 14, 15, 16 E

17, 18 F

19, 20 G

REVIEW SET

1 Calculate the simple interest on $850 at a flat rate of 7% pa over 3 years. 2 An a b c d

investment pays 10% interest pa. Express this as a decimal. Find the six monthly interest rate as a percentage and a decimal. Find the quarterly interest rate as a percentage and a decimal. Find the monthly interest rate as a percentage and a decimal.

3 Calculate the simple interest on $4000 at a flat rate of 7.92% pa over 17 months. 4 If $4200 is invested for 6 years the amount of interest earned is $1512. Calculate the annual simple percentage interest rate. 5 Draw a graph showing the amount of interest earned over a period of 10 years if $1000 is invested at 7% pa simple interest. a Find the interest after 6 12 years. b Find the time to earn interest of $250: 6


7 Use the compound interest formula to calculate the amount of a fixed term investment of $450 over 4 years at 2.9% pa interest compounding quarterly. 8 Calculate the amount that must be invested at 6% pa interest compounding annually to have $6000 at the end of 7 years.

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9 Calculate the amount that must be invested at 3.8% pa interest compounding monthly to have $1150 at the end of 5 years. 10 Calculate the total cost to purchase 900 MYOB shares with a market price of $9.75. Brokerage is 2.5% and stamp duty is 15 cents per $100 or part thereof.

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11 Calculate the amount obtained when 600 Wattyl shares are sold at $4.72. Brokerage is $32 fixed and stamp duty is 15 cents per $100 or part thereof.

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12 Calculate the amount of dividend on 900 shares with a face value of $1.00 if the dividend is 6.5%:

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13 Use a table from Section F to calculate the amount when $2880 is invested at 8% per month compounded monthly for 7 months.

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14 In three successive years the annual rate of inflation was 5.4%, 4.6%, and 3.8%. How much would you expect to pay after these 3 years, for a dozen eggs which cost $2.68 at the beginning of this period?

0

8B

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REVIEW SET

1 Calculate the simple interest on $3550 at a flat rate of 11% pa over 3 years. 2 An investment pays 13% interest pa. a b c d


3 Calculate the simple interest on $6000 at a flat rate of 9.74% pa over 13 months. 4 If $1590 is invested for 8 years the amount of interest earned is $699.60. Calculate the annual simple percentage interest rate. 5 Draw a graph showing the amount of interest earned over a period of 10 years if $1000 is invested at 5% pa simple interest. b Find the time to earn interest of $80: a Find the interest after 3 12 years. 6

a Use the compoundinterest formula to calculate the amount of a fixed term investment of $3580 over 5 years at 3.54% pa interest compounding yearly. b Find the total interest earned.

7 Use the compound interest formula to calculate the amount of a fixed term investment of $1700 over 6 years at 5.4% pa interest compounding quarterly. 8 Calculate the amount that must be invested at 6.2% pa interest compounding annually to have $10 000 at the end of 9 years. 9 Calculate the amount that must be invested at 3.5% pa interest compounding quarterly to have $1050 at the end of 6 years.

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10 Calculate the total cost to purchase 750 OPSM shares with a market price of $3.85. Brokerage is 2.5% and stamp duty is 15 cents per $100 or part thereof. 11 A dividend of 5.65 cents per share is paid on Rebel shares with a market value of $1.54. Find the percentage yield. 12 Calculate the amount of dividend on 3000 shares with a face value of 50 cents if the dividend is 6.5%:

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13 Use a table from Section F to calculate the amount when $7000 is invested at 1% per quarter compounded quarterly for 5 quarters.

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14 In three successive years the annual rate of inflation was 1.8%, 4.5%, and 3.1%. How much would you expect to pay after these 3 years, for a carton of beer which cost $29.99 at the beginning of this period?

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REVIEW SET

1 Calculate the simple interest on $1600 at a flat rate of 8.5% pa over 6 years. 2 An a b c d

investment pays 6% interest pa. Express this as a decimal. Find the six monthly interest rate as a percentage and a decimal. Find the quarterly interest rate as a percentage and a decimal. Find the monthly interest rate as a percentage and a decimal.

3 Calculate the simple interest on $9000 at a flat rate of 3.97% pa over 19 months. 4 If $11 000 is invested for 7 years the amount of interest earned is $4312. Calculate the annual simple percentage interest rate. 5 Draw a graph showing the amount of interest earned over a period of 10 years if $1000 is invested at 9% pa simple interest. b Find the time to earn interest of $160: a Find the interest after 5 12 years. 6


7 Use the compound interest formula to calculate the amount of a fixed term investment of $1700 over 6 years at 9.4% pa interest compounding quarterly. 8 Calculate the amount that must be invested at 9% pa interest compounding annually to have $6000 at the end of 4 years. 9 Calculate the total cost to purchase 1500 AMP shares with a market price of $17.05. Brokerage is 2.5% and stamp duty is 15 cents per $100 or part thereof. 10 Calculate the amount obtained when 3000 Prime Life shares are sold at $1.90. Brokerage is $27.50 fixed and stamp duty is 15 cents per $100 or part thereof. 11 Calculate the amount of dividend on 1800 shares with a face value of 50 cents if the dividend is 13.5%: 12 Use a table from Section F to calculate the amount when $5200 is invested at 2% per quarter compounded quarterly for 7 quarters. 13 In three successive years the annual rate of inflation was 7.4%, 4.6%, and 5.8%. How much would you expect to pay after these 3 years, for a block of chocolate which cost $4.70 at the beginning of this period?

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1 Calculate the simple interest on $700 at a flat rate of 7% pa over 7 years. 2 An investment pays 5% interest pa. a b c d

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3 If $9200 is invested for 3 years the amount of interest earned is $2263.20. Calculate the annual simple percentage interest rate. 4 Calculate the simple interest on $7000 at a flat rate of 5.53% pa over 11 months. 5 Draw a graph showing the amount of interest earned over a period of 10 years if $1000 is invested at 15% pa simple interest. a Find the interest after 9 12 years. b Find the time to earn interest of $850: 6


7 Use the compound interest formula to calculate the amount of a fixed term investment of $2950 over 4 years at 5.1% pa interest compounding quarterly. 8 Calculate the amount of dividend on 2500 shares with a face value of $1.00 if the dividend is 7.6%: 9 Calculate the amount that must be invested at 6% pa interest compounding annually to have $1900 at the end of 5 years. 10 Calculate the amount that must be invested at 3.2% pa interest compounding quarterly to have $10 000 at the end of 7 years. 11 Calculate the total cost to purchase 3000 Fosters shares with a market price of $5.85. Brokerage is 2.5% and stamp duty is 15 cents per $100 or part thereof. 12 A dividend of 28 cents per share is paid on Stockland shares with a market value of $3.96. Find the percentage yield. 13 Use a table from Section F to calculate the amount when $4000 is invested at 4% per month compounded monthly for 6 months.

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14 In three successive years the annual rate of inflation was 5.8%, 4.5%, and 3.1%. How much would you expect to pay after these 3 years, for a hamburger which cost $3.20 at the beginning of this period?

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CHAPTER

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Probability

AREA OF STUDY

This chapter deals with the language of chance, counting outcomes, describing the sample space of an event and comparing experimental probability with theoretical probability. The main mathematical ideas investigated in this chapter are: 8 the language of chance 8 the sample space for simple events 8 equally likely outcomes 8 the sample space for multi-stage events 8 methods of counting the number of outcomes 8 experimental probability 8 theoretical probability 8 comparing calculated probabilities with experimental results 8 complementary events 8 calculating the probability for multistage events.

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PROBABILITY (Chapter 9)

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THE LANGUAGE OF CHANCE

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There are many words in common language that are used to describe the chance, or likelihood, of an event happening - impossible, no chance, unlikely, certain, definite, probably, possibly, maybe, even chance, 50-50 chance etc. For example, “I’ve got no chance of finishing all my maths homework tonight.” “I’ll probably go to the party on Saturday night.” “I’m definitely going to buy a new pair of shoes.” “When you toss a coin there is an even chance of getting a head or a tail.” Can you think of any other words used to describe chance? Use the words in a sentence.

EXERCISE 9A 1 Use a c e g h i j

words like those in the introduction above to describe the chance of the following events the sun rising tomorrow b winning Lotto it will rain tomorrow d being struck by lightning getting homework today f living to 500 years of age snow falling at your school in December somebody in your class having a birthday today selecting a red card from a normal playing pack attending school tomorrow

2 Copy the scale below and place the events of question 1 on it.

Impossible

Even

Certain

3 The following is a list of colloquial terms and expressions used to describe chance. Research and discuss the meaning of these terms: pigs might fly, once in a blue moon, as scarce as hen’s teeth, a long shot, a sure thing, in the box seat, Buckley’s chance, bank on it, put your house on it, a one in 300 year flood, a snowflake’s chance in hell

THE SAMPLE SPACE 100

The sample space of an experiment is the set of all the possible outcomes.

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EXAMPLE 25

List the sample space when a a coin is tossed

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b

a die is rolled

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a When a coin is tossed, the possible outcomes are a head or a tail. This may be written as the sample space S = fH, T g.

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b When a die is rolled, the possible outcomes are a 1, 2, 3, 4, 5 or 6, i.e., the sample space is S = f1, 2, 3, 4, 5, 6g.

largest face

c If each of the 6 faces can be identified then the sample space is S = ftop, bottom, front face, back face, right end, left endg. If the 6 faces cannot each be identified i.e., then the matchbox can land on one of the largest faces, or on one of the side faces or on one of the end faces,

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i.e., the sample space is S = flargest face, side face, end faceg.

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side face end face

EXERCISE 9B 1 List a b c d e f g h i j

the sample space for the following a child is born a drawing pin is tossed a cylindrical can is tossed a marble is chosen from a bag containing 5 red, 3 blue and 2 white marbles the colour of a set of traffic lights as a car approaches a vowel is chosen from the English alphabet a letter is chosen from the word INSIGHT the day of the week a child is born the result (not the score) of a soccer match a card is chosen from a normal playing pack (there are several answers depending on what is being investigated.)

EQUALLY LIKELY OUTCOMES

Equally likely outcomes are outcomes which have exactly the same (equal) chance of occurring.

EXAMPLE

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State whether the outcomes of each of the sample spaces in Example 1 are equally likely to occur.

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a If the coin is “fair” (not biased in some way), then we would expect each outcome to occur an equal number of times.

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b Again, if the die is “fair” we would expect the outcomes to be equally likely.

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c The matchbox is more likely to fall onto one of its larger faces, i.e., the outcomes are not equally likely.

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2 State whether or not you think that the outcomes for each of the sample spaces in question 1 above are equally likely to happen (we will test some of these experimentally, later). Discuss any problems with the class.

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3 Comment on the statements: a Since it either rains or is fine then there is an even chance of it raining tomorrow. 25

b If there are six swimmers in a final, each has an equal chance of winning.

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c The chance that a person’s family name starts with Z is the same as the chance that it starts with any other letter of the alphabet.

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d Since there are only two candidates for the election, then each has a 50-50 chance of winning. e Each student in the class has an equal chance of getting their driving licence this year.

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THE SAMPLE SPACE FOR MULTI-STAGE EVENTS

In the language of chance, an event is an outcome or a collection of outcomes. For example, when a die is rolled “getting a 2” is one of the six equally likely outcomes. The event is “getting a 2”. “Getting an even number” includes the outcomes 2, 4 or 6. This event is a collection of outcomes. A multi-stage event is one which is made up of simpler events. When finding the sample space for multi-stage events, it is often useful to use a systematic method such as a list, table or tree diagram.

EXAMPLE

1

Tim has three pairs of trousers (blue, brown and grey) and two shirts (one long sleeved and one short sleeved) in his wardrobe. Find all the possible combinations of trousers and shirts he can . wear Using a table:

Shirt

Long Short

Blue Bl L Bl S

Trousers Brown Grey Br L G L Br S G S 100

From the table we can determine that there are six equally likely outcomes:

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Blue trousers with long sleeved shirt Brown trousers with long sleeved shirt Grey trousers with long sleeved shirt Blue trousers with short sleeved shirt Brown trousers with short sleeved shirt Grey trousers with short sleeved shirt

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Using a tree diagram yields the same results:

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shirt 75

trousers Blue

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Brown

5

Grey

Long

Bl and L

Short

Bl and S

Long

Br and L

Short

Br and S

Long

G and L G and S

Short

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This is called a two-stage event because it is made up of two simpler events; the colour of the trousers (with three possible outcomes) and the type of shirt (with two possible outcomes). Tables or tree diagrams are convenient ways of identifying the sample space for two-stage events.

EXAMPLE

2

Find all the possible ways you can answer the first three questions on a True/False test.

This is a three-stage event because it is the combination of the outcomes of three simpler events (the possible answers to three questions). It is not possible to use a table for three-stage events. Using a tree diagram, we can find the eight outcomes listed below.

1st Q T

2nd Q T F T

F

?

F

3rd Q T F T F T F T F

outcomes TTT TTF TFT TFF FTT FTF FFT FFF

EXERCISE 9C 1 Find all the possible ways you can answer the first two questions on a True/False test. (Use T for true and F for false.) 2 Find the sample space when two coins are tossed. It does not matter if the coins are tossed simultaneously or one after the other. (Use H for heads and T for tails.) Are all these outcomes equally likely? 3 What is the possible makeup (including the order of birth) of a family with three children. (Use B for boy and G for girl.) Are the outcomes equally likely?

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second die

6 Copy and complete the following table to find the sample space when two dice are rolled.

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1 2 3 4 5 6

1 2 3 (1, 1) (1, 2) (1, 3) (2, 1) (3, 1)

4

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second die

first die

7 1 2 3 4 5 6

1 2 3 4 5 6

2 3 4 5

3 4 5

4 5

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Use the table of question 6 to find the possible sums of the 2 numbers uppermost, when two dice are rolled. This has been started for you alongside.

second die

8 Use the table in question 6 to find the possible differences between the larger and smaller of the two numbers uppermost, when two dice are rolled. This has been started alongside.

first die

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5 Sarah travels through two sets of traffic lights on her way to the station each morning. What are all the possible sequences of coloured lights she could encounter? Do you think that all these outcomes are equally likely?

first die

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1 2 3 4 5 6

1 0 1 2 3 4

9 Find all the “words” that can be made using the letters shown.

2 1 0 1 2

3 2 1 0

4 3

5

6

b, c, d a, e x, y

First letter: Second letter: Third letter:

10 List all the ways in which John, Paul and George can be arranged in a line. 11 A special “Sports Pack” model of a small car is produced. It has a choice of 1:8 L or 2:2 L engines, comes in automatic or manual and is available in metallic orange, blue or red paint. List all the different combinations of features available in this model. 12 Cathy, Melinda and Rebecca are the only three runners in a race. Find all the possible orders in which they could finish. 13 A bag contains one red, one blue and one green marble. Two marbles are drawn from the bag in succession. List the sample space if the first marble is a replaced before the second marble is drawn b not replaced before the second marble is drawn.

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14 The numbers 5, 7, 9 are written on three cards and placed in a hat. Two cards are selected, in succession, from the hat and placed in order of drawing on a table. List all the two digit numbers it is possible to make if the first card is a replaced before the second is drawn b not replaced before the second is drawn.

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METHODS OF COUNTING

In Example 1 of 9C, Tim had 3 pairs of trousers and 2 shirts. The number of possible combinations of trousers and shirts was found to be 6. Note that this is the product (3 £ 2) of the number of choices at each stage.

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In Example 2 of 9C, the total number of outcomes for this three stage event was 8, which is again the product (2 £ 2 £ 2) of the number of choices (= 2) at each of the three stages. These results may be generalised.

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The Fundamental Counting Theorem states: The total number of outcomes for a multi-stage event can be determined from the product of the number of choices at each stage.

EXAMPLE

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A restaurant menu offers four entrees, five main courses and three sweets. How many three course meals is it possible to choose? Total number of meals = 4 £ 5 £ 3 = 60

?

EXERCISE 9D 1 A restaurant menu has a choice of five entrees, six main courses and four sweets. How many different three course meals are possible? 2 Ann has three pairs of slacks and four blouses. How many “mix and match” outfits can she make? 3 Two normal dice are rolled. Find the total number of outcomes. 4 A 4-sided die and a 6-sided die are rolled. How many outcomes are possible? 5 Find the number of possible outcomes if a 2 coins b 3 coins c

4 coins

6 Find the number of different responses to the first a 2 b 3 c 4

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5 coins are tossed. 100

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5 questions on a true/false test.

7 How many different sequences of lights are possible if you pass through a 2 sets b 3 sets c 4 sets of traffic lights?

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8 A brand of men’s jeans are made in eight waist measurements with three leg lengths. How many different sizes are available? 9 A combination lock has three wheels with ten numbers on each wheel. How many different combinations are there?



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10 Part of a maths test contains five multiple choice questions each having four possible responses. How many different ways are there of answering these five questions?

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11 In NSW the postcodes are four digit numbers starting with two, for example, 2087. How many different postcodes can be made using this method?

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12 Determine the number of car number plates possible using a 3 letters and 3 digits b 2 letters and 4 digits

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4 letters.

13 At present Sydney telephone numbers begin with a 9 followed by 7 digits. How many telephone numbers are possible? 14 Different key shapes are made by dividing the key into six sections and using one of two different patterns for each section. a

How many different keys is it possible to make in this way?

b

If there is a choice of three patterns for each section, how many keys are possible?

c

If the key is cut on both sides, how many keys are possible, given a choice of 3 patterns for each section?

EXAMPLE In how many different ways can four people stand in a queue? The first position in the queue can be taken by any of the four people, i.e., there are four choices for first position. The second position can be taken by any of the three remaining people, i.e., there are three choices for second position. The third position can be taken by any of the remaining two people, i.e., there are two choices for the third place. There is then only one person left to take the fourth place. Hence, number of different queues = 4 £ 3 £ 2 £ 1 = 24 100

15 In how many ways can a 3 people

b 5 people stand in a queue?

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16 There are four candidates for the local state election. In how many ways can the names be listed on the ballot paper? 17 There are 24 horses in the Melbourne Cup. In how many ways can the first three places be filled (if there are no dead-heats)?

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ESTIMATING CHANCE

One way to predict the likelihood of an event happening in the future is to investigate the occurrence of the event in the past. This assumes that patterns in the immediate past will not change greatly in the future.

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EXAMPLE

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The students of a class combined to toss two coins 2000 times. The results are recorded in the table alongside. Use this information to predict the chance of getting “2 heads” when you toss two coins.

Result 2 heads 1 head and 1 tail 2 tails

Frequency 563 987 450

From the data collected, 2 heads occurred 563 times out of the 2000 tosses, i.e., on 28:15% of the occasions. If we assume that the frequency of occurrence will be the same for future tosses, then we could predict that when two coins are tossed the chance of getting 2 heads is approximately 28%. (Similarly we could predict from the data that the chance of getting 1 head and 1 tail is wL_pK_pJ_p_ , or approximately 49%, and the chance of getting 2 tails is wF_pG_p:_p_ , or approximately 23%.) EXPERIMENTAL PROBABILITY

In Example 1, we have assigned a number (a percentage) to the chance of an event happening. The mathematics of chance is called probability. The number we have used above is called the relative frequency of the event. Relative frequency gives an estimate of the probability of an event occurring. Because of the way that it is determined, relative frequency is often referred to as experimental probability. i.e.,

Experimental probability = relative frequency f f or P £ 100% = P f f 100

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EXERCISE 9E

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1 A cylindrical can was tossed 200 times. The number of times it landed on its side and on an end is shown in the table alongside.

Outcome End Side

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Frequency 76 124

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a Find the relative frequency of each outcome. b In a future toss of the can, estimate the probability that it will land i


on an end

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Value <6 >6

Frequency 48 72

A card was selected at random from a normal playing pack and its value noted. This was repeated 120 times. The results were recorded as shown. (An ace was counted as a one and the court cards as 10.)

Using this data, estimate the probability that a card selected at random from a normal pack is a less than 6 b greater than or equal to 6. 3 Five hundred salesmen were randomly selected and asked the question “In what country was your car manufactured?” The results are shown in the table: a Find the relative frequency for each country of manufacture. b Using this data, estimate the probability that another salesman chosen at random will own a car manufactured in Australia.

Country Australia Japan Germany Korea Other

Frequency 169 153 77 62 39

c Estimate the probability that the salesman’s car has been manufactured in Korea. 4

a In a 1998 Federal government health survey of 10 000 people, 4578 people admitted to using illicit drugs. Based on this data, what is the probability that a person chosen at random has used illicit drugs? b Of the 1080 girls aged 14 - 19, in the survey, 264 said that they had used cannabis. What is the probability that a teenage girl (aged 14 - 19) chosen at random has used cannabis?

5 A restaurant sold 5284 bottles of wine last year. Of these 107 were returned because the wine was faulty. Based on this information, estimate the probability that a bottle of wine will be faulty when it is opened. 6 The table alongside summarises the answers to the question concerning religion in the last Australian census.

Religion Catholic Anglican Other Christian Non-Christian No stated religion

Estimate the probability that the religion of a person chosen at random will be Catholic

b

Number of days 0-2 3-5 6-8 9 - 11 12 - 14 15 - 17

Frequency 21 18 8 2 0 1

a 7

Anglican

c

Number (millions) 4:75 4:4 4:05 0:53 3:9

Non-Christian.

Data for the past 50 years giving the number of days, in the month of September, on which it rained, is shown in the table alongside. Estimate the probability that, next September, it will rain on a 12 - 14 days b more than 8 days c less than 3 days d 5 or less days

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8 One hundred batteries were selected at random from a day’s production and tested. The results are shown alongside. Determine the approximate probability that a battery selected at random from those produced on this day will have a life of a less than 10 hours b 40 or more hours c less than 20 hours d 20 or more hours

Battery Life (hours) < 10 10 - 19 20 - 29 30 - 39 40 - 49


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9 In a survey of 4000 marriages it was found that 1356 ended in divorce. Phil and Jenny are getting married next weekend. Based on the evidence from the survey, what is the approximate probability that their marriage will end in divorce? 10 Alongside is a life expectancy table similar to those used by insurance companies. It is the result of the collection of statistics on 100 000 males and 100 000 females. a

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LIFE EXPECTANCY TABLE Age

Number Surviving

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

100 000 98 809 98 698 98 555 98 052 97 325 96 688 96 080 95 366 94 323 92 709 89 891 85 198 78 123 67 798 53 942 37 532 20 998 8416 482 482

Male

Age

Number Surviving

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

100 000 99 307 99 125 98 956 98 758 98 516 98 278 98 002 97 615 96 997 95 945 94 285 91 774 87 923 81 924 72 656 58 966 40 842 21 404 7004 1953

Female

b Using the above procedure, estimate the probability that a female baby will survive to 15 years of age. c Estimate the probabilities of a male and a female living to age i 40 ii 60 iii 80 d i How many males were still alive at 50 years of age? ii How many males were still alive at 75 years of age? iii What percentage of males who survived to 50 were still alive at 75? iv Estimate the probability of a 50 year old male living to 75. e Repeat d for females. Experimental probability = Relative frequency 100

PRACTICAL ACTIVITIES 1

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a Roll a die 100 times. b Use tally marks to record the results. c When a die is rolled, what is the experimental probability of getting a i 3 ii 4 iii 6? d Type the results into a spreadsheet and use the CHART option to make a graph of the results.

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e Combine the results for the whole class. f Type the combined results into your spreadsheet and use the CHART option to compare the results of a and e. g Are the outcomes equally likely? Discuss with the class.

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a Make a spinner with four colours, each occupying 14 of the area of the spinner. b Spin the spinner 100 times. c Record the results in a frequency table. d Type the results into a spreadsheet and use the CHART option to make a graph of the results. e What is the experimental probability of getting each colour on your spinner?

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a Drop a drawing pin 100 times and record whether it lands point up or point down.

Outcome Point up

b What is the experimental probability of a drawing pin landing i point up ii point down?

Point down

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Tally

Frequency

Type the results into a spreadsheet and use the CHART option to make a graph of the results. d Combine the results for the whole class. e Type the combined results into your spreadsheet and use the CHART option to compare the results of parts a and d. c

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Are the outcomes equally likely? Discuss with the class.

a Toss an empty matchbox 100 times and record the number of times it lands on i one of its largest faces ii a side face iii an end face

Outcome Large

Tally

Frequency

Side End

b Write down the experimental probability of each outcome. Type the results into a spreadsheet and use the CHART option to make a graph of the results. d Combine the results for the whole class. e Type the combined results into your spreadsheet and use the CHART option to compare the results of a and d. f Are the outcomes equally likely? Discuss with the class. c

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5 Count and record the number of boys and girls born on a particular day by examining the birth notices in a daily newspaper. On the basis of this information, estimate the probability that a baby born will be a male b female.

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6 Equipment needed: Large sheet of paper, toothpick (or matchstick), ruler, pencil. a Measure the length of the toothpick (matchstick). b On the sheet of paper, draw parallel lines which are the length of the toothpick (matchstick) apart.

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i Drop the toothpick from a reasonable height above the sheet of paper and record whether or not any part of it touches one of the lines.

Outcome Touches a line Does not touch a line

Frequency

ii Repeat 100 times. d Calculate the experimental probability that the toothpick lands on a line. e It has been proven that the probability, p, of the toothpick landing on a line is given by p = ¼2 . Find an approximate value for ¼ using this formula and your answer from d. f Repeat d and e using the results of the whole class. g Compare your experimental values of ¼ with that from a calculator. Comment.

F

THEORETICAL PROBABILITY

If all the possible outcomes are eqully likely, then the theoretical probability of an event E happening is given by P(E) =

number of favourable outcomes total number of outcomes

The probability of an event may be expressed as a fraction, a decimal or a percentage.

EXAMPLE

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When a die is rolled, what is the probability of getting The sample space is S = f1, 2, 3, 4, 5, 6g and all these outcomes are equally likely.

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a a2

b an even number? 95

P(2) means the probability of getting a 2.

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Therefore the total number of outcomes is 6. a

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To get a 2 there is only one favourable

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outcome. Hence, P(2) = 16 .



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b

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To get an even number there are three favourable outcomes, a 2, 4 or 6. Hence, P(even number) =

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3 6

=

1 2

(or 0:5 or 50%).

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EXERCISE 9F 1 A die is rolled. What is the probability of getting a a5 b an odd number d a 5 or a 6 e a9

c f

a number greater than 4 a number less than 7?

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2 In Erin’s maths class there are 11 girls and 12 boys. If the teacher selects one student at random from the class, what is the probability that it is a Erin b a girl c a boy d a girl or a boy? 3 A 6 sided die has 3 faces painted black, 2 faces painted red and 1 face painted yellow. If the die is rolled, what is the probability that the uppermost face is a black b red c yellow d black or red e green?

8 67

6 or 7 a number > 5 a number < 9?

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c f i

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4 The spinner shown in the diagram is spun. What is the probability that it stops on a 6 b 7 d an even number e an odd number g a number < 3 h 9

5 A card is selected at random from a normal playing pack. Determine the probability that it is a the 7 of clubs b a7 c a club d a black 7 e a red 7 f black g red h a green 7 i either red or black 6 If there are 200 tickets in a raffle what is the probability of winning first prize if you buy a 1 b 2 c 5 d 10 e 20 f 200 tickets? 7 A letter is chosen at random from the word HIPPOPOTAMUS. What is the probability that it is a P b O c T d a vowel e a consonant f not an A g U or S h Q? 8 A marble is selected at random from a bag containing four red, three blue and two green marbles. What is the probability that it is a red b blue c green d red or blue e not red f not blue g neither red nor green h not red or blue i yellow j red, blue or green? 9 Discs numbered 1 to 20 are placed in a hat and one is chosen at random. Find the probability that the number on the disc chosen is a 7 b not 7 c even d not even e less than 10 f not less than 10 g divisible by 3 h not divisible by 3 i bigger than 20 j less than 20 k a 1 digit number l a 2 digit number



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You should have discovered from the above exercises that:

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If an event E is

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² ²

impossible, certain,

then then

P(E) = 0 P(E) = 1.

Hence, 0 6 P(E) 6 1. 25

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10 Write down an event for which i P(E) = 0 ii P(E) = 1 if a a die is rolled b a ball is chosen from a bag containing three red and two white balls c a card is selected from a normal playing pack. 11 In a class discussion, Matthew gave the following answers to three questions on probability: a 1:2 b ¡0:6 c 99% What could you tell Matthew about his answers? 12 The digits 0 to 9 are written on cards which are placed in a hat. One card is drawn at random from the hat. Write down an event which has a probability of 1 a 10 b 12 c 0 d 1 e

G

3 10

f

9 10

g

11 10

h

7 ¡ 10

COMPARING EXPERIMENTAL AND THEORETICAL PROBABILITY

EXERCISE 9G 1

a Toss a coin 100 times and record the results. b From your results: i Are the outcomes equally likely? ii Calculate the experimental probability of each outcome. c i Combine your results with the rest of the class, by writing them on the board. ii Calculate the experimental probability Number of trials = of each outcome using the combined number of times the results. d Compare the experimental probabilities with the expected (theoretical) probabilities.

experiment is repeated.

100

e What do you think would happen if you tossed a coin 1 000 000 times?

95

f Does the number of trials have an effect on the results? Comment on your findings. 2

75

a When a die is rolled, what is the theoretical probability of getting a number less than 3? b Roll a die 100 times and find the experimental probability of getting a number less than 3. c Compare the theoretical and experimental results of a and b. Comment on your findings.

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d Combine your results from b with the rest of the class by writing them on the board, and calculate the experimental probability using the combined results. e Compare the results of d and a. Comment.

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3 A cup contains five red, three green and two blue cubes. The cup is shaken and one cube is selected. a What is the theoretical probability that the cube is i red ii green iii blue? b Perform this experiment 100 times and find the experimental probability of getting a i red ii green iii blue cube c Compare the experimental results of b with the theoretical results of a. Comment on your findings. d Combine your results of b with the rest of the class by writing them on the board and calculate the experimental probability of each outcome using the results of the whole class. e Compare the results of d with the results of a. Comment. 4 A card is selected at random from a normal pack. a What is the theoretical probability that the card is a i heart ii diamond iii spade b i Select a card from a normal pack and record its suit. ii Replace the card and shuffle the pack. iii Repeat 50 times.

club?

iv

c Use your results to determine the experimental probability that a card chosen at random is a i heart ii diamond iii spade iv club? d i Combine your results with the rest of the class by writing them on the board. ii Repeat c using the class results. e Compare the experimental results with the theoretical results. Comment on your findings. From the above exercises you should have discovered that: The larger the sample size in an experiment the closer the experimental probability is to the expected (theoretical) probability.

H

COMPLEMENTARY EVENTS

100

An event E is an outcome or collection of outcomes. It is a subgroup of the sample space.

95

The remaining group of outcomes of the sample space which do not belong to E is called the e complement of E. The complement of E is written E.

e are called complementary events. From the definition it follows that E and E

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e = Total number of possible outcomes. Number of outcomes in E + number of outcomes in E

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EXAMPLE

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1

A die is rolled. a List all the possible outcomes.

75

b 25

5

0

What is the complement of the event i rolling a 6 iii rolling a number < 3

ii rolling an even number iv not rolling a 4?

a S = f1, 2, 3, 4, 5, 6g b

e = f1, 2, 3, 4, 5g. i If E = f6g then E i.e., the complement of “rolling a 6” is “rolling a 1, 2, 3, 4 or 5” or “not rolling a 6” e = f1, 3, 5g: ii If E = f2, 4, 6g then E The outcomes which do not result in an even number are 1, 3, 5, i.e., the complement of “rolling an even number” is “rolling a 1, 3 or 5” or “not rolling an even number” (or “rolling an odd number”) e = f3, 4, 5, 6g: iii If E = f1, 2g then E The outcomes which do not result in a number < 3 are 3, 4, 5, 6, i.e., the complement of “rolling a number < 3” is “rolling a 3, 4, 5 or 6” or “not rolling a number less than 3” (or “rolling a number > 2) e = f4g, iv In this case, E = f1, 2, 3, 5, 6g hence E i.e., the complement of “not rolling a 4” is “rolling a 4”.

?

EXERCISE 9H 1 A coin is tossed. a List the possible outcomes. b What is the complement of the event i tossing a head

ii tossing a tail?

2 The 5-sided spinner shown alongside is spun. a List all the possible outcomes. b List the outcomes and write down in words the complement of the event i spinning a 3 ii spinning an odd number iii spinning a number > 3 iv not spinning a 5 3 A box contains one red (R), one blue (B) and one green (G) counter. One counter is selected at random. Match the following events with their complement:



100

95

a b c d e f

Event R G R or B B or G not B R, B or G

A B C D E F

C

Complement G G or B not R, B or G R or B R B

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4 A card is selected from a pack. Write down the complement of the event a selecting a diamond b selecting a diamond or a heart c selecting a diamond, heart or club d selecting a red card e selecting an Ace f not selecting a King 5 List the sample space when two coins are tossed. Write down the complement of the event a two heads b head, tail in that order c exactly one head d no heads e at least one head f at least one tail

5

0

EXAMPLE

2

A die is rolled. Find the probability of a i rolling a 6 ii b i rolling an even number ii c i rolling a number less than 3 ii a

i ii

b

i

P(6) =

1 6

P(not 6) =

Note that P(6) + P(not 6) = 1 Hence, P(not 6) = 1¡ P(6) and P(6) = 1¡ P(not 6)

5 6

P(even number) = =

ii

P(not even) = =

c

i

3 6 1 2

Note again P(even) + P(not even) = 1 Hence, P(not even) = 1¡ P(even) and P(even) = 1¡ P(not even)

3 6 1 2

P(number < 3) = =

ii

not rolling a 6 not rolling an even number not rolling a number less than 3

2 6 1 3

P(number not < 3) = =

4 6 2 3

Note again P(number < 3) + P(number not < 3) = 1 Hence, P(number not < 3) = 1¡ P(number < 3) and P(number < 3) = 1¡ P(number not < 3)

In general, for complementary events

e =1 P(E) + P(E)

e Hence, P(E) = 1¡ P(E) e = 1¡ P(E) and P(E)

The last two results are often used when it is inconvenient or time consuming to count all the favourable outcomes for an event.

EXAMPLE

100

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75

3

25

e = 0:24 . a Calculate the probability of the event E if P (E) b Calculate the probability of the complement of the event E if P (E) = 56 .

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95

e a P (E) = 1 ¡ P (E)

75

b

e = 1 ¡ P (E) P (E)

= 1 ¡ 0:24

=1¡

= 0:76

=

5 6

1 6

25

5

6 Calculate the probability of the event E if a

0

e = P (E)

3 4

b

e = 0:3 P (E)

c

e = 15% P (E)

7 Calculate the probability of the complement of the event E if a

P (E) =

7 8

b

P (E) = 0:67

c

P (E) = 48%

EXAMPLE The numbers 1 to 200 are written on cards. The cards are shuffled and one card is selected at random. What is the probability that the number on the card is a 93

b not 93

a P(93) =

1 200

c divisible by 10

b P (not 93) = 1¡ P(93) 1 = 1 ¡ 200 =

c

d not divisible by 10?

P(divisible by 10) = =

20 200

199 200

d P(not divisible by 10) = 1¡ P(divisible by 10) =1¡

1 10

=

1 10

9 10

8 A pack consists of 100 cards numbered 1 to 100. The pack is shuffled and one card is selected at random. What is the probability that it is a c e g i k m

37 divisible by 10 a one digit number a multiple of 12 a number whose last digit is 9 less than 5 greater than 93

b d f h j l n

not 37 not divisible by 10 not a one digit number not a multiple of 12 a number whose last digit is not 9 not less than 5 not greater than 93?

100

95

75

25

9 A card is selected from a normal playing pack. Find the probability that it is a d

not the 6 of clubs not the Ace or King of diamonds

b e

not a black 6 not an Ace or King

c f

not a 6 not a heart

5

0

H:\...\NSWGM\NSWGM_09\315NG09.CDR Wed Feb 02 10:37:47 2000


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10 If three coins are tossed, P(3H) = 18 , P(exactly 2H) = 38 , P(exactly 1H) = 38 . a Write down the probability of i no heads ii no tails iv exactly two tails v three tails b i What is the complement of the event “at least one tail”? ii Find the probability of at least one tail.

95

75

25

c

iii

exactly one tail

i What is the complement of the event “at least one head”? ii Find the probability of at least one head.

5

0

I

PROBABILITY FOR MULTI-STAGE EVENTS

EXAMPLE

1

A shop sells jeans in three colours (blue, grey and white) with either front or side pockets. If there is an equal number of each style on the shelves a use a table to find the sample space. b Find the probability that a pair of jeans chosen at random from the shelves will i be blue with front pockets ii be grey with side pockets iii be white iv have side pockets a B G W

Colour

b

i P(BF) = ii P(GS) =

Pockets Front Side BF BS GF GS WF WS

There are six possible equally likely outcomes.

1 6 1 6

iii The favourable outcomes are WF, WS hence P(white) =

2 6

=

1 3

iv The favourable outcomes are BS, GS, WS, hence P(front pockets) =

EXAMPLE

3 6

=

1 2

2

100

95

a Find all the possible ways you can answer the first three questions on a True/False test. b Find the probability that i the first three answers are True ii the first two answers are True and the third is False iii two answers are True and one is False, in any order iv only one answer is True v all answers are False

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a

95

2nd Q T

1st Q T

75

F T 25

F F

5

Outcomes TTT TTF TFT TFF FTT FTF FTT FFF

From the tree diagram there are 8 possible equally likely outcomes.

i There is only one favourable outcome - TTT. Hence P(TTT) = 18 .

b

0

3rd Q T F T F T F T F

ii P(TTF) =

1 8

iii There are three favourable outcomes TTF, TFT, FTT. Hence P(two True, one False) = 38 . iv There are three favourable outcomes TFF, FTF, FFT. Hence P(1 True) = 38 . v P(FFF) =

?

1 8

EXERCISE 9I 1

a Find all the possible ways of answering the first two questions on a True/False test. b What is the probability that i the first two answers are True ii the first answer is True and the second False iii exactly one answer is True iv both answers are False?

2 A coin is tossed and a die is rolled. a Find the sample space. b Find the probability of getting i a head and a 6 iii a head and an even number

ii iv

a tail and a 3 a tail and a number less than 3

3 A spinner has the numbers 1 to 5 on each of its edges with equal probability of occurring. The spinner is spun and a coin tossed. a List the sample space. b Find the probability of getting i a head and a 5 iii a tail and an odd number 4

100

ii

a head and an even number 95

a List the sample space for a family with two children. b Assuming that the chances of having a boy or girl are the same, what is the probability of having i two boys ii a boy first and then a girl iii a girl first and then a boy iv a boy or a girl in any order v two girls?

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5 Two four-sided dice (regular tetrahedra) with the numbers 1 to 4 on their faces, are tossed. a Use a table to find all the possible outcomes. b Find the probability of getting i double 4 iii 1 and a 4, in any order v two odd numbers

ii iv vi

any double two even numbers a sum of 5?

c What is the probability that i exactly one of the numbers is a 4

ii

at least one of the numbers is a 4?

ii iv vi

any double two even numbers a sum of 5

ii

at least one of the numbers is a 2?

6 Two normal 6-sided dice are tossed. a Use a table to find all the possible outcomes. b Find the probability of getting i double 5 iii a 2 and a 3, in any order v two odd numbers vii a sum of 11 c Which sum is the most likely to occur? d What is the probability that i exactly oneof the numbers is a 2 7 The menu at Joe’s Cafe is shown: Entree

Pasta or BBQ Prawns

Main Course

Steak or Fish or Chicken

Sweets

Apple Cake or Fruit Salad

a List all the possible three course meals. b Mike decides to choose a three course meal at random. What is the probability that he chooses i Pasta, Steak and Fruit Salad ii BBQ Prawns, Fish and Apple Cake iii a meal containing Pasta and Chicken iv a meal which includes Steak v a meal not containing Steak vi a meal not containing BBQ prawns vii a meal which includes Pasta and Apple Cake? 8 A bag contains a red, a blue and a white marble. One marble is chosen at random from the bag and then replaced in the bag. A second marble is then chosen. a Find the sample space. b Find the probability that i both marbles are red iii both marbles are different colours v there is one blue and one white marble vii at least one of the marbles is red

100

95

75

ii iv vi

both marbles are the same colour the first marble is red exactly one of the marbles is red

25

5

9 Repeat question 5 if the first marble is not replaced in the bag before the second is chosen. 0



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25

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319

10 A normal pack of playing cards is shuffled and cut to show one card. The suit of the card is noted. This procedure is repeated once more. a Find all the possible outcomes. b What is the probability that the two cards will be i both diamonds ii a diamond followed by a heart iii a diamond and a heart, in any order iv the same suit v different suits vi both coloured vii both black viii a coloured card followed by a black card ix coloured and black, in any order? c Find the probability that i exactly one of the cards is a spade iii at least one of the cards is black 11

12

ii

a List all the possible outcomes for a family of three b Calculate the probability of having i three girls ii iii two girls and a boy, in any order iv v no girls vi vii at least two boys viii

at least one of the cards is a spade

children. two girls and a boy, in that order exactly one girl at least one boy three boys

a List all the ways that Kylie, Marie and Helen can form a queue in the canteen line. b What is the probability that Kylie will be i first in the queue ii second in the queue iii third in the queue iv in front of Helen v behind Marie vi in front of Helen and behind Marie vii not last in the queue?


You should be able to


2

use the language associated with chance

2

list the sample space of single and multi-stage events

100

2

use tables and tree diagrams to list the sample space

95

2

use the fundamental counting theorem to find the number of possible outcomes

2

determine experimental probability

2 2

calculate theoretical probability understand the relationship between experimental and theoretical probability

2

find complementary events

2

calculate probability for multi-stage events.

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9

100

95


This is a list of key words and phrases used in this chapter. Write a sentence to describe each one. chance, sample space, equally likely outcomes, event, multi-stage event, tree diagram, fundamental counting theorem, relative frequency, probability, experimental probability, complementary event

75

25

5

RESEARCH PROJECT

0

Research and describe how probability is used in some of the following areas: weather forecasting, insurance, medicine, genetics, astronomy, quality control, gambling.

9

?

DIAGNOSTIC TEST

1 Write down three words or phrases that are used to describe chance. 2 Comment on the statement “Since there are only two people in the final then each has a 50-50 chance of winning”. 3 Find all the possible ways you can answer the first three questions on a true/false test. 4 A cafe menu has a choice of two entrees, three main meals and five desserts. The number of different three course meals is A 10 B 30 C 11 D 17 5

Score Frequency A

6 20

1 3

2 5

3 6

4 2 B

5 4 6

The relative frequency of the score 3 is

C

14

14 20

D

4 5

7 If an event E is certain to happen, then A P(E) = 0 B P(E) = 12

0 < P(E) < 1

P(E) = 1

D

1

4

3

C

2

6 The spinner shown is spun once. The probability of getting an odd number is A 25 B 35 C 15 D

5

8 If E is the event a number less than 3 when a die is rolled, then the complement of the event e= is E A f1, 2, 3g B f3, 4, 5, 6g C f5, 6g D f1, 2, 3, 4g 9 Two three sided spinners with the numbers 1, 2, 3 on each are spun. If the probability of each number occurring is the same, then the probability of two 3’s is A

1 3

B

2 9

C

2 3

95

75

1 9

D

100

10 A bag contains one red, one black and one white marble. A marble is selected from the bag and not replaced and then a second marble is selected. The probability of getting a red and a white marble in any order is A 29 B 19 C 26 D 16

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If you have any difficulty with these questions, refer to the examples and questions in the exercise indicated.

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Question Section

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0

9A

1 A

2 B

3 C

4 D

5 E

6, 7 F

8 H

9, 10 I

REVIEW SET

1 Describe the chance of the following events a the sun will set tomorrow night

b

the temperature will be 30o tomorrow

2 List the sample space for a rolling a six sided die

b

a two child family

3 Jeans are made in seven waist sizes and three leg lengths. How many different sizes are possible? 4 A coffee tin was tossed 100 times and the way it landed is recorded alongside.

Landed on End Curved face

What is the experimental probability that this tin will land on one of its ends, when tossed?

Frequency 18 82

5 A card is chosen from a normal playing pack. What is the probability that the card is a the Jack of hearts b a Jack c a heart d a picture card e black? 6 Write down an example of an event E for which a P(E) = 0 b P(E) = 12

c

P(E) = 1

e 7 Write down the value of P(E) + P(E). 8 A coin is tossed and a die is rolled. What is the probability of getting a a head and a 3 b a head and an odd number c a tail and a 5 d a tail and a number less than 3?

?

9B

REVIEW SET

1 Percy, Ming and Yuki form a queue. List the sample space.

100

2 How many three digit numbers can be formed using the numbers 0 - 9?

95

3 Of the 24 students in a class, six watch the evening news. Find the probability that a student chosen at random watches the news. 4 A spinner is made using a regular octagon with the numbers 1 - 8 on each edge. If it is spun, find the probability of getting a a5 b a7 c an odd number d a9 e a number less than 9

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5 If P(E) = 35%, find the probability of the complementary event occurring.

95

6 Two normal six sided dice are tossed. a Use a table to list the sample space. b Find the probability of getting i double 6 iii a total of 7

75

25

ii iv

any double a total of 13

5

0

?

9C

REVIEW SET

1 A jar contains one red, one blue and one green disc. a A disc is selected, its colour noted and it is replaced. A second disc is selected. List the sample space. b List the sample space if the first disc is not replaced. 2 The numbers on a book of raffle tickets range from 1 to 100. What is the probability that a ticket chosen at random is a c e g i

number 56 divisible by 10 a three digit number a number less than 100 a number which is not divisible by 5

b d f h j

a one digit number not divisible by 10 a number less than 10 a number greater than 100 not an even number?

3 The results of a survey of the number of children in each of 100 families in a certain town is shown below. Number of children Number of families

0 11

1 24

2 34

3 19

4 8

5 3

6 1

What is the probability that a family chosen at random from this town will have a

no children

b

two children

c

more than six children?

4 Five cards labelled 1 - 5 are placed in a hat. Another five cards labelled 1 - 5 are put in another hat. A card is selected from each hat. a Use a tree diagram to find the sample space. b Find the probability of getting i double 5 ii a 3 and a 5, in any order iv a total of 5 v a total of 1

iii

a total of 2 100

95

?

9D

75

REVIEW SET

1 A letter is chosen at random from the word WOOLLOOMOOLOO. What is the probability that it is a a W b O c L d M e P?

25

5

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95

323

2 A bag contains three red, four blue and five green marbles. If one marble is chosen at random, what is the probability that it is a red b blue c green d red or blue e not blue?

75

25

5

0

3 A pack consists of 200 cards numbered from 1 - 200. The pack is shuffled and one card selected at random. Find the probability that it is a 53 b not 53 c divisible by 100 d not divisible by 100 e a one digit number f not a one digit number g less than 56 h not less than 56 i not less than 5 j not greater than 150 4 A multiple choice test has two questions, each with four alternative answers A, B, C, D. a Use a tree diagram to find all the possible ways of answering these two questions. b Find the probability that i the answers are A then B ii the answers are B then A iii the answers are both A iv neither answer is A v at least one of the answers is A

100

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100

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75

25

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100

95

75

25

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100

95

CHAPTER

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Taxation

10

AREA OF STUDY

This chapter deals with the calculation of tax. The main mathematical ideas investigated in this chapter are: 8 calculation of allowable deductions 8 calculation of taxable income 8 calculation of medicare levy 8 calculation of tax refund or payment 8 calculation of GST and VAT 8 taxation graphs.

100

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326 100

TAXATION FM3 (Chapter 10)

A

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TAXABLE INCOME

The taxable income is calculated using total income and allowable deductions. Total income includes income from all sources throughout the year. These may include wages and salaries, bonuses, interest earned, commissions, allowances etc. The total income may be subject to deductions that reduce the amount used to calculate the tax payable. Tax deductions may include: union fees, donations over $2, car travel expenses, uniforms and their cleaning, tools used for work, safety equipment, self education expenses, and tax agent fees. Depending on the profession, some or all of these may be allowable deductions.

25

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Taxable income = Total income ¡ allowable deductions. An extra tax added to all taxable incomes is the medicare levy. Medicare is the public hospital medical system available without charge for Australians. The Medicare levy is currently calculated at 1.4% of taxable income. A taxpayer only pays a Medicare levy if their taxable income exceeds $13 643. However, for incomes between $12 689 and $13 643 a reduced levy is payable.

EXAMPLE

1

Amita uses the following information to calculate her taxable income: wages $35 980, interest $568 (joint account), bonus $200, cost of uniforms $140, tax agent fee $80, use of car $298. Calculate Amita’s taxable income.

Interest in joint accounts is shared between both people.

Income = $35 980 + ($568 ¥ 2) + $200 = $36 464 Deductions = $140 + $80 + $298 = $518 Taxable income = $36 464 ¡ $518 = $35 946

?

EXERCISE 10A 1 Annie uses the following information to calculate her taxable income: wages $45 320, interest $665 (joint account), bonus $800, cost of uniforms $340, tax agent fee $60, use of car $45. Calculate Annie’s taxable income. 2 Gary uses the following information to calculate his taxable income: wages $42 330, interest $2355 (joint account), cost of uniforms $300, cleaning of uniforms $156, tax agent fee $100, use of car $3220. Calculate Gary’s taxable income. 3 Asha uses the following information to calculate her taxable income: wages $29 555, interest $243, tax agent fee $70, use of car $452, donations to charity $120. Calculate Asha’s taxable income. 4 Paula uses the following information to calculate her taxable income: wages $53 022, interest $3659, bonus $1000, tax agent fee $120, use of car $1256, donations to charity $500. Calculate Paula’s taxable income.



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TAXATION FM3 (Chapter 10) 100

327

5 Calculate the taxable income for the following people.

95

75

wage allowances interest bonus/tips uniforms cleaning tax agent fee union fee donations car use

25

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EXAMPLE

A Tasman $45 996 $590 $32 $100 $50 $35 $80 $300

B Polding $36 899 $100 $12

C Flockhart $72 665

$60

$200

$423

$19

$100 $2333

$366 $2000

D Gen $12 990 $390 $5

$20 $60 $2

E Old $26 773 $225 $15 $25 $150 $55 $45 $190 $20

2

Calculate the Medicare levy payable on a taxable income of $35 908. Medicare levy = 1.4% of taxable income = 1:4 ¥ 100 £ $35 908 = $502:71

6 Calculate the Medicare levy on these taxable incomes. a d

$39 021 $26 110

EXAMPLE

$21 906 $60 958

b e

c f

$43 879 $10 365

3

The Medicare levy on taxable incomes between $12 689 and $13 643 is 20 cents in the dollar for each dollar in excess of $12 689. Find the Medicare levy on $13 070. Excess = $13070 ¡ $12689 = $381

100

) Medicare levy = 0:20 £ $381 = $76:20

95

75

7 Calculate the Medicare levy on these taxable incomes. a

$12 905

b

$13 640

c

$12 450

d

25

$13 444

5

0



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100

RESEARCH PROJECT

95

75

1 Choose three different professions and find the deductions to which they are entitled. 2 Examine the Tax Pack to determine allowable deductions. 3 Visit the Taxation Office Internet Site at www.ato.gov.au 4 Write about the latest tax information. 5 What changes have been made to Medicare throughout its history?

25

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0

B

CALCULATING TAX

The calculation of income tax is performed using a tax table. Following is the table for calculating annual tax for the year 2000.

EXAMPLE

Calculating Gross Tax Payable for year 2000 $1 to $6000 No tax payable $6001 to $20000 17 cents per $ in excess of $6000 $20001 to $50000 $2380 plus 30 cents per $ in excess of $20000 $50001 to $75000 $11380 plus 42 cents per $ in excess of $50000 $75001 and over $21880 plus 47 cents per $ in excess of $75000 Apply these rates to your taxable income.

1

Use the tax table to calculate the tax payable on an income of $53 251

a

b

$26 784

c

$7342

Use only whole dollars in the calculation of tax.

a $53 251 is in the fourth row of the table. Excess = $53251 ¡ $50000 = $3251

Tax = $11380 + 0:42 £ $3251 = $12745:42

b $26 784 is in the third row of the table. Excess = $26 784 ¡ $20 000 = $6784

Tax = $2380 + 0:30 £ $6784 = $4415:20 100

c $7342 is in the second row of the table. Excess = $7342 ¡ $6000 = $1342

?

Tax = 0 + 0:17 £ $1342 = $228.14

95

75

EXERCISE 10B 1 Calculate the tax payable on these taxable incomes. a $58 921 b $30 911 c $8100



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$3100

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2 The table shows the pay rates that state MP’s receive. Calculate the amount of tax that would be payable on these annual salaries.

Office Premier Deputy premier Opposition leader Minister Speaker/president Parliamentary secretary Backbencher

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Pay $259 000 $227 000 $208 000 $217 000 $208 000 $144 000 $100 000

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Profession Dentistry Medicine Optometry Engineering Computer Science Veterinary Science Education Mathematics Psychology

Starting salary $45 000 $42 000 $39 500 $36 000 $34 000 $32 250 $31 500 $31 400 $29 400

The table shows the estimated starting salaries for new graduates in 2000. Calculate the amount of tax that would be payable on these annual salaries.

RESEARCH PROJECT Find the starting salaries for various apprenticeships. Work out the tax payable and the take home pay.

4 Mrs Jones has an annual salary of $33 700 and receives an income of $896 from other sources. a Calculate Mrs Jones’ total annual income. b Determine the amount of tax Mrs Jones will have to pay on her total income, using the tax table at the beginning of this section. 5 Joshua Eldom earns $47 500 per year. a Calculate the amount of tax he must pay. b If Joshua has already paid $12 674 in tax (PAYE - pay as you earn) throughout the year, calculate the amount of money he receives/pays in the form of a tax refund or tax payment.

Taxable income = Total income - deductions.

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6 Sam has a few odd jobs. In total, he earns $19500 per year. Fortunately, he receives a tax deduction of $ 2600. a Calculate his taxable income. b Calculate the amount of tax he must pay on his taxable income.

A refund is due if too much tax has been paid.

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EXAMPLE

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Sarah works as a secretary and receives a yearly salary of $28 479. She also receives an income of $1950 per year from baby-sitting. Her total tax deductions are $1570. During the year she paid tax instalments amounting to $6 850. Find a c

her total income tax payable on her taxable income

b d

her taxable income her tax refund or balance payable.

a

Total of all income = $28 479 + $1950 = $30 429

b

Taxable income = total income ¡ tax deductions = $30 429 ¡ $1570 = $28 859

c

Tax payable on taxable income = tax payable on $28 859 = $2380 + [30 cents per $ in excess of $20 000] = $2380 + [30 cents £ $(28 859 ¡ 20 000)] = $5037:70

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) the amount of tax payable on $28 859 = $5037:70 d

Tax refund/balance payable Sarah has already paid $6850. She only needed to pay $5037:70 ) she has paid too much and so receives a refund. Tax refund = $6850 ¡ $5037:70 = $1812:30

7 Mr Benton has a yearly salary of $29 760. He receives a salary of $2 500 from his hobby of wood carving. His tax deductions amount to $1090 and throughout the year he has paid a total of PAYE instalments of $8 640. Calculate a his total income b his taxable income c tax payable on his taxable income d his tax refund or balance payable.

PAYE Tax instalments means Pay As You Earn tax.

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8 Brittany earns $45 230 pa as a company manager and $5203 as a part-time singer. She has tax deductions of $2612 and throughout the year pays a total of $11 467 in tax instalments. Find a her total income b her taxable income c tax payable on her taxable income d tax refund or balance payable. 9 Celine earns $32 056 pa as an environmentalist and $8159 as a part-time analyst. She has tax deductions of $4903 and throughout the year pays a total of $10 856 in tax instalments. Find a her total income b her taxable income c tax payable on her taxable income d tax refund or balance payable.

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10 Ricky earns $84¡365 as a cabaret artist and $4620 from his share portfolio. He has tax deductions of $13¡884 and throughout the year pays a total of $ 22¡988 in tax instalments. Find a his total income b his taxable income c tax payable on his taxable income d tax refund or balance payable. 11 Use the following group certificate to find and calculate a gross income c tax paid b taxable income if there are deductions d tax payable on the taxable income of $1350 e tax refund or balance payable.

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Income Year 1 July 1998 to 30 June 1999

Employee’s Tax Return Copy

Australian Taxation Office 1999 Group Certificate Tax File Number

4 5 6 Employee’s Surname/Family Name

M U R P H Y

Suburb/Town/City/State/Postcode

1 5 - T H E - W A Y - C R S Allowances (whole dollars)

H O M E T O W N - N S W - 2 2 1 3

TEA ALW

$

2 3 6

Gross Salary, Wage, Bonus, etc. (whole dollars) (excluding amounts

$

1 2 9 3 4

TRVL AL

$

1 1 0

Tax Instalments Deducted (include cents.)

$

9 7 4. 5 0

Tax instalments deducted (whole dollars only in

Name of Employer

B E L L A - R E S T A U R . Employer’s Group Account

Thousands

ZERO

Hundreds

Tens

NINE

Units

SEVEN

FOUR

Cents

50

Signature of Authorised Person

1 2 3 7 8 9 4 5

12 Use a b c d e

7 8 9

J A N I C E · S O N I A

Street No. and Street Name

Allowances (give details)

1 2 3

Employee’s Given Name(s)

Date

A group certificate is 30/06/1999 issued by the employer showing income and tax paid.

J. Peace

the following group certificate to find and calculate gross income taxable income if there are deductions of $ 2854 tax paid tax payable on the taxable income tax refund or balance payable. Income Year 1 July 1990 to 30 June 2000

Employee’s Tax Return Copy

Australian Taxation Office 2000 Group Certificate Tax File Number

4 5 6 Employee’s Surname/Family Name

M U R P H Y

7 8 9

J A N I C E · S O N I A

Street No. and Street Name

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Suburb/Town/City/State/Postcode

1 5 - T H E - W A Y - C R S Allowances (give details)

1 2 3

Employee’s Given Name(s)

Allowances (whole dollars)

$ $ Name of Employer

A C C O U N T - W H I Z C O Employer’s Group Account

3 1 0 2 2 2 5 8

H O M E T O W N - N S W - 2 2 1 3 95

Gross Salary, Wage, Bonus, etc. (whole dollars) (excluding amounts

$

Tax Instalments Deducted (include cents.)

$ 9 8 3 1. 7 5

3 4 2 7 7 75

Tax instalments deducted (whole dollars only in Thousands

NINE

Hundreds

EIGHT

Tens

THREE

Units

ONE

Cents

75

Signature of Authorised Person

M.Y. Cash

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Date 5

23/07/2000

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This spreadsheet calculates the amount of tax payable on a given taxable income

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The formula typed into cell B5 is: =IF(B3 > 75000,(B3-75000)*0.47+21880,IF(B3 > 50000,(B3-50000)*0.42+11380,IF(B3> 20000,(B3-20000)*0.3 + 2380,IF(B3 > 6000,(B3-6000)*0.17,IF(B3 > 0,”NO TAX”))))) Explain why this formula works. Try it to calculate tax.

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GST AND VAT

GST stands for Goods and Services Tax. A GST is a tax on various goods like TV’s, prepared food and so forth, and on services like electrical work, plumbing, etc. The GST is set at a percentage of the cost or price of the goods or service. In year 2000, the GST in Australia is 10%. It is added on to the sale of goods and services. Most food items, including meat and vegetables, are GST free.

EXAMPLE

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Calculate the GST payable on a a restaurant meal of $84.00 a GST = 10% of $84 = 10 ¥ 100 £ 84 = $8:40

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b

a bill from an electrician for $368.00

b GST = 10% of $368 = 10 ¥ 100 £ 368 = $36:80 100

EXERCISE 10C 1 Calculate the 10% GST payable on goods priced at: a $46:00 b $150:00 c $8:40 e $975:50 f $10:99 g $1980

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$459:00 $15:42

d h

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2 Michael purchases a stereo listed at $654 plus GST. a Calculate the GST payable on the stereo. b Find the amount Michael pays for the stereo.

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3 Aaron has a new bathroom installed in his house. The bill is $8742.80, and is subject to GST of 10%. a Calculate the GST payable on the bathroom. b Find the amount Aaron pays for the bathroom. 4 Angelica buys a watch listed at $389.00 plus GST.

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a Calculate the GST payable on the watch. b Find the total amount Angelica pays for the watch. 5 Keong wants to buy a VCR that is listed at $634 plus GST. a Calculate the GST payable on the VCR. b Find the total amount Keong pays for the VCR.

GST CALCULATIONS

The easiest way to work out how much GST has been paid on a particular item is to divide the total price including tax, by 11. This is a GST rule of thumb.

EXAMPLE

2

Guido buys a lounge suite for $720 including GST. Find a the GST paid on the lounge. b the original cost of the lounge suite before the GST was added. GST paid = $720 ¥ 11 = $65.45 (to 2 decimal places)

a

b

Original cost = $720 ¡ $65:45 = $654:55 or using the unitary method The lounge price = the cost price plus 10% = 110% of the cost price. Now, ) )

110% = $720 1% = 6:54545::: 100% = $654.55 (to 2 dec pl)

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Note: The rule of thumb of dividing by 11, only works if the GST is 10%.

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6 Use the rule of thumb to find the GST already paid on goods and services costing a $458.00 b $264 c $34.56 d $219 e $19.80 f $1904 g $2980 h $14.90

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7 Find the cost of each item in question 6 before the GST was added.

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EXAMPLE

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Other countries have taxes similar to Australia’s GST. They are called VAT or Value added taxes. The table alongside shows the standard VAT rate for a number of countries. Find the amount of VAT imposed on a washing machine valued at $A850 in South Africa.

VAT rate is 14%,

Country % Rate Country % Rate Belgium 21.0 Italy 20.0 Cyprus 8.0 Netherlands 17.5 Germany 15.0 South Africa 14.0 France 20.6 Switzerland 6.5 Ireland 21.0 UK 0.7

) VAT = 14% of $A850 = 14 ¥ 100 £ $A850 = $A119

8 Calculate the amount of VAT on a stereo valued at $A480 if purchased in a France b Ireland c Italy d Germany

e

Cyprus

9 How much VAT is paid on a box of chocolates valued at $A15.00 if purchased in a UK b South c Belgium d Switzerland e France Africa 10 Use the unitary method to find the cost without VAT of a camera quoted at $A460 in these countries. a Belgium b Cyprus c Germany d France e Italy f Netherlands g South Africa h Switzerland

RESEARCH PROJECT 1 Investigate GST and VAT in other countries. 2 Visit the ATO Internet site at www.taxreform.ato.gov.au and other tax sites like www.mymoney.com.au 3 Contact the Australian Competition and Consumer Commission at www.accc.gov.au 4 Write a report on the benefits of GST or VAT. Research VAT on the internet using www.vatrefund.com.au or other sites. 100

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GRAPHING TAX RATES

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The amounts of tax payable may be found using graphs. Using a graph to find tax is an example of mathematical modelling. It is not only possible to find the amount of tax payable using a graph, but the taxable income given the amount of tax. There are a number of different types of graphs depending on the tax system.

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EXAMPLE

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Tax $ 30 $ 25 Tax payable ($'000)

Use the graph to find a the tax payable on an income of $35 000 b the taxable income with tax of $7000

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$ 20 $ 15 $ 10 $5 $0

EXERCISE 10D

Tax

1 From the graph find a tax paid on $22 000 b tax paid on $38 000 c tax paid on $44 000 d tax paid on $4500 e income giving tax of f income giving tax of g income giving tax of h income giving tax of

$ 30 $ 25 Tax payable ($'000)

?

$6000 $8500 $11 500 $18 500.

$ 20 $ 15 $ 10 $5 $0

44% tax $7¡000 threshold

2 $ 25 Tax payable ($'000)

$ 20 $ 30 $ 40 $ 50 Taxable income ($'000)

The taxable income with tax of $7000 is about $12 000.

b

a The tax payable on an income of $35 000 is about $20 000.

$ 10

$ 20 $ 15 $ 10

$ 10

$ 20 $ 30 $ 40 $ 50 Taxable income ($'000)

From the graph find a tax paid on $23 000 b tax paid on $39 000 c tax paid on $45 000 d tax paid on $5500 e income giving tax of f income giving tax of g income giving tax of h income giving tax of

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$2000 $7500 $10 500 $14 500.

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$ 10

$ 20

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$ 40

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Taxable income ($'000)

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3 Repeat question 2 using this tax graph.

Tax $ 30

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Tax payable ($'000)

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$ 20 $ 15 $ 10 $5

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$0

$ 10 $ 20 $ 30 $ 40 $ 50 $ 60 $ 70 $ 80 Taxable income ($'000)

EXAMPLE

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Draw the graph for tax versus income for a country that has a fixed rate of 35% on all income. Use the values from $0 to $50 000 for taxable income along the horizontal axis and values up to $18 000 for tax paid on the vertical axis. a Use the graph to find the tax on $24 000. b Use the graph to find the income that has a tax bill of $4000.

Some values need to be found to draw the graph. For an income of $0 there is $0 of tax. For an income of $10 000 there is 35% of $10 000 = 35 ¥ 100 £ 10 000 = $3500 tax For an income of $30 000 there is

35% of $30 000 = 35 ¥ 100 £ 30 000 = $10 500 tax 35% tax $ 20

Tax payable ($'000)

Plot these three points and draw the straight-line graph through them extending to include incomes of $50000.

$ 17¡500

$ 15

$ 10¡500

$ 10

100

$5

$ 3¡500

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$ 10

$ 20

$ 30

$ 40

$ 50

Taxable income ($'000) 25

a From the graph the tax on $24 000 is about $8500. b From the graph tax of $4000 is paid on an income of about $12 000.

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4 A country has a fixed tax rate of 32%. Draw a graph representing this information. Use a scale from $0 to $50 000 on the horizontal axis for the taxable income and a scale from $0 to $16 000 on the vertical axis for the tax payable. From the graph find a tax paid on $21 000 b tax paid on $37 000 c tax paid on $43 000 d tax paid on $2500 e income giving tax of $3000 f income giving tax of $4500 g income giving tax of $9500 h income giving tax of $12 500. 5 Using values up to $50 000 on the horizontal axis repeat question 4 for countries with a tax rate of a 28% b 43% c 18% d 63%

EXAMPLE

3

Some countries have a tax-free threshold. This is an amount that may be earned before any tax is paid. Draw the graph for a country with a flat tax rate of 42% on all taxable incomes greater than $8000. a Use the graph to find the tax on $37 000. b Use the graph to estimate the taxable income that pays tax of $8500. Some values need to be found to draw the graph. For an income up to $8000 there is $0 of tax. For an income of $20 000 there is 42% of ($20 000 ¡ $8000) = 42 ¥ 100 £ $12 000 = $5040 tax For an income of $50 000 there is

42% of ($50 000 ¡ 8000) = 42 ¥ 100 £ $42 000 = $17 640 tax Plot these three points and draw the straight-line graph through them extending to include incomes of $50 000. 42% tax $8¡000 threshold $ 20

$ 17¡640 Tax payable ($'000)

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$ 15

$ 10 100

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$ 5¡040

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$0

$ 10

$ 20 $ 30 $ 40 $ 50 Taxable income ($'000) 25

a The tax on $37 000 is about $12 000. b The taxable income with tax of $8500 is about $28 000.

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6 Draw the graph for a country with a flat tax rate of 52% on all taxable incomes greater than $9000. From the graph estimate the a tax paid on $17 000 b tax paid on $28 000 c tax paid on $47 000 d tax paid on $9500 e income giving tax of $2000 f income giving tax of $3500 g income giving tax of $10 500 h income giving tax of $14 500. 7 Repeat question 6 for countries with these tax rates. a b c d

29% 31% 63% 25%

tax tax tax tax

EXAMPLE

on on on on

incomes in incomes in incomes in incomes in

excess excess excess excess

of of of of

$5000 $9000 $25 000 $7000. Notice where these numbers come from.

4

Draw the graph for the tax rate in this table.

Calculating Gross Tax Payable for year 2000 $1 to $6000 No tax payable $6001 to $20000 17 cents per $ in excess of $6000 $20001 to $50000 $2380 plus 30 cents per $ in excess of $20000 $50001 to $75000 $11380 plus 42 cents per $ in excess of $50000 $75001 and over $21880 plus 47 cents per $ in excess of $75000 Apply these rates to your taxable income.

a Use the graph to estimate the tax on $28 000. b Use the graph to estimate the taxable income that pays tax of $7500.

When drawing the graph use these points from the table. Tax

income $6;000:00 $20;000:00 $50;000:00 $75;000:00 $80;000:00

tax $¡ $2;380:00 $11;380:00 $21;880:00 $24;230:00

$ 24¡230

$ 25 Tax payable ($'000)

100


$ 21¡880

$ 20 $ 15

$ 11¡380 $ 10 $5

100

$ 2¡380

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$0

$ 20 $ 40 $ 60 $ 80 Taxable income ($'000)

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a Tax on $28 000 is about $5000. b The taxable income for tax of $7500 is about $37 000.

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The above graph is a piecewise linear graph. We will examine these in another chapter. 0



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8 Draw the graph for a country with a tax system as shown in the table. Calculating Gross $1 to $8000 $8001 to $20 000 $20 001 to $60 000 $60 001 and over

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Tax Payable No tax payable 19 cents per $ in excess of $8000 $2280 plus 35 cents per $ in excess of $20 000 $16 280 plus 49 cents per $ in excess of $60 000 Apply these rates to your taxable income.

From the graph estimate the: a tax paid on $14 000 c tax paid on $53 000 e income giving tax of $3000 g income giving tax of $11 500

b d f h

tax paid on $29 000 tax paid on $9500 income giving tax of $5500 income giving tax of $24 500.

9 Repeat question 5 for countries with these tax tables. a

Calculating Gross $1 to $5000 $5001 to $30 000 $30 001 to $50 000 $50 001 and over

Tax Payable 2000 No tax payable 15 cents per $ in excess of $5 000 $750 plus 30 cents per $ in excess of $30 000 $6750 plus 42 cents per $ in excess of $50 000 Apply these rates to your taxable income.

b

Calculating Gross $1 to $3000 $3001 to $10 000 $10 001 to $30 000 $30 001 and over

Tax Payable 2000 No tax payable 24 cents per $ in excess of $3000 $1680 plus 41 cents per $ in excess of $10 000 $9880 plus 62 cents per $ in excess of $30 000 Apply these rates to your taxable income.


You should be able to: 2 2 2 2 2

10

calculate taxable income allowing for deductions calculate the medicare levy payable calculate tax refund or payment calculate GST and VAT interpret taxation graphs. 100


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1 Here is a list of terms used in this section. Write a description of each term in a sentence. group certificate, income tax, Medicare levy, tax deduction, taxable income 2 For what does GST stand? Write a paragraph about GST. 3 For what does VAT stand? Write a paragraph about VAT. 4 For what does PAYE stand? Write a paragraph about PAYE.

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DIAGNOSTIC TEST

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1 Erin earns $35 690 pa, has interest of $29, pays tax agent fee of $80, and $120 for the purchase of uniforms. Erin’s taxable income is A $35 461 B $35 919 C $35 610 D $35 519 2 Ossie has wage $23 879, bonus $800, tax agent fee $60, donations to charity $75, use of car $356. Ossie’s taxable income is A $24 415 B $25 397 C $24 188 D $23 961 3 The Medicare levy is 1.4% of taxable income provided your taxable income exceeds $13 643. The medicare levy on a taxable income of $34 677 is A $191:00 B $485:48 C $294:48 D $21 034 4 The Medicare levy on taxable incomes between $12 689 and $13 643 is 20 cents in the dollar for each dollar in excess of $12 689. The Medicare levy on $13 145 is B $2629 C $2537 : 80 D $0 A $91:20 5 The tax on $35 710 is $2380 plus 30 cents per $ in excess of $20 000. The tax on $35 710 is A $2380 B $10 713 C $4713 D $7093 6 The tax on $15 230 is 17 cents per $ in excess of $6 000. The tax on $15 230 is A $2589.10 B $1569:10 C $1020 D $8589:10 7 The tax on $100 666 is $21 880 plus 47 cents per $ in excess of $50 000. The tax on $100 666 is A $45 693.02 B $47 313.02 C $23 813.02 D $23 500 8 The 10% GST on a video recorder valued at $458 is A $412:20 B $41:63 C $41:64

D

$45:80

9 A stereo is listed at $863 plus 10% GST. The amount paid to purchase the stereo is A $86:30 B $949:30 C $78:45 D $863 10 Sharee buys a lounge for $576 including GST. The GST on the lounge is A $52:36 B $633:60 C $523:64 D

$57:60

11 The VAT rate in Cyprus is 8%. The VAT on a car valued at $A12 990 is A $1039:20 B $11 950.80 C $12 027.77 D

$12 027.78

12 The VAT rate in Italy is 20%. A camera is listed as $A846 including VAT. The cost of the camera without VAT is A $846 B $169.20 C $676.80 D $705 Use the tax graph in Example 1 to answer the next two questions. 13 Using the graph, the tax on $25 000 is closest to A $43 000 B $14 900 C

$11 400

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14 Using the graph, the taxable income that requires $12 000 in tax is closest to A $21 000 B $11 000 C $6000 D $7000

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Use the graph from Example 2 to answer the next two questions.

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15 Using the graph, the tax on $20 000 is closest to A $58 000 B $6000 C

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$7000

D

$8000

16 Using the graph, the taxable income that requires $10 000 in tax is closest to A $28 000 B $3500 C $5000 D $25 000

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If you have any difficulty with these questions, refer to the examples and questions in the exercises indicated.

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Question Section

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10A

1, 2, 3, 4, A

5, 6, 7 B

8, 9, 10, 11, 12 C

13, 14, 15, 16 D

REVIEW SET

1 Marsha uses the following information to calculate her taxable income: wages $36 522, interest $64, tax agent fee $65, use of car $1274, donations to charity $225. Calculate Marsha’s taxable income. 2 The medicare levy is 1.4% of taxable income over $13 643. Calculate the Medicare levy payable on a taxable income of $37 559. 3 The Medicare levy on taxable incomes between $12 689 and $13 643 is 20 cents in the dollar for each dollar in excess of $12 689. Find the Medicare levy on $13 122. 4 Use the tax table in Section B to calculate the tax payable on these taxable incomes. a $57 368 b $31 259 c $6422 d $1233 5 Gloria earns $47 366 pa as a company manager and $4865 as a part-time tax agent. She has tax deductions of $3698 and throughout the year pays a total of $12 699 in PAYE tax instalments. Find a her total income b her taxable income c tax payable on her taxable income. d tax refund or balance payable. 6 Mark has a new kitchen installed in his house. The bill is $12 546.30, and is subject to GST of 10%. a Calculate the GST payable on the kitchen. b Find the amount Mark pays for the kitchen. 7 Sara buys a gold chain listed at $1250 plus GST of 10%. 100

a Calculate the GST payable on the gold chain. b Find the total amount Sara pays for the gold chain.

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8 Use the rule of thumb to find the GST already paid on goods and services costing a $1235 b $359 c $42:18 d $266

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9 The VAT in Netherlands is 17 12 %. Find the VAT on an item costing $A429.

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10 Use the unitary method to find the cost without VAT of an item quoted at $A680 including VAT in France with a VAT rate of 20.6%:

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11 A country has a fixed tax rate of 37%. Draw a graph representing this information. Use a scale from $0 to $50 000 on the horizontal axis for the taxable income and a scale from $0 to $20 000 on the vertical axis for the tax payable. From the graph find a tax paid on $21 000 b tax paid on $37 000 c tax paid on $43 000 d tax paid on $2500 e income giving tax of $3000 f income giving tax of $4500 g income giving tax of $9500 h income giving tax of $12 500.

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10B

REVIEW SET

1 Giselle uses the following information to calculate her taxable income: wages $67 220, interest $985, meal allowance $612, bonus $5000, tax agent fee $180, use of car $1255, donations to charity $600. Calculate Giselle’s taxable income. 2 The medicare levy is 1.4% of taxable income over $13 643. Calculate the Medicare levy payable on a taxable income of $41 229. 3 The Medicare levy on taxable incomes between $12 689 and $13 643 is 20 cents in the dollar for each dollar in excess of $12 689. Find the Medicare levy on $13 525. 4 Use the tax table in Section B to calculate the tax payable on these taxable incomes. a $66 889 b $33 699 c $12 369 d $5522 5 Shania earns $38 996 pa as a cocktail attendant and $7560 as a part-time singer. She has tax deductions of $3561 and throughout the year pays a total of $8725 in tax instalments. Find a her total income b her taxable income c tax payable on her taxable income d tax refund or balance payable. 6 Ollie has a new billiard room built as an extension to his house. The bill is $18 360, and is subject to GST of 10%. a Calculate the GST payable on the billiard room. b Find the amount Ollie pays for the billiard room. 7 Harry wants to buy a camera that is listed at $849 plus GST (of 10%). a Calculate the GST payable on the camera. b Find the total amount Harry pays for the camera. 8 Use the rule of thumb to find the GST already paid on goods and services costing a $3556 b $859 c $27:65 d $199 9 The VAT in Germany is 15%. Find the VAT on an item costing $A356. 100

10 Use the unitary method to find the cost without VAT of an item quoted at $A952 including VAT in Belgium with a VAT of 21%: 11 Draw the graph for a country with a flat tax rate of 43% on all taxable incomes greater than $7000. From the graph estimate the a tax paid on $17 000 b tax paid on $28 000 c tax paid on $47 000 d tax paid on $4500 e income giving tax of $2000 f income giving tax of $3500 g income giving tax of $10 500 h income giving tax of $14 500.

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REVIEW SET

1 Vanja uses the following information to calculate his taxable income: wages $53 005, interest $135 (joint account), cost of uniforms $600, cleaning of uniforms $260, tax agent fee $100, use of car $2551. Calculate Vanja’s taxable income. 2 The medicare levy is 1.4% of taxable income over $13 643. Calculate the Medicare levy payable on a taxable income of $28 280. 3 The Medicare levy on taxable incomes between $12 689 and $13 643 is 20 cents in the dollar for each dollar in excess of $12 689. Find the Medicare levy on $12 777. 4 Use the tax table in Section B to calculate the tax payable on these taxable incomes. a $72 556 b $38 694 c $11 211 d $5006 5 Tania earns $48 259 pa as a country and western performer and $9634 in royalties from her records. She has tax deductions of $4980 and throughout the year pays a total of $12 099 in tax instalments. Find a her total income b her taxable income c tax payable on her taxable income. d tax refund or balance payable. 6 Cedric has a new roof put on his house. The bill is $22 586, and is subject to GST of 10%. a Calculate the GST payable on the roof. b Find the amount Cedric pays for the roof. 7 Lisa buys a bracelet listed at $425 plus GST (of 10%). a Calculate the GST payable on the bracelet. b Find the total amount Lisa pays for the bracelet. 8 Use the rule of thumb to find the GST already paid on goods and services costing a $7500 b $1268 c $25:99 d $401 9 The VAT in Belgium is 21%. Find the VAT on an item costing $A833. 10 Use the unitary method to find the cost without VAT of an item quoted at $A265 including VAT in Ireland with a VAT of 21%: 11 Draw the graph for a country with a tax system as shown in the table. Calculating Gross $1 to $8000 $7001 to $20 000 $20 001 to $60 000 $60 001 and over

Tax Payable No tax payable 15 cents per $ in excess of $7000 $1050 plus 33 cents per $ in excess of $20 000 $14 250 plus 45 cents per $ in excess of $60 000 Apply these rates to your taxable income.

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From the graph estimate the a tax paid on $14 000 c tax paid on $53 000 e income giving tax of $3000 g income giving tax of $11 500

b d f h


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10D

REVIEW SET

1 Ito uses the following information to calculate her taxable income: wages $28 369, interest $32 (joint account), bonus $300, cost of uniforms $250, tax agent fee $50, use of car $25. Calculate Ito’s taxable income. 2 The medicare levy is 1.4% of taxable income over $13 643. Calculate the Medicare levy payable on a taxable income of $72 122. 3 The Medicare levy on taxable incomes between $12 689 and $13 643 is 20 cents in the dollar for each dollar in excess of $12 689. Find the Medicare levy on $12 985. 4 Use the tax table in Section B to calculate the tax payable on these taxable incomes. a $55 233 b $28 775 c $7 998 d $2225 5 Natalie earns $38 555 pa as a recording artist and $23 997 in royalties. She has tax deductions of $8550 and throughout the year pays a total of $10 332 in tax instalments. Find a c

her total income tax payable on her taxable income

b d

her taxable income tax refund or balance payable.

6 Lui-Min has his house painted. The bill is $3258.50, and is subject to GST of 10%. a Calculate the GST payable on the painting. b Find the amount Lui-Min pays for the painting of his house. 7 Gunter wants to buy a TV that is listed at $520 plus GST (of 10%). a Calculate the GST payable on the TV. b Find the total amount Gunter pays for the TV. 8 Use the rule of thumb to find the GST already paid on goods and services costing a $9953 b $855 c $39.51 d $399 9 The VAT in France is 20.6%. Find the VAT on an item costing $A226. 10 Use the unitary method to find the cost without VAT of an item quoted at $A680 including VAT in South Africa with a VAT of 14%: 11 Draw the graph for a country with a flat tax rate of 55% on all taxable incomes greater than $10 000. From the graph estimate the a c e g

tax paid on $17 000 tax paid on $47 000 income giving tax of $2000 income giving tax of $10 500

b d f h


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6 – 10

CUMULATIVE REVIEW 1

a Name the solids with these nets. i ii

iii

b Find the area of these figures. i ii

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2.2 m 5.1 cm

15 m 7m

4.3 m 11 m

9.3 cm

iv

C 10.3 m B

A

AB = 33.2 m

15.1 m D

c Draw an isometric sketch of this solid.

d Calculate the surface area of each solid. i ii 20 cm 10 cm

12 cm

8 cm

6 cm

4 cm

5 cm

e Calculate the volume of these solids. i ii

5 cm

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9 cm

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iii

iv

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10 cm 5 cm 25

100 cm 7 cm A = 4 cmX


r = 15 cm


A = 20 cmX

1.2 m 8 cm

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f Find the capacity in kL of a rectangular tank length 10 m, breadth 4 m, height 3 m.

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2 The results of a survey of the hair colour of a group of people is shown below using the code: black (B), brown (b), blonde (w), red (r), grey (g). BBwwb bBBrw wbgBw BBbbw wwwbB gggBB Organise the information into a frequency distribution table. How many people were surveyed? What was the most common hair colour in this group of people? Add a relative frequency column to the table above and find the relative frequency of people with i black ii blonde hair e Display the data above using a i sector graph ii divided bar graph iii dot plot iv horizontal bar graph

a b c d

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3 The table shows the variation in the price of AMB shares for one week. Day Price ($)

M 6:80

T 7:05

W 7:45

T 6:42

F 6:55

Illustrate this information using a line graph. 4

a Calculate the simple interest on $4250 at a flat rate of 13% p.a. over 3 years. b An i ii iii iv

investment pays 11% interest p.a. Express this as a decimal. Find the six monthly interest rate as a percentage and a decimal. Find the quarterly interest rate as a percentage and a decimal. Find the monthly interest rate as a percentage and as a decimal.

c Calculate the simple interest on $4000 at a flat rate of 6:74% over 17 months. d If $1750 is invested for 6 years the amount of interest earned is $840. Calculate the annual simple percentage interest rate. e Draw a graph showing the amount of interest earned over a period of 10 years if $1000 is invested at 7:2% p.a. ii Find the time to earn interest of $300. i Find the interest after 3 12 years. f

Use the compound interest formula to calculate the amount of a fixed term investment of $7580 over 3 years at 4:54% p.a. interest compounding yearly. ii Find the total interest earned. i

g Use the compound interest formula to calculate the amount of a fixed term investment of $2300 over 5 years at 3:4% p.a. interest compounding quarterly. h Calculate the amount that must be invested at 3:2% p.a. interest compounding annually to have $10 000 at the end of 9 years. i

Calculate the amount that must be invested at 5:5% p.a. interest compounding quarterly to have $1040 at the end of 6 years.

j Calculate the total cost to purchase 1500 OPSM shares with a market price of $3:35. Brokerage is 2:5% and stamp duty is 15 cents per $100 or part thereof. k A dividend of 4:65 cents per share is paid on Rebel shares with a market value of $1:14. Find the percentage yield.



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l Calculate the amount of dividend on 5000 shares with a face value of 50 cents if the dividend is 4:5%.

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m In three successive years the annual rate of inflation was 2:8%, 2:5%, and 2:1%. How much would you expect to pay, after these 3 years for a carton of beer which cost $23:99 at the beginning of this period?

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a Describe the chance of the following events: i the sun will set tomorrow night ii the temperature will be 32o C tomorrow. b List the sample space for a 3 child family.

0

c Jeans are made in 7 waist sizes and 4 leg lengths. How many different sizes are possible? d A coffee tin was tossed 100 times and the way it landed is recorded alongside. What is the experimental probability that this tin will land on one of its ends, when tossed?

Lands on End Curved face

Frequency 23 77

e A card is chosen from a normal playing pack. What is the probability that the card is i the Jack of hearts ii a Jack iii a heart iv a picture card v black? f A coin is tossed and a die is rolled. What is the probability of getting i a head and a 3 ii a head and an odd number iii a tail and a 5 iv a tail and a number less than 3? 6

a Giselle uses the following information to calculate her taxable income: wages $47 220, interest $385, bonus $3000, tax agent fee $180, use of car $1255, donations to charity $600. Calculate Giselle’s taxable income. b The medicare levy is 1:4% of taxable income over $13 643. Calculate the Medicare levy payable on a taxable income of $41 229. c The Medicare levy on taxable incomes between $12 689 and $13 643 is 20 cents in the dollar for each dollar in excess of $12 689. Find the Medicare levy on $13 525. d Use the tax table to calculate the tax payable on these taxable incomes. Gross $1 to $6000 $6001 to $20 000 $20 001 to $50 000 $50 001 to $75 000 $75 001 and over i

$62 889

ii

Tax Payable 2000 No tax payable 17 cents per $ in excess of $6000 $2380 plus 30 cents per $ in excess of $20 000 $11 380 plus 42 cents per $ in excess of $50 000 $21 880 plus 47 cents per $ in excess of $75 000 $37 699

iii

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$19 369

iv

$3522

e Shania earns $29 996 p.a. as a cocktail attendant and $17 560 as a part-time singer. She has tax deductions of $3561 and throughout the year pays a total of $8725 in tax instalments. Find i her total income ii her taxable income iii tax payable on her taxable income iv tax refund or balance payable.

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f Ollie has a new billiard room built as an extension to his house. The bill is $14 360, and is subject to GST of 10%: i Calculate the GST payable on the billiard room. ii Find the amount Ollie pays for the billiard room. g Harry wants to buy a camera that is listed at $725 plus GST. i Calculate the GST payable on the camera. ii Find the total amount Harry pays for the camera. h Use the rule of thumb to find the GST already paid on goods and services costing $3254. i The VAT is Germany is 15%. Find the VAT on an item costing $656. j Use the unitary method to find the cost without VAT of an item quoted at $A452 including VAT in Belgium where a VAT is 21%. k Draw the tax graph for a country with a flat tax rate of 43% on all taxable incomes greater than $7000. From the graph estimate the i tax paid on $17 000 ii tax paid on $28 000 iii tax paid on $47 000 iv tax paid on $4500 v income giving tax of $2000 vi income giving tax of $3500 vii income giving tax of $10 500 viii income giving tax of $14 500

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CHAPTER

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Summary statistics

AREA OF STUDY

This chapter deals with the calculation of summar y statistics for single data sets and their use and interpretation. The main mathematical ideas investigated in this chapter are: 8 the mean, mode and median for ungrouped data 8 selection and use of the appropriate measure to describe features of a data set 8 comparison of the summary statistics of samples from the same population 8 the division of data into quartiles and deciles 8 range and interquartile range 8 box and whisker plots 8 population and sample standard deviation.

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SUMMARY STATISTICS (Chapter 11)

MEASURES OF CENTRAL TENDENCY

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In Chapter 8 we looked at ways to rearrange and condense raw data into tables and graphs which are more easily read so that patterns and characteristics may be identified, and conclusions drawn. Often it is convenient and useful to condense the data further to a single number, judged to be central or representative of all the data. This number is sometimes called an “average” but statisticians prefer to call it a measure of central tendency or measure of location. The three commonly used measures of central tendency are the mean, mode and median.

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THE MEAN

The mean of a set of scores is the sum of all the scores divided by the number of scores. x is the symbol we use to represent the mean.

EXAMPLE

The mean does not necessarily have the value of one of the scores.

1

Find the mean of the scores 3, 7, 8, 9, 9. x=

?

36 3+7+8+9+9 = = 7:2 5 5

EXERCISE 11A 1 Find a c e g i

B

the mean (to 1 d.p. if necessary) of the following sets of data 2, 4, 5, 6, 9, 9, 10 b 2, 3, 3, 4, 5, 6, 7, 8, 9 11, 13, 13, 16, 17 d 27, 28, 29, 27, 30, 31, 27, 31, 30 0, 2, 4, 5, 7, 6, 4, 5, 4, 0, 1 f 20, 20, 20, 23, 25, 27 51, 52, 54, 55, 57, 57, 58, 59 h 1, 1, 2, 4, 4, 4, 4, 7, 7, 8, 9, 10 240, 243, 245, 246, 244, 243 j 104, 101, 104, 102, 104, 105, 106, 101

USING YOUR CALCULATOR TO FIND THE MEAN

Given below is one way of finding the mean using a CASIO calculator. Check the instruction booklet to determine the appropriate steps for your calculator. Follow these steps: Step 1:

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Set the calculator to statistics mode (SD) by pressing MODE and the key for statistics functions.

Step 2:

Make sure the statistics memory is clear by pressing SHIFT AC .

Step 3:

Enter the first score and press the M+ key. Repeat for each score.

Step 4:

When all the scores have been entered, press the appropriate key for the mean x.

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SUMMARY STATISTICS (Chapter 11) 100

EXAMPLE

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1

Use your calculator to find the mean of 7, 5, 9, 6, 5, 7, 4.

25

The calculator steps are MODE

5

Then SHIFT AC

0

or MODE SD or the appropriate key on your calculator.

to clear the contents of the memory.

Then 7 M+ 5 M+ 9 M+ 6 M+ 5 M+ 7 M+ 4 M+

to enter the scores.

Press the keys for the mean, x = 6:1 (to 1 d.p.).

EXERCISE 11B 1 Use your calculator to check the mean of each part of question 1. 2 Below is an excerpt from a table of random numbers.

48047 08695 90070 10132 58547 85566 40942 80553 27328 64584

45381 58112 98873 27359 01331 81574 42373 58331 85758 20776

33232 96070 89846 13017 62538 71965 38710 62724 45342 86792

35178 91910 50953 41045 79181 20977 39916 74004 98884 32340

46971 18868 92529 13817 33071 48005 08187 09344 36034 83522

85879 52251 68249 65603 63766 83418 00133 91315 79836 62139

31458 99827 54949 87615 73613 58738 16288 25791 94902 14038

22016 32581 83829 55691 24470 98771 64277 40296 80442 88433

a Find the mean of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This is the theoretical mean of the one digit numbers in a table of random numbers since each occurs, theoretically, with the same frequency. b From the table of random numbers given above, choose a random sample of i 5 ii 10 iii 20 iv 50 one digit numbers and find the mean of each sample. Working in pairs will make this task simpler. c Compare these means with the theoretical mean found in a. Is it true that the larger the sample is, the closer the sample mean is to the population mean? Discuss with your class. d

i ii iii iv

Write the mean of your sample of size 10 on the blackboard. Have all the other students in the class do likewise. Now find the mean of these sample means. Compare the result of iii with the theoretical mean in a. Is it true that the mean of a large number of sample means approaches the value of the population mean? Discuss with your class.

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100

PRACTICAL ACTIVITIES – THE THEORY OF RUNS

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When a coin was tossed 20 times the following results occurred HTTTHHTHTTHTHTTHHHTH

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The “run length of tails” is the number of times in a row that a tail occurs.

Notice that the run lengths of Tails are 3, 1, 2, 1, 2, 1 and hence the average (mean) run length of Tails is 10 3+1+2+1+2+1 = + 1:67 6 6

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What to do: a Toss a coin 100 times and record the results in the order in which they occur. b Calculate the mean run length of Tails for the 100 tosses. c It can be shown that for a large sample of results, the expected mean run length (A) 1 is A = , where p is the probability of the event occurring at a single trial. 1¡p Calculate the mean run length of Tails using this formula.

1

d Does your answer in b agree with the result from the formula? a Roll a die 100 times and calculate the mean run length of 6’s. b Compare your answer with the answer from the formula.

2

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THE MEAN FROM A FREQUENCY DISTRIBUTION TABLE

EXAMPLE

The Greek letter

S

is used to mean the “sum of¡”.

1

Find the mean of the scores given in the following frequency distribution table

Score Frequency

4 3

5 2

6 4

7 8

8 6

For simplicity we will use x for the values of the scores and f for the frequencies. We add an f £ x column. 100

Score (x) 4 5 6 7 8 Totals

Frequency (f ) 3 2 4 8 6 P f = 23

f £x 3 £ 4 = 12 2 £ 5 = 10 4 £ 6 = 24 8 £ 7 = 56 6 £ 8 = 48 P f x = 150

this this this this this this and

is the is the is the is the is the is the 8’s

sum sum sum sum sum sum

of of of of of of

all all all all all all

the the the the the the

4’s 5’s 6’s 7’s 8’s 4’s, 5’s, 6’s, 7’s

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P f = the sum of the frequencies, i.e., the total number of scores. P f x = the sum of the sub-totals 12, 10, 24, 56 and 48 = the sum of all the scores.

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) mean, (x) = 25

sum of all scores number of scores

= 150 23 = 6:5 (to 1 d.p.)

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EXAMPLE

2

x f

Find the mean for the frequency distribution shown We add an fx column

?

x 23 24 25 26 27

f 5 8 9 12 3 P f = 37

23 5

24 8

fx 115 192 225 312 81 P f x = 925

25 9

26 12

27 3

P fx ) mean = P f 925 = 37 = 25

EXERCISE 11C 1 For each of the following frequency distribution tables i copy the table ii add an f x column a

x f

3 2

c

x f

50 3

e

x f

24 9

EXAMPLE

4 3

5 6

6 4

7 1

51 5

52 8

53 6

54 2

25 11

26 18

27 0

28 12

55 4

3

Use your calculator to find the mean of the following distribution

iii

find the mean

b

x f

12 4

13 6

14 5

15 3

16 2

d

x f

8 12

9 28

10 25

11 26

12 9

100

x f

43 6

44 8

45 11

46 9

47 7

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Possible calculator steps, using a CASIO calculator (check your instruction booklet), are 25

MODE and the appropriate key for statistics functions on your calculator, then 5

SHIFT AC to clear the contents of the memory.

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Press 43 £ 6 M+

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44 £ 8 M+

45 £ 11 M+

46 £ 9 M+

47 £ 7 M+ .

Press the keys for mean, x = 45:1 (to 1 d.p.)

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2 Use your calculator to find the mean of the data given in each part of question 1. 25

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THE MEAN FOR GROUPED DATA

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EXAMPLE

1

The grouped frequency table gives the heights of a sample of seedlings in a nursery. Find the mean height.

Height (mm) 300-324 325-349 350-374 375-399 400-424 425-449

Frequency 3 18 47 32 14 6

When we have data grouped into classes, we consider all the scores to be evenly distributed throughout the class and use the class centre as the “score”, or “x”, for the purpose of calculat-

Height (mm) 300-324 325-349 350-374 375-399 400-424 425-449

Class Centre (x) 312 337 362 387 412 437

Frequency (f ) 3 18 47 32 14 6 P f = 120

f £x 936 6066 17 014 12 384 5768 2622 P fx = 44 790

P fx mean + P f 44 790 = 120 = 373:25 mm

Notes: ² You can also use your calculator to do these questions, as for the ungrouped data. ² Because we do not know the exact scores and have assumed that the scores are evenly distributed throughout each class, the answer is an approximation for the real mean.

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EXERCISE 11D 1 For each of the following grouped frequency distributions i find the class centres ii add an fx column iii calculate an approximation for the mean of the distribution a

Mass (kg) 60 - 62 63 - 65 66 - 68 69 - 71 72 - 74

Frequency 2 5 7 4 2

b

Score 41 - 50 51 - 60 61 - 70 71 - 80 81 - 90 91 - 100

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Frequency 13 15 16 10 4 2

2 For the data given below i copy the table, adding a column for the class centres and ii use your calculator to find an approximation for the mean. a

Height 155 - 159 160 - 164 165 - 169 170 - 174 175 - 179


b

No. of arguments 0-2 3-5 6-8 9 - 11 12 - 14


3 The marks scored by 30 students in a test are shown below 25 35 8 61 45 70 47 21 52 72 56 36 20 44 54 51 32 60 16 61 44 34 42 60 41 47 61 76 53 62 a Calculate the mean of this data. b Organise the data into frequency distribution tables, using classes i 0 - 4, 5 - 9, 10 - 14, 15 - 19, .... ii 0 - 9, 10 - 19, 20 - 29, 30 - 39, .... iii 0 - 19, 20 - 39, 40 - 59, .... iv 0 - 39, 40 - 79, .... c Calculate an approximation for the mean of this data using each of the grouped frequency tables in b. d Compare the approximations for the mean, found in c, with the actual mean found in a. What is the effect of the size of the class interval on the approximations for the mean?

E

THE MODE

The mode is the score that occurs most often, i.e., it is the score with the highest frequency.

EXAMPLE

1

Find the mode of the scores 3, 7, 8, 9, 9.

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The mode is always one of the scores.

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The mode is 9, as it occurs more often than any other score. 25

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A set of scores may be bimodal.

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Bimodal means having 2 modes.

For example, 2, 3, 3, 3, 4, 4, 4, 5, 6, 8 is bimodal as it has 2 modes, namely 3 and 4.

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EXAMPLE

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2

Find the mode of the scores shown

0

a

Score 4 5 6 7 8

b

Frequency 3 2 4 8 6

Score 0-4 5-9 10 - 14 15 - 19 20 - 24

Frequency 1 3 6 4 2

a It is obvious from the table that the score with the highest frequency is 7, (7 occurs 8 times) i.e., mode = 7. b For grouped data the actual score with the highest frequency cannot be found unless we also have the raw data. However, in these cases, we can find the class with the highest frequency. It is called the modal class. Modal class = 10 - 14

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EXERCISE 11E 1 Find a c e g i

the mode of the following data sets 2, 4, 5, 6, 9, 9, 10 11, 13, 13, 16, 17 0, 2, 4, 5, 7, 6, 4, 5, 4, 0, 1 51, 52, 54, 55, 57, 57, 58, 59 240, 243, 245, 246, 244, 243

b d f h j

2, 3, 3, 4, 5, 6, 7, 8, 9 27, 28, 29, 27, 30, 31, 27, 31, 30 20, 20, 20, 23, 25, 27 1, 1, 2, 4, 4, 4, 4, 7, 7, 8, 9, 10 104, 101, 104, 102, 104, 105, 106, 101

2 Write down the mode, or modal class, for the following distributions b a x 3 4 5 6 7 x 12 13 14 f 2 3 6 4 1 f 4 6 5 c

e

x f x 24 25 26 27 28

50 3

51 5

52 8

53 6

54 2

55 4

f 9 11 18 0 12

d

f

x f

8 12

Mass (kg) 60 - 62 63 - 65 66 - 68 69 - 71 72 - 74

9 28

10 25

15 3

16 2

11 26

12 9

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Score 41 - 50 51 - 60 61 - 70 71 - 80 81 - 90 91 - 100


h


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Frequency 13 15 16 10 4 2

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F

THE MEDIAN

The median is the middle score after the scores have been arranged in ascending order (i.e., from smallest to largest). The median divides the data into two parts such that there are equal numbers of scores below it and above it.

The median strip is down the middle of the road.

1

EXAMPLE

Find the median of the scores a 7, 9, 3, 9, 8 b 3, 4, 8, 9, 8, 9, 3, 5

c

3, 4, 8, 2, 9, 8, 9, 10, 3, 8

a First arrange the five scores in ascending order: 3, 7, 8, 9, 9 For an odd number of scores there is one middle score. By observation, the middle score is 8 because there are two scores below it and two scores above it, i.e., median = 8. Notes: ²

²

The median can be found by crossing off equal numbers of scores from each end of the ordered arrangement:

3 7 8 9 9 " median = 8

If the number of scores n, is odd then the median is the value of the score in the n+1 2 th position. In this example, n = 5 hence

n+1 2

= 3,

i.e., the median is the 3rd score (= 8): b Arranging the scores in numerical order: 3, 3, 4, 5, 8, 8, 9, 9. Cross off scores from each end 3, 3, 4, 5, 8, 8, 9, 9. We find in this case that there are two middle scores, 5 and 8.

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) median = the number midway between 5 and 8 = mean of 5 and 8 = 5+8 2 = 6:5 i.e., 3 3 4 5 8 8 9 9 " median = 6:5

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Notes: ² ² ²

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There are four scores below 6:5 and four scores above 6:5: The median is not always one of the scores. In this example, n = 8 and n+1 2 = 4:5 This indicates that the median is the number midway between the 4th and 5th scores.

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c Arrange the scores in numerical order: 2 3 3 4 8 8 8 9 9 10 Crossing off scores from each end 2 3 3 4 8 8 8 9 9 10 the two middle scores are 8 and 8.

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) median = average of 8 and 8 =8 or in this case n = 10 and

i.e., 2 3 3 4 8 8 8 9 9 10 " median = 8 n+1 2

= 5:5

This indicates that the median is the number midway between the 5th and 6th scores, i.e., median = mean of the 5th and 6th scores = mean of 8 and 8 =8

?

EXERCISE 11F 1 Find the median of the following sets of data by i rearranging the scores, where necessary, into numerical order ii crossing off equal numbers of scores from each end. a

2, 4, 5, 6, 9, 9, 10

b

2, 3, 3, 4, 5, 6, 7, 8, 9

c

11, 13, 13, 16, 17

d

27, 28, 29, 27, 30, 31, 27, 31, 30

e

0, 2, 4, 5, 7, 6, 4, 5, 4, 0, 1

f

20, 20, 20, 23, 25, 27

g

51, 52, 54, 55, 57, 57, 58, 59

h

1, 1, 2, 4, 4, 4, 4, 7, 7, 8, 9, 10

i

240, 243, 245, 246, 244, 243

j

104, 101, 104, 102, 104, 105, 106, 101

2 For the following sets of data i rearrange the scores, where necessary, into numerical order ii calculate the value of n+1 2 , where n = number of scores. iii hence find the median.

100

a

6, 6, 7, 7, 8, 9, 10, 10, 11

b

20, 21, 21, 22, 22, 23, 23

c

15, 15, 16, 16, 17, 17, 18, 19

d

48, 49, 49, 49, 50, 50, 50, 51, 52, 52

e

4, 5, 7, 8, 2, 4, 6, 7, 7

f

12, 18, 14, 16, 12, 16, 13, 12

g

48, 49, 50, 51, 53, 49, 50, 50, 51, 55, 53

h

18, 17, 17, 20, 23, 17, 18, 20, 23, 21

i

2, 1, 0, 3, 4, 5, 4, 4, 3, 1, 0, 2

j

34, 35, 38, 40, 37, 36, 33, 38, 34, 39, 36, 37, 38, 37

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MEDIAN FROM A FREQUENCY DISTRIBUTION TABLE

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EXAMPLE

1

Score 4 5 6 7 8

Find the median of the scores in the frequency distribution table given

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Frequency 3 2 4 8 6

To find the median it is useful to add a cumulative frequency column to our table Score 4 5 6 7 8

Frequency 3 2 4 8 6

Cumulative frequency 3 5 9 17 23

Scores 1st, 2nd, 3rd 4th, 5th 6th, 7th, 8th, 9th 10th, 11th, 12th, ...., 16th, 17th 18th, 19th, 20th, ...., 22nd, 23rd

Note that the scores are already arranged in numerical order. P 23+1 From the table n = f = 23: Hence, n+1 = 12, 2 = 2 i.e., the median is the 12th term. (There are 11 terms below it and 11 terms above it.) From the cumulative frequency column, we can see that the 12th term is 7, i.e., median = 7.

EXAMPLE

2

x f

Find the median of the set of scores given

18 5

19 7

20 11

21 9

22 4

Add a cumulative frequency column to the table 100

x 20 21 22 23 24

f 5 7 11 9 4

From the table n =

Cumulative frequency 5 12 23 32 36 P

f = 36, hence,

n+1 2

Scores 1st, 2nd, 3rd, 4th, 5th 6th, 7th, 8th, ...., 11th, 12th 13th, 14th, 15th, ...., 22nd, 23rd 24th, 25th, 26th, ...., 31st, 32nd 33rd, 34th, 35th, 36th

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= 18:5

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This indicates that the median is the number midway between the 18th and 19th scores,

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i.e., the median = the mean of the 18th and 19th scores,

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From the cumulative frequency column, we can see that the 18th and 19th scores are both 22, ) the median =

22+22 2

= 22:

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EXERCISE 11G 1 Find the median of each set of scores below a b x f x f 3 2 12 4 4 3 13 7 5 7 14 5 6 4 15 3 7 1 16 2 d

x 33 34 35 36 37

H

e

f 2 4 9 10 5

x 8 9 10 11 12

f 12 28 25 26 9

c

x 50 51 52 53 54 55

f 3 5 8 6 2 4

f

x 24 25 26 27 28

f 9 11 18 0 12

THE MEDIAN FOR GROUPED DATA

EXAMPLE

1

Find the median of the scores given in the frequency distribution table

Height (mm) 300 - 324 325 - 349 350 - 374 375 - 399 400 - 424 425 - 449

Frequency 3 18 47 32 14 6 100

Add a cumulative frequency column.

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Height (mm) 300 - 324 325 - 349 350 - 374 375 - 399 400 - 424 425 - 449

Frequency 3 18 47 32 14 6

Cumulative frequency 3 21 68 100 114 120

n+1 2

= 60:5

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i.e., the median is the mean of the 60th and 61st scores.

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The 60th and 61st scores are in the 350 - 374 class but we do not know the exact values of these scores, i.e., from the table it is only possible to state that the median is in the 350 - 374 class. This is sometimes called the median class. We could estimate the median to be at the class centre (= 362) but a more accurate estimate can be found using the ogive (cumulative frequency polygon) for this distribution.

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120


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Using the class centres and cumulative frequency, we can draw a cumulative frequency histogram and polygon:

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To find the median using the ogive ² locate the point half the way up the cumulative frequency axis ( 12 £ 120 = 60) ² draw a horizontal line across to the polygon ² draw a vertical line, from this point, to the score axis and read off the median.

80

60

40

20

0 312 337 362 387 412 437 Height

In this example, median = 372.

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EXERCISE 11H 1 Estimate the median from the ogives drawn below. a

b

Cumulative frequency


100 80 60 40 20 0 7

60 40 20 0

Mass

2

80

Height

14 21 28 35 42 49

12 17 22 27

c

d 100



60 50 40 30 20 10 0

Mark

10 20 30 40 50 60 70

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120 75

100 80 60 40

25

20 0

Days absent

5

10 15 20 25 30 35 0



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2 For each of the grouped frequency distributions given i copy the table and add columns for class centre and cumulative frequency ii draw the ogive for the distribution iii find the median score using the ogive. a

Mass (kg) 60 - 62 63 - 65 66 - 68 69 - 71 72 - 74

c


e

No. of arguments 0-2 3-5 6-8 9 - 11 12 - 14

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Frequency 4 7 9 6 4

Frequency 13 15 16 10 4 2

b

Score 41 - 50 51 - 60 61 - 70 71 - 80 81 - 90 91 - 100

d

Height 155 - 159 160 - 164 165 - 169 170 - 174 175 - 179

Frequency 5 0 12 10 9 4 Frequency 19 21 24 19 17

Frequency 13 31 20 11 5

USING A STEM AND LEAF PLOT

EXAMPLE

1

For the stem and leaf plot given, find the a mean b mode c

median

Stem 2 3 4 5 6 7

Leaves 2336 1 1 1 0

23556668 557777 2355567 0

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a Using the statistics function on a CASIO calculator, press MODE and the appropriate

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key on your calculator, then press SHIFT AC to clear the contents of the memory. Press

22 M+

23 £ 2 M+

26 M+

41 M+

42 M+

43 M+

45 £ 2 M+

46 £ 3 M+

48 M+

......, 70 £ 2 M+

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to enter the scores. Press the keys for the mean, giving x = 51:3

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b The leaf which occurs the most is the 7 in the 50’s row. Hence mode = 57.

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c To find the median, cross off numbers in pairs from the top and bottom, starting with the lowest and highest numbers, 22 and 70. Stem Leaves 2 2336 3 There are 2 middle scores 55 and 55. 4 123556668 5 1557777 Hence, median = 55+55 2 6 12355567 = 55 7 00

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EXERCISE 11I 1 Find the i mean and leaf plots a Stem 5 6 7 8 9

J

Leaves 12 025 1455 67 3

ii mode b Stem 2 3 4 5 6

iii median of the data given in the following stem c

Leaves 466 0115 5889 168 7

Stem 10 11 12 13 14 15

Leaves 56888 2336 01 8 3556 1137

d Stem 1 1¤ 2 2¤

Leaves 2334 56 0113 66689

RELATIVE MERITS OF MEAN, MODE AND MEDIAN

The mean, mode and median were introduced in this chapter as measures of central tendency, i.e., single numbers which are central or typical of all the numbers under consideration. Which measure is the most appropriate in a given situation? The answer is that there is no general rule which can be used, rather it depends on the nature of the data and the relative merits of these measures. We will investigation some of these properties next.

EXAMPLE

1

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a Find the mode of the scores i iii

14, 15, 15, 16, 16, 16, 17, 18 14, 15, 16, 17, 26, 26, 26

ii

14, 15, 16, 17, 19, 21 25

b Discuss the relative merits of the mode, mean and median in each of the cases above.

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mode = 16

iii

mode = 26

a

i

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b

i The number 16 occurs the most often. It is central and typical of this set of scores. Note that the mean (= 15:9) and the median (= 16) are also good measures of central tendency.

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ii

no mode

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ii This set of scores has no score which occurs more often than any other. The mode is obviously not a satisfactory measure of central tendency for this set of scores. The mean (= 17) or the median (= 16:5) are good measures of central tendency.

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iii This set of scores does have a mode but it occurs at one end of the set, i.e., it is not a central value. The mean (= 20) and the median (= 17) are good measures of central tendency. Note that there are cases when the mode is important even if it is not a central value. For example, if the numbers are the sizes of hats sold by a shop then the size which is sold most often would be important to the shopowner and the manufacturer of hats.

EXAMPLE a Find the mean of the scores i 5, 5, 6, 7, 8 ii 5, 5, 6, 7, 80 b Discuss the relative merits of the mean, mode and median in each of these sets of data. i mean = 6:2

a

ii mean = 20:6

b In i the mean is central and typical of the data. It is a good measure of central tendency. The mode (= 5) is not a central value but the median (= 6) is both central and typical. In ii, four out of five scores are less than the mean, that is, it is not a central value and also, it is not typical of the size of any of the scores. The mean has the disadvantage of being affected by extremely large (or small) values, such as the score 80 in ii above. Very small or very large atypical scores are called outliers. In some cases it may be useful to ignore outliers when calculating the mean. The mode (= 5) is not a central value. The median (= 6) is unaffected by the extreme value and would be the best measure of central tendency in this case. Note that if we ignore the outlier 80, then the mean = 5:75 100

95

EXAMPLE

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a Find the median of the scores 1, 2, 2, 3, 6, 6, 6, 6, 6, 6. b Discuss the relative merits of the median, mode and mean for this data. a

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Median = average of 5th and 6th scores = 6

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b 95

For this pattern of scores the median happens to be the end score (which is also the mode). In this case the mean (= 4:4) would be a better measure of central tendency.

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EXERCISE 11J 1

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a Find the mode of the scores i 5, 6, 6, 7, 7, 7, 8, 9 iii 23, 24, 25, 26, 35, 35, 35

ii

2, 3, 4, 5, 7, 9

b Is the mode a satisfactory measure of central tendency for the data in i, ii and iii above? c By also finding the mean and median, discuss the relative merits of these measures for each of the above distributions. 2

a Find the mean of the scores i 8, 8, 9, 10, 11 ii 8, 8, 9, 10, 110. b Is the mean a satisfactory measure of central tendency for the data in i and ii? c By finding the median and mode, discuss the relative merits of these measures for each set of data.

3

a Find the median of the scores 2, 3, 3, 3, 5, 5, 5, 5, 5. b Is the median a satisfactory measure of central tendency for these scores? c Discuss the relative merits of the mode and mean for this data.

4 A shoe store had a special on ladies’ running shoes and sold the following shoe sizes: 6, 10, 4, 7, 8, 7, 6, 5, 7, 8, 7, 5, 6, 4, 3, 7. a Find the i mean ii mode iii median of the shoe sizes sold. b Which of these measures would be of most use to the shop owner? 5 The Beacon Lamp Company has a total of 30 employees whose annual salaries are listed below: Owner/General manager Marketing manager Warehouse manager 15 Production workers a b c d e f

$81 000 $50 000 $50 000 $25 000 ea.

$60 000 $50 000 $30 000 ea

Accountant Engineer 10 Tradesmen

What is the total annual wages bill for this company? Calculate the mean wage for the employees. How many employees earn i less than ii more than the average wage? What is the median wage? What is the modal wage? In a wage determination case for the employees of this company, which “average” would you use to support your argument if you were i

the general manager

ii

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the production workers’ representative?

g Which measure of central tendency is the most appropriate to represent the wages of the employees of this company? Give reasons.

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6 The times to the nearest minute, spent in the waiting room of a hospital by 10 patients, chosen at random, were 18, 19, 19, 20, 20, 27, 28, 29, 29, 29: a Find the mean, mode and median of these waiting times. b Discuss the relative merits of these three measures for this data. 7 Geoff scored the following marks in a series of topic tests: 6, 3, 6, 3, 6, 2, 6, 1 a Find the mean, mode and median for this set of marks. b Which of these “averages” would Geoff prefer to tell his parents?

0

8 The prices of houses which were sold by a real estate agent in a particular suburb one month were $255 000, $315 000, $480 000, $270 000, $280 000. a The real estate agent claimed that, for this month, the “average” price of houses in the suburb was $320 000. Which “average” has the estate agent used? b Is this an appropriate “average” to use? Give reasons, with reference to the other measures of central tendency. 9 The eye colours of a group of 10 students are recorded below: brown, blue, brown, green, blue, grey, brown, brown, blue, green. Which, if any, of the measures mean, mode and median can be used with this categorical (qualitative) data. 10 The number of mistakes made by 100 students in a spelling test is shown below (no student made more than 9 mistakes): 48047 45381 33232 35178 46971 85879 31458 22016 08694 58112 96070 91910 18868 52251 99827 32681 90070 98873 89846 50953 a

i Copy and complete the frequency distribution table shown for the first 20 students (start at the first number, 4, and work from left to right):

Number of mistakes 0 1 2 3 .. .

Tally

Frequency

9 ii Use the table and your calculator to find the mean, mode and median number of mistakes made by the first group of 20 students. b c d e f

C:\...\NSWGM\NSWGM_11\366NG11.CDR Mon Feb 07 09:45:42 2000

Repeat a for the second group of 20 students. Repeat a for the third group of 20 students. Repeat a for the fourth group of 20 students. Repeat a for the fifth group of 20 students. Summarise the results, 1st 20 from the samples of 20 Mean students taken above, in Mode a table like the one Median alongside:


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2nd 20

3rd 20

4th 20

5th 20

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g Which of these measures (mean, mode, median) varies the most between samples? h Which of these measures varies the least? SUMMARY – Choosing the appropriate measure

When selecting which of the three measures of central tendency to use as a representative figure for a set of data, consider the following characteristics of each measure: Mean ² ² ²

It is the most commonly used, is easy to understand and is easy to calculate. Its value depends on the value of every score in the data set. Its main disadvantage is that its value is easily distorted by very large, or very small, atypical scores (outliers).

²

It has the best sampling stability, i.e., if we choose random samples, of the same size, from a large population, then the mean is the measure which varies the least, from sample to sample.

²

It is not suitable for categorical data.

Mode ² ² ² ² ² ² ²

It is the most typical value within a set of data. Its value is easy to determine. It is not affected by outliers. It is the only measure suitable for categorical data. There may be no mode, or more than one mode, in a data set. It may not be a central value. When comparing random samples of the same size, taken from a large population, the mode is the measure which varies the most.

Median ² It is easy to understand - there are equal numbers of scores above and below it. ² It is not affected by outliers since its value depends on the number of scores above and below it, not the values of these scores. ² ² ²

It is not suitable for categorical data. For some unusual patterns of scores it may not be a central value. For example, 0, 1, 1, 5, 5, 5, 5. When comparing samples, it varies more than the mean but much less than the mode. 100

INVESTIGATION 1:

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MEASURES OF DISPERSION

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Consider the two sets of data 3, 4, 5, 6, 7 and 0, 1, 5, 9, 10. For both sets the mean is 5. However it is obvious that these sets of scores are quite different in nature. The first set of scores are closely grouped around the mean but the second set are quite scattered from the mean. Hence, as well as having a central value for a set of scores, it is useful to have some measure of the scatter, or spread, of the scores. Such measures of spread are called measures of dispersion and we will investigate some of these next.

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THE RANGE The range is the difference between the highest score and the lowest score.

EXAMPLE

1

Find the range of the scores a 3, 4, 5, 6, 7 a

?

Range = 7 ¡ 3 =4

b

b 0, 1, 5, 9, 10

Range = 10 ¡ 0 = 10

c 3, 4, 5, 6, 70 c

Range = 70 ¡ 3 = 67

EXERCISE 11K 1 Find a c e g

the range of the scores given 2, 4, 5, 6, 9, 9, 10 11, 13, 13, 16, 170 0, 2, 4, 5, 7, 6, 4, 5, 4, 0, 1 51, 52, 54, 55, 57, 57, 58, 59

2 Find the range for the following data b a x f 3 2 4 3 5 6 6 4 7 1

b d f h

x 12 13 14 15 16

2, 3, 3, 4, 5, 6, 7, 8, 9 27, 28, 29, 27, 30, 31, 27, 31, 30 20, 20, 20, 23, 25, 27 104, 101, 104, 102, 104, 105, 106, 101

c

f 4 6 5 3 2

x 50 51 52 53 54 55

f 3 5 8 6 2 4

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3 Can you find the range given the following information? Height (mm) 300 - 324 325 - 349 350 - 374 375 - 399 400 - 424 425 - 449



95

Frequency 3 18 47 32 14 6

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INTERQUARTILE RANGE

The range is used in some situations such as measurement of temperature, height, length, blood pressure etc. but has the major disadvantage that it depends only on the extreme values of the data. The extreme values can be atypical scores, such as the 70 in Example 1c above, often caused by an error of measurement. In these cases the range is of little use. To avoid this problem we sometimes find the difference between two numbers which are not at the extremes of the data. One such measure is called the interquartile range. We are already familiar with the concept of the median which divides the data into two parts, with an equal number of scores in each. In the same way we can find scores which divide the data into four parts, with equal numbers of scores in each. These scores are called quartiles. Each set of data has three quartiles, the lower, the middle (this is the median) and the upper, usually denoted by Q1 , Q2 , Q3 respectively. The interquartile range is the difference between the upper and lower quartiles, i.e., interquartile range = Q3 ¡ Q1 .

EXAMPLE

1

Find the lower, middle and upper quartiles of the scores given and hence find the interquartile range: a 2, 2, 3, 5, 6, 6, 7 b 1, 2, 2, 3, 4, 5, 5, 6, 6 c 1, 2, 2, 3, 4, 5, 6, 6 a Write down the scores in numerical order: First find the median (2nd quartile): n+1 2 =4 Hence, the median is the 4th score, i.e., Q2 = 5 We can show this on a diagram

2 2 3 5 6 6 7 " Q2

Note that the median divides the data into two parts with three scores in each. The lower part contains the scores 2, 2, 3. The lower quartile is the score which divides this set into two parts with the same number of scores in each, i.e., it is the middle score of this set. Hence, Q1 = 2. The upper part contains the scores 6, 6, 7. The upper quartile is the score which divides this set into two parts with the same number of scores in each, i.e., it is the middle score of this set. Hence, Q3 = 6. 2 2 3 5 6 6 7 It is useful to show this on a diagram " " " Q1 Q2 Q3

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We can see that the quartiles have divided the data into four parts with one score in each. 75

) interquartile range = Q3 ¡ Q1 = 6 ¡ 2 = 4: b The scores are already in numerical order. In this case n = 9, hence i.e., the median is the 5th score = 4

1 2 2 3

n+1 2

= 5,

4 5 5 6 6 " Q2

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100

2+2 2

=2

5 6 " Q3

6

From the diagram, Q1 is the middle score of the set 1, 2, 2, 3 i.e., Q1 =

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Q3 is the middle score of the set 5, 5, 6, 6 i.e., Q3 =

75

It is useful to show the information on a diagram

1

25

5+6 2

2 2 " Q1

= 5:5 3

4 5 " Q2

The quartiles have divided the data into four parts with two scores in each.

5

Interquartile range = Q3 ¡ Q1 = 5:5 ¡ 2 = 3:5

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c The scores are already in numerical order. In this case n = 8, hence The median is the mean of the 4th and 5th scores i.e., median = 1 2 2

3 4 " Q2

3+4 2

n+1 2

= 3:5

5 6 6

From the diagram, Q1 is the middle score of the set 1, 2, 2, 3 i.e., Q1 = Q3 is the middle score of the set 4, 5, 6, 6 i.e., Q3 = On a diagram

1

2 2 " Q1

= 4:5

3 4 " Q2

5 6 " Q3

6

5+6 2

2+2 2

=2

= 5:5

) interquartile range = Q3 ¡ Q1 = 5:5 ¡ 2 = 3:5

Note: ² For a relatively large number of scores 25% of the scores are less than Q1 and 75% are greater than it. 50% of the scores are less than Q2 and 50% are greater than it. 75% of the scores are less than Q3 and 25% are greater than it. Hence the interquartile range contains the middle 50% of the scores. ² The interquartile range gives an indication of the spread of the scores from the median. The greater its value, the greater is the spread from the median. A relatively smaller value indicates that the scores are more closely clustered around the median. ² The interquartile range has an advantage over the range in that it is not affected by atypical extreme scores. For example, in Example 1a, if the last score was 70 instead of 7 then the range changes from 5 to 68 whereas the interquartile range is unaffected by this outlier.

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EXERCISE 11L 1 Find a c e g h i j

the interquartile range of the following data 15, 16, 17, 20, 22, 23, 25 30, 32, 35, 35, 35, 37, 38 15, 15, 16, 17, 17, 18, 20, 21, 21, 22 33, 35, 38, 42, 43, 44, 52, 53, 55, 58, 61, 64, 11, 12, 14, 18, 18, 20, 22, 25, 25, 26, 30 46, 50, 50, 53, 54, 54, 58, 58, 58, 60, 62, 62, 42, 44, 45, 48, 53, 61, 64, 68, 71

100

b d f 66,

11, 13, 13, 14, 14, 15, 18 50, 50, 52, 55, 55, 57, 57, 58, 60, 60 23, 23, 23, 24, 25, 26, 28, 28, 29, 32 68

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170, 170, 170, 185, 188, 189, 194, 196, 203 2, 3, 5, 8, 8, 8, 9, 9, 9, 11, 15, 17, 18 95, 102, 95, 89, 92, 103, 90, 98

m o

15, 17, 20, 23, 28, 35, 42, 44 3, 4, 6, 5, 8, 4, 4, 2, 1, 3, 2, 7

EXAMPLE

5

Estimate the lower and upper quartiles for the data whose ogive is graphed below. Hence find the interquartile range of the scores.

For large numbers of scores the lower quartile can be estimated using the ogive as follows:

120 110

² locate the point one quarter of the way up the cumulative frequency axis ² draw a horizontal line across to the polygon

100 90 80 70

² drop a vertical line from this point to the score axis and read off the lower quartile. To estimate the upper quartile ² locate the point three quarters of the way up the cumulative frequency axis ² draw a horizontal line across to the polygon ² drop a vertical line from this point to the score axis and read off the upper quartile.

60 50 40 Cumulative frequency

0

30 20 10 0 2

7

12

17

Q1¡=¡15

22

27

32

Mark

Q3¡=¡25

From the ogive, Q1 = 15, Q3 = 25 (found from the actual point where the vertical line meets the scale axis.) .Hence, interquartile range = Q3 ¡ Q1 = 25 ¡ 15 = 10.

For ungrouped discrete data, the quantile is the score of the rectangle into which the vertical line falls.

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2 By first estimating the values of the lower and upper quartiles, find the interquartile range for the data whose ogives are drawn below: a

b

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100 80 60 40 20 0 7

60 40 20 0

Mass

2

80

Height

14 21 28 35 42 49

12 17 22 27

c

d 140 Cumulative frequency


60 50 40 30 20 10 0

Mark

120 100 80 60 40 20 0

Days absent

10 15 20 25 30 35

10 20 30 40 50 60 70

EXAMPLE Find the interquartile range of the marks given in the frequency distribution table shown.

4 5

Mark Frequency

5 9

6 12

7 8

8 6

First add a cumulative frequency column to the table Mark 4 5 6 7 8


Cumulative frequency 5 14 26 34 40

Now draw the cumulative frequency histogram and polygon (ogive). From the ogive, Q1 = 5 (data is discrete) Q3 = 7 Hence, interquartile range = 7 ¡ 5 = 2

40 35 30 100


100


20

95

15 75

10 5 25

0

Q1 4

5

Q2 6

7

8

Mark 5

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3 By drawing the ogive, find the interquartile range of the scores (discrete) given in the frequency distribution tables below a

25

5

x 33 34 35 36 37

b

f 2 4 9 10 5

x 8 9 10 11 12

c

f 12 28 25 26 9

x 24 25 26 27 28

f 9 11 18 0 12

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4 For the following grouped data i draw a cumulative frequency histogram and polygon ii estimate the interquartile range from the ogive. a

Score 41 - 50 51 - 60 61 - 70 71 - 80 81 - 90 91 - 100

c

Mass (kg) 60 - 62 63 - 65 66 - 68 69 - 71 72 - 74

EXAMPLE

Frequency 5 0 12 10 9 4

b

Number of patients / day 10 - 19 20 - 29 30 - 39 40 - 49 50 - 59 60 - 69

Frequency 13 15 16 10 4 2

Frequency 4 7 9 6 4

4

In the term test, the results of Kelly’s class had a median of 65 and an interquartile range of 20. In Sean’s class the median was 65 and the interquartile range was 12. From the information given, what conclusions could be made about the results of these two classes? The median mark in both classes is 65. Hence, half of each class scored less than 65 and half scored more than 65. However, the interquartile range of Sean’s class is smaller than that of Kelly’s. This indicates that the marks in Sean’s class are more closely clustered around the median than the marks in Kelly’s class. (Or, the marks in Kelly’s class are more spread out from the median.)

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5 In a test, the marks in Linda’s class had a median of 58 and an interquartile range of 21. In Lara’s class, the median was also 58 but the interquartile range was 10. From the information given, what conclusions could be made about the results of these two classes?

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6 At the end of the cricket season, Tom’s median batting score was 31 and the interquartile range of his scores was 8. Matthew’s median score was 31 and interquartile range was 24. Based on the information given a which batsman is more consistent b who is more likely to make a high score c who is more likely to make a low score?

25

DECILES

5

0

A natural and useful extension of the concept of quartiles is to divide the data into 10 parts with equal numbers of scores in each. The scores which separate these 10 parts are called deciles. Decile 1 is the score which has 10% of the scores below it and 90% above it. Decile 2 is the score which has 20% of the scores below it and 80% above it. Decile 3 is the score which has 30% of the scores below it and 70% above it. etc. (Decile 5 is the median) The concept of deciles is usually only applied to large sets of data.

EXAMPLE 120

Use the ogive shown to find the 1st, 4th and 9th deciles for the given data.

10 9

100

8 7


80

6

60

5 4

40

3 2

20

1

0

0

0

5

10 15 20 25 30 35 40 45 50

100

Score

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Add another scale to the vertical axis and divide this scale into 10 equal parts. The deciles can be found as shown on the graph.

75

From the ogive, decile 1 = 13, decile 4 = 27, decile 9 = 48 i.e., 10% of the scores are less than 13 (and 90% greater than 13) 40% of the scores are less than 27 (and 60% greater than 27) 90% of the scores are less than 48 (and 10% greater than 48)

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EXERCISE 11M 1 Use the ogives drawn below to find the 1st, 4th and 9th deciles for the given data. a b 110

25

80

100 70

90

5

60

80 0

70

50



60 50 40 30 20 10 0

40 30 20 10 0 0

0 10 20 30 40 50 60 70 80 Score

c

5 10 15 20 25 30 35 40 Score

d 100

140

90 120

80 70

100 80


50 40 30 20 10 0

60 40 20 0 10 30 50 70 90 110 130 150 170 190


60

15 20 25 30 35 40 45 50 55 Score

Score 100

BOX-AND-WHISKER PLOTS

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A useful way to display and compare sets of data is by means of a box-and-whisker plot. This is a diagram which illustrates the extremes (the highest and lowest scores), the median and the upper and lower quartiles of the data as follows.



Whisker

Box

Whisker 25

Lowest score

Qz Median

Qc

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Highest score

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EXAMPLE

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1

From the box-and-whisker plot shown below, find the a i highest score ii lowest score and hence iii the range of the scores b median c i upper quartile 5 10 15 20 25 30 35 40 45 50 55 ii lower quartile and hence iii the interquartile range.

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25

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Using the scale on the number line a i Highest score = 53 ii

Lowest score = 10

iii

Q1 = 18

iii

Range = 53 ¡ 10 = 43

b Median = 36 c

i

Q3 = 45

ii

Interquartile range = 45 ¡ 18 = 27

From these results we can say: The bottom 25% of the scores take values from 10 up to, but less than, 18. The top 25% of the scores take values from, but not including, 45 up to 53. The middle 50% of the scores lie between 18 and 45. Since the median is closer to the upper quartile than to the lower quartile, then the top half of the scores are clustered closer to the median than the bottom half.

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EXERCISE 11N

a

1 From the box-and-whisker plots shown, find the i ii iii iv v vi vii

highest score lowest score range of the scores median upper quartile lower quartile interquartile range

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50

60

70

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90

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70

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b

c 100

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EXAMPLE

2

Draw a box-and-whisker plot given the following information: highest score = 82 lowest score = 25 median = 64 lower quartile = 49 upper quartile = 71

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Draw a number line to include scores from 25 to 82. Mark the position of each of the five scores given and draw the box-and-whisker plot. 20

30

40

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60

70

80

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2 Draw box-and-whisker plots given

a b c d e

Lowest score 15 130 6 71 1

EXAMPLE

Highest score 40 153 28 83 9

Lower quartile 23 139 7 73 3

Median 28 141 10 78 5

Upper quartile 32 148 18 80 7

3

Draw a box-and-whisker plot for the following set of data 35, 35, 36, 36, 37, 38, 39, 39, 39, 40 lowest score = 35, 35 35

Using a diagram

36 " Q1

36

highest score = 40 37 38 " Q2

39

39 " Q3

39 40

) Q1 = 36, Q2 = 37:5, Q3 = 39.

Hence, 100

34

35

36

37

38

39

40

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41

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3 Draw box-and-whisker plots for the following sets of data: a c e

4, 5, 8, 8, 10, 12, 12, 14, 15, 19 89, 90, 92, 95, 95, 98, 102, 103 4, 7, 11, 5, 1, 3, 5, 15, 5, 4

b d

21, 21, 23, 24, 24, 24, 26, 28, 30 18, 20, 22, 23, 25, 29, 30, 30, 30, 31

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STANDARD DEVIATION

While the interquartile range is a better measure of the spread of a set of scores than the range, its weakness is that its value also depends on only two scores, the upper and lower quartiles. A more commonly used measure, whose value depends on all the scores, is the standard deviation.

25

5

Standard deviation is determined by finding the distances of all the scores from the mean and then calculating an “average” of these distances.

0

In the same way as the mean, mode and median are “typical” scores of the set of data, standard deviation is a “typical” distance of the scores from the mean. The smaller the standard deviation, the smaller the “typical” distance of the scores from the mean, i.e., the smaller the spread of the scores. Standard deviation can be calculated using an algebraic formula but it is simpler and quicker to use the statistics mode on a scientific calculator.

EXAMPLE

1

Find the standard deviation of the scores 2, 3, 5, 1, 4, 6, 5, 2, 1, 4. Possible calculator steps for a CASIO calculator are (check the instruction booklet for your calculator)

MODE and the appropriate key for statistics functions on your calculator then SHIFT AC to clear the contents of the memory then 2 M+ 3 M+ 5 M+ 1 M+ 4 M+

4 M+

6 M+

5 M+ 2 M+ 1 M+

to enter the scores.

To obtain the standard deviation of the scores press the key labelled

sn .

sn is pronounced sigma n.

In this case we get ¾n = 1:7 (to 1 d.p.) This is the typical distance of the scores from the mean (= 3:3). Note that there is another key on the calculator labelled sn-1 . This is another kind of standard deviation which we will investigate a little later.

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EXERCISE 11O 1 Use a c e g

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your calculator to find the mean and standard deviation of the scores 4, 5, 6, 7, 8 b 2, 4, 6, 8, 10 0, 3, 6, 9, 12 d 25, 26, 27, 28, 29, 33 88, 89, 89, 91, 91, 92 f 239, 245, 248, 250, 253, 253, 254, 258 7, 5, 9, 9, 3, 3 h 5, 5, 5, 5, 5, 5



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EXAMPLE

2

Find the mean and standard deviation of the following data

Score Frequency

5 3

6 4

7 6

8 5

9 2

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0

Possible calculator steps are:

MODE and the appropriate key for statistics functions on your calculator then SHIFT AC to clear the contents of the memory then 5 £ 3 M+

6 £ 4 M+

7 £ 6 M+

8 £ 5 M+ 9 £ 2 M+

Press the key for mean, x = 6:95 Press the key for standard deviation, ¾n = 1:20 (to 2 d.p.) 2 Obtain the mean and standard deviation of the data given b a Score 2 3 4 5 6 Score Frequency 5 7 8 4 1 Frequency c

Score Frequency

EXAMPLE

0 12

1 15

2 24

3 14

4 5

d

15 3

16 8

17 11

18 5

19 3

Score 121 122 123 124 125 Frequency 4 6 6 3 5

3

Find the mean and standard deviation for the following grouped data

For grouped data we again assume that the scores are evenly distributed throughout each class and use the class centres

Possible calculator steps are:

Height 300 325 350 375 400 425 -

Height 300 325 350 375 400 425 -

(mm) 324 349 374 399 424 449

Frequency 3 18 47 32 14 6

(mm) Class centre (x) Frequency (f ) 324 312 3 349 337 18 374 362 47 399 387 32 424 412 14 449 437 6

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MODE and the appropriate key for statistics functions on your calculator then.

5

SHIFT AC to clear the contents of the memory then

0



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312 £ 3 M+ 337 £ 18 M+ 362 £ 47 M+ 387 £ 32 M+ 412 £ 14 M+ 437 £ 6 M+ Press the key for mean, x = 373:3 (to 1 d.p.) Press the key for standard deviation, ¾n = 27:7 (to 1 d.p.)

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3 Find the mean and standard deviation for the following data: b a Mass (kg) Frequency Score 60 - 62 2 41 - 50 63 - 65 5 51 - 60 66 - 68 7 61 - 70 69 - 71 4 71 - 80 72 - 74 2 81 - 90 91 - 100 c

4

No. patients/day 10 - 19 20 - 29 30 - 39 40 - 49 50 - 59 60 - 69

Frequency 13 15 16 10 4 2

d

Height 155 - 159 160 - 164 165 - 169 170 - 174 175 - 179

a Find the mean and standard deviation of the scores b i Add 5 to each of the scores and recalculate the ii Add 2 to each of the scores and recalculate the iii What do you think would happen to the mean 10 to each score? Check by calculation. c

Frequency 3 0 10 8 7 2 Frequency 4 7 10 5 2

3, 3, 4, 5, 6, 6. mean and standard deviation. mean and standard deviation. and standard deviation if we added

i Multiply each score by 5 and recalculate the mean and standard deviation. ii Multiply each score by 2 and recalculate the mean and standard deviation. iii What do you think would happen to the mean and standard deviation if we multiplied each score by 10? Check by calculation.

From question 4 you should have discovered that: ²

If a constant k is added to each of the set of scores then i the mean is increased by k ii the standard deviation remains unchanged.

²

If each of a set of scores is multiplied by a constant k then i the new mean is k times the original mean ii the new standard deviation is k times the original standard deviation.

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5 For a set of scores the mean is 17 and the standard deviation is 4. Determine the new mean and new standard deviation if a 3 is added to each score in the set b 8 is added to each score c each score is multiplied by 3 d each score is multiplied 5 e 5 is subtracted from each score f each score is divided by 5



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ANOTHER TYPE OF STANDARD DEVIATION

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In the same way as there are different types of averages, there are also two types of standard deviation. 25

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The standard deviation we have been calculating above is called the population standard deviation and is obtained by using the *sn key on the calculator. It can be shown, however, that when ¾n is found for a sample taken from a large population, its value tends to underestimate the standard deviation of the whole population. For large numbers of scores there is very little difference between the values of ¾n¡1 and ¾n .

For this reason the algebraic formula for standard deviation is altered slightly and this value, denoted by ¾n¡1 and called the sample standard deviation, is used when determining the standard deviation for a sample. ¾n¡1 may be determined, on a calculator, using a similar method to that for ¾n .

EXAMPLE

1

A random sample of 50 sixteen year old boys was selected and their weights measured. The results are shown in the given table. Find the mean and standard deviation for this data.

Weight 35 - 39 40 - 44 45 - 49 50 - 54 55 - 59


Weight 60 - 64 65 - 69 70 - 74 75 - 79 80 - 84

Frequency 5 4 0 1 1

Find the class centres Weight 35 - 39 40 - 44 45 - 49 50 - 54 55 - 59

Class Centre 37 42 47 52 57


Weight 60 - 64 65 - 69 70 - 74 75 - 79 80 - 84

Class Centre 62 67 72 77 82

Frequency 5 4 0 1 1

100

Possible calculator steps are

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MODE and the appropriate key for statistics functions on your calculator then

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SHIFT AC to clear the contents of the memory then 37 M+

42 £ 8 M+

47 £ 11 M+

52 £ 12 M+

62 £ 5 M+

67 £ 4 M+

77 M+

82 M+

25

57 £ 7 M+

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Press the key for mean, x = 53: Because we are dealing with a sample from a larger population, press the key for the sample standard deviation, ¾n¡1 = 9:4 (to 1 d.p.) (Note that for this distribution, ¾n = 9:3)

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EXERCISE 11P 1 The shoe sizes of a random sample of Year 11 girls are recorded below. Find the sample standard deviation of this data. 5, 4, 7, 5, 6, 7, 4, 8, 5, 6, 4, 6, 7, 5, 9 2 A sample of 20 families is chosen at random and the number of children in each family is recorded as shown. Calculate the mean and sample standard deviation.

3

4

Time 24 - 26 27 - 29 30 - 32 33 - 35 36 - 38

45381 91910 68249 55691 81574 39916 91315 80442


A woman measures the time, to the nearest minute, it takes her to get to work each day. The times for a two week period are shown alongside. Determine the mean and sample standard deviation of these times.

Frequency 2 3 3 1 1

48047 96070 92529 87615 85566 38710 09344 94902

Number of Children 0 1 2 3 4 5

33232 18868 54949 58547 71965 08187 25791 64584

35178 52251 83829 01331 20977 00133 40296 20776

46971 99827 10132 62538 48005 16288 27328 86792

85879 32581 27359 79181 83418 64277 85758 32340

31458 90070 13017 33071 58738 80553 45342 83522

22016 98873 41045 63766 98771 58331 98884 62139

08695 89846 13817 73613 40942 62724 36034 14038

58112 50953 65603 24470 42373 74004 79836 88433

a From the table of random numbers above, choose an arbitrary starting point and select a random sample of 20 one digit numbers. Find the standard deviations, ¾n¡1 and ¾n , of this sample. b Put your results, with the results of all the other students in your class, in a table on the chalkboard. c For a table of random numbers, the theoretical value of the standard deviation of the total population is 2:87, correct to 2 decimal places. (This value is calculated on the assumption that each digit occurs the same number of times.) i Compare the results of b with the theoretical standard deviation of the population. ii Which standard deviation, ¾n¡1 and ¾n , is usually closer to the theoretical value? iii Discuss with the class.



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You should be able to 2 find the mean, mode and median for both ungrouped and grouped data 2 select the appropriate measure to use by recognising their merits and limitations 2 determine the range and interquartile range for a set of data 2 find deciles for a data set 2 interpret and draw box-and-whisker plots 2 calculate and use both the population and sample standard deviation.

The measures used for analysing data are: ² Central tendency: mean, mode, median. ² Dispersion (spread): range, interquartile range, standard deviation.

INVESTIGATION 2:

MAJOR STATISTICAL PROJECT

The process of statistical enquiry includes the following steps: Step Step Step Step Step Step

1: 2: 3: 4: 5: 6:

posing questions (designing questionnaires) collecting data (sampling techniques) organising data (frequency tables, stem and left plots, etc.) summarising and displaying data (summary statistics, graphs) analysing and drawing conclusions writing a report.

Use the above outline and the content and skills you have learnt in previous chapters on statistics to investigate and write a report for a major statistical investigation. Some suggestions are made below. 1 Find the characteristics of the average person in your year. You could consider personal characteristics such as sex, height, weight, etc. as well as social interests, income, family background, drive, etc. Does this average person exist? 2 Investigate how Year 11 students spend their leisure time. You could consider what, where, male/female, when, how often, cost, etc.

100

3 Conduct the appropriate research to find the viability of setting up a small business. You could consider need, location, cost, competition, etc. Use information supplied by the Australian Bureau of Statistics. 4 Investigate the future aspirations of Year 11 students at your school. You could consider expected HSC results, vocations, full-time/part-time work, type of tertiary institution, area of study, expected income in 1, 5, 10 years, live at home/move out, etc.

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The following is a list of key words and phrases used in this chapter. Write, in a sentence, a description of each term. average, measure of central tendency, mean, mode, median, representative, typical, outlier, spread, measure of dispersion, range, interquartile range, decile, box-and-whisker plot, standard deviation

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11

DIAGNOSTIC TEST

1 For the scores A mean = B mean = C mean = D mean = 2 For the data

7, 8, 8, 8, 8, 9, 10, 11, 12, 12 9:3 mode = 8 median = 8:5 9 mode = 8:5 median = 8 9:3 mode = 8 median = 9 9 mode = 9 median = 8:5 Score 23 24 25 26

Frequency 4 11 9 6

3 Find the mean of the scores

Class 0-4 5-9 10 - 14 15 - 19

A B C D

mean mean mean mean

= = = =

36:9 36:9 24:6 24:6

mode mode mode mode

= = = =

11 24 11 24

Frequency 3 4 6 2

4 The modal class of the data in the table above is A 0-4 B 5-9 C

10 - 14

D

15 - 19

5 The median of the data in question 2 is A 24 B 25

24:5

D

10

6 The median of the data whose ogive is graphed alongside is A 64:5 B 62 C 67 D 63:5

C


0

70 100

60 50

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40 75

30 20 10 0

25

52 57 62 67 72 77 Weight (kg)

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For A B C D

Leaf 024 13666 0488

Stem 3 4 5

the data alongside mode = 6 median mode = 6 median mode = 46 median mode = 46 median

= = = =

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6 4 46 4

8 The most appropriate measure of central tendency for the scores 5, 5, 5, 9, 10, 44 is the A mode B mean C median D range

25

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9 If we choose random samples from a large population, then the measure which varies the most is the A mean B mode C median D they all vary the same amount

0

10 For the ogive drawn in 6 the interquartile range is approximately A 15 B 40 C 69

10

D

11 Find the 3rd and 7th deciles for the data graphed in 6. Given the box-and-whisker plot alongside, the A range = 20 interquartile range = 23 B range = 27 interquartile range = 15 C range = 10 interquartile range = 5 D range = 15 interquartile range = 27

12

20

25

30

35

40

45

50

13 Find the population and sample standard deviation of the data

5 3

Mark Number of students

6 5

7 8

8 7

9 2

If you have any difficulty with these questions, refer to the examples and questions in the exercises indicated. Question Section

11A

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1 A, E, F

2 C, E

3 D

4 E

5 G

6 H

7 I

8, 9 J

10 L

11 M

12 K, N

13 O, P

REVIEW SET

1 Find the mean, mode, median and range of the scores 6, 6, 6, 7, 8, 8, 9, 9, 10. 2

Class 1-5 6 - 10 11 - 15 16 - 20 21 - 25

Frequency 12 8 18 31 11

For a b c

the grouped data alongside copy the table and add a cumulative frequency column draw a cumulative frequency histogram and polygon. From the ogive determine the i median ii fourth decile

3 Find i the 1st, 2nd and 3rd quartiles a 9, 9, 10, 11, 12, 12, 13 c 8, 9, 9, 10, 11, 12, 12, 12, 14, 14

ii the interquartile range for the scores b 8, 9, 9, 10, 11, 12, 12, 13, 14 d 8, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15

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4 Draw a box-and-whisker plot given the highest score = 42, lowest score = 25, upper quartile = 36, lower quartile = 30, median = 32.

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5 Find the mean and population standard deviation of the scores 17, 25, 19, 20, 23, 19, 21.

25

?

5

REVIEW SET 1 Find the mean, mode, median and range for the data

0

2

i ii

86 15

4, 4, 4, 5, 6, 6, 7

b

Use a b c d e

120 100 80

Cumulative review

85 9

87 24

88 12

Find the mean, mode and median of the following sets of scores. Discuss the relative merits of each to represent the data.

a 4, 5, 5, 6, 100 3

Score Frequency

60

c

1, 1, 1, 1, 5, 6

the ogive drawn alongside to find the median upper quartile lower quartile interquartile range 2nd decile

40 20 0

30 40 50 60 70 80 90 100 110 Mark

4 The weights of a sample of bags of potatoes are given: a Which type of standard deviation is appropriate for this situation? b Calculate the standard deviation of these scores.

Weight 4:3 4:4 4:5 4:6 4:7

Frequency 6 9 4 5 6

5 Which of the following measures are suitable for analysing categorical data? A range B mean C mode D median

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REVIEW SET 100

1

Class 30 - 39 40 - 49 50 - 59 60 - 69 70 - 79 80 - 89

Frequency 5 10 14 13 0 8

a b c

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Determine the modal class. Copy the table and add a column for the class centres. Find the mean.

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a Organise the following data into a stem and leaf plot: 56, 58, 74, 62, 54, 50, 51, 73, 60, 71, 59, 65, 49, 78, 62, 56, 48, 50, 64, 50 b Use the stem and leaf plot to determine the i range ii mode iii median

3 Draw a box-and-whisker plot for the scores 6, 6, 7, 8, 8, 9, 10, 10, 11, 11, 11, 12

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5

4 For the data given in question 1 a draw the ogive. b Estimate from the ogive the i lower quartile iii interquartile range v 9th decile

0

ii iv

upper quartile median

5 The mean of a set of data is 12:7 and the standard deviation is 4:5. What is the new mean and standard deviation if a 5 is added to each score b each score is multiplied by 5?

?

REVIEW SET 1

a Find the mean, mode, median, range and standard deviation (¾n ) of the scores 2, 2, 3, 3, 4, 5, 5, 5, 6. Put the results in the first line of the table below. Mean

Mode

Median

Range

¾n

a b

By b c d e f g h i j k l m

completing the table show how each of these measures changes if the score 6 is added to the set of data the score 2 is added to the set of data the score 4 is added to the set of data the score 5 is added to the set of data the score 100 is added to the set of data 3 is added to the value of each score 5 is added to the value of each score 100 is added to the value of each score 2 is subtracted from the value of each score each score is doubled each score is multiplied by 3 each score is multiplied by 100.

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CHAPTER

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12

Similar figures

AREA OF STUDY

This chapter is about the properties of similar figures. The main mathematical ideas investigated in this chapter are: 8 the enlargement transformation 8 the properties of similar figures 8 scale factor 8 finding sides of similar figures 8 applying properties of similar figures to practical problems 8 development of scale drawings 8 obtaining measurements from building plans and elevations 8 calculating dimensions and areas from floor plans 8 interpretation of commonly used symbols on home plans.

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SIMILAR FIGURES (Chapter 12)

A

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THE ENLARGEMENT TRANSFORMATION

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Two figures are similar if one is an enlargement of the other. 25

EXAMPLE

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Enlargement factor is the same as scale factor .

1

Using grid paper, enlarge the trapezium given using an enlargement factor of 2.

A

B

D

C

Step 1:

Choose any convenient point O as the centre of enlargement. O may be inside or outside the trapezium.

Step 2:

Draw lines from O through each of the vertices of the trapezium.

Step 3:

Produce OA to OA0 such that OA0 = 2 £ OA. Produce OB to OB0 such that OB0 = 2 £ OB. Produce OC to OC0 such that OC0 = 2 £ OC. Produce OD to OD0 such that OD0 = 2 £ OD.

Step 3:

Join A0 B0 C0 D0 to form the enlargement (image) of trapezium ABCD.

A' A O

D

B'

B

C C'

D'

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Note: ²

Trapezium A0 B0 C0 D0 and trapezium ABCD are similar figures.

²

Each angle and its image, ]ABC and ]A0 B0 C0 , ]BCD and ]B0 C0 D0 corresponding angles.

²

Each side and its image, AB and A0 B0 , BC and B0 C0 etc., are called corresponding sides.

25

etc., are called

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EXERCISE 12A 1 Using grid paper, copy and enlarge each of the figures below using the centre of enlargement and scale factor (= e) given. a b O

25

O

e=2

5

e=3

0

c

O

e = 0.5

Another method of enlarging figures, especially those that are drawings rather than geometrical shapes, is as follows:

EXAMPLE

2

Enlarge the given picture to twice its original size.

Step 1:

Draw a 5 mm by 5 mm grid over the original picture. 100

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Step 2:

75

To enlarge the picture to twice the original size (i.e., enlarge with a scale factor of 2), copy each part of the picture in the 5 £5 grid squares onto 10 £ 10 grid squares. The enlarged picture is started alongside:

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a Copy and complete the enlarged picture in Example 2. b Enlarge the picture using a scale factor of 12 by using a 2 12 mm by 2 12 mm grid. (This is of course a reduction in size).

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B

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PROPERTIES OF SIMILAR FIGURES

5

INVESTIGATION 1: SIDES AND ANGLES OF SIMILAR FIGURES

0

What to do: In Example 1 of 12A: 1 Use a protractor to measure and record the sizes of the angles of trapezium ABCD. 2 Measure and record the sizes of the angles of trapezium A0 B0 C0 D0 . 3 Comment on your findings. 4 By measurement and division copy and complete the following statements a

A0 B0 = AB

=

C0 D0 = = CD 5 Comment on your results. c

b

B0 C0 = BC

=

d

D0 A0 = DA

=

From the above investigation you should have discovered that ² ²

the corresponding angles of similar figures are equal the lengths of the corresponding sides of similar figures are in the same ratio.

“In the same ratio” is sometimes written as “in proportion”.

The value of this ratio is equal to the scale factor.

EXAMPLE

1

For the following pairs of figures a

b 100

Q

B

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A R

i ii

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determine by measurement whether or not they are similar. If similar, find the scale factor.

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a

95

Measuring the angles: ]ABC = ]PQR (= 53o ), ]BCA = ]QRP (= 90o ) and ]CAB = ]RPQ (= 37o )

i

75

The pairs of corresponding sides are opposite the pairs of equal angles.

25

5

Measuring the pairs of corresponding sides:

0

RP PQ QR = 96 = 1:5 = 12 = 15 8 = 1:5 10 = 1:5 BC CA AB i.e., the angles are equal and the corresponding sides are in proportion (in the same ratio). Hence the figures are similar. QR PQ RP = 1:5 (or or ) BC AB CA This means that the length of each side in the second figure is 1:5 times the length of each corresponding side of the first. Scale factor =

ii

Note that this is the scale factor from the first triangle (on the left) to the second triangle. The scale factor from the second triangle to the first is scale factor =

BC = QR

2 3

(or

AB CA or ) PQ RP

i.e., each side of the first figure is 23 times each corresponding side of the second. Either ratio is satisfactory as long as the meaning is made clear. b

?

The angles are equal but the sides are not in proportion ( 32 6= 43 ). The figures are not similar.

i

EXERCISE 12B 1

i ii a

Determine whether or not the following pairs of figures are similar. Give reasons. If they are similar, find the scale factor. b 100

95

75

c

d 25

5

0



C

M

Y

K


394 100


e

f

g

h

95

75

25

5

0

Figures which have a scale factor of 1 are called congruent figures.

2 The following phrase has been typed in 12, 18, 24, 30 and 36 point size: i ii

iii

iv

100

v

95

75

a What is the scale factor from i i to ii ii i to iii vi ii to iv vii ii to v

iii viii

b What is the scale factor from i ii to i ii iii to i vi iv to ii vii v to ii

iii viii



i to iv iii to iv

iv ix

iv to i iv to iii

iv ix

i to v iii to v

v x

v to i v to iii

v x

C

M

ii to iii iv to v iii to ii v to iv

Y

K

25

5

0



95

75

395

3 The design on the right has been enlarged below using a photocopier. Calculate the enlargement factor in each case, correct to 1 d.p.

25

a

5

b

c

0

EXAMPLE

2

Given that the following pairs of figures are similar, find the scale factor. a b P

A

D

5 cm 2 cm

8 cm

C R

20 cm

8 cm

8 cm

C S

a

A

B

5 cm

P

Q

6 cm

B R

Q

PS and AD are corresponding sides (PS is the enlargement of AD) Hence, the scale factor, e = 85 = 1:6 i.e., each side of the second figure is 1:6 times each side of the first figure (or AD is a reduction of PS, hence e = 58 i.e., each side of the first figure is 58 times each side of the second).

b

100

PQ and AB are corresponding sides. PQ Hence, the scale factor e = = 20 8 = 2:5 AB i.e., each side of the second triangle is 2:5 times each side of the first AB 8 (or e = = 20 = 0:4 PQ

95

i.e., each side of the first triangle is 0:4 times each side of the second).

5

75

25

0



C

M

Y

K


396 100

95


4 Given that the following pairs of figures are similar, find the scale factor. a b

75

3 cm

3 cm

2 cm

5 cm

5

6 cm

12 cm

25

c

d

0

18 cm

12 mm

24 mm

8 mm

15 cm

18 cm

e

f 18 m 6m

30 cm

15 cm

20 cm

24 m

g

h 15 m

18 m

6m

i

3 cm

4.5 cm

13.5 cm

j 2 cm

8 cm 4 cm

4 cm

6 cm

3 cm 12 cm

EXAMPLE Determine the scale factor of the following pairs of similar figures. (In a, rectangle EFGD is an enlargement of rectangle ABCD and in b, triangle ABC is an enlargement of triangle ADE.) a b E

F

A

100

95

75

5 cm

4 cm D A 2 cm

E

B

25

4 cm

5

D


C

G


B

C

C

M

0

Y

K


397


95

a

75

Separate the rectangles and redraw them next to each other: From the diagram ED = 6 cm and AD = 2 cm. Hence scale factor, e = 62 = 3

25

E A

B

2 cm D

C

F

6 cm

D

G

5

0

b

A

Separate the triangles and redraw them next to each other:

A 5 cm

9 cm

AC = 9 cm and AE = 5 cm Hence e =

9 5

D

or 1:8

E B

C

5 Determine the scale factor for the following pairs of similar figures. a b

6 cm

8 cm

2 cm

2 cm

c

d

6 cm

3 cm 5 cm

11 cm 100

e

95

4 cm

75

25

6 cm

5

0



C

M

Y

K


398


C

100

95

FINDING SIDES OF SIMILAR FIGURES

75

EXAMPLE 25

1

In the following pairs of similar figures i find the scale factor ii find the length of the unknown sides. b a

5

0

z cm

9 cm

y cm

15 cm

7.5 cm

5 cm 8 cm

3 cm

y cm

5 cm

a

i The scale factor, from the first to the second triangle, is e =

7:5 5

= 1:5

i.e., each side in the 2nd triangle is 1:5 times the corresponding side in the 1st. ii Hence y = 1:5 £ 8 = 12 and z = 1:5 £ 9 = 13:5 i.e., the unknown sides are 12 cm and 13:5 cm long. b

i The scale factor from the second to the first figure is e = i.e., each side of the 1st figure is ii Hence, y =

?

3 5

£ 15 = 9

3 5

3 5

(or 0:6)

times the corresponding side in the 2nd.

i.e., the unknown side is 9 cm long.

EXERCISE 12C 1 In the following pairs of similar figures i find the scale factor. ii find the length of the unknown sides. 7 cm a b y cm 8 cm

10 cm

z cm

12 cm

21 cm 6 cm

5 cm

100

y cm

95

c

d 75

18 cm z cm 13 m 12 cm

wm

7 cm

25

6m 5

15 m



0

C

M

Y

K


399


e v mm

12 mm

17 mm

20 mm

t cm

f

3 cm

95

75

18 cm

8 cm

25

5

g

h

19 m

8 cm

pm

0

30 cm

25 m

11 m k cm 12 cm

EXAMPLE In the following pairs of similar triangles find the length of the unknown sides.

y cm

z cm

7 cm

12 cm

10 cm 15 cm

The scale factor from the first triangle to the second is e=

10 15

= 23 :

Hence, z =

2 3

£ 12 = 8

The scale factor from the second triangle to the first triangle is e=

15 10

= 1:5

Hence, y = 1:5 £ 7 = 10:5

2 In the following pairs of similar triangles, find the length of the unknown sides. 7 cm a b y cm

18 cm

y cm

z cm

12 cm

4 cm z cm 100

16 cm

20 cm

c

9 cm

d

12 cm

6 cm

9 cm

8 cm

95

75

z cm

y cm

18 cm 11 cm

15 cm

14 cm

y cm

25

z cm

12 cm

5

0

H:\...\NSWGM\NSWGM_12\399NG12.CDR Fri Feb 04 12:38:42 2000


C

M

Y

K


400


100

e y cm

95

16 cm

10 cm

8 cm

75

12 cm

z cm

25

3

5

0

EXAMPLE

In the following pairs of similar figures a find the scale factor b find the length of the unknown side.

P

6 cm

A Q

4 cm 2 cm

z cm

B C

R

3 cm

a

P A 4 cm 2 cm

6 cm B

C

Q

3 cm z cm R

The corresponding sides are opposite the pairs of equal angles, i.e., the side PR is the enlargement of side CB. Hence, the scale factor is e =

6 3

= 2:

b QR is the enlargement of AB. Hence, z = 2£AB = 2 £ 4 = 8

3 In the following pairs of similar triangles i find the scale factor ii find the length of the unknown side. a b 3 cm

100

2 cm

z cm

95

3 cm

4 cm

75

y cm 4 cm

6 cm 2 cm

25

8 cm 5

0



C

M

Y

K


401


c

d

8 cm w cm

95

6 cm

4 cm

5 cm

y cm

20 cm

12 cm

75

6 cm 12 cm

25

e

f

v cm 10 cm

5

0

9 cm

8 cm

15 cm

9 cm

y cm

15 cm

18 cm 8 cm

g

h

z cm 10 cm 4 cm

5 cm k cm 7 cm

12 cm

15 cm 6 cm 12 cm

15 cm

i

j 9 cm

t cm

y cm

w cm

27 cm

30 cm

20 cm

25 cm

w cm

6 cm 12 cm

D ?

PROBLEMS INVOLVING SIMILAR FIGURES

EXERCISE 12D 1 A building casts a shadow 5:2 m long. At the same time a metre rule casts a shadow 0:65 m long. a Find the scale factor. b Hence find the height of the building.

100

1m

shadow 5.2 m

95

0.65 m 75

2 A man 1:8 m tall casts a shadow 3 metres long. At the same time his son’s shadow is 2:5 metres long. a Draw a diagram. b Find the scale factor. c How tall is the son? 3 The shadow cast by a flagpole is 11 m long. At the same time the shadow of a 30 cm ruler is 55 cm. Calculate the height of the flagpole.

25

5

0



C

M

Y

K


402 100

95

75

25

5

0


4 Practical activity Use the method of the previous questions to find the height of buildings, trees, the flagpole etc., in your school grounds. Method: a Measure the length of the shadow of the building, tree, flagpole, etc. b Measure the length of the shadow of a metre rule, at the same time. c Draw a diagram to illustrate the data. d Find the scale factor. e Calculate the unknown height. 5 A 129 mm £ 90 mm photograph is enlarged to 172 mm £ 120 mm. What is the enlargement factor? 6 A 15 cm £ 10 cm photograph is enlarged so that the new length is 18 cm. What is the new width of the photograph? tree

7 river

w peg peg 8 m

peg 15 m

Jane wanted to find the width of the river alongside without crossing it. She did this by placing pegs and measuring distances as shown in the diagram. Find the width of the river.

30 m peg

8 Practical activity Use the method shown in 7 to find the width of the road at the front of your school, without crossing the road. Method (read in conjunction with the diagram in question 7.): a Select an obvious feature on the opposite side of the road. b Place a peg or marker directly opposite this feature. c Walk (and measure) a convenient distance along the edge of the road and place another peg. d Walk (and measure) another convenient distance further along the road and place another peg. e Walk at right angles to the road until the feature on the opposite side of the road is in line with the second peg (step c). Measure the distance walked to this point. f Draw a sketch and put all the measurements on it. g Use the properties of similar figures to calculate the width of the road. 100

w

9 At another point on the river Jane made the following measurements:

95

12 m

Calculate the width of the river at this point.

river

75

7m 25

15 m

5

0



C

M

Y

K


403


A street lamp is 5 metres high. When a girl stands 6 metres from the base of the lamp her shadow is 1:8 metres long. Calculate the height of the girl, to the nearest centimetre.

10

95

5m

75

h 1.8 m

6m

25

5

0

11 The supports on a straight section of a big dipper are placed 5 metres apart, as shown in the diagram. If the length of the first support is 4 metres, find the lengths of the next 3 supports.

4m 5m 5m

xm 5m

ym 5m

zm

5m

12 Practical activity Equipment: Overhead projector, transparency sheets. a Draw a line 100 mm long on a transparency sheet and place it on the OHP. Focus the image on the wall. b Measure the length of the image on the screen, in millimetres. length of image c Calculate the scale factor e = length of object length of image = 100 = d Measure the distance of the object from the wall. e Move the OHP further away from the wall and focus the image. f Repeat steps b to e five times and record your results in a table like the one below. Distance from wall Scale factor g How is the scale factor related to the distance of the object from the wall?

E

SCALE DRAWINGS 100

The object of making a scale drawing is to draw a figure similar to the original, i.e., the shape of the original is maintained by making the angles the same and the lengths in the same ratio as the original. The scale factor from the length on the drawing to the real length is simply called the scale of the drawing. The scale may be expressed as a ratio such as 1 : 1000, or a fraction such as words such as 1 centimetre to 10 metres.



C

1 1000 ,

or given in

95

75

25

5

0

M

Y

K


404


100

EXAMPLE

95

What length on a scale drawing would represent a distance of 85 metres on land if the scale a 1 : 10 000 b 1 : 5000 c 1 : 1000? was

75

25

a

5

0

c

Scaled length = 10 1000 £ 85 m = 0:0085 m = 0:85 cm or 8:5 mm

b

1 Scaled length = 5000 £ 85 m = 0:017 m = 1:7 cm or 17 mm

1 Scaled length = 1000 £ 85 m = 0:085 m = 8:5 cm or 85 mm

EXAMPLE What distance on land would be represented by a map distance of 28 mm on a map with a scale of a 1 : 500

b 1 : 1000

c 1 : 20 000?

a

Real length = 500 £ 28 mm = 14 000 mm = 14 m

c

Real length = 20 000 £ 28 mm = 560 000 mm = 560 m

If the scale is 1 : n then

?

b

Real length = 1000 £ 28 mm = 28 000 mm = 28 m

scaled length

=

1 £ real length n

real length

=

n £ scaled length

100

EXERCISE 12E

95

1 Using a scale of i 1 : 10 000 ii 1 : 5000 drawing would represent a distance on land of a 90 m b 65 m c 160 m

iii 1 : 1000 what length on a scale

2 Using a scale of i 1 : 500 ii 1 : 1000 would be represented by a map distance of a 17 mm b 35 mm c 4:2 cm

iii 1 : 20 000 what distance on land



d

180 m

e

75

240 m? 25

5

d

6 cm

e

C

2:6 cm?

M

Y

0

K


405


95

75

EXAMPLE

3

Express the scale a 1 cm to 5 m

b 1 mm to 25 m as a

i ratio ii fraction.

Expressing both measurements in the same units 25

5

a

i

1 cm to 5 m = 1 : 500

ii

1 : 500 =

b

i

1 mm to 25 m = 1 : 25 000

ii

1 : 25 000 =

1 500 1 25 000

0

3 Express each of the following scales as a i ratio a 1 cm to 10 m b 1 cm to 4 m d 1 mm to 5 m e 1 mm to 20 m

EXAMPLE

a b

Scale

or

c

1 cm to 50 m

4

Find the scale on a drawing given

a

ii fraction

Scaled length 35 cm 32 cm

Real length 350 m 1600 m

Scale

b

=

Scaled length Real length

=

Scaled length Real length

=

35 35 000

=

32 160 000

=

1 1000

=

1 5000

= 1 : 1000

or

= 1 : 5000

The fraction key on your calculator makes this easy.

100

95

4 Find the scale on a drawing given a b c d e

Scaled length 52 cm 52 cm 16 mm 15 mm 11:2 cm

Real length 520 m 5200 m 64 m 7:5 m 56 km

75

25

5

0



C

M

Y

K


406


100

95

75

25

EXAMPLE Below is a scale drawing of a building. Use your ruler and the scale given to find the real a height of the building b width of the building c height of the first floor windows (to the bottom sill).

5

scale 1:200

0

a

By measurement on the diagram scaled height = 75 mm Hence, real height = 200 £ 75 mm = 15 000 mm = 15 m

c

By measurement on the diagram scaled height = 22 mm Hence, real height = 200 £ 22 mm = 4400 mm = 4.4 m

b

By measurement on the diagram scaled width = 25 mm Hence, real width = 200 £ 25 mm = 5000 mm =5m

5 The diagram alongside is a scale drawing of a rectangular field. Using your ruler and the scale given, find the real length of the breadth of the field.

100

scale 1:2500

95

75

25

5

0



C

M

Y

K


407


95

75

a

6 This is a scale drawing of a parking area. Use the scale given to find the real dimensions (a, b, c, d) of the area. d

b

scale 1:1000

25

5

c 0

EXAMPLE

6

This is a scale drawing of a truck. The actual length of the truck is 5:4 m. What is the scale used in this drawing?

Scale =

Scaled length 54 mm = = Real length (5:4 £ 1000) mm

1 100

or

1 : 100

7 The diagram is a scale drawing of a car. a b c

If the real length of the car is 5:12 m, find the scale used in the drawing. Calculate the height of the car above the ground. What is the diameter of the wheel? The actual length of the plane shown in the scale diagram is 55 m. Find

8

a b

the scale used in the diagram the actual wingspan of the plane.

100

95

9 A surveyor marked the following sketch with measurements to find the distance across a swamp. a Make a scale drawing of the figure using a scale of 1 : 1000. b From the scale drawing, calculate the distance across the swamp.

75

90 m

110 m 25

5

75°

0



C

M

Y

K


408 100


When standing 20 m from the base of a tower, the angle of elevation of the top of the tower is 65o , as shown in the sketch.

10

95

a Make a scale drawing using a scale of 1 cm to 4 m. b Find the actual height of the tower.

75

25

65° 20 m

5

0

BUILDING PLANS Building plans are scale drawings of houses or other buildings. They include a site plan, a floor plan and side elevations so that all construction details can be determined. A site plan is a drawing of the block of land showing the position of the residence and any other buildings, or main features, on the block.

EXERCISE 12F 1

N

2m

16 m

4m

8m

street

6m

pool

pool fence 20 m

?

house

4m garage 4m

drive way

7.2 m

scale 1:400

a b c d e f g h i j k

What are the dimensions of the block? Find the length of the driveway to the garage. Determine the distance between the house and the fence on i the northern boundary ii the southern boundary How far is the house from the street? How far is the garage wall from the southern boundary? Calculate the area of the garage, to the nearest square metre. Find the total length of the pool fences. Calculate the area of land taken up by the pool, to the nearest square metre. If the pool has an average depth of 1:2 metres, calculate the volume of the pool. Find the amount of water in the pool, in litres. How many blocks of land of this size would fit into an area of 1 hectare?



1 m 3 = 1 kL

100

95

75

25

5

0

C

M

Y

K



409

100

95

75

EXAMPLE Draw the plan, front elevation and side elevation of the solid shown.

25

front (rear) elevation

5

0

e sid ion at v ele

The plan of a solid is the view seen from directly above it. The front elevation is the view from directly in front of it and the side elevation is the view from the side. The plan and elevations for this solid are shown alongside.

plan

2 Draw the plan, front elevation and side elevation of the following solids. a b c

d

e

3 On the next page is the floor plan of a 4 bedroom house drawn to a scale of 1 : 200. a How many i windows (shown as ii hinged doors (shown as iii sliding doors (shown as

100

)

95

75

) ) does the house have?

25

b Find the following symbols on the plan and discuss the meaning of each: P, WO, F, WC, ENS, WIR., LIN.

5

0



C

M

Y

K


410


100

95

T 75

S/D L'DRY BED 3

RUMPUS

WC BATH

25

LIN

S/D

5

0

S/D

FAMILY

DECKING

BED 2

S/D DINING

KITCHEN BED 4 F

WO P

LIVING ROOM

ENS W.I.R

BED 1 PORCH

CONCRETE PATH

scale 1:200

c By measurement and calculation, find the dimensions of the i Rumpus room (assume rectangular) ii Bedroom 1 iii Bedroom 2 iv Bedroom 3 v Bedroom 4 vi Living room vii Porch d

i What is the area of land occupied by the whole house, including the porch and decking? ii A “square” (= 100 square feet) is an old imperial unit used for measuring the area of a house. If 1 square + 9:29 m2 , calculate the area of the house to the nearest square. e The porch is a reinforced concrete slab 100 mm thick. How much concrete, in cubic metres, was needed to make the porch? (It is easier if you first convert all measurements to metres.) f The owners decide they want to put a picture rail along the windowless internal wall in the living and dining rooms. i What is the total length of picture rail needed? ii If the picture railing is sold in 1800 mm lengths which cost $12:80 each, how much will it cost to purchase the quantity needed?

100

95

75

25

5

0



C

M

Y

K


411


g

95

75

25

i Determine the thickness of the i internal walls ii external walls. ii Discuss why they are different. iii Research and list the different ways in which the walls of a house could be constructed. How is this shown on a building plan?

Investigation - modelling h

5

0

i Find the dimensions of the laundry. ii If the floor of the laundry is to be tiled with 250 £ 250 mm square tiles, calculate the number of tiles needed. iii The tiles come in boxes of 16. How many boxes must be bought?

i Carpet comes in rolls which are 3:6 m wide. i Calculate the number of metres of carpet needed to cover the floor in bedroom 2 if the carpet is laid as shown in diagram A. ii Calculate how much carpet is needed if it is laid as in diagram

3.4 m

3.4 m

bed 2

bed 2

4m

4m

3.6 m

3.6 m diagram A

diagram B

iii Which method of laying would be the cheapest? j Repeat the method above to determine the number of metres of carpet needed for bedrooms 1, 3 and 4. k The decking is to be covered with metal roofing 75 mm which comes in 825 mm widths with the profile shown alongside. How many of these sheets would be needed to cover the decking?

825 mm

l The ceilings in the house are 2:7 m above floor level. Calculate the cost of painting the four bedrooms. 4 The diagram on the next page shows the floor plan of a kit home. a Find the following symbols on the plan and discuss the meaning of each: WIR, LIN, ENS, WC, V, WM. b i How many hinged doors are shown on the plan? ii How many windows does the house have?

100

How many vertical posts support the verandah roof¡? Find the number of rainwater pipes (r.w.p.) or downpipes (d.p.) needed. Write down the length and width of the house (including the verandah). Guttering is needed around the four sides of the house. Calculate the total length of guttering needed. g Find the area of the house, including the verandah, to the nearest square metre. h If built with weatherboard cladding, the house costs $149 300 to complete. What is the cost per square metre of the house? c d e f

95

75

25

5

0



C

M

Y

K


412


WC

V ENS

Master bed 4.3 × 4.2

75

Bed 2 3.0 × 4.2

Bed 3 3.0 × 3.9 v Robe

WIR

WM

Bed 4 3.0 × 3.9

Laundry

Lobby

Lin

Cbd

Kitchen 4.0 × 4.6 7.5 × 4.3 Lounge 6290

0

Family 3.5 × 4.6

Dining

fireplace

Entry

5440

5

10¡590

25

Robe

95

Robe

100

(optional)

Verandah 20¡890

i If constructed on a concrete slab, 250 mm thick, what volume of concrete, in cubic metres, is needed? Ceiling

j A double railing made of timber is built around the verandah, as shown alongside.

Railings

Calculate the number of metres of timber needed (leave a gap in front of the house entry and ignore the width of the posts).

Floor

k Bedrooms 2, 3 and 4 are to be carpeted. If carpet comes in rolls 3:6 m wide, how many metres of carpet will be needed? l The master bedroom is also to be carpeted. i If the carpet is laid as in diagram A following, how many metres of carpet would be needed?

4.3 m

4.3 m

4.2 m

4.2 m

3.6 m 3.6 m Diagram A

Diagram B

ii If laid as in diagram B, how many metres would be needed? iii What features of the carpet or the house might affect which of the methods A or B are used?

100

m Find the total cost of carpeting the four bedrooms if the carpet costs $95 per metre.

95

n Investigation - Modelling The lounge, dining, entry, kitchen and family rooms are to be tiled.

75

i Calculate the area to be tiled. ii 300 £ 300 ceramic tiles are to be laid on the floors. Calculate the number of tiles needed. Explain any assumptions you make in your calculations. iii Find the total cost of tiles for this area.



C

M

Y

K

25

5

0



413

5 The diagrams are the plan and elevations for the proposed extension to a house.

95

5486

250

75

W5

W4

d.p.

250

Remove walls

25

Dining

Laundry

Kitchen (to be extended)

0

Patio

Proposed Family Room

W3 6340

5

Lounge 250

Bed 3 W1

W2

Bath

Press

Patio

remove bricks from outer wall Bed 1 Bed 2 FLOOR PLAN (1 : 200)

2740

K-M aluminium awning windows

a b c d e f g h i j k l

NE ELEVATION Existing Residence

SE ELEVATION

Existing Residence

SW ELEVATION

Aluminium sliding door

Aluminium awning window

Aluminium awning windows

Floor ventilation

What is the scale on the plan and elevations? What are the internal dimensions of the proposed family room? Find the thickness of the walls of the extension. What length of existing wall has to be removed? Determine the dimensions of the new kitchen. What length of guttering is needed across the rear of the family room? What length of downpipe is needed at the rear of the extension? Find the height of the ceiling above the floor. Use the plan and elevations to determine the dimensions of windows W1, W3 and W4. What is the size of the sliding door? Calculate the area of the new room. Calculate the area of the new kitchen.

Investigation - Modelling m How much would it cost to cover the floor of the new kitchen and family room with cork tiles? n How many steps would be needed to the rear sliding door?

100

95

75

25

5

0



C

M

Y

K


414 100


6 The diagram shows the floor plan and elevations of a small holiday house.

95

75

R.W.P

W1

R.W.P Ceiling level

Family 25

5

Scale 1 : 200 W9

Kitchen

W2

Floor level

R.W.P EAST ELEVATION

0

W3 W8

Bathroom

W7

Bed 2

R.W.P

W6

Bed 1

8000

W4

W5 NORTH ELEVATION

a How many i windows ii hinged doors b How many windows can be seen in the i north ii east

iii

sliding doors are there?

iii

south

iv

west elevations?

c What is the scale used? d Find the dimensions of i the bathroom ii bedroom 1 iii bedroom 2 e What is the size of windows W2, W5, W6 and the sliding door? f Find the height of the ceiling above the floor. g What is the maximum height of the roof above floor level? h How far do the eaves extend beyond the walls of the house in the East elevation? i Draw the south elevation. j What area of land does the house occupy? k If the house is built on a reinforced concrete slab of depth 250 mm, calculate the amount of concrete needed for the slab, to the nearest cubic metre. Investigation - Modelling l Find the cost of painting the exterior of the house. 7 Make a list of all the symbols and abbreviations, with their meanings, used in this exercise. 100

INVESTIGATION 2:

95

MODELLING

75

Is it cheaper to build a new house or buy an existing house? You could consider the price of land, the cost of building a home and the price of existing properties. Gather information from newspapers about current prices. Investigate the cost of different house plans - are extras such as carpet, fences and paths included in the price? Investigate council charges, solicitor’s fees, etc. Write a short report on your preferred option.

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You should be able to: 2 enlarge figures given the enlargement factor and the centre of enlargement 2 enlarge drawings using a grid 2 determine whether or not figures are similar 2 find the scale factor of similar figures 2 calculate the dimensions of similar figures 2 solve practical problems involving similar figures 2 interpret and construct scale drawings 2 obtain measurements from building plans and elevations 2 calculate dimensions and areas from floor plans 2 recognise commonly used symbols and terms used with house plans.

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12


The following is a list of key words used in this chapter. Write, in a sentence, a description of each term. enlarge, similar, corresponding angles, corresponding sides, ratio, proportion, scale factor, scale drawing, site plan, floor plan, elevation

?

12

DIAGNOSTIC TEST

1 Copy each figure below onto grid paper and enlarge (or reduce) each one using the centre of enlargement and scale factor given.

O e=2

O e =\ Qw_

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2 In similar figures

i the corresponding angles are equal ii the corresponding angles are in proportion iii the corresponding sides are equal iv the corresponding sides are in proportion. Which of the above statements are correct? A i and iii B i and iv C ii and iii

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D

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none

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3 Determine by measurement which of the following pairs of figures are similar. a b

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25

5

0

4 Given that the figures below are similar, the enlargement factor is A

6 5

B

6 4

C

6 6

D

5 6

5 cm 4 cm 6 cm

A

6 cm

5 cm

5 Triangle ABC is an enlargement of triangle ADE. The scale factor is A 85 B 13 C 86 D 13 5 6

D

E

6 cm

8 cm

B

6 In this pair of similar figures the length of the unknown side is A 27 m B 9 m C 12 m D 6:75 m

C xm

9m 3m

4m

6 cm

7 In the given similar triangles y 6 y 6 A = B = 5 4 4 5 y 6 y 6 C = D = 6 5 6 4

6 cm

5 cm

y cm

4 cm

8 Similar figures related by a scale factor of 1 are A complementary B scalene C

right-angled

D

congruent.

9 A 16 cm £ 10 cm photograph is enlarged so that the new length is 20 cm. The new width is A 32 cm B 12:5 cm C 8 cm D 14 cm 10 Using a scale of 1 : 10 000, what length on a scale drawing would represent a length on land of 120 m? A 12 mm B 12 cm C 1:2 mm D 0:12 mm 11 The scale 1 cm to 5 m is equivalent to A 1:5 B 1 : 50

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C

1 : 500

D

1 : 5000

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12 The actual length of the boat shown in the scale drawing is 11:2 m. The scale used is A 1 : 200 B 1 : 2000 C 1 : 500 D 1 : 112

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If you have any difficulty with these questions, refer to the examples and questions in the exercises indicated. Question 1 2, 3, 4, 5 6, 7, 8 9 10, 11, 12 Section A B C D E

5

0

?

12A

REVIEW SET

1 Copy each figure below onto grid paper and enlarge (or reduce) each one using the centre of enlargement and scale factor given. a b

O O e = 0.6 e=3

2 The diagram has been drawn on a 1 cm grid. Reduce it using a scale factor of 12 by copying it onto a 5 mm grid.

100

3 Determine by measurement which of the following pairs of figures are similar. a b

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4 Given the following pair of figures are similar, find the scale factor. E a b

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75

F

5 cm 36 mm A

24 mm

25

B

2 cm 5

12 mm

0

D

C

G

5 In the following pairs of similar figures find i the scale factor ii the length of the unknown sides. 8 cm a b 11 m z cm

5 cm

xm

6m

13 cm

10 cm

15 m y cm

?

12B

REVIEW SET

1 Copy the figure given onto grid paper and enlarge it using the centre of enlargement and the scale factor given.

O e =\\ Qe_

2 Determine by measurement which of the following pairs of figures are similar. a b 4 cm 6 cm

3 cm

2.5 cm 4 cm 8 cm

6 cm

7 cm 5 cm

4 cm

3 cm

2 cm

3.5 cm

8 cm

100

3 In the following pairs of similar figures, find i the scale factor ii the length of the unknown sides. a b

95

75

1 cm 4 cm 20 cm y cm

2.7 cm

12 cm

25

11 cm x cm

5

x cm

10 cm

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2 cm

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3 cm

75

4 cm

25

x cm

5

4 A tree casts a shadow 5:6 m long. At the same time a metre rule casts a shadow 1:6 m long. Calculate the height of the tree.

0

5 Using a scale of 1 : 10 000, what real length would a length of 3:6 cm on a scale drawing represent?

12C

?

REVIEW SET

1 In the following pairs of similar figures find i the scale factor ii the length of the unknown sides. a b 6 cm

10 cm

8 cm

y cm

xm

5m

x cm

9m 4m

14 cm ym

15 m

2

a Make a scale drawing of the field shown, using a scale of 1 : 1000. b What is the perimeter of the field?

95 m

45 m 80 m

3 Write the scale 2 cm to 5 m in the form 1 : n. N

Find the dimensions of the garage. Find the area of the driveway. Find the actual dimensions of the house. Find the area of the block of land. If I walk from the front of the driveway to the garage, in which direction do I walk?

100

95

Driveway

a b c d e

Garage

4 For this site plan answer the following questions. Scale 1 : 400.

A

A

75

House

25

Scale 1 : 400

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5 Draw the plan, side elevation and front elevation of the solid below.

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25

5

0

?

12D

REVIEW SET

1 In the following pairs of similar figures find i the scale factor ii the length of the unknown sides a b

xm

10 cm 15 m 7 cm 8 cm

y cm

4m

16 m 12 m

z cm

ym

5 cm

2 A rough sketch of the site plan of a house is shown alongside. a Draw a scale diagram, using a scale of 1 cm to 2 m. b What is the area of the block of land? c What area of land is taken up by the house? d What percentage of the block of land is occupied by the house?

28 m 2m

16 m 3m 2m 2m 4m

3 Here is the North Elevation of a house. Scale 1 : 100. a Find the dimensions of the window. b Find the dimensions of the door. c Find the height of the top of the roof above the ground. d Find the length of the roof. e How far past the walls do the eaves extend?

12 m

100

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Window

Door 75

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4 Here is an elevation 3 from a plan. The scale is 1 : 100.

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25

5

0

a b c d

What is the length of the building? What is the height of the top of the roof above the ground? Find the dimensions of the two different sized windows. Find the height of the wall.

5 A street lamp is 4 metres high. When a man stands 5 metres from the base of the lamp his shadow is 4:2 metres long. Calculate the height of the man, to the nearest centimetre.

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CHAPTER

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13

Modelling linear relationships AREA OF STUDY

This chapter deals with straight line graphs and their interpretation. The main mathematical ideas investigated in this chapter are: 8 modelling quantities that vary over time or with each other graphically 8 calculating and interpreting gradient 8 finding and interpreting the vertical intercept 8 using and interpreting graphs of the form y = mx + b 8 modelling using step and piecewise linear functions 8 recognising the limitations of models 8 interpreting linear simultaneous equations 8 drawing the line of best fit.

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424 100

MODELLING LINEAR RELATIONSHIPS AM2 (Chapter 13)

A

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SKETCHING GRAPHS

There are many situations that are modelled using straight-line graphs. Sometimes points are plotted on a number plane and a line drawn through them. Other times a rule or relation is given and ordered pairs are calculated.

25

5

EXAMPLE

0

1

Bulk washing powder is sold for $1.40 per kilogram. This table shows weight versus cost for the washing powder . number of kg cost in dollars

1 1:40

2 2:80

5 7:00

10 14:00

15 21:00

20 28:00

a Draw the graph of weight versus cost. b Use the graph to find the cost of 12.5 kg of washing powder. Washing powder 30.0 25.0

Cost ($)

As the cost depends on the number of kilograms then it is the dependent variable and so is used on the vertical (y) axis. The number of kilograms is the independent variable and so is on the horizontal (x) axis. a The graph is a straight line. b 12.5 kilograms would cost about $17.50

20.0 15.0 10.0 5.0 0.0 0

?

5

10 15 Kilograms

20

EXERCISE 13A 1 Bulk laundry liquid is sold for $3.50 per litre. This table shows capacity in litres versus cost for the laundry liquid. number of litres cost in dollars

1 3:50

2 7:00

3 10:50

4 14:00

5 17:50

6 21:00

a Draw the graph of capacity versus cost. Which is the dependent variable? b Use the graph to find the cost of 2.5 litres of laundry liquid.

100

95

2 Chocolate frogs are sold for $15.00 per kg. This table shows weight in kg versus cost for the chocolate frogs. number of kilograms cost in dollars

1 15

2 30

3 45

4 60

5 75

6 90

25

a Draw the graph of weight versus cost. Which is the dependent variable? b Use the graph to find the cost of 4.5 kg of chocolate frogs.



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MODELLING LINEAR RELATIONSHIPS AM2 (Chapter 13) 100

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3 Pistachio nuts are $22.50 per kg. This table shows weight in kg versus cost for the nuts. 1 22:5

number of kilograms cost in dollars

95

2 45

3 67:5

4 90

5 112:5

6 135

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25

5

0

a Draw the graph of weight versus cost. Which is the independent variable? b Use the graph to find the cost of 3.8 kg of nuts. c How many kg of nuts can be purchased for $50? 4 Mobile telephone calls cost 65 cents per minute. This table shows time versus cost for the calls. number of minutes cost in dollars

1 0:65

2 1:3

3 1:95

4 2:60

5 3:25

6 3:9

a Draw the graph of time versus cost. Which is the independent variable? b Use the graph to find the cost of 2.5 minutes of calls. c How long can someone talk for $3.00? 5 Coffee beans cost $25.80 per kilogram. This table shows weight in kg versus cost for the coffee beans. number of kg cost in dollars

1 25:8

2 51:6

3 77:4

4 103:2

5 129

6 154:80

a Draw the graph of weight versus cost. Which is the dependent variable? b Use the graph to find the cost of 4.3 kg of coffee beans. c How many kg can be purchased for $90?

EXAMPLE

2

The cost of travelling in a hire car is $10 booking fee plus $3.20 per kilometre. a Complete this table of values for the hire car. kilometres cost in dollars

0

10

20

30

40

b Sketch the graph of cost per kilometre. c Find the cost of a journey of 25 km. d How far can you travel for $50? a For For For For

a a a a

journey of journey of journey of journey of

10 20 30 40

kilometres, kilometres, kilometres, kilometres,

cost = $10 cost = $10 cost = $10 cost = $10

+ + + +

$3:20 £ 10 $3:20 £ 20 $3:20 £ 30 $3:20 £ 40

= = = =

$42 $74 $106 $138

100

95

75

) the table of values for the hire car are: kilometres cost in dollars

0 10

10 42

20 74

30 106

25

40 138

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b


Plotting these points and drawing a straight line through them gives the graph alongside.

75

25

c The cost of a journey of 25 km is about $90. d For $50 you can travel about 12 km.

Hire car charges 140.0 120.0

Cost ($)

426

100.0 80.0 60.0 40.0 20.0

5

0.0 0

0

10

20 30 Kilometres

Remember to label the axes and use equal divisions on the scale.

40

As the distance increases the cost increases.

As a change in the number of kilometres travelled causes a change in the cost, then distance travelled (in kilometres) is the independent variable and the cost is the dependant variable. This means that a change in distance travelled means there must be a change in cost. 6 The cost of hiring a taxi is $3.00 flag fall and $2.40 per kilometre travelled. a Copy and complete the table of costs of taxi hire. b Draw the graph showing the cost of hiring the taxi. c How much does it cost to travel 35 km? d How far can you travel for $15? 7 The cost of hiring a night-time taxi is $5.00 flag fall and $3.20 per kilometre travelled. a Copy and complete the table of values of taxi hire. b Draw the graph showing the cost of hiring the taxi. c How much does it cost to travel 15 km? d How far can you travel for $75? 8 The distance d kilometres, travelled by a train over time t hours, is given by d = 90t. a Copy and complete this table of values.

kilometres cost in dollars

0 0

10 27

20

30

40

kilometres cost in dollars

0

10

20

30

40 100

95

75

time in hours distance in km

0 0

1

2 180

3

4

25

5

0

C:\...\NSWGM\NSWGM_13\426NG13.CDR Fri Mar 10 15:19:28 2000


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b Draw the graph of distance travelled. c How far away is the train after 3 12 hours? d When is the train 100 km away?

75

9 A person drives non-stop from Taree to Sydney, a distance of 300 km. The function giving the distance from Sydney is d = 300 ¡ 100t where d is distance in km and t is time in hours. 25

5

0

a Copy and complete this table of values. b Draw the graph. c How far is it to Sydney after 1 12 hours?


0 300

1

2 100

3

10 At a particular time $50 Australian buys $32 US. Use this and the fact that $0 Aus = $0 US to draw a conversion graph for Australian and US dollars. Put Australian dollars on the horizontal axis (values up to $100) and US dollars on the vertical axis. How much is $50 US? 11 Use $50 Australian buys 3500 yen to draw a conversion graph for dollar values up to $100. How much is 5000 yen?

B

GRADIENT AND INTERCEPT

The words slope and gradient are used when talking about the steepness of a line graph. Gradient or slope is associated with house roofs, escalators, hills etc. The steeper the slope the larger the gradient. In order to compare the slopes in different situations there is a standard approach

i.e., slope =

vertical rise horizontal run


The following illustrations indicate slopes of varying amounts. House Roof

Escalator

Leaning Tower of Pisa

2m 8m

56 m

4m

=

2 8 1 4

slope = =

95

4m

10 m

slope =

100

4 10 2 5

slope =

56 4

75

= 14

Notice: For a horizontal line the vertical rise is 0, therefore the slope is 0.

25

When line segments are drawn on graph paper we can easily determine the slope of the line segments by drawing horizontal and vertical lines to complete a right-angled triangle.

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EXAMPLE

95

1

a Find the slope of AB.

75

b

Find the slope of BC. C

B 25

A

B

5

0

C

B

3

2 A

B 5

a slope of AB = =

6


b

slope of BC =

2 5

= =

?

vertical rise horizontal run 3 6 1 2

EXERCISE 13B 1

a Copy and complete this table to find the slope of each of these line segments (part lines). N

L

Line segment

J

F K

IG E C A

y-rise

slope

AB CD EF GH IJ KL MN

H

M

x-run

D B

b Copy and complete: ii The slope of a vertical line is ......

i The slope of a horizontal line is ....... iii As the line segments become steeper, their slopes ......

100

2 Find the slope of the following. a b

95

c

10 m 150 m

4m

1 km road up hill

75

6m

3m

25

4m 5

barn roof

‘slippery-dip’ 0



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3 Determine the slope of a OA b OB c OC d OD e OE f OF g OG

100

95

75

25

G

F E

D

C B A

O

5

4

0

a Copy and complete: Line segment x-run y-rise 2

BC

1

y-rise x-run

F

E

1 2

D

DE

C

AC

B

BE

A

AE AF b State, in sentence form, any conclusions drawn from the graph and table. We can determine slopes, using directed line segments, by adopting the following procedure: ²

Choose two points on the graph (at lattice points).

²

From one of the points move horizontally and then vertically to determine the directed steps to the other point. vertical rise Determine the slope using : horizontal run

²

EXAMPLE

2

Find the gradient (slope) of these lines. a

+4

b

+2 4 100

+2

95

75

rise a gradient = run +4 = +2 = +2

rise b gradient = run ¡4 = +2 = ¡2

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100

EXAMPLE

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75

3

B

Determine the slope of AB and CD in the following.

C

25

D A

5

0

B C

+2

+6 A

slope of AB

rise run +6 = +4

rise run ¡5 = +2

=

slope of CD

3 2

= ¡ 52

5 Determine the slope of a b c d e f

D

+4

=

=

5

A

P

AP AQ AR AS AT AU

Q

R

S

T U

6

U S 100

Q P

R T

O

V

95

W

75

Imagine that you are walking across the countryside from O to W, (i.e., from left to right). a c e

Indicate when you are going uphill. Where is the steepest positive slope? Where is the slope 0?

b d f

Indicate when you are going downhill. Where is the steepest negative slope? When is the slope not zero but least?

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4

100

95

75

EXAMPLE

y

Find the gradient of this line.

(3, 9)

8 6 4

25

2 x

5

2

0

4

6

‘uphill¡’ slopes are positive and ‘downhill¡’ lines have negative slopes.

8

Draw a triangle showing the length of the sides. Note that the slope is positive. rise run +9 = +3 = +3

gradient =

7 Find the gradient of these lines. a b y (1, 10)

c

y

y x

8

8 -2

6

2

4

-2

2

-4 2

4

6

4

(2,-4)

4

(2,-8) y

f 24

-2 x 2

-6

y 2

2

4

6

x

16

(4,-3)

12

(4, 24)

6

-4

6

x

-6 y

g

h 2

4

6 x

-2

2

(4,-2)

-4

4

6

y

100

95

6

2

6

-4

2

(2, 4)

4

-2

-8

e

6

x

4

x 8

y

d

2

75

(5, 4)

4 2

25

x 2

4

6

5

0



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75

EXAMPLE

y 4

a Find the gradient of this line. b Find where the graph cuts the vertical axis.

x

4

25

5

0

This line does not pass through the origin. Draw a triangle in the same way as for other lines. Note that the gradient is negative.

y

rise run ¡3 = +2 = ¡ 32

a gradient =

-3

x +2

b The line cuts the vertical axis at 3.

8 Find the gradient of these lines and where they cut the vertical axis. a b c y y 4

2

x -8

-4

4

-4

-2

4

6

2

x

-4 (4,-11)

-12

-2

e

4

f

y x -2

2

4

2

4

-2

-6

2 -2

2 -2

-8

y

4

x

8

-4

d

y

2

y -2

4

2

x

-2

x

-2

-2

-4

-4

-4

-6

9 Find the slopes from each of these graphs. Be careful as the scales are not the same. a b 300

y

160

95

y

75

120

200 100 100

100

x 5

10 15 20

80 40

25

x 5

10 20 30 40 50 60

0



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c

95

y

d

y 60

0.2

30

0.1

75

x

25

2

4

8

6

10

x

12

25 20 15 10

5

5

5

e

f

24

1200

20

1000

16

Calories

metres per second

0

12

800 600

8

400

4

200

0

20

40

60 80 km per hour

0

1000

2000

3000 4000 kilojoules (kJ)

10 Find where the graphs from question 9 cut the vertical axis.

INVESTIGATION 1:

ROAD SIGNS

Another word for slope is gradient. One application of gradient is to give an indication to drivers of the steepness of the road going up or down a steep incline. Examples of steep inclines include Mount Victoria Pass, Mount Ousley and Bulli Pass. Some examples of signs used include:

TRUCKS USE LOW GEAR

GRADIENT IS 1 IN 8

STEEP GRADES

What to do: 1 Find examples of road signs that express gradient and copy them down. 2 Find the steepest gradient that can be driven up and driven down by a trucks b cars c four wheel drives.

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C

LINEAR GRAPHS 25

The rules or equations for straight-line graphs may all be written in a similar form. 5

The convention is to write the rule for straight-line graphs in the form: y = mx + b.

0



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This relationship connects every point on a straight-line graph.

95

x y m b

75

is is is is

the the the the

independent variable as any value may be used. dependent variable as it depends on the value of x. gradient. intercept with the vertical axis, the y intercept in this case.

25

5

EXAMPLE

0

1

a y = 2x + 3

Draw the linear graph modelled by the equation a Use a table of values x y

¡2 ¡1

¡1 1

0 3

b Use a table of values 1 5

2 7

Plot the points and draw a straight line through them. 8

b y = ¡3x + 5

x y

¡2 11

¡1 8

0 5

1 2

2 ¡1

Plot the points and draw a straight line through the points.

y

12

y

10

6

8

4

6 4

2 -2

2

x -2

2 -2

?

2

x

-2

EXERCISE 13C 1 Copy and complete this table of values to sketch

y = 3x + 1 x y

2 Copy and complete this table of values to sketch

¡1 ¡2

0

1

2 100

95

y = 2x + 2 x y


¡2

¡2

¡1 ¡2

0

1

2

75

y = ¡2x ¡ 1 x y

¡2

¡1 ¡2

25

0

1

2

5

0



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y = 5x + 2


x y

95

¡2

¡1 ¡2

0

1

2

75

5

0

y = ¡3x ¡ 1 y = ¡3x + 2

c g

d h

y = 1x ¡ 1 y = ¡2x ¡ 2

6 Write the y (vertical axis) intercept for each of the graphs in question 5. What do you notice?

D

DIRECT VARIATION

The gradient and intercept of linear graphs have meaning in practical situations. This section examines these practical applications. A relationship that gives a straight line when graphed is a direct relationship between the two variables.

EXAMPLE

1

At the athletics carnival Celine runs the 100m in 16 seconds. It is assumed that she runs at a constant rate. a Sketch a straight-line graph to model Celine’s run. b Find the gradient. c Explain the meaning of the gradient. She runs 0 metres 0 in seconds.

a Use two points to draw the graph. They are (16, 100) and (0, 0). This is the graph.

Celine’s 100 m run 100

b Draw in a triangle. rise gradient = run 100 = 16 = 6:25

Distance (m)

25

5 Sketch these straight-line graphs. a y = 2x + 7 b y = 4x ¡ 6 e y = 2x + 1 f y = ¡x + 3 1 j y = ¡ 34 x + 1 i y = 2x ¡ 1

80 60 100

40

100

20 95

0

16 0

5

10 Time in seconds

75

15

The units for the gradient in this case are m/s.

25

c The gradient is Celine’s speed in metres per second. She runs at an average speed of 6.25 m/s.

5

0



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EXERCISE 13D 1 Christopher runs 100 metres in 25 seconds. a Draw a graphical model of Christopher’s run. b How long would it take to run 30 metres? c Find the gradient and hence Christopher’s speed in metres per second. 2 Craig drives 200 km in 4 hours. a b c d

Draw a graphical model of Craig’s drive. How far does Craig drive in 1 12 hours? Find the gradient. What are the units of the gradient in this case?

Extending a graph and using this extension is called extrapolation .

3 On a particular day one hundred Australian dollars buys 78 Euros. a b c d e

Draw a conversion graph with Australian dollars on the x-axis. How many Australian dollars are needed to purchase 50 Euros? Find the gradient. What is the meaning of the gradient in this context? Extend your graph up to $500 Australian. Is it still accurate as an exchange model?

4 Use the information in question 1 and extend the graph to cover a distance of 500 m. a Is the gradient still the same? Why, why not? b The graph is no longer accurate as a model for the running race. Explain, including some reference to the gradient. 5 Use the information in question 2 and extend the graph to cover a distance of 1000 km. a Is the gradient still the same? Why, why not? b Is the graph still accurate as a mathematical model for Craig’s drive?

[Hint: Consider petrol.]

EXAMPLE

2

Printing costs

Use this graph of printing costs to find a the cost to print 500 books b the number of books that can be printed for $3000. c Find the gradient. What is its meaning? d Find the y intercept. What is its meaning?

Cost in dollars

100


6000 5000 4000

3200

3000 2000 1000

400

0 0

100

200 300 400 500 Number of books

600

100

95

a 500 books would cost about $4800. b $3000 would buy about 280 books c Draw in a triangle to find the gradient. gradient =

3200 400

75

=8

25

The gradient is 8. The units are dollars per book. This means that the gradient is the cost per book.

5

0



C

M

Y

K


437


The y intercept is approximately 800. The units are dollars. This means that the cost to print 0 books is about $800. This is the set up cost.

d

95

75

d

Cost in dollars

100 80 60 40 20 0

Find the intercept on the vertical axis. What is its meaning?

7 300 200 100 0 0

10

20 30 40 Number of people

0

5

10

15 20 25 Kilometres

30

35

Here is a graph modelling catering charges. a How much would it cost for 35 people? b How many people could eat for $300? c Find the gradient. What is the meaning of the gradient?

Catering charges 400

50

60

d


8 Here is a graph modelling plumber charges. a b c d

E

Find the cost for the plumber for 4 12 hours. Find the number of hours worked if the charge is $250 Find the gradient. What is the meaning of the gradient?

Plumber charges Cost in dollars

0

400 200 0 0


1

2

3 4 Hours

5

6

OTHER LINEAR FUNCTIONS

A step graph is another type of line graph. It looks like a set of steps.

EXAMPLE

1

Shopping Centre Parking Fees

is included not included in answer

20 16 Cost ($)

5

Cost in dollars

25

Taxi hire charges

6 Here is a graph modelling taxi charges a Find the cost of travelling 25 kilometres. b How far can you travel for $15? c Find the gradient. What is the meaning of the gradient?

100

95

12

75

8 4 0

25

0

1

2

3 4 5 6 7 Hours Parked

8

9

5

10

0



C

M

Y

K


438 100

a

95

75


Find the cost of parking for i 2 12 hrs ii 3 hrs

iii

3 12 hrs

5 hrs

iv

b

What is the maximum time a car can park for i $10 ii $14 iii $18?

c

What is the range in time which a car can park for a cost of i $0 ii $18 iii $6?

25

v

5 hrs 10 mins.

v

$14

5

0

i

$0

ii

$0

iii

$6

b

i

5 hrs

ii

6 hrs

iii

10 hrs

c

i ii iii

$10

iv

0 - 3 hours. More than 6 hours and up to and including 10 hours. More than 3 hours and up to and including 4 hours.

EXERCISE 13E 1

a From the graph, find the cost of a service which takes i 35 min ii 12 min iii 46 min. b What is the maximum length of a service costing i $65 ii $85 iii $125? c If the calling fee is $45, how much does the company charge for each 15 minutes or part thereof of service? d Copy the graph and extend the horizontal axis to 105 minutes. What is the cost of a service which takes i 70 min ii 85 min iii 100 min?

Refrigerator Repair Costs 130 120 110 Cost ($)

?

a

100 90 80 70 60 50 0

15

30 45 60 Time (mins)

75

90

Daily Day Care Fees for under 5 years

2

62

Cost ($)

52

42 100

32 7am

95

9am

11am

1pm Time

3pm

5pm

75

7pm

a Find the regular daily fee for day care. b What is the late fee charged for children collected after 5:30 pm? c Find the daily fee charged if a parent picks up his/her child at i 5:30 pm ii 5:40 pm iii 5:50 pm.

25

5

0



C

M

Y

K



d In a particular week, Louise collects her toddler from the day care centre at these times: 5 pm, 5:30 pm, 4 pm, 5:45 pm, 6 pm. How much does Louise pay in day care fees for this week?

100

95

75

439

Cost of House Contents Insurance

3

200

25

190

Cost ($)

5

0

180

170

160

4

5

6

7 8 9 10 11 12 House Contents Insurance ($'000s)

13

14

15

16

a Find the cost of insuring house contents valued at i $6900 ii $9000 iii $12 310: b What is the range in value of the house contents which can be insured for i $182 ii $191 iii $172? c Can we find the range in value for contents valued at $196? Give reasons for your answer. 4

a Use the step graph below to complete the table.

i ii iii iv v

Weight 0 < weight of letter 6 20 g 20 g ...... weight of letter ...... 50 g 50 g < weight of letter 6 ...... ...... < weight of letter 6 ...... 250 g < weight of letter 6 ......

Cost ...... $2:80 $4:80 ...... $15:00

Air Mail Postage Rates to Japan, Nepal & Hong Kong 8

Cost ($)

6 4 100

2

95

Weight (g) 75

0 100

200

300

400

500

b Find the cost of sending a letter weighing i 5g ii 125 g iii 251 g. c What is the heaviest letter you can send for i $2:40 ii $4:80 iii $15:00?

25

5

0



C

M

Y

K


440

d Rikako has two letters to send to the same address in Japan, one weighing 125 g and the other 130 g. Is it cheaper to send them separately or together in the one envelope? How much can she save? e Michiko wishes to send two letters to the same address in Hong Kong; each letter weighs 125 g. If the letters were sent together in the one envelope would it cost more or less than if they were sent separately? By how much? f Examine the price difference between consecutive steps on the graph. Do you think it would be more economical to send two letters weighing 200 g and 500 g to the same address separately, or together in one envelope? Give reasons for your answer.

100

95

75

25

5

5

PARKING COSTS Times 1 hour or less more than 1 hour and up to 2 hours more than 2 hours and up to 3 hours more than 3 hours and up to 4 hours more than 4 hours

6

Cost $0:00 $5:00 $7:50 $10:00 $12:50

BOBCAT HIRE Hire Time 4 hours or less for each hour or part thereof after 4 hours 8 hours or more

d

Draw a step graph to represent this information.

Tax payable ($'000)

$ 30

Tax

$ 25 $ 20 $ 15 $ 10 $5 $0

$ 20 $ 40 $ 60 Taxable income ($'000)

$ 80

Find the gradient of the section between taxable incomes of $60 000 and $70 000. What does the gradient represent? Tax

8

$ 24¡230

$ 25

$ 21¡880

$ 20 $ 15

$ 11¡380

$ 10 $5

$0


Draw a step graph to represent the information in the table. On the horizontal axis let 1 cm represent 1 hour and on the vertical axis 1 cm represent $1.

Fee $200 $50 $500

7 The taxation graph from Chapter 10 is a piecewise linear function. This means that it is made up of several joined straight lines. It is not one straight line only. The tax values where the graph changes are shown. a Find the tax payable on $40 000. b Find the income that pays tax of $20 000. c Find the gradient of the section between taxable incomes of $30 000 and $40 000. What does the gradient represent?

Tax payable ($'000)

0


$ 2¡380 $ 20 $ 40 $ 60 $ 80 Taxable income ($'000)


Here is another piecewise linear function. It is a tax function from Chapter 10. The tax values where the graph changes are shown. a Find the tax payable on $45 000. b Find the income that pays tax of $15 000. c Find the gradient of the section between taxable incomes of $30 000 and $40 000. What does the gradient represent? d

What is the marginal tax rate for taxable incomes between $20 000 and $50 000?

C

M

Y

K

100

95

75

25

5

0


441


e

Find the gradient of the section between taxable incomes of $60 000 and $70 000. What does the gradient represent?

f

What is the marginal tax rate for taxable incomes over $50 000?

95

75

SIMULTANEOUS EQUATIONS

25

5

When more than one graph is drawn on the same set of axes comparisons can be made. These equations are called simultaneous equations. These graphs can be used to find information.

0

EXAMPLE cost (dollars)

Bianca is deciding which mobile phone plan to use. She is comparing the Plan 30 and Plan 60 rates. Plan 30 has minimum $30 per month with $5 of free calls. Calls cost 24 cents/30 seconds. Plan 60 has minimum $60 per month with $40 of free calls. Calls cost 19 cents/30 seconds. Here is the graph showing these plans.

Mobile phone charges 120

Plan 30

100 80

Plan 50

60 40 20 0 0

50

100 150 Minutes of call

200

a How many minutes of calls can be made on Plan 30 before paying more than the minimum? b How many minutes of calls can be made on Plan 60 before paying more than the minimum? c After how many minutes of calls does Plan 60 become cheaper? How is this shown on the graph? a The graph is horizontal for about about 10 minutes. It then starts to rise. This is when the minimum is exceeded. b about 105 minutes c After about 75 minutes the two graphs intersect. This is where the charges are equal. After that Plan 30 is more expensive. The Plan 60 line is then always below the Plan 30 line. So, Plan 60 is cheaper if there are more than 75 minutes of calls per month. 100

EXERCISE 13F 1 Rebecca is deciding which mobile phone plan to use. She is comparing the Plan 30 and Plan 50 rates. Plan 30 has minimum $30 per month with $5 of free calls. Calls cost 24 cents/30 seconds. Plan 50 has minimum $50 per month with $40 of free calls. Calls cost 19 cents/30 seconds. Here is the graph showing these plans.

Mobile charges Cost (dollars)

?

120 100 80 60 40 20 0

95

75

plan 30 plan 50

0

50 100 150 Time in minutes

25

200

5

0



C

M

Y

K


442 100


After how many minutes of calls are the costs of each program equal? Lorena averages 35 minutes of calls per month. Which plan should she use? Why? Jenny uses her phone to receive calls, and makes at most three 1 minute calls per month. Which plan should she use? Explain.

a b c

95

75

Mobile phone charges

2

150

Cost (dollars)

25

5

0

Plan 55 100

Plan 60

50 0 0

50


200

Therese is deciding which mobile phone plan to use. She is comparing the Plan 55 and Plan 60 rates. Plan 55 has minimum $55 per month with $55 of free calls. Calls cost 33 cents/30 seconds. Plan 60 has minimum $60 per month with $40 of free calls. Calls cost 19 cents/30 seconds. Here is the graph showing these plans.

After how many minutes of calls are the costs of each program equal? If Therese averages 60 minutes of calls per month, which plan should she use? Why? If Therese averages 100 minutes of calls per month, which plan should she use? Why? How much would Therese save by using the cheapest plan for 2 hours of calls per month?

a b c d

EXAMPLE Company Costs and Income on Daily Production 6

'000 Dollars

5 4

Production Costs

3 Income

2 1 0 0

a b c d

a

200

400

600 800 Number of Items

1000

1200

Use the graph to determine the number of items which need to be sold to break even (i.e., Profit = Expenses). How much profit or loss is made when i 100 items are sold ii 1000 items are sold? How many items were sold for the company to make a i $1000 profit ii $1000 loss? Find the i initial daily production costs of the company ii production cost of each item iii selling price of each item.

100

95

75

25

5

800 items (the Cost and Income lines intersect at this point).

0



C

M

Y

K


443


b

i

95

75

For 100 items:

Cost = $2200 Income = $500 Since Production Cost > Income there is $1700 loss.

Cost = $4450 Income = $5000 Production Cost < Income ) $550 profit. ii

25

c

i

5

0

ii

d

For 1000 items:

For profits, we look at items > 800 (breakeven). At 1200 items, cost = $5000 Income = $6000 )

$1000 profit.

For loss, we look at the number of items < 800: At 400 items, the cost = $3000 Income = $2000 )

$1000 loss.

i

When number of items equals zero, cost = $2000.

ii

Production Cost of 200 items = $2500 Initial Production Cost = $2000 200 items cost $500 ) 1 item costs $2:50 to produce.

iii

200 items sell for $1000 ) 1 item sells for 1000 200 = $5:

3 The production costs of printing a book are initially $8000 plus $3000 per thousand books printed. The books are to be sold at $6:50 each. a Copy and complete the table. b Copy the graph alongside. Graph production costs on this graph using the table in a. c How many books need to be sold to break even? d How much profit or loss is made when i 1000 books are sold ii 2500 books are sold iii 1200 books are sold? e How many books were sold for the company to make i $3000 loss ii $2500 profit?

Number of books Production costs

0

1000

2000

3000

Production of books 35 30 In c

25 $'000s

100

om

e

20 15 10 100

5

95

0 0

1

2 3 4 Number of books ('000s)

5 75

25

5

0



C

M

Y

K


444 100

95


4 A plastics company has an initial cost of $3000 per day plus $8 per item produced. The selling price of the items is $15 each, and the maximum daily production is 1000 items. a Copy and complete these tables:

75

25

i

Number of items Production costs

0

100

300

500

700

ii

Number of items Income

0

100

300

500

700

5

b Choosing a suitable scale, graph on the number plane i production cost versus number of items ii income versus number of items.

0

c How many items need to be sold for the company to break even?

EXAMPLE

3

By drawing the graphs of y = 9 ¡ x and y = 7 ¡ 23 x on the same number plane find the solution to the simultaneous equations y = 9 ¡ x and y = 7 ¡ 23 x. Copy and complete these tables to draw the graphs. y =9¡x

x y

0

2

y = 7 ¡ 23 x

5

0

x y

2

5

y 9

y = 9 -x

7

(6, 3) is the point that is common to both graphs, so it is the solution.

y = 7- We_\x

(6, 3)

3

This is the solution to the simultaneous equations.

10\Qw_ x 6

9

100

5

a Copy and complete the table for y = 4 ¡ x.

b Copy and complete the table for y = 2x ¡ 5.

x y

0

x y

0

2

4

95

75

2

4 25

c Graph both lines on the same number plane and find their point of intersection.

5

0



C

M

Y

K



95

75

25

5

6 Use the same method as question 5 to solve these pairs of simultaneous equations. a y = 6 ¡ x and y = 10 ¡ 2x b y = x ¡ 3 and y = 1 ¡ x c y = x ¡ 1 and y = 2x d y = 4 ¡ x and y = 5 ¡ 2x e y = 6 ¡ x and y = 10 ¡ 2x f y = x + 1 and y = ¡ 23 x + 2

445

Graphics calculators or spreadsheets make this easy!

0

G

LINE OF BEST FIT

Sometimes, especially in scientific experiments, the information in a table may not lie exactly on a straight line. If the information appears to be linear then we can draw the line of best fit for the information. If the data is entered into a graphics calculator then the calculator will find the line of best fit with the gradient and intercept. A spreadsheet will do this as well.

EXAMPLE

1

Graph the information in this table. Draw the line of best fit.

x y

1 4

2 6

3 12

4 13

5 16

6 17

7 23

The graph with the line of best fit appears alongside. The spreadsheet has drawn the line of best fit and shows the equation of this line. The gradient is 2:96 and the intercept is 1:14, so the equation is y = 2:96x + 1:14 . If a spreadsheet or a graphics calculator are not available, then find the gradient using a triangle and rise over run. The intercept can be read off the y-axis after extrapolating (extending) the line. In this case the answer is: gradient is 3, intercept is 1, and so the equation is y = 3x + 1.

100

95

75

25

5

0



C

M

Y

K


446 100

95

75

?


EXERCISE 13G 1 Copy these graphs and draw the line of best fit. a b y

y 25

30

20 15

25

20

10 5

10

5

x

0

0

0 0

c

1

2

3

4

5

6

7

8

1

2

3

4

5

3

4

6

7

8

9

9

d

y

y 1

8

2

5

6

7

0

6

x

-5

4 -10 2 0

e 3 2 1 0 -1 -2 -3

-15

x

0 1

2

3

4

5

6

7

-20

f

y

20

y

15

x 1

2

3

4

5

6

7

10 5 0

x

1 2

3

4

5

6

7

c

equation of the line.

-5

2 For each of the lines of best fit in question 1, find a the gradient b y-intercept

SPREADSHEET APPLICATION Here are the instructions for EXCEL to draw a line of best fit and find the linear equation. We will use the information from the example at the beginning of this section. 100

1 2 3 4 5 6 7 8

Enter the table into the spreadsheet. Highlight the cells. Select the chart wizard. Draw an XY scatter graph. Click on one of the points with the right mouse button. Select insert trendline, select linear and OK. Place cursor on the trendline and click the right mouse button. Select display equation on the chart.

95

75

25

5

0



C

M

Y

K



447

100

Try these tables. 95

a

x y

1 ¡3:1

2 ¡0:9

3 1:1

b

x y

1 ¡1:8

2 ¡5:2

3 ¡8:1

c

x y

1 12

2 16:5

3 22:6

d

x y

1 4:8

2 4:1

3 2:9

75

25

5

0

13

4 3

5 4:8 4 ¡11:3

4 27 4 1:8

6 7:1 5 ¡14:2

5 33

6 36

5 1:1

6 0:2

6 ¡17


1 Here is a list of terms used in this section. Explain each in a sentence. horizontal, vertical, relation, dependent variable, independent variable, axes, gradient, increasing function, decreasing function, step graph, simultaneous equations, line of best fit 2 Write a paragraph about some of the mathematical modelling from this chapter.


You should be able to: ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤

sketch straight line graphs calculate and interpret gradient find and interpret the vertical intercept complete tables, plot points, and draw graphs of the form y = mx + b graph and interpret quantities that vary directly model situations using step and piecewise linear functions recognise the limitations of models solve linear simultaneous equations and interpret the solution draw and interpret the line of best fit given quantities that vary directly.

100

95

75

25

5

0



C

M

Y

K


448 100

95

?


13

DIAGNOSTIC TEST

25

The following graph shows the distance travelled by a car over a period of time. Use this graph to answer the next four questions.

5

Distance in km

75

500 400 300 200 100 0

0

1 The time taken to travel 200 km is closest to A 2 hours B 2 12 hours

0

1

2 3 4 Time in hours

5

C

3 hours

D

3 12 hours

2 The distance travelled after 3 34 hours is closest to A 5 km B 250 km C

300 km

D

6 km

3 The slope of the graph is closest to B 54 A 45

C

100

D

80

4 The slope represents the A Speed B

C

Distance travelled

D

All of these

Time taken

Use this graph to answer the next two questions.

y 5 -1

1

2

3

x -5

5 The gradient of this line is A 3 B

1 3

C

¡ 13

D

¡3

6 The y-intercept is closest to A 2 B

3 4

C

¡3

D

¡2

7 The graph of y = 2x ¡ 1 is A B

100

C

y

x 2 -2

4

y

-1

2 -4 -2

-2

x

75

1

4

x

2

2

95

y

y

4

4

-4 -2

D

-1 -2

2 -4

x

-2

25

2

-2

5

0



C

M

Y

K


449


Use the tax graph to answer the next two questions.

Tax

$ 30

Tax payable ($'000)

95

75

25

5

$ 25

$ 20 $ 15 $ 10 $5 $0

$ 20

$ 40

$ 60

$ 80

Taxable income ($'000) 0

8 The tax payable on $65 000 is closest to A $20 000 B $21 000

C

$23 000

D

$25 000

9 The income that pays tax of $10 000 is closest to A $2000 B $36 000 C

$30 000

D

$2500


Use the graphs of mobile phone charges to answer the next three questions.

Plan 30

Cost of call

100 80

Plan 50

60 40 20

Minutes of call

0 0

50

100

150

10 The number of minutes of calls that can be made on Plan 50 before paying more than the minimum is closest to A 100 B 105 C 110 D 10 11 The number of minutes of calls needed to make Plan 50 cheaper than Plan 30 is closest to A 50 B 30 C 100 D 0 12 The number of extra minutes of calls $70 will get on Plan 50 rather than Plan 30 is closest to A 30 B 50 C 70 D 100 13 Which of A B C D

the following statements about lines of best fit is true? The line of best fit passes through all the points. The line of best fit does not have to pass through any points. The line of best fit always passes through (0,0). The line of best fit must join the first and last points.

100

95

75

Question Section

1, 2 A

3, 4 B

5 C

6, 7 D

10, 11, 12 E

13 G 25

5

0



C

M

Y

K


450 100

95

75

?


13A

REVIEW SET

1 Bulk laundry liquid is sold for $4.50 per litre. This table shows capacity inlitres versus cost for the washing powder. number of litres cost in dollars

25

5

0

1 4:50

2 9:00

3 13:50

4 18:00

5 22:50

6 27:00

a Draw the graph of capacity versus cost. Which is the dependent variable? b Use the graph to find the cost of 3.5 litres of laundry liquid. 2 A person drives non-stop from Condoblin to Sydney, a distance of 400 km. The function giving the distance from Sydney is d = 400 ¡ 100t where d is distance in km and t is time in hours. a Copy and complete this table of values.

time in hours 0 distance in km 400

1

2 200

3

4

b Draw the graph. c How far is it to Sydney after 1 12 hours? 3 Find the gradient and y-intercept of this straight line graph.

y 4 3 2 1

x

-3 -2 -1 -1

1

-2

4 Sketch the graph of y = 3x + 5. 5 On a particular day one hundred Australian dollars buys 68 Euros. a b c d e


6 For this scatter plot a b c d

draw the line of best fit find the gradient find the y-intercept find the equation of the line.

4

100

2 95

0 -2

2

4

6

8 75

-4 -6 -8

25

-10 5

-12

0



C

M

Y

K


451


75

25

13B

?

REVIEW SET

1 Red frogs are sold for $10.00 per kg. This table shows weight in kg versus cost for the red frogs. number of kilograms 1 2 3 4 5 6 cost in dollars 10 20 30 40 50 60

5

a Draw the graph of weight versus cost. Which is the dependent variable? b Use the graph to find the cost of 4.5 kg of red frogs.

0

2 The distance d kilometres, travelled by a train over time t hours, is given by d = 70t. a Copy and complete this table of values.


0 0

b Draw the graph of distance travelled. c How far away is the train after 3 12 hours? d When is the train 100 km away?

1

2 140

3

4

y 5

(1, 5)

4

3 Find the gradient and y-intercept of this straight line graph.

3 2 1

x -1

1

4 Sketch the straight line graph y = 3 ¡ 2x. Catering charges 400

5 Here is a graph modelling catering charges. a How much would it cost for 35 people? b How many people could eat for $300? c Find the gradient. What is the meaning of the gradient? d Find the intercept on the vertical axis. What is its meaning? 6 Cost in dollars


a

Plan 40 100

Plan Yes

b

50

c

0 0

50 100 150 Minutes of call

Cost in dollars

95

300 200 100 0 0

10

20 30 40 50 Number of people

60

How many minutes of calls can be made on Plan 40 before paying more than the minimum? How many minutes of calls can be made on Plan Yes before paying more than the minimum? After how many minutes of calls does Plan Yes become cheaper? How is this shown on the graph?

100

95

75

25

5

0



C

M

Y

K


452 100

95

75

?


13C

1 Cashew nuts are $15.00 per kg. This table shows weight in kg versus cost for the nuts.

0

1 15

number of kilograms cost in dollars

25

5

REVIEW SET

2 30

3 45

4 60

5 75

6 90

a Draw the graph of weight versus cost. Which is the dependent variable? b Use the graph to find the cost of 3.8 kg of nuts. c How many kg of nuts can be purchased for $50? 2 The cost of hiring a night-time taxi is $7.00 flag fall and $3.80 per kilometre travelled. a Copy and complete the table of costs of taxi hire. kilometres cost

0

10

20

30

40

b Draw the graph showing the cost of hiring the taxi. c How much does it cost to travel 15 km? d How far can you travel for $75? 6

3 Find the gradient and y-intercept of this straight line graph.

4

y 5

2 -3

-2

-1

x 1 -2 -4

4 Sketch the straight line graph y = 3x ¡ 4.

5 On a particular day one hundred Australian dollars buys 87 Euros. a b c d e


6 For this scatter plot a b c d

draw the line of best fit find the gradient find the y intercept find the equation of the line.

3

y

100

2 1

95

0

x

75

-1 -2 -3 2

4

6

25

8

5

0



C

M

Y

K


453


95

75

?

13D

1 Mobile telephone calls cost 85 cents per minute. This table shows time versus cost for the calls. number of minutes cost in dollars

25

5

0

REVIEW SET

1 0:85

2 1:70

3 2:55

4 3:40

5 4:25

6 5:10

a Draw the graph of time versus cost. Which is the dependent variable? b Use the graph to find the cost of 2.5 minutes of calls. c How long can someone talk for $3.00? 2 The cost of hiring a taxi is $10 flag fall and $3.40 per kilometre travelled. a Copy and complete the table of values of taxi hire. kilometres cost

0 0

10 27

20

30

40

b Draw the graph showing the cost of hiring the taxi. c How much does it cost to travel 35 km? d How far can you travel for $15? 3 Find the gradient and y intercept of this straight line graph.

y

6 4

(1' 2\Qw_\)

2 -1

1

2

x

-2

4 Sketch the straight line graph y = ¡2 ¡ 4x. 5 Here is a graph modelling taxi charges. a Find the cost of travelling 65 kilometres. b How far can you travel for $40? c Find the gradient. What is the meaning of the gradient? d Find the intercept on the vertical axis. What is its meaning?

Taxi hire charges

100

40

Cost ($)

100

95

30

75

20 10 0 0

20

40 60 Kilometres

25

80

5

0



C

M

Y

K


454


100

6 95


25

5

Cost ($)

75

120 100 80 60 40 20 0

Plan 40 Plan Yes

0 0

50


These graphs compare the Plan 40 with the Plan Yes. a How many minutes of calls can be made on Plan 40 before paying more than the minimum? b How many minutes of calls can be made on Plan Yes before paying more than the minimum? c After how many minutes of calls does Plan Yes become cheaper? How is this shown on the graph?

100

95

75

25

5

0



C

M

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K


100

95

CHAPTER

75

25

5

0

14

Right-angled triangles AREA OF STUDY

This chapter is about the solution of practical problems involving right-angled triangles. The main mathematical ideas investigated in this chapter are: 8 Pythagoras' Theorem 8 the definition of the trigonometric ratios – sine, cosine, tangent 8 finding sides using trigonometry 8 solving problems involving angles of elevation and depression. 100

95

75

25

5

0



C

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456

RIGHT-ANGLED TRIANGLES (Chapter 14)

A

100

95

PYTHAGORAS’ THEOREM

75

In a right-angled triangle, the length of the hypotenuse squared is equal to the sum of the squares of the lengths of the other two sides. 25

5

a2 = b2 + c2

a b

0

The hypotenuse is opposite the right-angle. It is always the longest side in a right-angled triangle.

c

The value of this theorem, known to the ancient Greeks, is that if we know the lengths of any two sides of a right-angled triangle then we can calculate the length of the third side. Also, if we know the lengths of the three sides then we can determine whether or not the triangle is right angled.

EXAMPLE

1

y cm

4 cm

Find the length of the hypotenuse in the given triangle. 5 cm

y2 = 42 + 52 ) y2 = 16 + 25 ) y2 = 41 p ) y = 41 = 6:4 (to 1 d.p.) i.e., the length of the hypotenuse is 6:4 cm.

?

EXERCISE 14A 1 Find the length of the hypotenuse, correct to 1 decimal place, in the following triangles a b c 18 m y cm

5 cm

k cm

7 cm

100

15 m 6 cm

d

5.6 km

xm

12 cm

e

75

f 94 cm

z km

4.8 km

t mm

156 cm 25

73 mm

5

w cm

65 mm


95


0

C

M

Y

K


457

RIGHT-ANGLED TRIANGLES (Chapter 14) 100

95

75

EXAMPLE

2

Find the length of the unknown side in the given triangle.

25

5

0

15 cm

11 cm

p cm

Using Pythagoras’ theorem in this triangle p2 + 112 = 152 i.e., p2 + 121 = 225 ) p2 = 225 ¡ 121 ) p2 = 104 p ) p = 104 = 10:2 (to 1 d.p.) i.e., the length of the unknown side is 10:2 cm.

2 Find the length of the unknown side, correct to 1 decimal place, in the following right-angled triangles a b c qm 21 cm 12 cm

8 cm

26 cm

v cm

4.9 m

5.6 m

p cm

d

e

f 23.6 km

84 m wm

x mm

87 mm

y km

18.4 km

112 mm 75 m

EXAMPLE i ii a

3

State whether or not the following triangles are right-angled. If right-angled, sketch the triangle and mark the right-angle on it. b

100

95

75

7m 3.6 cm

4.8 cm

5m 25

5

6m 6 cm

0



C

M

Y

K


458


100

a

Longest side, squared = 62 = 36 Sum the squares of the other two sides = 4:82 + 3:62 = 23:04 + 12:96 = 36

i

95

75

i.e., 62 = 4:82 + 3:62 i.e., Pythagoras’ theorem is true for this triangle, hence it is right-angled.

25

5

The right-angle is opposite the longest side, hence

ii

0

4.8 cm

3.6 cm

6 cm

b

i

3

2

Longest side, squared = 7 = 49 Sum the squares of the other 2 sides = 52 + 62 = 25 + 36 = 61 2 i.e., 7 6= 52 + 62 i.e., Pythagoras’ theorem is not true for this triangle, hence it is not right-angled.

i ii

State whether or not the following triangles are right-angled. If right-angled, sketch the triangle and mark the right-angle on it.

a

b 8 cm

c

6 cm

d

6 cm

15 m

10 cm

8 cm

5m

12 m

e

f

26 m 15 cm

4 cm

2.4 cm

8 cm 4.5 cm

24 m

10 m

5.1 cm 17 cm

4 Find the length of the diagonal of a 6 cm £ 4 cm rectangle. 5 Find the length of the diagonal of a 8 cm £ 8 cm square.

100

6 A rectangular gate of height 1:2 m has a diagonal brace of length 1:6 m. Calculate the width of the gate.

95

1.2 m

75

1.6 m

25

5

wm 0



C

M

Y

K



3.6 m

7

Kylie measures out an area of ground to be concreted for the floor of a tool shed, as shown in the diagram. To check that the floor will be rectangular she measures the diagonal and finds it to be 4:5 m. Is the area of ground rectangular?

95

2.7 m

2.7 m

4.5 m

75

459

3.6 m 25

5

wall

8 A ladder 6 m long reaches 5:2 m up a wall. How far is the foot of the ladder from the base of the wall?

6m

5.2 m

0

A kite is flying with 28 m of string let out. At this time the horizontal distance to the kite is 18 m. Calculate the height of the kite.

9 28 m

18 m

10 A ship leaves Sydney Heads and travels 45 km due east. It then turns and travels 86 km due north. How far is it from its starting point?

EXAMPLE

4

Find, to the nearest metre, the perimeter of

a

80 m

b 56 m

75 m 36 m

82 m

a Let the lengths of the unknown sides be y and z as shown.

120 m

ym

zm

56 m 1

In triangle 1, y 2 = 822 + 562 = 9860 p ) y = 9860 = 99 m (to nearest m)

2

36 m

82 m

z 2 = 362 + 562 = 4432 p ) z = 4432 = 66 m (to nearest m)

In triangle 2,

) perimeter = 36 + 82 + 99 + 66 = 283 m b Divide the figure as shown

100

95

75

80 m

k2 = 402 + 752 = 7225 p ) k = 7225 = 85 m

In the triangle, 75 m

75 m 80 m

km 40 m

25

) perimeter = 120 + 75 + 80 + 85 = 360 m

5

0



C

M

Y

K


460 100

95


11 Find the perimeter of the following figures, to the nearest metre a b c

75

96 m

64 m

36 m 110 m

125 m

25

d

38 m

e

63 m

0

92 m

48 m

29 m

68 m 180 m

90 m

g

26 m

f

55 m

h

45 m

39 m

16 m

28 m

72 m

76 m

54 m

32 m

95 m

12 A surveyor makes the following measurements of a field. Calculate the perimeter of the field to the nearest metre. a b c 28 m 16 m

27 m 21 m

30 m 32 m 18 m

48 m

36 m

25 m

47 m 68 m

23 m

33 m

52 m

NAMING THE SIDES OF RIGHT-ANGLED TRIANGLES In a right-angled triangle, the side opposite the right-angle is called the hypotenuse. For further work, in trigonometry, the other sides are given names in relation to the other angles of the triangle as shown:

100

95

Relative to the angle marked xo : AB is the side which is opposite the angle. It is called the opposite side. CB is the side which is adjacent to the angle. It is called the adjacent side.

Adjacent means “next to”.

A

hy po ten us e

5

48 m

75

opposite side 25

x° C

adjacent side

5

B

0



C

M

Y

K



EXAMPLE

95

1

Name the i hypotenuse ii opposite side iii adjacent side for the angle marked xo in each of the following right-angled triangles. a b A C R

75

461

x°

25

P

5

0

x°

Q

B

a b

?

hypotenuse is AB hypotenuse is RP

i i

opposite side is AC opposite side is RQ

ii ii

iii iii

adjacent side is BC adjacent side is QP

EXERCISE 14B 1 Name the i hypotenuse ii opposite side iii adjacent side for the angle marked xo in each of the following right-angled triangles. a

b B

C

c M

P

K

x°

x°

L

A

d

e D

E

W

x°

V

U

F

T

h

Q

f

x°

g

x°

R

L

Q

x°

S

R

J x°

100

95

W

75

25

x° 5

V



K 0

C

M

Y

K


462 100

95

75

25


C

DEFINING THE TRIGONOMETRIC RATIOS

EXERCISE 14C (INVESTIGATION) 1 On grid paper draw three different right-angled triangles having an angle of 40o as shown below.

5

0

1

40°

3

2

40°

30 mm

40° 40 mm

50 mm

2 Carefully measure the length of the hypotenuse (H), opposite side (O) and adjacent side (A) for the angle of 40o , in each triangle. O A O 3 Copy the table shown. By calculation, correct Triangle H H A to 1 decimal place, complete the table. 1 2 3 4 Draw any other right-angled triangle with a 40o angle and use it to calculate the ratios in the table above. Add these results to your table. 5 Discuss the answers in each column of the table. From the investigation above, you should have discovered that for a 40o angle in a right-angled triangle, the ratio of the length of the opposite side to the length of the hypotenuse (expressed as a decimal) is always the same, irrespective of the size of the triangle. (The value of this ratio is approximately 0:6) This result is true for any given angle, i.e., if we drew right-angled triangles with a 56o angle then the ratio of the length of the opposite side to the length of the hypotenuse would always be the same (approximately 0:8 for a 56o angle). Since the ratio is constant for a given angle then it is given a name. It is called the sine (abbreviated sin) of the angle. In a right-angled triangle, the sine of an angle xo is the ratio of the length of the opposite side to the length of the hypotenuse. This may be written sin xo =

opposite hypotenuse

&=

O * H

100

95

Similarly, from above, the ratio of the length of the adjacent side to the length of the hypotenuse is constant for any given angle. This ratio is called the cosine (abbreviated cos) of the angle.

75

In a right-angled triangle, the cosine of the angle xo is the ratio of the length of the adjacent side to the length of the hypotenuse. This may be written 25

cos xo =

adjacent hypotenuse

&=

A * H

5

0



C

M

Y

K



463

Again, from above, the ratio of the length of the opposite side to the length of the adjacent side is constant for any given angle. This ratio is called the tangent (abbreviated tan) of the angle.

100

95

In a right-angled triangle, the tangent of an angle xo is the ratio of the length of the opposite side to the length of the adjacent side. This may be written

75

25

Can you work out what the letters stand for?

O (= ) A

opposite tan xo = adjacent

5

To help remember the above definitions, try using the following mnemonic: SOHCAHTOA

0

D

FINDING VALUES OF TRIGONOMETRIC RATIOS

EXAMPLE

1

a Measure accurately the length of the opposite side, adjacent side and the hypotenuse, for the 50o angle in the triangle given. b Calculate the value of sin 50o , cos 50o and tan 50o .

50° 40 mm

a hypotenuse = 63 mm, opposite = 48 mm, adjacent = 40 mm b sin 50o =

O H

cos 50o =

= 48 63 = 0:76

A H

tan 50o =

= 40 63 = 0:63

O A

= 48 40 = 1:2 100

95

?

EXERCISE 14D 1

a Measure accurately the length of the opposite side, adjacent side and the hypotenuse, for the 30o angle in the triangle alongside. b Calculate the value of sin 30o , cos 30o and tan 30o .



75

25

5

30° 60 mm

C

0

M

Y

K


464 100


a Draw any right-angled triangle with a 65o angle. b Measure accurately the length of the opposite side, adjacent side and the hypotenuse, for the 65o angle. c Hence, calculate the sine, cosine and tangent of 65o .

2

95

75

3 Repeat 2 for a 25o angle. 25

5

0

EXAMPLE

2

Calculate the value of sin xo , cos xo and tan xo in the triangle alongside, which is not drawn to scale.

25 m

15 m

x° 20 m

sin xo = = =

O H

cos xo =

15 25 3 5 or

= =

0:6

A H

tan xo =

20 25 4 5 or

= =

0:8

O A 15 20 3 4 or

0:75

4 Write down the value of the sin xo , cos xo and tan xo in the triangles below, which are not drawn to scale. 15 cm a b c 20 m 16 m

24 cm

12 m

d

e 41 m

8 cm

17 cm

x°

x°

x°

f x°

x°

40 m

25 cm

20 m

21 m

9m

E

x°

26 cm

10 cm

29 m

7 cm 24 cm

TRIGONOMETRIC RATIOS BY CALCULATOR

In the last exercise we found the values of the sine, cosine and tangent of some angles by measurement and calculation. Because this method is quite time consuming and because the values of these ratios are so useful they are stored, for ready access, in scientific calculators.

100

95

²

Make sure your calculator is in degrees mode.

²

Find the

²

Depending on the model of calculator, the sequence is either trigonometric ratio key followed by the angle and equals or, angle followed by the trigonometric ratio key. Determine the sequence for your calculator.

sin ,

cos

and

tan

75

keys on your calculator. 25

5

0



C

M

Y

K



EXAMPLE

95

1

a sin 48o b cos 53:6o c tan 74:29o correct to 3 decimal places.

Determine the value of

75

a

465

sin 48o = 0:743

b cos 53:6o = 0:593

c

tan 74:29o = 3:555

25

5

0

?

EXERCISE 14E 1 Use a e i

your calculator to find, correct to 3 decimal places, the value of b cos 42o c tan 56o sin 37o tan 12:6o f cos 67:8o g tan 64:3o o o cos 8:62 j tan 36:47

Degrees with minutes are entered using the degrees, minutes, seconds key (

EXAMPLE

2

d h

o

sin 87:2o sin 27:84o

, ,, or D.M'S ).

Remember 1 degree = 60 minutes.

Change to degrees 26o 510 . Possible sequences of keys are

, ,, (or D.M'S ) 51 o , ,, (or D.M'S ) 26

o

The display shows 26:85, i.e., 26o 510 = 26:85o Check the correct sequence for your calculator. 2 Change to degrees a d g j

34o 240 51o 420 80o 390 36o 170

EXAMPLE

b e h

49o 120 67o 540 44o 570

c f i

76o 180 14o 210 18o 200

Clear your calculator after each question.

3

Find sin 38o 510 , correct to 3 decimal places.

100

Enter the angle as for question 2 with the trigonometric ratio key, in the correct order for your calculator. sin 38o 510 = 0:627 (to 3 dec. pl.)

3 Use a e i


your calculator to find, correct to 3 decimal places, the value of b cos 74o 210 c tan 23o 540 sin 42o 480 o 0 o 0 tan 61 42 f sin 50 38 g tan 45o 360 o 0 o 0 cos 13 59 j tan 83 28


d h

95

75

cos 5o 450 sin 69o 430

25

5

0

C

M

Y

K


466 100


F

95

USING THE CALCULATOR TO FIND THE ANGLE

Given the value of the trigonometric ratio, it is possible to find the angle, using a calculator.

75

EXAMPLE

25

1

Find the size of the angle xo if sin xo = 0:64

5

0

Enter 0:64 Press SHIFT

sin

The calculator displays 39:7918::::

0:64

sin

or press SHIFT

i.e., xo = 39:8o

=

to the nearest 0:1 of a degree.

Check the correct sequence of keys for your calculator.

?

EXERCISE 14F 1 Find a e i

the size of the angle xo , to the nearest 0:1 of a degree, if sin xo = 0:7 b cos xo = 0:4 c tan xo = 0:5 o o f tan x = 1:25 g cos xo = 0:28 cos x = 0:96 j cos xo = 0:86 sin xo = 0:45

EXAMPLE

sin xo = 0:68 tan xo = 0:46

d h

2

Change 26:54o to degrees and minutes, to the nearest minute. A possible sequence is Enter 26:54 Press SHIFT

o

, ,, (or SHIFT D.M'S )

The display shows 26o 32o 24. This is the notation the calculator uses for 26o 320 2400 , i.e., 26 degrees, 32 minutes and 24 seconds. Since 1 minute = 60 seconds, 26o 320 2400 = 26o 320 , to the nearest minute. Check the correct sequence for your calculator.

If the number of seconds is less than 30 round down. If the number of seconds is greater than 30, round up.

100

95

75

2 Change to degrees and minutes, to the nearest minute. a 24:6o b 87:3o c 47:2o e 16:8o f 35:85o g 54:25o o o i 36:598 j 63:293 k 21:44o



80:9o 17:75o 9:689o

d h l

C

M

25

5

0

Y

K



95

75

EXAMPLE

3

Find the size of the angle µ, to the nearest minute, if cos µ = 0:56 :

Enter 0:56 then press

25

5

467

cos

SHIFT

The calculator displays 55:9442:::::

This is the answer in degrees.

0

Press

o

SHIFT

, ,,

The calculator displays 55o 560 39:1300

i.e., µ = 55o 570

to the nearest minute.

3 Find the size of the angle µ, to the nearest minute if a

sin µ = 0:7

b

cos µ = 0:3

c

tan µ = 0:9

d

sin µ = 0:28

e

cos µ = 0:87

f

tan µ = 1:45

g

cos µ = 0:13

h

tan µ = 0:68

i

sin µ = 0:2587

j

cos µ = 0:6541

EXAMPLE

q is the letter “theta” of the Greek alphabet. It is often used as a pronumeral in trigonometry.

4

Find the size of the angle x if tan x = 57 . Enter 5

7 or 5

÷

7

=,

then press

SHIFT

tan

The calculator displays 35:537:::: Press SHIFT

o

, ,,

This gives x = 35o 320

to the nearest minute.

4 Find, to the nearest minute, the size of the angle x, if a

sin x =

e

tan x =

i

tan x =

3 4

b

cos x =

5 8

c

tan x =

3 7

d

cos x =

9 15

8 3

f

sin x =

5 6

g

tan x =

6 11

h

cos x =

4 9

23 48

j

sin x =

56 75

100

95

75

5 Find, to the nearest minute, the size of the angle µ if a

sin µ =

2:5 3:6

b

cos µ =

7:8 10:7

d

cos µ =

48:1 69:4

e

tan µ =

64:5 37:8



c

tan µ =

25

12:9 15:2

5

0

C

M

Y

K


468


100

FINDING ANGLES GIVEN TWO SIDES

95

75

EXAMPLE 4 7

For the following triangles, state whether

25

sin x

A a

5

cos x

B b

0

4m

7m

?

4 7

=

c

4 7

=

opposite = sin xo hypotenuse adjacent =none of these opposite

none of these ratios

D d

4m x°

x° 7m

7m

x°

x°

a

tan x

C c

7m

4m

=

4m

b

4 7

=

opposite = tan xo adjacent

d

4 7

=

adjacent = cos xo hypotenuse

Remember SOHCAHTOA.

EXERCISE 14G 1 For the following triangles, state whether sin x

A

b

c

5m

d

5m x°

3m x° 5m

7 9

=

cos x

B

a

3m

x°

2 For the following triangles, state whether sin x

tan x

C

b

D

c

7m


d

x° 9m


D

3m x°

A

tan x

C

5m

3m

=

cos x

B

a

3 5

x° 9m 9m

x°

9m

7m x°

100

7m

7m

3 For the following triangles, state whether A

sin x

a

B

cos x

b

x°

6 11

95

= tan x

C

D

c

11 cm x°

6 cm

25

6 cm 6 cm

x°

x°

5

11 cm

11 cm

11 cm


75

d

6 cm



0

C

M

Y

K


469


4 For the following triangles, state whether sin x

A

95

a

75

5m

5 6

=

B

cos x

C

b

5m x°

c

tan x

D none of these ratios d

6m

6m

6m

6m

x°

x°

25

x°

5m

5m

5

0

5 For the following triangles, state whether A

sin x

=

cos x

B

a

5 7

b

tan x

C

D none of these ratios

c

7 cm

d 7 cm

x° 7 cm

x°

7 cm

5 cm

5 cm

x°

x° 5 cm

5 cm

EXAMPLE Find the size of angle µ, to the nearest degree.

a

q

b

7m 5m

9.8 cm

6.5 cm

q

a

The known sides are the opposite side and the hypotenuse. The trigonometric ratio that connects these two sides is the sine ratio. Hence sin µ = ) )

b

5 7

= 0:7142::::::

µ = 45:58::::o µ = 46o to the nearest degree.

The known sides are the opposite side and the adjacent side. The trigonometric ratio that connects these two sides is the tangent ratio. Hence tan µ = ) )

9:8 6:5

= 1:5076::::

µ = 56:445::::o µ = 56o to the nearest degree. 100

95

6 Find the size of the angle µ, to the nearest degree a b

75

c

q 4m

3m

q



6 cm

1.5 m

10 cm

q

25

5

2.9 m

0

C

M

Y

K


470 100


d

e

8 cm

95

q 4.3 m

f

15.6 cm q

5.6 m

12 cm

12.8 cm

75

q 25

g

h

14 m

15 mm q

i 28.5 cm 16.4 cm

5

15 m

q

0

j

35 mm

q

k q

6.1 km

l

46 cm q

36 cm

75 cm q

28 cm

4.5 km

m

n 100 mm

5.2 cm

3.6 cm

o 17 km

q

q

q

64 mm

p

16 km

q

r

q

q 148 m

18 cm

15.6 m

12.8 m q 196 m

14 cm

7 Find the size of the angle µ, to the nearest minute a b

c

7 cm

7.2 m

15 cm

6m

8m q

q

q 5.8 m

100

d

e

f

95

q

q

5m

75

q

25 cm

24 cm

14.7 m 25

3.6 m

5

12.3 m 0



C

M

Y

K


471


5.9 m

g

h q

95

75

36 m

7.5 m

q

47 m 25

5

H

FINDING SIDES OF TRIANGLES

0

EXAMPLE

1

Find the length of the unknown side in the triangles below. a b 18 cm

c

20°35' 35 m

zm xm

ym 52° 158 m

48°23'

a Relative to the given angle, the unknown side (x) is the opposite side and the known side (35 m) is the hypotenuse. The trigonometric ratio which connects these two sides is the sine ratio. Hence opposite x = 35 hypotenuse x i.e., = sin 48o 230 35 x or x = 35 £ sin 48o 230 ) = 0:747:::: 35 ) x = 35 £ 0:747:::: ) x = 35 £ 0:747:::: ) x = 26:166:::: ) x = 26:166:::: ) x = 26:2 m to the nearest 0:1 m. b Relative to the given angle, y is the opposite side and 18 cm is the adjacent side. The trigonometric ratio which connects these two sides is the tangent ratio. Hence opposite y = 18 adjacent y i.e., = tan 20o 350 18 y = 0:375:::: or y = 18 £ tan 20o 350 ) 18 ) ) )

y = 18 £ 0:375:::: ) y = 18 £ 0:375:::: y = 6:759:::: ) y = 6:759:::: y = 6:8 cm to the nearest 0:1 cm (to the nearest mm).

100

95

75

25

c Relative to the given angle, z is the adjacent side and 158 m is the hypotenuse. The trigonometric ratio which connects these two sides is the cosine ratio. Hence

5

0



C

M

Y

K


472


z 158 z i.e., 158 z ) 158 ) z ) z ) z

100

95

75

25

5

adjacent hypotenuse

=

= cos 52o = 0:615::::

or

= 158 £ 0:615 = 97:274:::: = 97:3 m

) ) )

z=

158 £ cos 52o

z= z= z=

158 £ 0:615:::: 97:274 97:3 m

0

?

EXERCISE 14H 1 Find the length of the unknown side, correct to 1 decimal place, in the following triangles a b c 38 m zm

26° 26 m

xm

ym

62°

54°

140 m

d

e

f

36 mm 64°

48° w cm

56 cm

16.8 m

km

h mm 72°

g

h

i 75°39'

84 cm

xm

ym

186 m

54°35'

9.2 m

zm

42°18'

j

k 63 mm

wm 24°43'

y cm

t mm 265 m

36°54'

m

n xm

l

76 m

15.6 cm

52°16'

o zm

y km

7.3 km

100

42° 95

72°50'

126 m

64°

75

p

48 cm

q

64°17'

z mm

r 44°

xm 25

w cm

18°36'

126 mm

5

14.8 m

0



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75

EXAMPLE

2

Find the value of z, correct to 1 decimal place. a b 7.6 cm

c

63°19' zm

25

5

54 m

zm 71° 15.8 m

z cm

47°

0

a Relative to the given angle, 54 m is the opposite side and z is the hypotenuse. The trigonometric ratio that connects these two sides is the sine ratio. Hence

) ) ) )

54 opposite = = sin 47o z hypotenuse z 1 = finverting both sidesg 54 sin 47o 1 z = 54 £ sin 47o 54 z= sin 47o z = 73:835:::: z = 73:8 m to 1 d.p.

Invert means turn upside down.

b Relative to the given angle, 7:6 cm is the adjacent side and z is the hypotenuse. Hence

) ) ) )

7:6 adjacent = = cos 63o 190 z hypotenuse z 1 finverting both sidesg = 7:6 cos 63o 190 1 z = 7:6 £ cos 63o 190 7:6 z= cos 63o 190 z = 16:924:::: z = 16:9 cm

c Here, 15:8 m is the opposite side and z is the adjacent side. Hence 15:8 = tan 71o z 1 z = 15:8 tan 71o ) ) ) )

100

finverting both sidesg 95

1 z = 15:8 £ tan 71o 15:8 z= tan 71o

75

25

z = 5:440:::: z = 5:4 m

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2 Find the length of the unknown side, correct to 1 decimal place, in the following triangles wm a b c 12 cm zm

75

62°

38°

56°

16.3 m

y cm 34 m 25

d

e

f

5

24.9 cm km

0

t cm

86 m

x km 46°53' 17.6 km

67°42' 75°

g

8.6 m 24°27'

h 28°15'

z mm

p cm

93 cm

i 83 mm

ym 31°49'

ANGLES OF ELEVATION AND DEPRESSION The angle of elevation of an object, from an observer, is the angle between the horizontal and the line of sight up to the object. The angle of depression of an object, from an observer, is the angle between the horizontal and the line of sight down to the object. observer

t f sigh line o observer

object

angle of elevation horizontal

horizontal angle of depression lin eo f si ght

object

EXAMPLE The angle of elevation of the top of a building from a point 50 metres from its base is 48o . Draw diagram to illustrate this information.

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EXAMPLE

95

475

2

The angle of depression of a ship at sea, from the top of a vertical cliff 56 m high, is 27o . Draw adiagram to illustrate this information.

75

25

270 5

56m

0

?

ship

EXERCISE 14I 1 Draw diagrams to illustrate the following information. a The angle of elevation of the top of a building from a point 60 metres from its base is 65o . b The angle of depression of a ship at sea, from the top of a vertical cliff 36 m high, is 45o . c The angle of elevation of the top of a tree from a point 20 m from its base is 32o . d From the top of a 36 m building, the angle of depression of a car on the street below is 54o . e A flagpole casts a shadow 15 m long when the angle of elevation of the sun is 67o .

EXAMPLE

3

The angle of elevation of the top of a flagpole, as observed from a point 15 m from its base, is 40o . Find the height of the flagpole. Draw a diagram and let the height of the pole be h. By trigonometry,

h = tan 40o = 0:839:::: 15

)

h = 15 £ 0:839::::

)

h = 12:6 m to 1 d.p.

h

40° 100

15 m

95

2 The angle of elevation of the top of a flagpole, as observed from a point 19 m from its base, is 42o . Find the height of the flagpole. 3 From a point 30 m from the base of a building, the angle of elevation of the top of the building is 65o . How high is the building? 4 When the angle of elevation of the sun is 38o , a tree casts a shadow 23 m long. Find the height of the tree.

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100

EXAMPLE

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4

The angle of depression from the top of a cliff 84 m above sea level, to a ship, is 22o . Find the distance of the ship from the base of the cliff. Draw a diagram and let the distance be d.

x°

In the diagram, the angle xo = 90o ¡ 22o i.e., xo = 68o

22°

84 m

d By trigonometry, = tan 68o = 2:475:::: 84 Hence, d = 84 £ 2:475:::: ) d = 208 m to the nearest metre.

ship dm

5 The angle of depression from the top of a cliff 76 m above sea level, to a ship, is 18o . Find the distance of the ship from the base of the cliff. 6 From the top of a building 96 m high, the angle of depression of a parked truck in the road below is 33o . How far is the truck parked from the base of the building? 7 From a plane approaching Sydney at a height above ground of 3000 m, the angle of depression of the Harbour Bridge is 15o . Find the horizontal distance (i.e., the land distance) of the plane from the bridge. 8 Sophia measures a point 20 m from the base of a tree. Using a clinometer, she measures the angle of elevation of the top of the tree, from this point, to be 39o . Her height, to eye level is 1:64 m. a Calculate the distance d in the diagram. b Hence find the height of the tree.

dm 39° 1.64 m 20 m

Practical tasks 9 Using the method of 8, find the height of some of the buildings, trees or the flagpole at your school. 10

hm

a

Choose one of your school buildings which is on level ground.

b

Walk away from the base of the building and, by trial and error, find the point on the ground where the angle of elevation of the top of the building is 45o .

Hm

c

Measure this distance, shown as d on the diagram.

d e

Hence find the height h on the diagram. Measure your height to eye level and calculate the height H of the building.

45° dm

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MISCELLANEOUS PROBLEMS

EXERCISE 14J 1 A 6 m ladder makes an angle of 71o with the ground.

25

a How far does the ladder reach up the wall? b How far is the foot of the ladder from the base of the wall?

5

0

6m

wall

71°

2

shadow

A flagpole casts a shadow 12:6 m long when the angle of elevation of the sun is 43o . Find the height of the flagpole.

43° 12.6 m

3 The foot of a ladder 6:4 m long is placed 2:2 m from the base of a wall. Calculate the angle the ladder makes with the ground, to the nearest degree.

6.4 m

x° 2.2 m

4

shadow

1m

A metre rule casts a shadow 0:86 m long. What is the angle of elevation of the sun, at this time?

0.86 m

5 The anchor rope of a boat is 45 m long. When it is let out fully, it makes an angle of 58o with the surface of the water. Calculate the depth of the water at this point.

58° dm

45 m

6

100

2.8 m p° 11.6 m

7 A ski slope falls 196 m over a run of 370 m. Find the angle the slope makes with the horizontal.

The diagram shows the dimensions of the gable roof at the end of a house. Calculate the angle of pitch, po , of the roof.

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370 m

196 m 5

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8° 95

2000 m

The pilot of a plane flying at an altitude of 2000 m notes that the angle of depression of the airport tower is 8o . How far, in a straight line, is the plane from the tower?

dm

75

Airport

9 A ramp, inclined at 15o to the ground rises 1:3 m vertically to an entrance. How long is the ramp?

25

lm

5

0

A boat is tied to a wharf which is 1:65 m above the boat, as shown in the diagram. The rope makes an angle of 36o with the horizontal. How long is the rope?

1.65 m

10

1.3 m

15°

36°


You should be able to 2 use Pythagoras’ Theorem to find sides of right-triangles 2 test whether or not triangles are right-angled 2 name the sides of a right triangle in relation to one of its angles 2 define the trigonometric ratios - sine, cosine, tangent 2 determine the value of the trigonometric ratios 2 find angles using trigonometry 2 find sides using trigonometry 2 solve problems involving angles of elevation and depression.

14


The following is a list of key words used in this chapter. Write a description of each term. Pythagoras’ theorem, hypotenuse, adjacent side, opposite side, diagonal, sine, cosine, tangent, trigonometry, angle of elevation, angle of depression

14

? 1

DIAGNOSTIC TEST

a The length of the unknown side is A 8 m B 23 m C 32 m

100

17 m

D

15 m

514 m

95

xm 79 m

b xm

56 m

The A B C D

75

length of the unknown side is 3105 m 56 m 97 m 135 m

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2 The A B C D

side adjacent to xo is KM LM KL KLM

K

479

L

x°

M

3 The sine ratio is the ratio of adjacent opposite A B hypotenuse adjacent

adjacent opposite

C

D

opposite hypotenuse

0:85

D

1:18

21o

D

69o

4 In the triangle, tan xo = A

12 9

9 12

B

9 15

C

12 15

D

15 m

9m

x° 12 m

5 Correct to two decimal places sin 58o = A 1:60 B 0:53

C

6 If cos Á = 38 , then, to the nearest degree, Á = A

22o 5 8

7 In the triangle, A

68o

B

sin xo

B

C

= cos xo

tan xo

C

1 tan xo

D

x°

8 cm

5 cm

8

In the triangle, to the nearest degree, x = A 43 B 47 C 34 D 56.

15 m 22 m

x°

9 To the nearest 0:1 m, the unknown side y = A 11:5 m B 16:4 m C 14:0 m D

24:4 m

ym 35° 20 m

plane

10

The angle of depression from the top of the building to the car is A 52o

15° 52° 38° car

B

48o

C

15o

100

D

67o

95

75

If you have any difficulty with these questions, refer to the examples and questions in the exercises indicated. Question Section

1 A

2 B

3 C

4 D

5 E

6 F

7, 8 G

9 H

10 I

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14A

?

REVIEW SET

1 Find the length of the unknown side a

b

7m 25

9.2 m

6.8 m

6.4 m

5

xm xm

0

2 For the angle marked µ in this triangle, name the a opposite side b adjacent side

X

q

Z

3 13 m

5m

x°

Y

Write down the value of a sin x b cos x c tan x

12 m

4 Find correct to 3 decimal places b cos 18o 190 a sin 56o 420

c

tan 75o 510

5 Find x, to the nearest degree, if b cos xo = 0:426 a sin xo = 0:314

c

tan xo = 1:12

6 Find, to the nearest degree, the angle xo in the triangles a b 4.3 km 7m x°

x° 5.2 km

10 m

7 Find the length of the unknown side (answer correct to 1 decimal place) a b 52 mm

x mm

15 cm

x cm

43°15'

100

75° 95

8 Find the length of the unknown side (answer correct to 1 decimal place) a b xm

75

61°43' x cm

24 cm

18 m

25

56°

5

0



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14B

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481

REVIEW SET

1 Find the value of x in the following triangles a b

25

xm

4.2 m

5

3.6 m

4.9 m

0

xm 8 = 2 In the triangle, 15 A sin µ B cos µ C tan µ

D None of these 8 cm q° 15 cm

3 Change a 47:3o into degrees and minutes 4 Find Á to the nearest minute if a sin Á = 0:92 b cos Á = 0:145 5

8 cm

y°

10 cm

6 cm

6 Find the unknown side in the triangles a

16o 410 into degrees

b

c

tan Á = 0:5

In this triangle, cos y o = 8 6 B 68 C 10 D A 10

8 6

b x cm

zm 21°

53°19' 24.7 cm

154 m

7 What is the length of the unknown side in the triangles? a

b z km

100

x cm

47°55'

95

75

25° 36 km

84 cm

8 The angle of elevation of the top of a building, from a point 50 m from its base, is 58o . Find the height of the building.

25

5

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95


14C

?

REVIEW SET

75

25

5

0

c cm

1 In the triangle PQR A b2 + c2 = a2 B a2 = b2 ¡ c2 C b2 = a2 + c2 D a2 + b2 = c2 2

a cm b cm

In the diagram, x = A 5 B 12 C 13 D 1

12 cm x cm 13 cm 7 cm

3 Which statement is true? A cos x < sin x B

sin x < sin y

C

tan y < tan x

D

cos y > tan y

y° 10 cm 6 cm x° 8 cm

4 Change a

68:84o to degrees and minutes

b

46o 270 to degrees

c

tan 76o 140

c

tan Á =

5 Find, correct to 4 decimal places, a

sin 54o 170

b

cos 18o 240

6 Find Á, to the nearest minute, if a

sin Á =

3 5

b

cos Á =

7 8

5 12

7 A tree casts a shadow 5:5 m long when the angle of elevation of the sun is 43o . What is the height of the tree?

100

95

8 The foot of a 6 m ladder is placed 2:5 m from the base of a wall. What angle does the ladder make with the ground, assumed horizontal?

75

9 From the top of a lighthouse, 53:2 m above sea level, the angle of depression of a ship at sea is 6o . What is the distance of the ship from the base of the lighthouse, assumed to be at sea level?

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5

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75

?

14D

483

REVIEW SET

1 Calculate the perimeter of the field. 18 m

25

5

36 m

45 m

0

2 Write down the value of a sin x b cos x c tan x

8m 15 m

17 m x°

3 Find Á, to the nearest degree, if a

sin Á = 0:512

4 In the triangle, A B C D

21 25

b

cos Á =

7 9

c

tan Á =

1:2 0:86

=

sin µ = cos µ = tan µ = None of these

5 In the triangle, x = A 92 sin 57o B 92 tan 57o C 92 cos 57o 92 D tan 57o

q 21 m

25 m

xm

92 m

57°

6 A ramp which is 5:6 m long, rises 1:2 m vertically to an entrance. What angle does the ramp make with the ground, assumed horizontal? 7 A pilot flying at an altitude of 1500 m, notes that the angle of depression of a bridge is 25o . What is the horizontal distance (distance across land) of the plane from the bridge? 8 A boat drops its anchor and anchor rope, which is 72 m long. When the rope is inclined at an angle of 38o to sea level, 7 m of the rope is above the water. Calculate the depth of the water.

100

7m

95

38°

75

dm 25

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CUMULATIVE REVIEWS (Chapters 11 - 14) 100

95

?

11 – 14

CUMULATIVE REVIEW

75

a Find the mean, mode, median and range of the scores 6, 6, 6, 7, 8, 8, 8, 8, 10.

1

b

Class 1-5 6 - 10 11 - 15 16 - 20 21 - 25

25

5

0


For the grouped data alongside i copy the table and add a cumulative frequency column. ii Draw a cumulative frequency histogram and polygon. iii From the ogive determine the (a) median (b) fourth decile

c For the scores 8, 9, 10, 10, 10, 11, 11, 12, 12, 12, 12, 13, find i the 1st, 2nd and 3rd quartiles ii the interquartile range d Organise the following data into a stem and leaf plot: 56, 58, 74, 62, 54, 50, 51, 73, 60, 71, 59, 65, 49, 78, 65, 56, 48, 48, 64, 57 Use the stem and leaf plot to determine the i range ii mode iii median e Draw a box-and-whisker plot for the scores 6, 6, 7, 8, 8, 9, 10, 10, 10, 10, 11, 12 f The mean of a set of data is 11:3 and the standard deviation is 2:9. What is the new mean and standard deviation if i 5 is added to each score ii each score is multiplied by 5? g Find the mean, mode, median, range and standard deviation (¾n ) of the scores 2, 2, 3, 3, 3, 5, 5, 5, 6 2

a Copy this figure and enlarge it using O as the centre of enlargement and the scale factor is 3.

2 cm

O

3 cm

b A tree casts a shadow 3:8 m long. At the same time a metre rule casts a shadow 0:73 m long. Calculate the height of the tree. c Using a scale 1 : 100 000, what real length would a length of 2:3 cm on a scale drawing represent? d Write the scale 4 cm to 5 m in the form 1 : n. e A 16 cm £ 12 cm photograph is enlarged so that the new length is 40 cm. Find the new width.

100

95

3

a Mobile telephone calls cost 75 cents per minute. This table shows time versus cost for the calls. number of minutes 1 2 3 4 5 6 cost in dollars 0:75 1:50 2:25 3:00 3:75 4:50

75

25

i Draw the graph of time versus cost. Which is the dependent variable? ii Use the graph to find the cost of 2:5 minutes of calls. iii How long can someone talk for $2:50?



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CUMULATIVE REVIEWS (Chapters 11 - 14)

b Find the gradient and y-intercept of the given straight line graph.

100

95

y 12 8 4

75

-2 25

-1

1

c Sketch the straight line graph y = ¡3 ¡ 4x. d Here i ii iii

is a graph modelling taxi charges. Find the cost of travelling 75 kilometres. How far can you travel for $50? Find the gradient. What is the meaning of the gradient? iv Find the intercept on the vertical axis. What is its meaning?

0

4

50 40 30 20 10 0 0

a Find the value of x in the following triangles. i ii 6m

2

Taxi hire charges

Cost ($)

5

x

20

40 60 80 Kilometres

100

10 m

xm xm

14 m

7m

b For the angle marked Á in this triangle, name the i opposite side ii adjacent side.

P f

R

c Write down the value of i sin x ii cos x iii d Find correct to 3 decimal places ii cos 71o 150 iii i sin 73o 120 e Find x, to the nearest degree, if ii cos xo = 0:389 iii i sin xo = 0:271 f Find the length of the unknown side i ii 47°

13

5

tan x

Q

x 12

tan 18o 110 tan xo = 1:41

12 m 18°

17 m 100

xm xm

95

g Find the value of µ to the nearest minute i q

75

ii

14.1 m

15.3 m

25

6.2 m 5

q

11.8 m

0



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75

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ANSWERS

5

0

100

95

75

25

5

0

C:\...\NSWGM\NSWGM_AN\487NGAN.CDR Thu Feb 17 16:43:10 2000


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ANSWERS

EXERCISE 2A 1 2 3 4 5 6

a a a a a a f k 7 a f j 8 a e j 9 a b c d

80 b 470 c 30 d 42 590 e 0 400 b 34 500 c 2000 d 100 e 0 6000 b 24 000 c 764 000 d 0 e 1000 13:7 b 3:7 c 0:3 d 5:1 e 16:0 6:73 b 0:09 c 32:71 d 285:12 e 5:00 40 000 b 400 c 60 d 5 e 100 0:7 g 0:03 h 0:008 i 0:0004 j 0:03 1000 l 0:1 290 000 b 490 c 4000 d 25 e 2700 8:6 g 0:82 h 0:049 i 0:000 16 0:0040 k 0:30 l 1:0 3690 b 4 560 000 c 20 700 d 154 000 813 f 14:3 g 1:05 h 0:003 51 i 0:651 0:0391 k 2:00 l 1:00 i 20 ii 17 iii 17:3 i 0:5 ii 0:45 iii 0:451 i 500 000 ii 520 000 iii 522 000 i 8 ii 8:0 iii 8:05

EXERCISE 2B a Y b N c N d Y e N f N g N h Y i N j Y 2 a 100 101 102 103 104 1 10 100 1000 10 000 105 106 100 000 1 000 000

1

b

3

4

5

6

7

a d g j a d g j a d h l a e i l a d

10¡1 1 101 0:1 10¡6 1 106 0:000 001

10¡2 1 102 0:01

10¡3 1 103 0:001

10¡4 1 104 0:0001

10¡5 1 105 0:000 01

EXERCISE 2C

8 £ 104 b 2 £ 105 c 5 £ 103 4 £ 106 e 4:6 £ 104 f 4:3 £ 103 7:9 £ 105 h 3:5 £ 105 i 7:54 £ 104 2:354 £ 106 4 £ 10¡3 b 7 £ 10¡5 c 2 £ 10¡4 9 £ 10¡2 e 5:3 £ 10¡4 f 6:2 £ 10¡2 5:17 £ 10¡3 h 8:5 £ 10¡4 i 5:62 £ 10¡3 2:95 £ 10¡4 5 000 000 b 30 000 c 6000 800 000 e 43 000 f 5600 g 290 000 9 400 000 i 35 700 j 8 720 000 k 2080 705 000 0:000 06 b 0:03 c 0:0005 d 0:009 0:000 042 f 0:0083 g 0:034 h 0:000 58 0:000 249 j 0:006 29 k 0:000 016 7 0:0706 5:26 £ 105 b 2:8 £ 104 c 7 £ 106 4:98 £ 104 e 2:8 £ 107 f 6:03 £ 108

C:\...\NSWGM\NSWGM_AN\488NGAN.CDR Tue Feb 15 15:21:40 2000

9:1 £ 105 h 1:32 £ 1010 i 7:2 £ 1013 3:659 £ 1011 8 4:3 £ 10¡4 b 8:21 £ 10¡3 c 7 £ 10¡6 2:9 £ 10¡5 e 6:5 £ 10¡2 f 3:87 £ 10¡4 8:2 £ 10¡6 h 6 £ 10¡5 i 7:9 £ 10¡7 6:58 £ 10¡11 9 3 400 000 b 830 000 000 c 29 400 000 258 000 e 526 000 3 020 000 000 000 g 29 000 000 875 000 000 i 70 000 000 000 4 835 000 000 10 0:000 59 b 0:000 003 2 0:000 000 071 d 0:002 0:000 000 8 f 0:000 026 4 0:000 000 008 67 h 0:000 002 97 0:000 351 2 j 0:000 000 000 605 11 2:29 £ 108 b 2:54 £ 10¡11 c 3 £ 1027 1:3 £ 105 e 1 £ 1013 12 31 600 000 b 2 600 000 30 000 000 000 d 0:000 26 0:000 000 89 13 1:07 £ 1016 b 4:76 £ 1022 3:22 £ 1017 d 1:39 £ 10¡14 3:36 £ 1011 f 1:25 £ 1020 6:28 £ 103 h 2:40 £ 108 8:25 £ 108 j 2:86 £ 1042 2:62 £ 10¡55 l 4:68 £ 1024 14 3 £ 105 i 1:8 £ 107 km ii 1:08 £ 109 km iii 2:592 £ 1010 km iv 9:4608 £ 1012 km c 7:38 £ 107 km 15 a 5:1 £ 108 km2 b 1:1 £ 1012 km3 16 9:4 £ 108 17 a 10 000 000 000 b 280 km g j a d g j a d f h j a c e g i a d a c e a c e g i k a b


2 a e i m 3 a b 4 a f k o s 5 a e i 7 a e i m 8 a b c

3600 b 8400 c 3482 d 560 290 f 964 g 65:8 h 452 15 680 j 36:9 k 16 370 l 4265 80 n 106:5 o 75 i 100 000 ii 1 £ 105 i 1 000 000 ii 1 £ 106 7 b 5:94 c 8:93 d 6 e 4 0:085 g 8 h 32:8 i 0:62 j 14:3 0:86 l 0:63 m 9:4 n 0:07 24:895 p 23 q 14:96 r 162:7 36 t 72:945 metres b cm or mm c cm d cm km f cm or mm g mm h cm m j km 2700 b 4500 c 23 920 d 340 5600 f 1758 g 1538 h 23 490 58 j 23 490 k 800 l 4050 875 n 2050 o 50 i 1 000 000 ii 1 £ 106 i 1 000 000 000 ii 1 £ 109 i 1000 ii 1 £ 103

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ANSWERS 100

a 4 b 8:5 c 1:65 d 8:365 e 0:75 f 0:045 g 1:48 h 1:9 i 0:4 j 0:95 k 0:09 l 0:08 m 0:009 n 0:005 o 0:2 a kg b g c g d kg e t or kg f t or kg g mg h g i mg j kg a 35 000 b 15 900 c 1650 d 850 e 60 f 1080 g 15 h 5 i 2804 j 400 000 1 000 000 = 106 a 15 b 8 c 7:6 d 0:8 e 9:28 f 0:725 g 0:095 h 0:04 i 0:006 j 2:108 a mL b kL c L d L e L f kL g L h L i L j kL a 72 b 120 c 240 d 720 e 420 f 900 g 2880 h 10 080 i 10 800 j 43 200 a 206 b 351 c 136 d 737 e 56 f 143 g 3940 h 16 698 i 1705 j 5330 k 276 l 144 m 195 n 312 o 114 p 468 q 114 r 57 s 20 952 t 18 864 a i 2:8 ii 2 h 48 min b i 3:2 ii 3 h 12 min c i 4:35 ii 4 h 21 min d i 5:65 ii 5 h 39 min e i 4:45 ii 4 h 27 min a i 1:9 ii 1 min 54 s b i 3:6 ii 3 min 36 s c i 2:55 ii 2 min 33 s d i 6:85 ii 6 min 51 s e i 4:05 ii 4 min 3 s

9

95

75

10 12

25

5

13 14

0

15 17

18

19

20

EXERCISE 2D 1 a 75% b 150% c

62:5% d 66 23 %

e 80% f 283 13 % g 34% h 70% i 129% j 58:7% k 202:4% l 0:3% 2 a f 3 a f j 4 a e 5 a

12 25 21 400

b g

1 45

1 40

c 2 14

d

h

i

1 8

37 50 1 24

e j

19 20 39 250

0:16 b 0:8 c 1:37 d 0:03 e 2:08 0:452 g 0:089 h 1:645 i 0:1791 0:004 1:98 m b $29:40 c 43:2 kg d $552 1:4 m f $47:10 g 75 min h $450 68% b 87 12 % c 83 13 % d 3:2%

e 8 13 %

6 a 32% b 68% c 188 89 % d 12:25% 38 13 %

e 7 a 179:2 b 324 c 918 d 559:2 e 570 8 a 64:5 b 210 c 655:2 d 80:5 e 364:8 9 a $576 b 26:25 m c 2:03 t d $18 e 65 s 10 a 3:888 km b 104:16 kg c 38 s d $417:24 e $2932:50 11 a $12 880 b $11 040 c $9976 d $9900

489

e $9900 f $9975 12 $14 787:55 13 a 21:4% b 8:3% c 7:9% d 168:8% e 100% 14 a 25:8% b 7:3% c 1:9% d 10:5% e 66:7% 15 a $2760 b $2980:80 c $580:80 d 24:2% 16 a i $2880 ii 28:8% b i $1040 ii 10:4% c i loss $24 ii decrease of 0:24% d i loss $100 ii decrease of 1% e i loss $100 ii decrease of 1% f i loss $25 ii decrease 0:25% 17 a loss $14 112:45 b decrease of 48:8% EXERCISE 2E 1 a 13 cm b 14 cm c 14 cm d 15 cm 2 a no b 13:5 and 14:5 cm c 0:5 cm d ruler with smaller units. EXERCISE 2F

3 a b c d e

i 1 cm ii 0:5 cm b i 1 g ii 0:5 g i 1 m ii 0:5 m i 1 min ii 0:5 min i 1 L ii 0:5 L i 0:1 kg ii 0:05 kg i 0:1 s ii 0:05 s i 0:1 m ii 0:05 m i 0:1 L ii 0:05 L i 0:01 m ii 0:005 m i 1 mm ii 0:5 mm iii 11:5, 12:5 mm i 1 g ii 0:5 g iii 347:5, 348:5 g i 1 mL ii 0:5 mL iii 374:5, 375:5 mL 0:1 km ii 0:05 km iii 8:15, 8:25 km i 0:1 s ii 0:05 s iii 18:35, 18:45 s i 0:1 kg ii 0:05 kg iii 4:85, 4:95 kg i 0:01 m ii 0:005 m iii 2:365, 2:375 m i 0:01 L ii 0:005 L iii 5:805, 5:815 L i 20 g ii 10 g iii 410, 430 g i 10 mL ii 5 mL iii 370, 380 mL i 5 m2 ii 2:5 m2 iii 1282:5, 1287:5 m2 i 1000 ii 500 iii 37 500, 38 500 i 12 h ii 14 h iii 6 14 , 6 34 h

4 a c d e f g h i j

i i i i i i i i i

1 a c d e f g h i j 2 a b c d e f g h

0:5 cm ii 5% b i 0:5 s ii 1:6% 0:5 g ii 0:2% 0:5 mL ii 0:7% 0:5 min ii 3:6% 0:5 L ii 8:3% 0:05 kg ii 2:1% 0:05 L ii 0:2% 0:05 s ii 0:4% 0:005 m ii 0:04%

100

95

75

25

5

0



C

M

Y

K


490 100

95

75

EXERCISE 2G 1 2

3 25

4 5

0

ANSWERS

5 6 7 8

a c a b c a c a b c a c a c a c a c

28 cm b 8:5, 9:5 and 4:5, 5:5 26 cm, 30 cm d 2 cm 292 cm 163:5, 164:5 and 127:5, 128:5 cm 291 cm, 293 cm d 1 cm 99 km b 46:5, 47:5 and 51:5, 52:5 kg 98, 100 kg d 1 kg 99:3 kg 47:35, 47:45 and 51:85, 51:95 kg 99:2, 99:4 kg d 0:1 kg 3:8 m b 5:55, 5:65 and 1:75, 1:85 m 3:7, 3:9 m d 0:1 m 15 m2 b 4:5, 5:5 and 2:5, 3:5 m 11:25, 19:25 m2 d 4:25 m2 17:28 m2 b 5:35, 5:45 and 3:15, 3:25 m 16:8525, 17:7125 m2 d 0:4325 m2 642:4 cm2 b 28:55, 28:65 cm 640:2, 644:7 cm2 d 2:3 cm2

3

EXERCISE 2H 1

2

3

4 5 7 8 9 10 11 12 13 14 15

$4:36/kg b 60 w/min c 19 L/100 km $14:20/h e $0:46/call f 4 deg/h 0:4 kg/m2 h 11:2 cents/kwh i 78 km/h 3:4 L/min i 16 000 kg/ha ii 1:6 kg/m2 i 375 c/h ii 6:25 c/min i 180 c/m ii 1:8 c/cm iii 0:18 c/mm i 3:75%/quarter ii 1:25%/month i 2400 g/L ii 2:4 g/mL i 0:75 L/min ii 750 mL/min iii 12:5 mL/s g 1:2 c/g h 5 m/s i 7:5 mL/s j 5 g/min a i 360 m/min ii 21 600 m/h iii 21:6 km/h b i 180 c/h ii 4320 c/day iii $43:20/day c i 5000 g/L ii 5 kg/L d i 800 c/kg ii $8/kg e i 7500 kg/Ha ii 7:5 t/Ha f 1:44 L/h g $8/km a 150 km b 3:75 h a 120 L b 37:5 min 6 a 34 500 b 87 min a 120 kg b 3 c 5000 m2 a 520 mL b 7:5 h c 20 drops/min a $177 b 40 h c 26 h a $US1787:50 b $A2000 c $2769:23 a 198 m2 b 18 L c 5 a $118 750 b 29 m2 a 7:5 L/100 lm b 37:5 L c 600 km a $39 000 b $152 000 a 9:36 £ 108 km b i 107 000 km/h ii 29:7 km/s a d g j a b c d e f

EXERCISE 2I 1

a b c d

i i i i

0:35 g/g ii 35 g/100 g 0:085 mL/mL ii 8:5 mL/L 0:58 g/mL ii 116 g/100 mL 0:09 mg/mg ii 0:09 g/g



i 0:45 mL/mL ii 90 mL/200 mL i 500 g/L ii 0:5 g/mL iii 50 g/100 mL i 600 mg/g ii 0:6 mg/mg i 50 mg/L ii 0:05 mL/mL i 8000 mg/L ii 8 g/L i 0:18 g/g ii 180 mg/g i 0:03 mL/mL ii 30 mL/L i 0:2 g/mL ii 200 g/L iii 200 mg/mL i 0:2 mL/mL ii 200 mL/L i 0:35 g/g ii 350 mg/g i 0:06 g/mL ii 60 g/L i 6:5% w/w ii 75% w/w iii 48:5% w/w iv 0:8% w/w v 6:5% w/w b i 56% v/v ii 3:5% v/v iii 0:6% v/v iv 4% v/v v 22:5% v/v a i 0:15 kg/kg ii 0:15 g/g iii 150 mg/g b i 0:28 kg/kg ii 280 g/kg iii 0:28 g/g iv 280 ,mg/g c i 0:16 L/L 0:16 mL/mL iii 160 mL/L d i 0:35 L/L ii 350 mL/L iii 0:35 mL/mL a 0:6 mg/mg b 60 mg/100 mg c 600 mg/g d 60% w/w a 0:05 g/g b 50 mg/g c 5% w/w a i 0:218 g/g ii 21:8% w/w b i 21:8 g ii 10:9 kg a i 0:008 mg/mg ii 0:8% w/w b 1:6 g a i 30 mL ii 75 mL b i 0:015 mL/mL ii 1:5% v/v c i 7:5 mL ii 11:25 mL iii 12:75 mL a i 0:322 mL/mL ii 32:2% v/v b i 322 mL ii 193:2 mL a 7:5 g b i 0:15 g/g ii 150 mg/g iii 150 g/kg a 0:225 g b i 0:003 g/g ii 3 mg/g iii 3 g/kg a 12 mL b i 24 mL/L ii 2:4 mL/100 mL iii 0:024 mL/mL a i 0:55 g/mL ii 550 mg/mL b i 55 g ii 2:75 kg c 50 L a i 0:1 g/mL ii 100 mg/mL b i 50 g ii 500 g c 25 L a i 0:1 kg/L ii 100 mg/mL b 1:5 g a 650 mg/mL b i 325 g ii 780 g a i 175 mg ii 245 mg iii 875 mg iv 420 mg b i 8 mL ii 15 mL iii 18 mL a i 9 g/L ii 9 mg/mL b i 9 g ii 45 g iii 5:4 g c i 4 L ii 6 L iii 2:5 L a 0:5 kg/L b 50% w/v a i 0:26 g/mL ii 0:26 kg/L b 26% w/v a A 2 g/L, B 1:2 g/L b 3:2 g c 16 g/L a 3 g b 3 g c 6 g d 15 g/L a 3 g b 5 g c 6 g d 60 g

e f g h i j k l m n o 2 a

4 5 6 7 8

9 10 11 12

13 14 15 16 17

18

19 20 21 22 23

C

M

Y

K

100

95

75

25

5

0


ANSWERS 100

95

75

25

5

0

24 a 1 g/mL b 5 g c 5 mL 25 a 0:1 g/mL b 10 g c 10 g/L d 20 mL EXERCISE 2J 54 : 13 b 13 : 54 c 7 : 54 d 13 : 4 54 : 13 : 7 f 13 : 7 : 4 5:7 b 3:2 c 7:4 d 2:4:3 3 : 4 : 6 f 16 : 19 g 8 : 9 h 7 : 4 31 : 20 j 8 : 25 k 3 : 2 l 9 : 2 3 : 5 n 15 : 16 o 1 : 10 p 5 : 22 44 : 17 r 15 : 18 s 9 : 4 t 2 : 5 : 7 3:5 : 1 b 1:8 : 1 c 3:59 : 1 0:4 : 1 e 0:9 : 1 5 : 1 b 1:5 : 1 c 5 : 1 d 2:6 : 1 0:8 : 1 1 : 1:174 b 1 : 2:25 c 1 : 0:8 1 : 0:064 e 1 : 0:77 f 1 : 100 1 : 1000 h 1 : 250 i 1 : 50 j 1 : 20 b Males who currently smoke. Ratio c Females who have never 1 : 76 smoked. 1 : 12 1 : 4:5 7 351 8 $41 600 1 : 157 9 $48 600 10 423 1 : 23 11 215 cm 12 a 24 b 45 1 : 8:8 $18 750, $11 250 14 $1125 20 mL, 160 mL 16 6 kg Katrina $47 000, Paul $21 000 Alison 56 kg, Vincent 42 kg, Matthew 70 kg a Property $900 000, shares $540 000, bonds $360 00

1 a e 2 a e i m q 3 a d 4 a e 5 a d g 6 a

13 15 17 18 19

REVIEW SET 2A 1 a e 2 a 3 a 4 a c

1 5 9 12 13

$22:50 2 $12:45 740 6 360 000 7 $12 000 10 $680 a 875 b 3875 a i 900 g pork 4 12 tbsp oil

3 $25:20 4 $999 80 kg 8 30 000 11 a 40% b $1280 c 3875 d 12 500 1 12 cloves garlic 3 tbsp lime juice 1 2

12 onions 3 34 tsp ginger ii

1350 g pork 6 34 tbsp oil 18 onions 3 4

cup stock 2

14 a 200 m

cup stock

300 g beans 2 14 4 12

(2 or 3) cloves garlic tbsp lime juice

5:625 (5 12 or 6) tsp ginger 450 g beans

b 24 kg

DIAGNOSTIC TEST 1 8 13 19 25

B D B D B

2 C 3 B 4 B 5 B 6 D 7 C 9 D 10 A 11 B 12 C 14 C 15 B 16 B 17 B 18 B 20 C 21 D 22 A 23 C 24 A

3700 b 3659 c 3659:06 d 3659:1 f 4000 No b No c Yes 1:05 £ 108 b 6:2 £ 10¡5 2:46 £ 1014 b 1:4 £ 10¡10 4:096 £ 1023 d 2:9 £ 10¡6

3660

5 a 62 12 % b 166 23 % c 6:5% 6 a i c i 7 8 9 11 12 13 14 15 16 17 18 19 20

21

b 9:7:2 EXERCISE 2K

491

22 23 24

25 26 27

27 100 3 125

9 ii 0:27 b i 1 20

ii 1:45

29 400

ii 0:024 d i

ii 0:0725

a 9:57 m b $81 a 7% b 22:7% c 36 23 % a 82:4 kg b $246:50 10 a 6:25% b 4% a $25 650 b 2:6% increase a 5240 m b 306 cm c 159 mm d 2100 mm a 1:078 kg b 14:6 m c 329:5 d 56 cm a 7080 kg b 5654 g c 4200 mg a 19:3 t b 2:085 kg c 0:6 g a 7800 L b 4260 mL a 15:5 kL b 0:07 L a 210 b 230 a 4:3 b 4 mins 18 s a i 0:1 m ii 0:05 m iii 7:45, 7:55 m iv 0:7% b i 10 g ii 5 g iii 275, 285 g iv 1:8% a 20 cm b 5:5, 6:5 cm and 3:5, 4:5 cm c 18 cm, 22 cm d 2 cm e 24 cm2 f 19:25 cm2 , 29:25 cm2 g 5:25 cm2 18:4 L/100 km a 1:2 kg/m2 b 32:4 km/h a i 0:28 g/g ii 28 g / 100 g iii 28% w/w b i 250 mL/L ii 0:25 mL/mL iii 25% v/v c i 0:15 g/mL ii 0:15 kg/L iii 15% w/v 20 g a 5 : 3 b 8 : 15 c 6 : 13 d 3 : 8 e 15 : 8 a i 53 : 1 ii 1 : 35 b i c i d i e i

8 :1 15 6 : 1 13 3 : 1 8 15 :1 8

15 8 13 1: 6 1 : 83 8 1 : 15

ii 1 : ii ii ii

100

95

75

28 288 29 27 30 $18 000, $30 000 31 $3:95 32 $890 33 $480 REVIEW SET 2B 1 a 1470 b 1500 c 1472:63 2 a 7:49 £ 105 b 3 £ 10¡6 3 6:3 £ 1015

25

5

0

C:\...\NSWGM\NSWGM_AN\491NGAN.CDR Mon Feb 21 14:02:00 2000


C

M

Y

K


492 100

4

ANSWERS 1 2

1 3

1 4

1 5

1 6

95

0:5

0:3_

0:25

0:2

0:16_

75

50%

33 13 %

25%

20%

16 23 %

1 7

1 8

1 9

1 10

25

_ 857_ 0:142

0:125

0:1_

0:1

5

14:3%

12:5%

11:1%

10%

0

5 6 7 9

10 a d 11 a d 12 a b c d 13

28:8% a 1 cm b 0:5 cm c 153:5, 154:5 cm d 0:3% $35:60/m 8 a 1640 L b 8 h 50 min a 4 : 5 b 5 : 8 c 5 : 8 10 $39:12

5 6 7 9

a 0:0051 b 0:01 2 3% a 537 cm b 19 600 g c 70 mL a 100 kg b 48:5, 49:5 kg, 50:5, 51:5 kg c 99 kg, 101 kg d 1 kg a 16 23 m/s b 500 kg/Ha a 0:01 g/g b 400 mL/L c 560 mL a 131:25 mL b 35% v/v 8 $5625 320 g 10 75 kg

a 42 12 % b 19:6% c 300% Double ´ 100% increase a 2 b 2:1 c 2:07 d 2:070 a 1365 cm b 3:46 t c 4 min 36 s a 0:015 b 1:4 £ 1010 6 8 years a 96:25 cm2 b 95:55 375 < A < 96:94 875 c 0:698 75 8 20:2% 9 a 12:8 L/100 km b 83:2 L c 437:5 km 10 a i 3 mg/mL ii 0:003 kg/L iii 0:3% w/v b i 22:5 mg ii 600 mg

1 2 3 4 5 7

6

7 8

$1671:62 a i $3241:97 ii $748:13 iii $1496:27 b i $2258:83 ii $521:27 iii $1042:57 c i $2508:17 ii $578:81 iii $1157:62 d i $915:58 ii $211:29 iii $422:58 e i $423 ii $97:62 iii $195:23 f i $4249:17 ii $980:58 iii $1961:15 a $19 708 b $1642:33 $22 046:44, $26 547:56, $16 394:04, $14 829:88 a $411:51 b $676:92 c $1801:43 d $3019:23 e $1661:36 f $1547:75 a Stacey earns $2230:60 more than Kylie b Paul earns $8393 more than Roland c Aaron earns $4726:40 more than Joshua a A by $35:83 b B by $78:36 c B by $74:15 $306:25 9 $300:20


Weekly Wage

25 12

$11:70

$298:35

D De Souza

32

$10:15

$324:80

A Bentley

19

$13:10

$248:90

C Johnson

17

$12:17

$206:89

K Kernell

20

$8:74

$174:80

P Patrick

23:5

$16:83

$395:51

S Smith

14 16 19 21

1 2 3 7 10

EXERCISE 3A

3 4 5

Hourly Rate

a 15 b 10 c $209:25 15 $56 118:40 $26 030:16 17 $22 211:28 18 $13 977:60 weekly $1212:08, $23:31/hour 20 $21:37 a $1517:38 b $33:72

EXERCISE 3B

REVIEW SET 2D

1 2

Hours Worked

Name

REVIEW SET 2C 1 3 4

$179:20 b $268:80 c $392 $459:20 $133:74 b $260:05 c $297:20 $356:64 Rodney i 37 hours ii $310:80 Anastasia i 36 hours ii $302:40 Sue i 46 12 hours ii $390:60 Zoltan i 41 hours ii $344:40


a $11:75 b $21:53 c $17:39 d $28:05 a $16:70 b $39:16 c $25:30 d $18:86 $395:74 4 $542:72 5 $327:03 6 $371:93 $262:60 8 $429:55 9 $92:80

a

Normal 20

b c d e f

20 20 12 18 20

Hours time and a half 23 12

Wage $519:35

25 12 30 20 12 14 31

$547:55 $611 $401:85 $366:60 $625:10

$375:25 12 $518:50 13 $418 $391:10 15 $573:75 16 $419:44 $449 18 $4457:60 a $395:60 b $930:06 c $0 d $1257:32 20 a $3827 b $4057 c $5073 d $2795 21 a $257:86 b $183:37 c $0 d $539:60

11 14 17 19

EXERCISE 3D

100

95

75

1 a $2650 b $7400 c $11 112:12 d $19 632 e $25 345 f $40 664:40 2 a $619:50 b $660:80 c $708 d $672:60 3 $2528:60 4 a $2871:70 b $2476:90 c $1509:17 5 a $1997:50 b $1692 c $2021

C

M

Y

25

5

0

K


ANSWERS 100

95

75

6 7 8 10 11

d $1762:50 e $2574:43 $12 522:25 a $2550:08 b $446:26 c $2996:34 $400:40 9 a $1295:61 b $36 895:61 a $394:80 b $11:28 a $475 b $12:50

EXERCISE 3E 25

5

0

$1159:20 2 $645 3 $100:50 4 $659:40 $2056:92 6 $700:80 7 a $720 b $15 a $900 b $22:50 a $517:50 b $17:25 a $837:40 b $26:17 c $550:10 a $542:60 b $33:91 a $192:01 b $196:34 c $231:78 d $224:95 13 a $3792:33 b $26 798:15 c $2979:64 d $37 684:50 e $1494:89 f $135:28

1 5 8 9 10 11 12

EXERCISE 3F 1 2 3 4 5 9 12

a $267:40 a $146:40 a $176:00 a $267:40 $123:90 6 $71:40 10 $107:50

b $230 b $230 b $60 b $60 $141 7 $147:20

c $634:90 c $462 c $672:30 c $292 $166 8 $121:40 11 $237:65

EXERCISE 3G 1 2 3 4 5 6 7 8 9

10 11

12 13 14 15 16

$335:50 a $295:52 b $306:62 c $177:10 $73:29 a $464:95 b $609:61 c $443:47 d $569:56 a $11:34 b $49:10 c $147:40 d $579:55 a $5:83 b $303:16 c $5:46 d $2:04 a $8:86 b $10:13 c $7:93 Intermediate Hospital level 1 excess - $150 a Top hospital level 3 excess $500 b Intermediate hospital level 2 excess $250 c Basic hospital level 2 excess $500 d Top hospital level 2 excess $250 e Basic hospital level 2 excess $500 a $2934:54 b $1676:88 c $40 245:12 d $4611:42 a $2293:48, $327:64, $32 436:36, $2621:12 b $1973:51, $1973:51, $26219:24, $3947:02 c $906:01, $258:86, $12 684:14, $1164:87 d $2876:86, $2054:90, $39 043:10, $4931:76 e $1605:80, $0, $22 940, $1605:80 f $2373:07, $2034:06, $31 866:94, $4407:13 Income = $532:53 Super $54:90 $35 056:70 a $14 004:72 b $2334:12 c $22 563:16 d $433:91 a $140 943 b $156 972 c $57 973:76 a $26 162:64 b $8633:67 c $17 528:97



493

d $1460:75 17 $43 990:40 18 a $93 840 b $7820 c $4926:60 EXERCISE 3H 1 a $1:56 b $2:33 c $0:90 d $3:54 e $0:03 f $15:59 2 0:23 + 0:81 + 0:41 + 0:67 = $2:12 3 $0:38 4 a $3 b $1:50 c $0:30 d $0:70 e $0:30 f $4 5 a $6:00 b $2:50 c $4:00 d $7:90 6 a $2:40 b $4:80 c $12:50 d $19:85 EXERCISE 3I 1 Monthly Item

Budget Annual amount

Income Harry’s wage Rose’s wage Interest Total income

19 760 18 980 500 $39 240

Expenditure Home loan payment Home insurance Council rates Water Food Electricity Telephone Car registration Green slip Car Repayments Car insurance Car running Clothes Entertainment Other expenses Total expenses

10 140 390 654 384 7228 1368 654 165 405 3240 458 2860 1440 4160 2600 36 146

Monthly difference Cumulative difference 3094

expected surplus 2 Budget Income Expenses board fares and lunches entertainment clothes others total expenses expected surplus

9880 100

1820 1560 2600 1820 1040 8840 1040

95

75

25

5

0

C

M

Y

K


494 100

3

95

75

25

5

0

ANSWERS

NSW 2751 g 0247327676 h 131 991 anytime i 9:5% p.a.

a Budget Income

9620

Expenses rent car running entertainment car expenses car repayments total expenses expected surplus

2080 1560 1560 725 900 6825 2795

DIAGNOSTIC TEST 1 C 2 A 3 B 4 A 5 B 6 D 7 A 8 C 9 B 10 A 11 D 12 D 13 A 14 C 15 B 16 D 17 D 18 C REVIEW SET 3A

b Clothes, food, entertainment etc. 4

Annual Item Income Nick’s wage Daniella’s wage Total income Expenditure Home loan payment Home insurance Council rates Water Food Electricity Telephone Car Registration Green slip Car Repayments Car insurance Car running Clothes Entertainment Other expenses Total expenses

Budget Annual amount 14 820 15 080 29 900 9000 255 320 300 4940 640 450 165 405 1440 290 1820 960 3640 2600 27 225

expected surplus 2675 Insufficient funds by $3325. Reduce entertainment and other expenses by this amount. EXERCISE 3J 1 a 32 519 930 b Saturday 25 September 1999 c $210:55 d $183:15 e Increased by $27:40 f 13% g 25:3 h 23:9 i March 2 a 62 341 999 b 24=11=99 c $195:20 d $169:10 e Increased by $26:10 f 13:4% g 114 h $72:60 i $102:60 j less by 465 L/day k 132 092 l 90 cents/kilolitre 3 a 8 567 947 300 b T667 122 251 ¡ 9 c 27=9=99 d $98:46 e lower by $50:99 f lower g $149:45 h $81:65 i $16:81 j service and equipment charge 4

a 31 498 382 b 194 213 c 30=11=99 d $164:70 e PO Box 60 Penrith NSW 2751 f Agency payments Locked bag 636 Penrith



1 4 7 10 11 14 17

$1394:10 2 $57 876 3 $23:16 $347:17 5 $442:50 6 $3318 $2331:20 8 $81:40 9 $6866:64 a $267:40 b $60 c $672:30 $236:15 12 $286:52 13 $538:31 $6:88 15 $419:04, $43:20 16 32 cents 70 cents

REVIEW SET 3B 1 3 4 7 9 10 11 14 16

$1201:44 2 $473:76 a $1114:08 b $31:83 $549:70 5 $293 6 $1605:38 a $550:62 b $17:21 8 $866:80 $79 774:92 a $146:40 b $230 c $462 $112:50 12 $237:94 13 $511:95 $13:17 15 $706:16, $72:80 46 cents 17 $3:00

REVIEW SET 3C 1 4 7 10 11 14 17

$574:63 2 $2356:48 3 $578:47 $603:20 5 $347:10 6 $304:50 $2599:10 8 $476 9 $26 048:40 a $146:40 b $60 c $292 $105 12 $316:45 13 $636:91 $5:83 15 $456, $57:60 16 32 cents $1:50

REVIEW SET 3D 1 3 4 7 9 11 14 17

$817:90 2 $13 057:20 a $1200:15 b $31:58 $349:36 5 $597:70 6 $74:88 a $465:04 b $12:92 8 $1866:15 $12 367 10 a $176 b $230 c $504 $243:90 12 $371:30 13 $452:23 $9:50 15 $481:28, $66:56 16 45 cents $3:00 100

EXERCISE 4A 1 a 17, 20, 23, add three b 18, 22, 26, add 4 c 14, 10, 6, subtract 4 d 14, 19, 24, add 5 e 100, 75, 50, subtract 25 f ¡1, 2, 5, add 3 g ¡12, ¡17, ¡22, subtract 5 h 18, 10, 2, subtract 8 i 58, 69, 80, add 11 j ¡30, ¡25, ¡20, add 5 k 3, 2 12 , 2, subtract l 3,

3 23 ,

4 13 ,

add

2 3

2 a 16, 32, 64, multiply by 2 b 64, 256, 1024, multiply by 4

C

95

75

1 2

25

5

0

M

Y

K


ANSWERS 100

95

75

25

5

0

c e f g h

3, 1, 13 , divide by 3 d 80, 16, 16 , divide by 5 5 8, 4, 2, divide by 2 1296, 7776, 46 656, multiply by 6 25, 125, 625, multiply by 5 1 1, 17 , 49 , divide by 7

i 3 a c d e f g

100, 1000, 10 000, multiply by 10 13, 16, 19, add 3 b 9, 27, 81, multiply by 3 5, 52 , 54 , divide by 2 ¡1, ¡5, ¡9, subtract 4 ¡36, ¡47, ¡58, subtract 11 27, ¡81, 243, multiply by ¡3 1 , 1 , 1 , divide by 6 h 4, 2, 1, divide by 2 6 36 216

i k l 4 a b c d e f g 5 a d f h

47, 39, 31, subtract 8 j 4, 9, 14, add 5 0:04, 0:008, 0:0016, divide by 5 40, 16, 6:4, divide by 2:5 9, 11, 13, add 2, 21, 101 14, 16, 18, add 2, 26, 106 60, 65, 70, add 5, 105, 305 15, 14, 13, subtract 1, 6, ¡34 170, 160, 150, subtract 10, 80, ¡320 22, 27, 32, add 5, 56, 252 3, 6, 9, add 3, 21, 141 5, 7, 9, 11 b 6, 4, 2, 0 c 5, 10, 20, 40 ¡15, ¡9, ¡6, 3 e 12, 7, 2, ¡3 32, 16, 8, 4 g 1000, 200, 40, 8 1 , 1 , 5, 50 i 13 , 1, 3, 9 20 2

6 a 22 b 42 c ¡10 d ¡34 e 2560 f 4 782 969 g 2560 h 14 7 a 3, 6, 9, 12, 15 b 4, 8, 12, 16, 20

c 5, 10, 15, 20, 25 d 4, 7, 10, 13, 16 e 5, 9, 13, 17, 21 f 8, 15, 22, 29, 36 g 6, 12, 18, 24, 30 h 7, 10, 13, 16, 19 i 4, 10, 16, 22, 28



495

j 6, 11, 16, 21, 26 k 16, 19, 22, 25, 28 l 13, 19, 25, 31, 37

Three times the number of triangles. Four times the number of matches on each side. Five times the number of pentagons. Three times the number of squares plus one. Four times the number of squares plus one. Seven times the number of octagons plus one. Six times the number of hexagons. Three times the number of squares horizontally plus four. i Three times the number of squares plus one. j Five times the number of pentagons plus one. k Three times the number of squares plus one. l Three times the number of squares plus one. 9 a 90 b 120 c 150 d 91 e 121 f 211 g 180 h 94 i 178 j 151 k 103 l 187 10 a 13 triangles, 1 match left b 10 matches per side, 0 matches left c 8 hexagons, 0 matches left d 13 squares, 0 matches left e 9 squares, 3 matches left f 5 octagons, 4 matches left g 6 hexagons, 4 matches left h 12 squares horizontally, 0 matches left i 7 squares on base, 0 matches left j 7 pentagons, 4 matches left k 11 squares on base, 0 matches left l 11 squares on top, 3 matches left

8 a b c d e f g h

EXERCISE 4B 1 a 7 b 3 c 19 d ¡8 e 25 f 50 g 100 h 125 i 13 j 28 k 49 l 29 m 12 n 95 o 20 p 2 2 a g l 3 a g 4 a c e g

¡3 b 7 c ¡21 d 16 e ¡1 f ¡ 32 49 h ¡21 i ¡80 j ¡3 k ¡16 ¡21 ¡2 b ¡4 c ¡8 d 4 e 3 f ¡13 3 h 11 1, 4, 7, 10, 13 b 9, 7, 5, 3, 1 4, 3, 4, 7, 12 d ¡5, ¡1, 3, 7, 11 ¡1, 2, 3, 2, ¡1 f ¡4, 0, 5, 21, 77 1 , ¡1, 3, 53 , 75 h ¡27, ¡5, 0, 1, ¡20 3

5 a d g j l n

¡2, ¡1, 1, 2 b 6, 5, 4, 3 c 13, 14, 15, 16 4, 3, 2, 1 e 3, 6, 9, 12 f 6, 9, 12, 15 6, 9, 12, 15 h 9, 12, 15, 18 i 4, 8, 12, 16 6, 10, 14, 18 k 12, 16, 20, 24 12, 16, 20, 24 m 3, ¡1, ¡5, ¡9 13, 9, 5, 1 o ¡7, ¡11, ¡15, ¡19

C

M

Y

K

100

95

75

25

5

0


496 100

95

75

25

5

0

ANSWERS

21, 17, 13, 9 q 1, 4, 9, 16 r 2, 4, 8, 16 5, 25, 125, 625 t ¡1, 5, 15, 29 32, 16, 8, 4 v 81, 27, 9, 3 2, 6, 12, 20 x 6, 24, 60, 120 6 ¡31 b 424 c 599 d ¡174 e 10 000 7 Yes b No c No d No e No f Yes No h No i No j Yes 8 ¡4 b ¡4 c ¡4 d ¡4 e all equal ¡4 no g a and c only equal expressions 9 75 b 110 c 150 d 15:36 10 905 cm3 b 2572 cm3 c 9:2 m3 11 1:2 b 58:9 12 23:9 b 26:3 c 22:2 13 2:1 mL b 2 mL c 2:3 mL About 2 mL, measuring instruments not that accurate. 14 a 4 mL b 3:7 mL c 5:1 mL d About 4 mL, middle result. p s u w a a g a f a a a a a d

EXERCISE 4C 1

2 3

4

5

6

a g l q w a f a e i m q u x a f k a e i m q a e i l

9a b 5x c 3x d 2b e 3x f 4x x h 2x i 2a2 j 7x + 3 k 11x2 17x ¡ 7 m 16x n x2 + x o 8b p 6b 2b2 r 5ab s 0 t 4ab u 3xy v p 7 x 2abc 10x b 2y c 5a2 d 16ab e 9x2 y 2q g 0 h 8ab i p2 q 9x ¡ 2 b m + 11 c 2x + 6 d 3y + 3 4p ¡ 5 f 7 + 3x g 3t + 4 h 21n ¡ 16 5p2 + p j 8k ¡ 8 k 3a2 ¡ 7a l 7cd ¡ 2 m2 +5n n 8x+2y o 2a+2b p 2ab+3b2 10x ¡ 10 r 9a ¡ ab s 2n + 2n2 t 4 + 2a 3x ¡ x2 v mn ¡ 8m w 5a2 b + 2ab2 2x2 + 6x3 ¡3x b ¡11x c 3x d 5d e ¡7d ¡5d g ¡7n h 7n i ¡15n j ¡3a + 2 ¡3 ¡ d l 3x m ¡2g n 4a o ¡ac 3a + 10 b 6a + 8 c 4a + b d 5a + 4b 4a2 + 3a f 3ab + 3b2 g 8y + 2 h 2n2 7c + 1 j 7x k ¡3 l ¡2p ¡ 5 10 ¡ 12x n 14p ¡ 8 o 9x ¡ 2 p 6 ¡ 7x ¡6x ¡ 13 r x2 ¡ 4x + 2 ¡5l ¡ 10 b ¡3x c 4x ¡ y d ¡ab ¡ 2b ¡3x ¡ 8 f 5t + 6 g a + 1 h ¡a2 b + 7b 6d ¡ 3c + 2 j 3p5 ¡ p4 k 4m2 ¡ 2n2 ¡2s2 t ¡ 4s2

EXERCISE 4D 1

2

a f k o s w a f j

12x b 28x c ¡16x d ¡20x e 3x2 21x2 g ¡6x3 h 2x2 i ¡27x3 j 2x3 ¡6d2 l 6a4 m ¡3a4 n ¡gh2 r xy 3 z 2 p p2 qr 2 q 35m2 n r ¡27pq 5a t 9x2 u 4xy 2 v ¡35p2 q 18m3 n x 20x3 y 84x b 90xy c 72x d 45y 2 e 140xyz 48x2 y g 120a2 b h ¡36m2 n2 p i ¡60x2 y 2 z 2 ¡42a2 b2 c2 k ¡18x3 y l 24p3 r2

4x2 b 4x2 c 2a3 d 4a3 e ¡8b2 64b2 g x2 h ¡x3 i ¡30x j 2a2 b a2 b2 l 6x3 m ¡8n3 n a3 o ¡3x4 16x4 a 4 a ¡3 b ¡a c d 2 e ¡8 f ¡2a 2 3a a i a j 1 k ¡ l 3a g a h ¡ 2 10 m ¡9x n ¡ab o 2a 3 a f k p

5 a 2x b 4y

¡5p d 5r e 3q f 2a 5b ¡7p ¡4a g ¡2t h 4m i j k 3 6 3 r z x 1 1 l m 5 n ¡ o ¡5b p q 2 3 2z x 1 1 r ¡ s t 1 2q 2m a3 6

b

g a3

h

6 a

m 7 a g

a4 a4 d e 3a f 3 2 3 c4 1 ab2 ¡3a2 i l j k 5 5c x2 2 m 3m o p m 2 1 a 1 2 c3 e c d f 2 3b n2 c 35 ac m3 ab i l j 2m2 k 2 x bd n2 20y 2 3x2 2b c aq2 d e y x c 3a 2

a2 n b2 3a b 2 c3 h 15

8 a 2q

b

f

15xy 2 2z

k

3x2 z 2ay

c

g l

c

ng 4

h

98s2 27rt

i

15xz2 8y 2

j

a b

3g 4

d2 a

9mp n2

EXERCISE 4E 1 a e i m p r 2 a e h k n 3 a e h 4 a d g i 5 a e i

3x + 6 b 5y ¡ 10 c 2a + 4 d 5b ¡ 15 6m + 24 f 9ab + 9c g 4a + 2 h 4 ¡ 8c 14a + 49 j 7 ¡ 21d k 10e + 45 l a2 + 4a 4x ¡ x2 n 3x2 + 12x o 6x2 ¡ 3x 8x + 12y ¡ 4 q 42x ¡ 28y ¡ 14 14x2 ¡ 35xy ¡ 28x ¡3x¡9 b ¡20+4x c ¡8x¡4 d ¡8+2x ¡6p ¡ 9 f ¡6a ¡ 18 g ¡5m ¡ 10n ¡x2 ¡ xy i ¡3ab ¡ 3ad j ¡7b + 2b2 ¡xy ¡ 3y l ¡3a2 + 5a m 6x ¡ 6y + 12z ¡4a + 4b + 4 o ¡x2 ¡ 3x ¡ xy ¡x ¡ 3 b ¡5 + x c ¡x ¡ 8 d ¡7p + 7 ¡2x ¡ 4 f ¡5 + 3x g ¡8y ¡ 4x ¡7q + 8p i ¡5a ¡ 7b ¡6a + 3ab b ¡4xy ¡ 12y c ¡16x + 4x2 ¡5mp ¡ 5m2 e ¡x3 ¡ 7x2 f ¡12c2 d + 4ct ¡3a2 ¡ 3ab + 3a h ¡5x2 + 5xt ¡ 10x ¡7q 2 ¡ 21qr + 14q ¡3x ¡ 1 b 5 + 4x c 7 ¡ 5x d 10x ¡ 4 ¡5x ¡ 5 f 1 + 10x g x + 7 h 13 ¡ 4x 20 ¡ 18x j 24 ¡ 14x k 7x ¡ 20 l ¡18

100

95

75

25

5

0

C:\...\NSWGM\NSWGM_AN\496NGAN.CDR Wed Feb 23 11:11:27 2000


C

M

Y

K


497

ANSWERS 100

95

75

6 a e i m p s

7x + 9 b 6y + 10 c 4p + 13 d 6 ¡ 6x 2d2 f 8d g 3n2 + 8n h ¡n2 + n 24x + 29 j 4x2 k ac ¡ bc l 3x2 + 6x + 6 5x2 + 10x ¡ 25 n ¡3 o ¡2x2 + 4x + 12 ¡2x2 ¡3x¡12 q ¡5x2 ¡9x¡17 r ¡8x¡12 ¡3x ¡ 30 t b2 ¡ b + 3 u ¡2x ¡ 9 3

25

2

3

7 a ¡3x + 16x ¡ 2x b ¡12x + 44x ¡ 5x c 9x3 + 19x2 + 4x d ¡8x3 + 18x2 ¡ 16x e 15x3 ¡ 40x2 + 35x f 15x3 ¡ 34x2

5

d x = ¡42 e x = 42 f x = 50 5 a No b Yes c Yes d No 6 a x = ¡3 b x = 94 c x = ¡ 15 d x = e x=

1 a e i m q u y 2 a e i m q u x 3 a e i m q u x 4 a e h k 5 a g

p = 8 b d = 10 c d = 16 d L = 5 a = 15 f q = 46 g c = 10 h x = 29 f = 64 j p = 22 k y = 21 l d = 56 d = 22 n f = 26 o g = 32 p x = 27 x = 8 r y = 40 s p = 82 t z = 71 f = 32 v p = 58 w L = 46 x r = 1 T = 30 z s = 19 m=5 b p=5 c m=9 d b=5 m = 7 f x = 9 g p = 12 h g = 12 a=5 j N =7 k z=4 l d=4 q = 21 n m = 10 o w = 21 p c = 9 L = 3 r z = 6 s T = 20 t a = 6 y = 12 v d = 9 w p = ¡8 q = ¡5 y r = ¡14 z c = ¡8 x = 14 b x = 8 c p = 66 d m = 70 p = 16 f d = 70 g y = 36 h q = 21 d = 30 j x = 24 k d = 14 l c = 110 m = 64 n p = 60 o f = 70 p d = 36 y = 49 r x = 68 s q = 30 t e = 152 a = 24 v s = 6 w t = 135 d = 270 a = 36 b d = 6 c x = 33 d m = 21 d = ¡42 f p = 11 g x = 100 y = ¡77 i y = ¡19 j n = ¡48 b = 12 l q = 2 Yes b Yes c No d No e Yes f No Yes h No i Yes j No k Yes l Yes x = 4 b p = 4 c x = 14 d x = 9 p = 10 f y = 10 g x = 24 h p = 4 m = 6 j d = 9 k x = 10 l c = 16 p = 20 n a = 5 o m = 12 p p = 12 r = 18 r q = 14 s q = 34 t s = 33 r = 13 v T = 5 w s = 9 x m = 16 Yes b No c Yes d Yes e Yes f No No h Yes i Yes x = ¡2 b x = ¡2 c x = ¡2

d x = ¡ 23 h l

x = ¡ 13 3 x = ¡ 72

e x = ¡ 65 i x=

¡ 87

f x=

5 2

j x=

¡ 23

g x = ¡ 12 k x=1

m x = 11 n x = ¡3

o x = ¡2 p x = 3 q x = s x = ¡4 t x = 8 u x =

5 7

r x=9

¡ 52

v x = ¡1

w x = 3 x x = ¡5 4 a x = 10 b x = 15 c x = ¡25


7 3

1 a e i 2 a

x = 4 b x = 1 c x = 1 d x = ¡3 x = ¡2 f x = ¡2 g x = 9 h x = ¡10 x=3 x = ¡ 92 b x = 43 c x = 65 d x = ¡ 10 3

e x = ¡ 72

f x = ¡ 17 2

h x = ¡14 i x = k s = ¡ 11 5 o

p=

3 a Yes b 4 a x=3 e x=0 5 a x=4 e x= i x= 6 a x= e x=

l x=

15 11

1 8 19 8 8 3 6 7

¡ 15 7 4

g x=

j a = ¡ 32 11 3

m a=

Yes c No d Yes b x = ¡2 c x = 2 f x = 6 g x = ¡5 b x = 45 c x = 13 3 f x=

3 2

4 5

g x=

n y = ¡ 98 Yes f No x=0 x = ¡1 x=2

e d h d

7 2

h x=

j x=2 k x=7 l x= b x = ¡ 25 4 f x=

27 10

13 10

c x=

d

5 4 31 16

x=

29 6

1 g x = 2 h x = ¡ 10

7 a x = 21 b x = ¡15 c x = ¡10 d x = ¡4 e x = 17 f x = ¡20 g x = ¡5 h x = ¡ 73 l x= 8 a x= e x= h x= 9 a x=

9 2 1 8 7 15

¡ 75

20 7

b x= f i

42 5

7 9

c x=

x = ¡ 11 4 x = ¡ 52

2 35

x=

g

k

18 13

x=

d x= 4 ¡ 11

b x = ¡5 c x = 8 d

e x=0 f x= i x=

1 2

i x = 24 j x =

14 3

11 8

x=0

g x = 20 h x = ¡8

EXERCISE 4I

EXERCISE 4G 1 a e i m q u 2 a g 3 a

f x=

5 3

EXERCISE 4H

EXERCISE 4F 0

10 7


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

a a a a a a 15 2

a a

a a a a a a a

26 b 35 s = 15 b a = 6 c n = 20 d R = 25 8 68 b 95 c 212 24:5 cm b 942:5 cm 59 cm b 23:9 m 120 km/h b 292:5 km c 8 hrs 39 mins cm, 196:875 cm2 1:68 cm b 3:2 cm c 0:58 m 21o b 42o 60 kg b 65 kg c 59 kg 1:55 secs b 25 cm 13:8 km b 126 m 78:4 m b 4:5 secs 145 cm2 b 8:9 m 5498 cm3 b 1:02 cm c 0:018 cm 866 cm2 b 0:282 m

100

95

75

25

5

0

C

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498 100

95

75

ANSWERS

DIAGNOSTIC TEST

e

1 B 8 D 14 A 20 B

i

2 D 3 B 4 D 5 D 6 A 7 B 9 B 10 D 11 C 12 B 13 A 15 D 16 B 17 D 18 B 19 D 21 C 22 B 23 B

m

5

0

1 2 4 5 6 7

8

9

a 19, 22, 25, 28 b start at 7 and add 3 c 154 8, 13, 18, 23, 28 3 ¡18 ¡13, ¡10, ¡7, ¡4, ¡1, 8 a 70, 65, 60 b ¡105 a 1437 cm3 b 180 cm3 c 0:9 m3 a 5p b 7m c 5b ¡ 4a d 6a ¡ 5b e 16x2 a2 a f g 3L h 15 2 a 3x ¡ 27 b ¡3x ¡ 15 c ¡5 + x d ¡3 ¡ 2x e 12x ¡ 40 f 9 g 47x2 ¡ 10x3 ¡ 30x h 6x3 + 12x2 + 4x a d = 44 b x = 11 c x = 576 d x = 5 e x = 23 f x = 24 g n = 31 h c = ¡ 73 2 4 6 i x=

1 6

j x=4 k x= x = 19 o ¡6 23

m x=2 n 10 11

o

9 5

o x= o

5 3

l x = 5 13

8

9

j x=

m x=

n x=

13 2

10

a 48 89

b ¡12 29

11

a u = 113 b d R = 5:7

11 3 1 2

k x = ¡ 11 25

o x=

l x=4

5 4

a = 4:4 c n = 6

a 1, 5, 9 b add 4 c 181 2 16, 12, 8, 4, 0 52 4 19, 1, ¡5, 1, 19, 145 5 a 8, 4, 2 b 2356 cm2 a 19p b 15m c ¡5a + 7b d 13a + 4b a3 g 4L 28 14x¡35 b ¡15x¡25 18x ¡ 30 f 20p ¡ 20 18x3 + 24x2 + 33x d = 20 b x = 10 c

e 121x2 8

9

a e h a

f


n x = ¡ 17

l x=

o x = ¡ 34

34 5

REVIEW SET 4D a ¡19, ¡25, ¡31, ¡37 b subtract 6 c ¡709 12, 9, 6, 3, 0 3 908 4 15, 3, ¡1, 3, 15, 99 a 27, 81, 243 b 6561 6 11 310 cm3 a 2p b ¡9m c ¡13a + 7b d 11a + 7b a5 4a g 5L h e 9x2 f 21 3 8 a 8x ¡ 36 b ¡32x ¡ 56 c ¡11 + 4x d 13 ¡ 5x e ¡7x ¡ 32 f ¡6p ¡ 24 g ¡45x3 + 47x2 ¡ 25x h ¡21x3 ¡ 51x2 + 70x 9 a d = 16 b x = 7 c x = 12 d x = 11 e x = 32 f x = 23 g n = 6 h c = ¡ 49 8

1 2 5 7

7 i x = ¡ 12

3 j x = 2 k x = ¡ 13

m x=

n x=

¡ 13

1 7

o x=

l x=

7 9

44 5

1 a d g j m p s v y

discrete b discrete e categorical discrete k categorical continuous discrete t discrete w continuous

continuous c categorical discrete f categorical h discrete i continuous continuous l categorical n discrete o categorical q categorical r continuous discrete u categorical discrete x categorical

EXERCISE 5C 1 a h o v

C S C S

b i p w

C C S C

c S d j C k q S r x S y

S e S f C g S C l C m S n C S s C t C u C S

EXERCISE 5E

c ¡17 79

REVIEW SET 4C 1 3 6 7

n=3 h c=3

g

x = 4 k x = ¡ 15

EXERCISE 5B

a 10, 12, 14, 16 b add 2 c 60 ¡2, 1, 4, 7, 10 3 ¡31 4 19, 14, 9, 4, ¡1, ¡16 a 44, 36, 28 b ¡316 a 201 cm2 b 707 cm2 c 8:0 m2 a 16p b 21m c ¡3a + 3b d 12a + b a2 3L a g h e 49x2 f 12 2 3 a 7x ¡ 63 b ¡5x ¡ 25 c ¡8 + x d 5 ¡ 2x e 3x ¡ 25 f ¡13p ¡ 25 g ¡35x3 + 57x2 ¡ 20x h 9x3 + 6x2 + 27x a d = 32 b x = 61 c x = 15 d x = 11 e x = 37 f x = 27 g n = 22 h x = ¡ 31 3 4 3 9 i x = ¡ 31 14

31 7

10 A = 113 11 a u = 150 b a = 4:2 c n = 10 d R = 2:1

a 93 13 b c 12 79 a u = 70 b a = 6:4 c n = 36 d R = 4

REVIEW SET 4B 1 2 5 6 7

f x=

10 a 3:1 sec b 16:2 11 a u = 20 b a = 6 c n = 12 d R = 1:35

REVIEW SET 4A 25

20 3 1 x= 4 j x = 17 5

x=

1 4

4a 5 c ¡12+x d 9¡3x g ¡36x3 + 51x2 ¡ 40x h

x=6 d x=7


1 a d 2 a c 3 a b c d 7 a b 8 a b

2, 0, 9, 7, 4 b 2, 5, 6, 9, 1 c 2, 3, 7, 9, 4 2, 7, 4, 5, 9 20, 97, 74, 80, 5 b 20, 65, 19, 47, 57 20, 39, 74, 98, 42 d 20, 79, 41, 50, 91 209, 774, 800, 583, 418 209, 965, 471, 157, 668 209, 399, 740, 988, 423 209, 791, 410, 509, 919 27, 1, 42, 58, 20, 38, 11, 35, 33, 57 27, 1, 20, 11, 33, 13, 29, 16, 32, 30 196, 272, 534, 679, 235, 91, 650, 520, 39, 474 196, 272, 235, 91, 39, 242, 178, 45, 181, 4

EXERCISE 5H

100

95

75

25

5

1 b 24, 4, 13, 5, 9

0

C

M

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499

ANSWERS 100

v 120, 132, 144, 156, 168, 180, 12, 24, 36, 48, 60, 72, 84, 96, 108 5 Start at any number and select every a 6th b 9th c 12th d 24th e 19th name 6 a i 40 ii 17

2 b 49, 24, 63, 67, 4, 13, 49, 26, 61, 27 7 b i 18 ii 1440

95

EXERCISE 5I 75

1 750 2 625 4 4000 5 200 EXERCISE 5J

25

1 a B 2 Yr

5

0

Yr

15 , 28

G

DIAGNOSTIC TEST 13 28

7 122 , Yr 683 81 12 683

8

b B 131 , 683

19 , 45

Yr 9

135 , 683

156 184 , Yr 8 1072 , Yr 1072 175 162 11 1072 , Yr 12 1072

120 , 683

G

15 29

Yr 11

9

196 , 1072

Yr 10

94 , 683

199 , 1072

9

98 , 459

Yr 10

78 , 459

125 , Yr 8 142 , Yr 859 859 134 11 859 , Yr 12 123 859

9

175 , 859

Yr 10

160 , 859

4 a Yr 7 Yr

Yr 10

14 , 29

86 88 , Yr 8 459 , Yr 459 61 48 11 459 , Yr 12 459

b Yr 7 Yr

c B

b 9, 10, 12, 11, 9, 9 c

5 a

Choose at random nine students from Year 7, ten from Year 8, 12 from Year 9, 11 from Year 10, nine from Year 11, nine from Year 12 96 , 84 , 78 , 65 , 56 , 60 439 439 439 439 439 439

b 9, 8, 7, 6, 5, 5

c Choose at random number of students from each Year 6 a B

51 , 120

G

69 120

b 17, 23

c Choose at random 17 boys and 23 girls 7 a Exotic 35 , Natives

2 5

b 36, 24

c Random samples of 36, 24 8 a

180 , 835 , 260 , 350 , 280 , 250 , 200 , 120 2475 2475 2475 2475 2475 2475 2475 2475

b 3, 13, 4, 6, 5, 4, 3, 2 c 9 a 10 a

Random samples

2500 , 4800 , 3400 , 6300 17 000 17 000 17 000 17 000 87 23 30 , Ride 150 , Car 150 , Bus 150

b 7, 14, 10, 19 Walk

10 150

b Bus 17, Ride 5, Car 6, Walk 2 EXERCISE 5K 1 a 12 b 6 c 30 d 20 2 a 12 b 6 c 8 d 30 3 a Select every 20th student b i 10, 30, 50, 70, 90, 110, 130, 150, 170, 190 ii 15, 35, 55, 75, 95, 115, 135, 155, 175, 195 iii 30, 50, 70, 90, 110, 130, 150, 170, 190, 10 iv 45, 65, 85, 105, 125, 145, 165, 185, 5, 25 v 113, 133, 153, 173, 193, 13, 33, 53, 73, 93 4 a Select every 12th name i 5, 17, 29, 41, 53, 65, 77, 89, 101, 113, 125, 137, 149, 161, 173 ii 10, 22, 34, 46, 58, 70, 82, 94, 106, 118, 130, 142, 154, 166, 178 iii 25, 37, 49, 61, 73, 85, 97, 109, 121, 133, 145, 157, 169, 1, 13 iv 50, 62, 74, 86, 98, 110, 122, 134, 146, 158, 170, 2, 14, 26, 38


1 B 2 A 7 D 8 B 9 12 D

C 10 A 11

B

REVIEW SET 5A

3 a Yr 7 Yr

26 45

G


1 Advantages: Good response rate. Questions can be clarified, if necessary. Disadvantages: Time-consuming. Expensive. 2 a discrete b discrete c discrete or categorical 3 a census b sample c census 4 a Only people at home on Saturday night are surveyed. b Only people on electoral roll are surveyed. 7 250 85 82 89 96 8 a Yr 7 500 , Yr 8 500 , Yr 9 500 , Yr 10 500 , Yr 11

75 , 500

Yr 12

73 500

b 17, 16, 18, 19, 15, 15 9 Start at any number and select every 15th name. REVIEW SET 5B 1 Advantage: Time efficient. Disadvantage: Response rate not good. 3 a sample b census 4 a Members of the local netball club b Senior students 5 a Members of another sporting club may want their own facilities improved. b Junior students 6 86, 70, 101, 190, 13 7 a V

56 , 200

S

43 , 200

T

51 , 200

M

50 200

b V6, S4, T5, M5

8 Stratified Random Sample REVIEW SET 5C 1 Advantages: Can be distributed quickly over a large area. People have time to consider their answers. Disadvantages: Poor response rate. Only people with very strong views may respond. No help to clarify questions. 2 a categorical b continuous c discrete 6 2400 7 a i 15 ii 25 8 Capture-recapture Technique

100

95

REVIEW SET 5D 1 Advantages: Cost efficient. Time efficient. Disadvantages: Only people watching that station at that time can respond. Only people with strong views may respond. 3 Advantages: Detailed. Accurate. Disadvantages: Expensive. Time-consuming. Often impractical.

75

25

5

0

C

M

Y

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500 100

95

ANSWERS

CUMULATIVE REVIEW CHAPTERS 2 - 5 1

75

25

a i v b i c i d i e i f i

4300 ii 4263 iii 4263:09 iv 4260 4263:1 vi 4000 no ii no iii yes 1:05 £ 105 ii 6:27 £ 10¡5 2:87 £ 1014 ii 5:12 £ 1017 37:5% ii 133 13 % iii 3:7% 43 ii 0:43 100

g i 6:525 m ii $114:75

5

5 6 7 11

e 224 cm2 f 65 cm2 g 45 12 m2 h 750 cm2 i 57 m2 a 28 cm2 b 71:5 cm2 c 308 m2 d 1254 m2 e 517 m2 f 5217 m2 a 5025 m2 b 964 m2 c 16 625 m2 100 m2 8 44 m2 9 $1311 10 4800 kg decrease by 1:6 cm 12 7:18 L

EXERCISE 6B 1 a B b C c D d A 2 a b

2 % iii 23 13 % h i 5:25% ii 15 15

i 84 kg j 6%

0

k i 6730 m ii 17:5 m iii 4300 mg iv 47t v 6350 mL vi 0:09 L vii 257 minutes l i 0:1 m ii 0:05 iii 6:75 to 6:85 iv 0:7% 2

3

4

5

6

$692:85 b $10 264:80 c $592:80 $471:30 e $387:50 i $465:04 ii $14:53 per hour $1733:40 h Income = $469:48, super = $48:40 i 0, 4, 8, 12 ii start at ¡16 and add 4 iii 180 b 14, 8, 2, ¡4, ¡10 c 52 d i 128, 64, 32 ii 2 e 1508 cm2

a d f g a

a 18p b 14m c ¡9a + 7b d 19a + 4b a3 2a g 5L h e 100x2 f 42 3 a 18x ¡ 45 b ¡24x ¡ 40 c ¡15 + x d 10 ¡ 3x e 8x ¡ 20 f ¡21p ¡ 25 g 48x2 ¡ 32x3 ¡ 40x h 21x3 + 15x2 + 33x a d = 21 b x = 8 c x = 18 d x=5 e x=9 f x=4 g n=5 5 h c = 94 i x = 1 j x = 12 k x = ¡ 47 5 37 5 5 = 12

l x= o x 7

m x=

31 22

c

d

e

f

g

h

1 n x = ¡ 14

a telephone interview: cheap, can hang up, face to face: instant response, expensive postal: cheap, non response b i discrete ii discrete iii discrete c line in alphabetical order and select every 12th student, or select 10 numbers randomly from 120. d 750 e A sample that is biased is not random, bias is unfair influence.

i

3 a

b

c

100

95

EXERCISE 6A 1 2

3 4

a e a e h a e a

2

2

2

2

63 cm b 168 m c 70 cm d 64 m 60:63 cm2 f 9:1 cm2 g 38:11 cm2 36 cm2 b 90 cm2 c 128 mm2 d 8 m2 10 cm2 f 24:8 cm2 g 98:58 m2 65:54 cm2 i 7:31 mm2 j 6:16 cm2 4 cm2 b 9 cm2 c 24 cm2 d 17 12 km2 18 m2 f 5 cm2 g 800 m2 h 20 cm2 110 cm2 b 96 cm2 c 44 m2 d 22 m2



d

e

f

75

25

5

0

C

M

Y

K


ANSWERS 100

4 a

8 a

b

b

501

c

95

cube

75

c

cylinder d 9 a A, 2 b B, 5 c C, 4 d e D, 3

25

5

sphere

rectangular prism e

f

E, 1

EXERCISE 6C 2 a

b

c

d

e

f

0

g

hexagonal prism

triangular-based pyramid

cone h

rectangular-based pyramid

5 a cube, cylinder, rectangular prism, hexagonal prism b

c triangular-based pyramid, rectangular-based pyramid d

e 6 a e g 7 a

sphere, cone sphere b cylinder c cylinder d cube rectangular prism f cone triangular-based pyramid h cone b c

d

e

4 a

b

c

d

e

f

5 a

b

c

d

e

f

f 100

EXERCISE 6D

g


h


1 a 6 b 6 c 5 d 5 e 6 f 6 2 a 120 cm2 b 432 cm2 c 704 cm2 d 656 cm2 e 314 cm2 f 508:8 cm2 3 a 486 cm2 b 216 cm2 c 150 cm2 d 54 cm2 e 1350 cm2 f 864 cm2 4 a 138:24 cm2 b 2:94 m2 c 1734 mm2 d 489:2454 cm2 e 0:5766 m2 f 1244:16 cm2 5 a 276 cm2 b 424 cm2 c 598 cm2 d 432 cm2 e 8970 mm2 f 8440 m2

C

M

Y

K

95

75

25

5

0


502 100

95

75

6 7 8

9 25

10 5

0

11 12 13 14

ANSWERS

a 510 cm2 b 750 cm2 c 280 m2 d 684 mm2 e 382:46 m2 a 96 cm2 b 156 cm2 c 96 cm2 a 600 m2 b 224 cm2 c 1490 cm2 d 300 cm2 e 3150 cm2 f 241:6 cm2 g 432 cm2 h 1382:4 cm2 i 907:52 cm2 a 920 cm2 b 614 cm2 c 1146:16 cm2 d 1050:18 cm2 e 11 820 cm2 f 1156:40 cm2 a 175 cm2 b 280 m2 c 47:52 cm2 d 156:8 cm2 e 244:8 cm2 f 752:83 cm2 a 1120 cm2 b $14 a 38:28 m2 b 34:4 m2 c Belinda d i 3:44 ) 4 tins ii $139:80 Frosty flakes 2352 cm2 a 313:4 m2 b 5 cans c $412:50

EXERCISE 6E 1 2 3 4 5

a d a d a A a

75 cm3 b 815:4 cm3 126 m3 b 288 cm3 e 571 cm3 b 4188:8 cm3

b 348 cm2 b 1005:3 cm3

718 mL

8

REVIEW SET 6B 1 a 12 cm2 2

b 6:24 cm2 c 54 m2 3 cylinder

4 cone

d 270 m2

5

6 a 899:84 cm2 b 256 cm2 7 a 360 cm3 b 274:6 m3 8 60 ML

115:2 cm3 c 315 cm3 e 17:48 cm3 f 125 cm3 343 cm3 c 395:8 cm3 3392:9 cm3 f 324 cm3 5089 cm3 c 40 cm3 b 3053:6 cm3

6 a 180 cm2 7 a 240 cm3

REVIEW SET 6C

c 2395:1 cm3

1 a 3:04 m2 b 67:24 cm2 d 1312:5 m2 2 3

c 90 cm2

4 rectangular pyramid

5

triangular prism

EXERCISE 6F 1 2 3 4 5

a 256 cm3 b 80 cm3 c 261:8 cm3 d 314:2 cm3 e 26:7 cm3 f 28 cm3 a 500 cm3 b 523:6 cm3 c 2309:1 cm3 a 400 m3 b 60 cm3 c 1539 cm3 d 268 cm3 e 804 cm3 f 144 cm3 a 100 cm3 b 240 cm3 c 1125 cm3 d 6807 cm3 e 302 cm3 f 9:4 m3 10 800 cm3 6 $30 7 $1080

6 a 529:74 m2 b 184 cm2 7 a 1376:1 cm3 b 297 cm3 REVIEW SET 6D

EXERCISE 6G 1 5 7 8 10 13

1 a 166:77 mm2 d 4500 m2 2

141:4 kL 2 90 kL 3 40 ML 4 393 mL a 181 m3 b 181 000 L 6 43 825 L a 14 844 L b 6715 L, A by 8129 L 4072 L 9 a 6233 L b + 1039 L 32 kL 11 24 kL 12 700 000 kL 1493 kL

4 Cylinder

DIAGNOSTIC TEST 1 8

4

b 12:74 m2 3

c 100 cm2

cube

5

A 2 C 3 C 4 A 5 B 6 D 7 C A 9 C 10 D 11 D 12 B 100

REVIEW SET 6A 1 2

201 kL

8

2

a 21 cm

2

b 30 m 3

triangular prism

2

2

c 2000 cm d 405 m rectangular prism

5

6 a 294 m2 b 673 cm2 7 a 1481:544 cm3 b 2345:8 m3 EXERCISE 7A 1 a Colour W B G R O

8 192 kL

95

75

50 white W 28%, B 24%, G 18%, R 22%, O 8%

b c d

No. of cars 14 12 9 11 4

25

5

0



C

M

Y

K


ANSWERS 100

2 a

95

75

25

5

3 a

0

4 a

Temp. 17 18 19 20 21 22 23 Colour B b G g Mark 0 1 2 3 4 5 6 7 8 9 10

30 1 7

b c d

No. of days 1 2 4 7 6 6 4

3 a

30 brown

b c

No. of students 10 11 3 6

f 3 9 7 4 2

b i 19 ii 23 c i d i 36% ii 8% 2 a

No. of goals 5 6 7 8 9 10 11 12

f 4 3 7 5 0 3 2 1

b i 14 ii 19 c i d i 12% ii 8%

r:f: 12% 36% 28% 16% 8% ii

c:f: 4 7 14 19 19 22 24 25 1 5

4 25

r:f: 16% 12% 28% 20% 0 12% 8% 4% ii

Mark 0-4 5-9 10 - 14 15 - 19 20 - 24 25 - 29 30 - 34 35 - 39

f 0 1 0 0 4 1 3 3

Mark 40 - 44 45 - 49 50 - 54 55 - 59 60 - 64 65 - 69 70 - 74 75 - 79

f 4 3 4 1 4 1 0 1

ii

Mark 0-9 10 - 19 20 - 29 30 - 39

f 1 0 5 6

Mark 40 - 49 50 - 59 60 - 69 70 - 79

f 7 5 5 1

iii

Mark 0 - 19 20 - 39 40 - 59 60 - 79

f 1 11 12 6

iv

40 2 1 70%

b c d e

c:f: 3 12 19 23 25 7 25

c:f: 9 21 39 49 55 60

r:f: 15% 20% 30% 16 23 % 10% 8 13 %

b i 18 ii 21 iii 39 iv 21 c i 20% ii 30% iii 16 23 % iv 65% v 65%

1 a i

No. of students 1 2 3 4 2 5 4 6 5 6 2

No. of children 1 2 3 4 5

f 9 12 18 10 6 5

EXERCISE 7C

EXERCISE 7B 1 a

No. of matches 48 49 50 51 52 53

503

4 25

Mark 0 - 49 50 - 99

f 19 11

c i classes overlap ii classes 10 - 19 missing iii classes are different sizes 2 a i

ii

Time (£100 h) 0-4 5-9 10 - 14 15 - 19

No. of Globes 3 12 23 2

Time (£100 h) 1-5 6 - 10 11 - 15 16 - 20

No. of Globes 5 16 18 1

b i no, classes overlap ii yes iii yes, but class size too big

100

95

75

c 40 d i 15 ii 19 iii 6 25

5

0



C

M

Y

K


504 100

3

ANSWERS

Distance 180 - 189

95

Tally

f 2

c:f: 2

r:f: 3 13 %

8

10

13 13 %

11

21

18 13 %

8

29

13

42

21 23 %

230 - 239

© © © jjj © jjjj jjjj © © © © jjjj jjjj

13 13 %

10

52

16 23 %

240 - 249

jjjj

4

56

6 23 %

250 - 259

jjjj

4

60

6 23 %

jj

© jjj © jjjj © © © j © jjjj jjjj © jjj © jjjj

190 - 199

75

200 - 209 210 - 219 220 - 229

25

5

0

b 60 d i 10 ii 42 4 5

e i 6 23 % ii a 3 b 40 c a 25 b 120 e Height 300 325 350 375 400 425

-

324 349 374 399 424 449

21 23 % iii 16 23 % 20% d 10% e no f no c no d no f c:f: r:f: 3 3 2:5% 18 21 15% 47 68 39:2% 32 100 26:7% 14 114 11:7% 6 120 5%

b Stem 4 5 6 7

6 a 40 b Stem 1 2 3 c

EXERCISE 7D

2

3

a

a

b c e a

b c e 4

Stem 3 4 5 6

Leaves 2367 257778 00179 02358

Stem 5 6 7 8 9 lowest = i 3 ii 14 f i Stem 12 13 14 15 lowest = i 1 ii i 7 ii

a Stem 0 1 2

b

lowest = 32 highest = 68

c d f

i 2 ii 0 47 e 5 10

Leaves 00014444456788 001233578899 445788 115577 00 50, highest = 90 2 iii 0 iv 0 d 54 14 ii 8 Leaves 4566688 00029 00125555689 0144688 124, highest = 158 2 iii 0 d 145 7

0122358 11 49

c Stem 25 26 27 28

5 a 25 b least = 3, greatest = e Time f Time 1-5 2 1-9 6 - 10 4 10 - 19 11 - 15 6 20 - 29 16 - 20 1 30 - 39 21 - 25 5 26 - 30 3 31 - 35 3 36 - 40 1

f i 21 ii 100 g 5% h 2:5% 1

Leaves 002377889

Stem 1 1¤ 2 2¤ 3 3¤

Leaves 1468 0335 0044 0001

9 5 8 3

36 c 5 d 4 f 5 8 7 5

Leaves 8 67777778888889999999 0001111111222222233

Leaves 8 67777778888889999999 0001111111222222233

EXERCISE 7E 1 a b c 2 a b 3 a b c

Blue 180o , Brown 90o , Grey 60o , Green 30o 1 i 12 ii 14 iii 16 iv 12 i 12 ii 6 iii 4 iv 2 Westinghouse 90o , Dishlex 100o , ASEA 20o , Bosch 70o , Hoover 80o Westinghouse 9, Dishlex 10, ASEA 2, Bosch 7, Hoover 8 Force window Force window 144o , force door 126o , open door 32o , open window 14o , other 54o i 35% ii 8 89 % d i 4200 ii 408

4 a beef

16 , 45

lamb o

4 , 15

chicken 15 , pork o

o

8 45

b beef 128 , lamb 96 , chicken 72 , pork 64o c Meat sales

100

95

pork

75

beef chicken

Leaves 24578889 012346 023446

25

lamb

5

0



C

M

Y

K


95

5 a Agriculture 18o , Manufacturing 94o , Construction 43o , Hospitality 126o , Finance 58o , Public Administration 22o b

75

Ch 9

Ch 7

Ch 10

ABC

(67 mm)

(58 mm)

(41 mm)

(28 mm)

Industry workforce Public admin.

EXERCISE 7G

Agriculture

Finance

25

ii

SBS

100

(6 mm)

505

ANSWERS

1

No. of heads

2

No. of children

Manufacturing

5

1

0

2

4

2

5

Eye colour

3

1

0

Getting to school

4

Construction

Hospitality

6 a

3

Sydney TV watching

6

b

Car

Train

5

Ch 7

Ch 9

Bus

Walk

Blue Brown Green Grey

Cycle

SBS Ch 10

Mark (/10)

ABC

TV audience nationally Ja

ra st Au

Ch 7

0 1 2 3 4 5 6 7 8 9 10

lia pa n Ko G rea er m an y O th er

SBS Ch 10

8 No EXERCISE 7H 100 80

%

EXERCISE 7F 1 a beef 35:6%, lamb 26:7%, chicken 20%, pork 17:8% b

beef

lamb

(36 mm)

(27 mm)

% protein in food

1

ABC

Ch 9

chicken

pork

(20 mm)

(18 mm)

60 40 20 0 Be

ef

Ric

e

Eg

gs

Fis

lk

h

Mi

100

2

Ch 7

(72 mm)

(53 mm)


($millions)

Fi na nc e

Ch 10 (43 mm)

ABC (26 mm)


SBS

Ch 9

Real estate sales

(16 mm) (6 mm)

Ad m in P is u tra b tio lic n

(35 mm)

H os pi ta lit y

(12 mm)

(6 mm)

3 i

(26 mm)

C on st ru ct io n

Ag ric ul tu re

(5 mm)

M an uf ac tu rin g

2

95

6

75

4 2 25

0

eb

n/F

Ja

pr

r/A

Ma

n

u y/J

Ma

C

M

t/O

p Se

Y

ec

ct

ug

l/A

Ju

v/D

5

No

0

K


506

100 50

70+

50 - 59

40 - 49

30 - 39

25 20 15 10 5

Time

0 7

4

8

9 10 11 12 1

2

Sporting injuries/year

3

4

5

Sales ($000’s) 100

150

200

7

50 40 30 20

Month

0 50

6

Computer firm’s sales

10 0

250

J

300

F

M

A

3

M

J

J

A

S

O

N

D

Overweight males 60 50

% Overweight

Beef Chicken

40 30 20 10

Duck

Age group

150

English

80

Maths

60

Science

40 20

14 12 10 8 6 4 2 0 5

10

15

20

25

ce

nt

n Vi n

Life saving week absences 7 6 5 4 3 2 1 0

CHP share prices over a fortnight

5 Price ($)

H

as

sa

dy

m Ja

W en

es

ko ei

9

Age (months) 0

0

7

-5

Baby’s weight over time Weight (kg)

Exam results

100

9 -3

9

4

50

100

30

50

20

0

10

-1

Prawns

-2

9

0

9

Lobster

-6

Cholesterol in food (mg/100g)

60

5

6

2

60

AFL Soccer Cricket Rugby League Rugby Union Netball Hockey Indoor Cricket Basketball

R

0

20 - 29

10 - 19

0-9

5

60 - 69

0 25

30

9

75

Summer temperatures in Sydney

1

150

-4

95

EXERCISE 7I

Age of vehicle occupants killed on NSW roads 200

40

3

Temperature (°C)

100

ANSWERS

8.40 8.20

100

8.00

Day

7.80 M

T

W

T

F

M

T

W

T

75

11-R1 11R-2

Mon

Tue


Wed Thurs Fri


95

F

EXERCISE 7J 1 a b c d e

i $25 000 ii $20 000 i $25 000 ii $17 500 Burwood Dec and Jan, Parramatta Dec Burwood July, Parramatta Sept Jan, Mar, May, Sept, Nov, Dec

25

5

0

C

M

Y

K


ANSWERS

2 Monthly rainfall (mm)

100

D

95

F

80 60

N

75

EXERCISE 7K

J

M

40 20

O

A

0

25

5

S

M

0

A

5 a b c 6 a d 7 a

J J

3

J

Computer firm’s monthly sales ($000’s)

80

D

F

60

N

M

40

b d

20 0

O

1 a b c e g 2 a 3 a e 4 a

A

Scale on vertical axis does not start at zero. Scale on vertical axis is irregular. No scale on vertical axis. d Use of area. Use of volume. f Use of volume. Scale on horizontal axis is irregular. A b No c No Sydney b $145 000 c + 25% d False Vertical axis does not start at zero. Percentages do not add up to 100%. Sector sizes are not in correct proportion. Yes. Scale on vertical axis is irregular To emphasise changes in rates No. The graph would show more subtle changes. Yes. Scale on vertical axis does not start at zero. per property Yes, horizontal axis does not start at zero. Vertical axis - months are not evenly spaced. Horizontal bar graph Change is not as obvious.

EXERCISE 7L 1 a

M A

Frequency

S J J No. of cloudy days

J

0

10 8 6 4 2 0

D N

O

F

b

M

Frequency

4

10 8 6 4 2 0

A

1 2 3 4 5 Number of snacks

6

20 15 10 5 0

S

M A

10

2 a

5 Sydney summer’s

Frequency

J AM 11 10

12 30

1 2

25

9

14

10 5 0

3

20 8

0 4

15 10

7 6

6

5

7 8

4 9

3 10

2 12

11

100

95

b

0

1

1 2 3 4 Number of goals

5

5

PM

12 13 Score

10 75

5 0

25

15 00 0 20 00 0 25 00 0 30 00 0 35 00 0 40 00 0

temperatures (°C)

11

J

Frequency

100

507

5

Salary ($) 0



C

M

Y

K


ANSWERS

3

10 6 4 2 0 14

15

16

17 18 Mark

25

a

Frequency

8

20 10 0

6

10

11

12 13 Score

14

0

1

2 3 Goals

4

4 2

2 a ii

0

30 25 20 15 10 5

95.5


73

85.5

75.5

65.5

0

Score

15

b ii

64.5

54.5

14.5

44.5

5

No. of patients per day

EXERCISE 7M a ii 50

20 15 10 5

0

0

Salary

3 a ii

30



40

00

0

0 00

20

15

00

0

0 00

1

25

25

0


10

34.5

Frequency

20

40

c

0

12 10 8 6 4 2 0

67 70 Mass (kg)

55.5

Frequency

b

64

45.5

61

00

0

20

30

00

4

24.5

5

19

40

35

75

50

8

30

95

b ii Cumulative frequency

Frequency

100

508

20

10

100

20 95

15 75

10 5 25

0

0 0

1

2 3 4 5 6 Number of snacks

61

64

67 70 Mass (kg)

73

5

0

C:\...\NSWGM\NSWGM_AN\508NGAN.CDR Thu Feb 17 14:00:38 2000


C

M

Y

K


509

15

5

b Jan/Feb Mar/Apr May/Jun Jul/Aug Sep/Oct Nov/Dec

60

0

14.5 24.5 34.5 44.5 54.5 64.5 Number of patients per day

d

DIAGNOSTIC TEST

2

Sep/Oct

Mar/Apr

1 0

May/Jun

Nov/Dec Jul/Aug

B 12 D 13 B

14

C 7 D 8 B 9 D

9 7899 7889 789

100

30

95

20

75

10 25

0 3.7

5 D 6

8 6 5 6

40

3.2

Leaves 4667 1344 0113 0245

50

2.7

11

4 Stem 0.¤ 1.¤ 2.¤ 3.¤

8 6 4 6

3

b 20

b 40

f 1 7 4 7 6 5 3 7 8 5 3 5

Jan/Feb

1.7

Distance 0 - 0:4 0:5 - 0:9 1:0 - 1:4 1:5 - 1:9 2:0 - 2:4 2:5 - 2:9 3:0 - 3:4 3:5 - 3:9

4

2.2

No. of students 1 1 4 6 3 3 1 1


Mark 3 4 5 6 7 8 9 10

Sales (× $1million)

Nov/Dec

10

3 2 1 0 Sep/Oct

20

4

4

Jul/Aug

c

30

Mar/Apr

40

0

2 C 3 a

1 2 3 Sales (× $1million)

50 Sales (× $1million)


70

May/Jun

45.5 55.5 65.5 75.5 85.5 95.5 Score

0

1 a

0

10

0

c ii

1

Jan/Feb

5

20

2

1.2

25

25

3

0.7

75

30

4

0.2

95

10 a

35

Ja n/ F M eb ar /A M pr ay /J Ju un l/A Se ug p/ O N ct ov /D ec

b ii Cumulative frequency

100

Sales (× $1million)

ANSWERS

5

Distance 0



C

M

Y

K


510

95

REVIEW SET 7A 1

a

0

f 8

r:f: 26 23 %

brown

8

blonde red grey

30 blonde

1, 2

f 1

c:f: 1

r:f: 3 13 %

26 23 %

10

2

3

6 23 %

9

30%

11

3

6

10%

2

6 23 %

12

2

8

6 23 %

3

10%

13

3

11

10%

d see table above e i 26 23 % ii 30%

14

6

17

20%

15

4

21

f i

13 13 %

16

2

23

6 23 %

17

2

25

6 23 %

18

4

29

13 13 %

19

1

30

3 13 %

25

Colour black

b c

Mark 9

75

5

REVIEW SET 7B

Hair colour

grey red

black

3 a 17 b 21 4 a 20% b 3 13 %

blonde brown

ii

(27 mm)

(27 mm)

6

Frequency

Hair colour brown blonde

black

5

red grey

(30 mm) (7 mm) (10 mm)

Cumulative frequency 35

6

30

5

25

iii

Hair colour

4 20 3 15 2

black

brown blonde

red

10

grey 1

5

Hair colour

iv black

0

0

brown

9

blonde

11 13 15 17 19 10 12 14 16 18

9

11 13 15 17 19 10 12 14 16 18

Mark

red

Mark

REVIEW SET 7C

grey 0

2

2

4 6 Frequency

8

10

1 a 50 b Sony

(16 mm)

Pioneer

(20 mm) Phillips

National

(24 mm)

Aiwa (10 mm)

ABM share prices

100

(30 mm)

Price ($)

7.5

95

7

c

20 Number

100

ANSWERS

6.5

75

15 10 25

5

6 M

T

W T Day of week

F

5

0 Sony

National

Pioneer

Aiwa

Phillips 0



C

M

Y

K


ANSWERS 100

2

Monthly rainfall (mm)

95

J 80 D

F

6

60 N

75

M

40

7

20 O

A

0

$5605:60 e $79 460 f $1010:62 g $87:08 $927:42 i $1713:78 0:08 b 4%, 0:04 c 2%, 0:02 _ 0 :6%, 0:066::: 0:03 b 1:5%, 0:015 c 0:75%, 0:0075 0 :25%, 0:0025

d h a d a d

8 a

annual rate 9%

six monthly rate 4:5%

quarterly rate 2 14 %

b

10%

5%

2:5%

25

S

5

M A

0

J J

3 a b c d

Vertical axis does not start at zero. Vertical axis does not start at zero. Irregular scale on vertical axis. Use of area.

REVIEW SET 7D 1 a Cumulative frequency

5 % 6 1 14 %

c

15%

7:5%

3 34 %

d

4:5%

2:25%

1:125%

e

7:6%

3:8%

1:9%

0:375% _ 0:63%

f

10:4%

5:2%

2:6%

_ 0:86%

g

15:6%

7:8%

3:9%

1:3%

h

0:78%

0:39%

0:195%

0:065%

i

3:90%

1:95%

0:975%

0:325%

40

a $160 b $630 c $6825 d $1435 e $271:40 f $558:31 g $24:73 h $40:28 i $1576:26 10 8% 11 4% 12 6:5%

30

EXERCISE 8B

50

20

1 a

years 0 1 2

10 0 10

11

12 13 Mark

14

15

Dollars

500

200

Years

c $380 d years 0 1 2

10 5 0 15

b Dollars

14

EXERCISE 8A 1 $54 2 $135 3 $825 4 a $1072 b $3354:40 c $15 120 d $3300 e $4500 f $657 g $5049 h $6480 i $34 385:12 5 a $365:94 b $1723:78 c $8382


300

0

2 a

12 13 Mark

interest 200 250 500

Interest

400

0

15

11

years 4 5 10

100

20

10

interest 0 50 100

b

b i 16 ii 34 iii 26 c d i 8 13 % Mark f ii 5% 10 5 11 11 12 18 13 15 14 8 15 3

Frequency

monthly rate 3 % 4

9

60

e

511


600 500 400 300 200 100 0

2

4

3 12

6

years

interest 0 60 120

8

years 4 5 10

10

interest 240 300 600

100

95

Interest

75

25

Years 0

c $570 d

2

4

6

8

10

5

about 4:2 years 0

C

M

Y

K


512 3

Interest ($)

100

ANSWERS

95

75

1200 1000 800 600 400 200 0

Interest

Years 0

25

5

11% 9% 8.5%

2

4

6

8


4

3 a

10

Interest ($)

2500

Start Start Start Start Start

12% 10.5%

2000

Interest

1500


8%

1000 500

Years

0 0

2

4

6

8

10


a A $1600, B $2400, C $2100 b A + $1050, B + $1550, C + $1370 c $800 d A + 6 14 years, B + 4 years 1 month, C + 4 years, 9 months a Start Start Start Start Start Start Start Start Start Start Start Start

4 a

Time of first year of second year of third year of fourth year Time of first year of second year of third year of fourth year Time of first year of second year of third year of fourth year

balance + interest $2400 + $144 $2544 + $152:64 $2696:64 + $161:80 total balance $2400 $2544 $2696:64 $2858:44 interest $144 $152:64 $161:80

b $458:44 2

a Start Start Start Start Start Start Start Start

Time of first year of second year of third year of fourth year Time of first year of second year of third year of fourth year


interest $180 $189 $198:45

Time first year second year third year fourth year fifth year Time of first year of second year of third year of fourth year of fifth year Time of first year of second year of third year of fourth year of fifth year of of of of of

balance + interest $980 + $29:40 $1009:40 + $30:28 $1039:68 + $31:19 $1070:87 + $32:13 total balance $980 $1009:40 $1039:68 $1070:87 $1103:00 interest $29:40 $30:28 $31:19 $32:13

b $123

EXERCISE 8C 1

Time first year second year third year fourth year

b $567:45

a A $900, B $1100, C $850 b A + $410, B + $495, C + $380 c $250 d A + 5:5 years, B + 4:5 years, C + 5:9 years

0

of of of of

balance + interest $3600 + $180 $3780 + $189 $3969 + $198:45 total balance $3600 $3780 $3969 $4167:45


Time first year second year third year fourth year fifth year sixth year Time Start of first year Start of second year Start of third year Start of fourth year Start of fifth year Start of sixth year Time Start of first year Start of second year Start of third year Start of fourth year Start of fifth year Start of sixth year Start Start Start Start Start Start

of of of of of of

balance + interest $7890 + 702:21 $8592:21 + $764:71 $9356:92 + $832:77 $10 189:68 + $906:88 $11 096:56 + $987:59 total balance $7890 $8592:21 $9356:92 $10 189:68 $11 096:56 $12 084:16 interest $702:21 $764:71 $832:77 $906:88 $987:59

100

95

b No, not enough money, only $12 084:16 c Needs another $905:84 5 a $6173:21 b $2173:21 6 a $7459:49 b $1006:49 7 $2732:35

75

8 $1073:67

9 a $731:97 b $670:43 c $13 245:75 d $894:66 e $12 573:94 f $2566:36 g $26 481:84 h $856:87 i $5227:82

C

M

Y

25

5

0

K


ANSWERS 100

95

75

EXERCISE 8E

10 a $5300 b $5301:59 c $5300:47 d by a maximum of $1:59 11 $4277:92 12 $1137:09 13 $5288:97 14 $1276:53 15 $1120:05 16 $6683:89 17 $2470:30 18 $663:24 19 $985:90 20 $2155:66 21 $16 427:35

1 4 7 10 11

EXERCISE 8D 25

1 a

5

0

year 1 2 3 4 5

amount $1070:00 $1144:90 $1225:04 $1310:80 $1402:55

year 6 7 8 9 10

amount $1500:73 $1605:78 $1718:19 $1838:46 $1967:15

12 16 20 21 22

Investment ($)

Compound interest 2000 1500 1000 500

Years 0 0

2

4

6

8

10

b + $1550 c + 5 12 years 2 a

year 1 2 3 4 5

amount $2186:00 $2389:30 $2611:50 $2854:37 $3119:83

year 6 7 8 9 10

amount $3409:97 $3727:10 $4073:72 $4452:58 $4866:67

Investment ($)

Compound interest 4000 3000 2000 1000

Years

0 2

4

6

8

10

b + $3000 c + $1000 d + 7:8 years 3

a b c

1000 800 600 400 200 0

Years 0

2

4

6

8

a y = 500 (1.065)x b y = 500 (1.0155)4 x

10

c y = 500 (1.004917)

The best investment is a


6:

23

$4908:75 2 $2863:95 3 $4634:78 $1316:51 5 $2987:25 6 $3845:15 $1444:75 8 $1669:95 9 $1725:30 $3411:75 a $7351:80 b $700:80 c $3694:43 d $3212:55 e $2569:95 f $6790:13 g $2768:55 8:66% 13 7:58% 14 5:11% 15 3:23% $66 17 $50 18 $42 19 $161 a $9982:83 b $13 117:88 c $3135:05 d 31:4% 5%, 18 cents, 3:23%, 12 cents, 80 cents Amcor a $980 b $1003:06 c $22 d 2:24% e loss $7:15 BHP a $995 b $1018:39 c $90 d 0:90% e Profit $21:53 BOREL a $1080 b $1105:41 c $36 d 3 13 % e loss $255:90 PMG a $550 b $563 c $60 d 10:9% e profit $47:20 RBC a $560 b $573:22 c $160 d 28:6% e profit $129:74 a + $28 b + $7:50 c + $5 d + $17

EXERCISE 8F

5000

0

513

12 x

annually


1 a $9264 b $9745 c $21 264 d $992:25 e $1189:44 f $11581:50 g $1836:54 h $8664 i $1312:50 j $1661:60 k $81 635:22 l $127 600 2 a $7060 b $5579 c $11 280 d $2331 e $1599 f $7432:40 g $3745:90 h $25 340 i $811:30 j $1059:99 k $12 975:31 l $246 000 3 a $7302 b $5264 c $5168 d $1353:60 e $2776:50 f $3529:30 g $2578:40 h $22 560 i $1428:30 j $1269:60 k $7941:12 l $292 500 4 a $9261:00 b 9743:59 c $21 258:74 d 991:88 e $1188:90 f $11 576:25 g $1835:98 h $8662:85 i $1311:75 j $1661:81 k $81 604:17 l $127 628:16 The table is only accurate to 3 decimal places. EXERCISE 8G 1 a $2:15 b $2:26 c $2:37 d $2:49 e $2:62 2 Microwave oven $548:45, Camera $912:87, Cordless phone $267:37, office chair $155:22, VCR $421:66, coffee $4:87, ice cream $4:36, potatoes $4:11 3 $3:00 4 $22:14 5 $16:38 6 a $3882:42 b $6029:29 c $22 581:59 7 a $4057:46 b $4703:71 c $6321:39 8 $1794:78 9 exceeded by $361:21 10 exceeded by $81 631:82 11 did not exceed by $438:66 12 2:53%

C

M

Y

K

100

95

75

25

5

0


514 100

13 15

ANSWERS

7:24% 14 8:51% 8:89%, 14:87%, 4:56%

5

1000

Dollars

DIAGNOSTIC TEST 75

25

1 B 2 D 3 B 4 A 5 A 6 D 7 B 8 C 9 D 10 A 11 B 12 A 13 C 14 C 15 A 16 D 17 B 18 D 19 C 20 A

0

Dollars

600 400 200 0 0

a + $450

1 12

6

8

10

years

6 a $2811:53 b $353:53 7 $2968:64 8 $4250:55 9 $26 252:78 10 $5663:80 11 $121:50 12 $5974:80 13 $5:59

2

4

6

Years

2000

8

1500

Interest

10

b + 3 12 years

REVIEW SET 8B $1171:50 a 0:13 b 6:5%, 0:065 c 3:25%, 0:0325 _ d 1:083%, 0:010 83_ $633:10 4 5:5%

1000 500

Years

0 0

2

a + $1450 b 6 8 11 14

4

5 12

6

8

10

years

a $11 019:67 b $4019:67 7 $3612:92 $190 9 $1419:79 10 $8000:28 $18 015:15 12 7:07% 13 $5060 $3:65

EXERCISE 9B

600

Interest

400 200

Years

0 0

a + $180 6 a $4260:14 8 $5819:42 9 11 3:67% 12 14 $32:89

2

4

6

8

10

b + 1:8 years b $680:14 7 $2345:41 $851:89 10 $2964:04 $97:50 13 $7357

1 a c d f h i 2 a f 3 a b c d

REVIEW SET 8C

3

4

a + $490 b +

6 a $3577:56 b $577:56 7 $505:14 8 $3990:34 9 $951:29 10 $9007:58 11 $2795:65 12 $58:50 13 $4936:32 14 $3:07

1 2

2

1 $343 2 a 0:05 b 2 12 %, 0:025 c 1:25%, 0:0125 _ d 0:416%, 0:004 16_ 3 8:2% 4 $354:84 5

Interest

3 5

Years

REVIEW SET 8D

800

1 2

400 0

Dollars

3 5

$178:50 a 0:1 b 5%, 0:05 c 2:5%, 0:025 _ d 0:83%, 0:0083_ $448:80 4 6%

Dollars

0

1 2

600 200

REVIEW SET 8A 5

Interest

800

95

$816 a 0:06 b 3%, 0:03 c 1:5%, 0:015 d 12 %, 0:005 $565:73 4 5:6%

e

fboy, girlg b fpoint up, point downg flands on an end, lands on curved faceg fred, blue, whiteg e fred, green, amberg fa, e, i, o, ug g fI, N, S, G, H, Tg fMon, Tues, Wed, Thurs, Fri, Sat, Sung fwin, lose, drawg j discuss with class yes b don’t know c no d no e no no g no h yes i don’t know False. The prevailing weather pattern determines the chance of rain. False. Not all swimmers are of equal ability. False. The letter z is more common (or less common) in different cultures. False. The candidates do not necessarily have the same qualifications, financial backing, etc. False. Not all students may be old enough, have access to a car, etc.

EXERCISE 9C


95

75

25

1 fTT, TF, FT, FFg 2 fHH, HT, TH, TTg , Yes 3 fBBB, BBG, BGB, BGG, GBB, GBG, GGB, GGGg 4 fH1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6g


100

C

M

Y

K

5

0


515

ANSWERS 100

95

75

25

5

5 fRR, RG, RA, GR, GG, GA, AR, AG, AAg 1 2 3 4 5 6 6 1 (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) 2 (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) 3 (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) 4 (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) 5 (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) 6 (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) 1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

1 2 3 4 5 6

1 0 1 2 3 4 5

2 1 0 1 2 3 4

3 2 1 0 1 2 3

4 3 2 1 0 1 2

7

0

8

5 6 7 8 9 10 11 5 4 3 2 1 0 1

EXERCISE 9E

2 3

4

6 7 8 9 10 11 12

5

6 a 26:9% b 25:0% c 3:0% 3 or 6% c 21 or 42% 7 a 0 b 50 50 d 8 a 9

6 5 4 3 2 1 0

10 fJohn Paul George, John George Paul, Paul John George, Paul George John, George John Paul, George Paul Johng

339 1000

3 a 4 a

6 a

g

14 a f55, 57, 59, 75, 77, 79, 95, 97, 99g b f57, 59, 75, 79, 95, 97g

7 a

15

8 a

g h M V M V

9 a g


98 555 100 000


iii 98:555%

b b b b

1 2 11 23 1 3 1 8

1 3

c c

1 6 1 4

c c

1 3

d 12 23

e 0 f 1

d 1 5 6 1 2

d d

e 0 1 2

e

f

3 8

g

1 26

f

1 2

1 4

1 52 1 2 1 200 1 4 1 6 4 9 2 9 1 20 3 10

1 13

b

1 4

c

1 26

d

e

0 i 1

h

1 100

b b

1 6

h

0

b

1 3

c

c

1 12

c

2 9

1 40

d

5 12 7 9

d

1 20

d

e 7 12

e 5 9

e

f

1 10

f 2 3

f 1 11 12

g

1 3

i 0 j 1 b h

19 20 7 10

c

1 2

d

i 0 j

1 2 19 20

9 20 9 20

e k

f l

11 20 11 20

10 a i P(7) = 0 ii P(odd or even) = 1 b i P(yellow) = 0 ii P(red or white) = 1 c P(17) = 0 ii P(red or black) = 1

EXERCISE 9D 120 2 12 3 36 4 24 a 4 b 8 c 16 d 32 a 4 b 8 c 16 d 32 a 9 b 27 c 81 8 24 9 1000 1024 11 1000 a 17 576 000 b 6 760 000 c 456 976 10 000 000 14 a 64 b 729 c 531 441 a 6 b 120 16 24 17 12 144 18 840

38 50

d

h 0 i 1

13 a fRR, RB, RG, BR, BB, BG, GR, GB, GGg b fRB, RG, BR, BG, GR, GBg

3rd set won by

12 50

c

or 33:9%

1 6 1 23 1 2 1 8

2 a

12 fCMR, CRM, MCR, MRC, RCM, RMCg

1 5 6 7 10 12 13 15

1 100

b

EXERCISE 9F

5 a

2nd set won by M V V M M V

or 78%

b 98:956% c i M 95:4%, F 97:6% ii M 85:2%, F 91:8% iii M 37:5%, F 59:0% d i 92 709 ii 53 952 iii 58:2% iv 58:2% e i 95 945 ii 72 656 iii 75:7% iv 75:7%

1 a

f1:8 AO, 1:8 AB, 1:8 AR, 1:8 MO, 1:8 MB, 1:8 MR, 2:2 AO, 2:2 AB, 2:2 AR, 2:2 MO, 2:2 MB, 2:2 MRg

1 st set won by M M M V V V

39 50 1 50

10 a i 98 555 ii

9 fbax, bay, bex, bey, cax, cay, cex, cey, dax, day, dex, deyg

11

76 ii 124 200 200 b i 19 or 38% ii 31 or 62% 50 50 a 25 or 40% b 35 or 60% 77 , Japan 153 , Germany 500 , a Australia 169 500 500 62 39 Korea 500 , Other 500 31 b 169 or 33:8% c 250 or 12:4% 500 2289 or 24:4% a 5000 or 45:78% b 11 45 107 or 2% 5284

1 a i

11

a wrong, 0 6 P(E) 6 1 b wrong, 0 6 P(E) 6 1 c possible

12 a P(3) =

1 10

b P(odd) =

1 2

f P(number < 9) =

9 10

95

75

c P(10) = 0

d P(number < 10) = 1 e P(number divisible by 3) =

100

25

3 10

g impossible

h impossible

5

0

C

M

Y

K


516 100

95

EXERCISE 9H

6 a

1

a fH, Tg b i fTg ii fHg

2

a f1, 2, 3, 4, 5g b i fnot spinning a 3g = f1, 2, 4, 5g ii fnot spinning an odd no.g = fspinning an even no.g = f2, 4g iii fnot spinning a no. > 3g = fspinning a no. 6 3g = f1, 2, 3g iv fspinning a 5g = f5g

75

25

5

0

ANSWERS

3 4

5

fHH, HT, TH, TTg a fHT, TH, TTg b fHH, TH, TTg c fHH, TTg d fHT, TH, HHg e fTTg f fHHg a

7

a

9 10

b i

a B b D c A d E e F g C a fnot selecting a diamondg = fselecting a heart, club or spadeg b fnot selecting a D or Hg = fselecting a C or Sg c fnot selecting a D, H or Cg = fselecting a Sg d fnot selecting a red cardg = fselecting a black cardg e fnot selecting an Aceg f fselecting a kingg

6

8

1 2 3 4 5 6

1 4 1 8

b 0:33 c

a

b

99 100

f

91 100

g

2 25

l

24 25

m

7 100

a

51 52

b

25 26

1 8

a i

ii

3 8

7 8

b i no tails ii

b i vi

9 10

j

1 25

k

25 26

iv

11 13

e

b i

2

3 8

1 8

v

c i 11

c i no heads ii

7 8

1 10

1 2

iii

1 12

ii

1 4

iii

1 5

ii 1 4

ii

1 4

1 (1, 1) (2, 1) (3, 1) (4, 1)

a 1 2 3 4 b i

1 16

c i

3 8

2 (1, 2) (2, 2) (3, 2) (4, 2) 1 4

ii ii

1 4

iii

iii

iv

1 2

3 (1, 3) (2, 3) (3, 3) (4, 3) 1 8

iv

1 4

v

1 9

vi

vii

1 6

iii

1 3

iv

2 3

v

1 3

ii

2 3

iii

1 3

iv

2 9

v

4 9

vi

1 3

1 3

v

2 3

vi

3 8

1 16

ii

1 4

viii ii

7 16

1 8

iii

1 4

ix 3 4

iii

1 4

iv

3 4

v

1 4

vi

1 2

a fBBB, BBG, BGB, BGG, GBB, GBG, GGB, GGGg 1 8 1 2

1 8

ii

iii

3 8

iv

3 8

1 8

v

vi

7 8

1 8

viii

v

ii

1 3

iii

1 3

iv

1 2

1 2

v

vi

1 6

1 2 3 5

v

a certain b discuss a f1, 2, 3, 4, 5, 6g b fBB, BG, GB, GGg 21 4 0:18 1 1 3 a 52 b 13 c 14 d 13 e 12

7 1 8 a 1 4

100

REVIEW SET 9A

1 4

4 (1, 4) (2, 4) (3, 4) (4, 4) 1 4

vii

1 3 2 3

2 Unlikely that both have same ability, qualifications etc. 3 fTTT, TTF, TFT, TFF, FTT, FTF, FFT, FFFg 4 B 5 A 6 B 7 C 8 B 9 D 10 C

vi

1 4

7 16

H:\...\NSWGM\NSWGM_AN\516NGAN.CDR Thu Feb 17 11:36:19 2000

6 6) 6) 6) 6) 6) 6)

DIAGNOSTIC TEST

3 10

iii

(1, (2, (3, (4, (5, (6,

11 36

ii

1 12 1 4

ii

1 16

b i

1 6

iv

1 4

iv

5 5) 5) 5) 5) 5) 5)

12 a fKMH, KHM, MKH, MHK, HKM, HMKg

1 4

iv

(1, (2, (3, (4, (5, (6,

5 9

vii

a fBB, BG, GB, GGg b i

5

1 12

1 9

b i

a fH1, H2, H3, H4, H5, T1, T2, T3, T4, T5g b

4

1 4

ii

1 12 1 2

b i

a fH1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6g b i

3

1 4

1 18

iii

4 4) 4) 4) 4) 4) 4)

10 a fHH, HD, HS, HC, DH, DD, DS, DC, SH, SD, SS, SC, CH, CD, CS, CCg

3 4

f

a fTT, TF, FT, FFg b i

(1, (2, (3, (4, (5, (6,

8 a fRR, RB, RW, BR, BB, BW, WR, WB, WWg

EXERCISE 9I 1

3 3) 3) 3) 3) 3) 3)

7 a fPSA, PSF, PFA, PFF, PCA, PCF, BSA, BSF, BFA, BFF, BCA, BCFg

vii d

iii

(1, (2, (3, (4, (5, (6,

9 a fRB, RW, BR, BW, WR, WBg

9 100

e

1 10

i

12 13

c

1 6 5 18

c 7 d i

93 100

n

1 8

9 10

d

23 25

2 2) 2) 2) 2) 2) 2)

b i 0 ii 0 iii 1 iv vii 23

1 10

h

(1, (2, (3, (4, (5, (6,

ii

1 18

vii

52% c

1 36

vii

b 0:7 c 85%

1 100

1 (1, 1) (2, 1) (3, 1) (4, 1) (5, 1) (6, 1)

b

1 4

c

1 12

75

1 6

d

25

REVIEW SET 9B 1 fPMY, PYM, MPY, MYP, YMP, YPMg 2 1000 3


1 12

95

5

1 4

0

C

M

Y

K


ANSWERS 100

95

4 a 6 a

1 2 3 4 5 6

75

25

b i

5

0

1 8

1 8

b

1 (1, 1) (2, 1) (3, 1) (4, 1) (5, 1) (6, 1) 1 36

1 2

c (1, (2, (3, (4, (5, (6,

2 2) 2) 2) 2) 2) 2)

1 6

ii

0 e 1 5 65%

d 3 3) 3) 3) 3) 3) 3)

(1, (2, (3, (4, (5, (6,

(1, (2, (3, (4, (5, (6,

1 6

iii

4 4) 4) 4) 4) 4) 4)

(1, (2, (3, (4, (5, (6,

5 5) 5) 5) 5) 5) 5)

(1, (2, (3, (4, (5, (6,

6 6) 6) 6) 6) 6) 6)

iv 0

d 8 a d 9 a d 10 a d 11 a d 12 a d

517

$2909 refund $50 433 b $47 821 c $10 726:30 $740:70 refund $40 215 b $35 312 c $6973:60 $3882:40 $88 985 b $75 101 c $21 927:47 $1060:53 $12 934 b $11 584 c $974:50 $949:28 e $25:22 refund $34 277 b $31 423 c $9831:75 $5806:90 e $4025:75 refund

REVIEW SET 9C 1 a fRR, RB, RG, BR, BB, BG, GR, GB, GGg b fRB, RG, BR, BG, GR, GBg 2 a f 3 a

1 100 9 100

b g

11 100

9 100 99 100

c

d

h 0 i

17 50

b

1 10

9 10

e

j

1 2

4 5

1 100

c Don’t know - there were no

families in the survey with more than 6 children. 4 1 2 3 4 5 1 (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) 2 (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) 3 (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) 4 (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) 5 (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) b i

1 25

ii

2 25

1 25

iii

4 25

iv

v 0

REVIEW SET 9D 1 a 2 a 3 a f

1 13 1 4 1 200

b b b

191 200

g

8 13 1 c 3 199 200 11 40

c

3 13 5 12

d

1 100

c h

d

29 40

1 13 7 12

d i

49 50

e 0 e 99 100

e

9 200

3 4

4 a fAA, AB, AC, AD, BA, BB, BC, BD, CA, CB, CC, CD, DA, DB, DC, DDg b i

1 16

ii

1 16

iii

1 16

iv

9 16

v

7 16

EXERCISE 10A $46 007:50 2 $39 731:50 3 $29 156 $55 805 $45 830, $36 932, $72 398, $13 303, $26 578 a $546:29 b $306:68 c $614:31 d $365:54 e $853:41 f $0 7 a $43:20 b $190:20 c $0 d $151

1 4 5 6

EXERCISE 10B 1 a $15 126:82 b $5653:30 c $357 d $0 2 $108 360, $93 320, $84 390, $88 620, $84 390, $54 310, $33 630 3 $9880, $8980, $8230, $7180, $6580, $6055, $5830, $5800, $5200 4 a $34 596 b $6758:80 5 a $10 630 b $2044 refund 6 a $16 900 b $1853:00 7 a $32 260 b $31 170 c $5731

H:\...\NSWGM\NSWGM_AN\517NGAN.CDR Thu Feb 17 11:37:20 2000

1 a e 2 a 3 a 4 a 5 a 6 a e 7 a e 8 a e 9 a e 10 a e h

$4:60 b $15 c 0:84 cents d $45:90 $97:55 f $1:10 g $198 h $1:54 $65:40 b $719:40 $874:28 b $9617:08 $38:90 b $427:90 $63:40 b $697:40 $41:64 b $24 c $3:14 d $19:91 $1:80 f $173:09 g $270:91 h $1:35 $416:36 b $240 c $31:42 d $199:09 $18 f $1730:91 g $2709:09 h $13:55 $98:88 b $100:80 c $96 d $72 $38:40 $2:63 b $2:10 c $3:15 d $0:98 cents $3:09 $380:17 b $425:93 c $400 d $381:43 $383:33 f $391:49 g $403:51 $431:92

EXERCISE 10D

2 3

j

EXERCISE 10C


1 a e 2 a e 3 a e 4 a e

$11 000 b $18 000 c $23 000 d $2500 $12 000 f $17 000 g $23 000 h $37 000 $7000 b $14 000 c $18 000 d $0 $12 000 f $23 000 g $30 500 h $39 500 $5000 b $11 000 c $14 000 d $3000 $12 000 f $28 000 g $38 000 h $49 000 $6800 b $12 000 c $13 800 d $800 $9400 f $14 000 g $29 700 h $39 000

5 a i $5800 ii $10 400 iii $12 000 iv $700 v $10 700 vi $16 000 vii $33 900 viii $44 600 b i $9000 ii $16 000 iii $18 500 iv $1100 v $7000 vi $10 500 vii $22 100 viii $29 100 c i $3800 ii $6700 iii $7700 iv $450 v $16 700 vi $25 000 vii $52 800 viii $69 400 d i $13 200 ii $23 300 iii $27 100 iv $1575 v $4800 vi $7100 vii $15 100 viii $19 800 6 a $4200 b $9900 c $19 800 d $260 e $12 800 f $15 700 g $29 200 h $36 900 7 a i $3500 ii $6700 iii $12 200 iv $1300 v $11900 vi $17 100

C

M

Y

K

100

95

75

25

5

0


518 100

95

75

25

5

8 0

9

ANSWERS

vii $41 200 viii $55 000 b i $2500 ii $5900 iii $11 800 iv $150 v $15 500 vi $20 300 vii $42 900 viii $55 800 c i $0 ii $1900 iii $13 900 iv $0 v $28 200 vi $30 600 vii $41 700 viii $48 000 d i $2500 ii $5300 iii $10 000 iv $630 v $15 000 vi $21 000 vii $49 000 viii $65 000 a $1100 b $5400 c $13 800 d $300 e $22 100 f $29 200 g $46 300 h $76 800 a i $1400 ii $3600 iii $8000 iv $700 v $38 000 vi $45 800 vii $61 300 viii $92 300 b i $3300 ii $9500 iii $24 140 iv $1600 v $13 200 vi $19 300 vii $32 600 viii $53 600

a $1100 b $4000 c $12 000 d $400 e $25 900 f $33 500 g $51 700 h $62 800

11

REVIEW SET 10D 1 $28 360 2 $1009:71 3 $59:20 4 a $13 577:86 b $5012:50 c $339:66 d $0 5 a $62 552 b $54 002 c $13 060:84 d $2728:84 payable 6 a $325:85 b $3584:35 7 a $52 b $572 8 a $904:82 b $77:73 c $3:59 d $36:27 9 $46:56 10 $596:49 11 a $3900 b $9900 c $20 400 d $0 e $13 600 f $16 400 g $29 100 h $36 400 CUMULATIVE REVIEW CHAPTERS 6 - 10 1 a i cube ii square pyramid iii cylinder b i 23:175 cm2 ii 9:46 m2 iii 121 m2 c iv 421:64 m2

DIAGNOSTIC TEST

REVIEW SET 10A $35 022 2 $525:83 3 $86:60 a $14 474:56 b $5757:70 c $71:74 d $0 a $52 231 b $48 533 c $10 939:90 d $1759:10 refund 6 a $1254:63 b $13 800:93 7 a $125 b $1375 8 a $112:27 b $32:64 c $3:83 d $24:18 9 $75:08 10 $563:85 11 a $7800 b $13 700 c $16 000 d $900 e $8100 f $12 200 g $25 700 h $33 800

1 4 5

REVIEW SET 10B $71 782 2 $577:21 3 $167:20 a $18 473:38 b $6489:70 c $1082:73 d $0 a $46 556 b $42 995 c $9278:50 d $553:50 payable 6 a $1836 b $20 196 7 a $84:90 b $933:90 8 a $323:27 b $78:09 c $2:51 d $18:09 9 $53:40 10 $786:78 11 a $4300 b $9000 c $17 200 d $0 e $11 700 f $15 100 g $31 400 h $40 700

d i 168 cm2 ii 640 cm2 e i 40 cm3 ii 70 686 cm3 iv 2400 m3 f 120 kL 2 a

Colour

number

r.f.

b

30

10

10 30

c

black

d

B

e i

iii 140 cm3

b

6

6 30

w

9

9 30

r

1

1 30

g

4

4 30

1 3 3 10

i ii

Hair colour

Grey

1 4 5

Red Black

Blonde Brown

ii

Hair colour (5 cm)

(3 cm)

(4.5 cm)

(0.5 cm)

1 D 2 C 3 B 4 A 5 D 6 B 7 A 8 D 9 B 10 A 11 A 12 D 13 B 14 A 15 C 16 A

100

(2 cm) 95

REVIEW SET 10C 1 4 5 6 7 8 9

$49 561:50 2 $395:92 3 a $20 853:52 b $7988:20 a $57 893 b $52 913 c d $504:46 payable a $2258:60 b $24 844:60 a $42:50 b $467:50 a $681:82 b $115:27 c $174:93 10 $219:01


$17:60 c $885:87 d $0 $12 603:46

Black

Brown

Blonde

Red Grey

75

iii Hair colour 25

5

$2:36 d $36:45


Black

Brown

Blonde

Red

Grey 0

C

M

Y

K


ANSWERS

iv

k

Hair colour

95

75

44% tax $7000 threshold

Black

20

Brown

15

Tax ($)

100

Blonde Red Grey

25

3

4

6

8

4.00 2.00

20

30

40

50

EXERCISE 11A

Day Thu

Wed

Tue

Mon

1 a 6:4 b 5:2 c 14 d 28:9 e 3:5 f 22:5 g 55:4 h 5:1 i 243:5 j 103:4

8%

c $381:93 d e

Simple interest at 7.2%

800

Interest ($)

10

i + $4300 ii + $9000 iii + $17 000 iv $0 v + $12 000 vi + $15 000 vii + $31 000 viii + $41 000

6.00

4 a $1657:50 b i 0:11 ii 5 12 %, 0:055 iii 2:75%, 0:0275 iv 11 %, 0:009 16_ 12

600 400 200

Years

0 0

1

2

3

4

5

6

7

8

9 10

about $250 ii about 4:2 years $8659:98 ii $1079:98 g $2724:24 $7531:52 i $749:36 j $5158:28 + 4:1% l $112:50 m $25:81 1 ii depends on season fGGG, GGB, GBG, GBB, BGG, BGB, BBG, BBBg 23 c 28 d 100

i f i h k 5 a i b

f i

Taxable income 0

10

Fri

Price ($)

2

0.00

e i

5

AMB share prices

8.00 0

10

0 0

5

519

1 52 1 12

ii ii

1 13 1 4

iii iii

1 4 1 12

iv iv

3 13 1 6

v

1 2

6 a $48 570 b $577:21 c $167:20 d i $16 793:38 ii $7689:70 iii $2272:73 iv 0 e i $47 556 ii $43 995 iii $9578:50 iv balance payable of $853:50 f i $1436 ii $15 796 g i $72:50 ii $797:50 h $295:82 i $98:40 j $373:55

EXERCISE 11C 1 a iii 4:9 b iii 13:65 c iii 52:4 d iii 9:92 e iii 25:9 EXERCISE 11D 1 a iii 66:9 b iii 72:8 c iii 31:7 2 a ii 165:9 b ii 4:8 3 a 46:2 b i ii Class Class f 0-9 0-4 0 10 - 19 5-9 1 20 - 29 10 - 14 0 30 - 39 15 - 19 1 40 - 49 20 - 24 2 50 - 59 25 - 29 1 60 - 69 30 - 34 2 70 - 79 35 - 39 2 iii Class 40 - 44 4 0 - 19 45 - 49 3 20 - 39 50 - 54 4 40 - 59 55 - 59 1 60 - 79 60 - 64 6 65 - 69 0 iv Class 70 - 74 2 0 - 39 75 - 79 1 40 - 79

f 1 1 3 4 7 5 6 3 f 2 7 12 9 f 9 21

c i 46:7 ii 47:5 iii 48:2 iv 47:5 d The smaller the class interval, the closer the approximation to the real mean. EXERCISE 11E 1 a g 2 a g

25

5

1 a 6 b 5 c 13 d 29 e 4 f 21:5


95

75

9 b 3 c 13 d 27 e 4 f 20 57 h 4 i 243 j 104 5 b 13 c 52 d 9 e 26 f 66-68 61-70 h 30-39

EXERCISE 11F


100

C

M

Y

K

0


520 100

2 95

75

ANSWERS

g 56 h 4 i 243:5 j 104 a 8 b 22 c 16:5 d 50 e 6 f 13:5 g 50 h 19 i 2:5 j 37

c i

Class centre 14:5 24:5 34:5 44:5 54:5 64:5

EXERCISE 11G 1

a 5 b 13 c 52 d 35:5 e 10 f 26

EXERCISE 11H a 14 b 29 c 41 d 23

2

a i

Class centre 61 64 67 70 73

0

c:f: 4 11 20 26 30

ii iii 67 Cumulative frequency

5

1


ii

60 50 40 30 20 10 0 14.5 24.5 34.5 44.5 54.5 64.5

30 25 20

d i

15

Class centre 157 162 167 172 177

10 5 0 61

b i

70

35

Class centre 45:5 55:5 65:5 75:5 85:5 95:5

64

67

c:f: 5 5 17 27 36 40

70

73

iii 73

iii 167

c:f: 19 40 64 83 100

ii


25

iii 32

c:f: 13 28 44 54 58 60

ii

120 100 80 60 40 20 0


157

162

167

172

177

50 40

e i

30 20 10 0 45.5 55.5 65.5 75.5 85.5 95.5

Class centre 1 4 7 10 13

iii 6:5

c:f: 13 44 64 75 80

100

95

75

25

5

0



C

M

Y

K


ANSWERS

ii

100


95

75

25

5

10 a i

No. mistakes 0 1 2 3 4 5 6 7 8 9

f 1 2 2 5 3 2 0 2 3 0

b i

No. mistakes 0 1 2 3 4 5 6 7 8 9

f 1 3 2 1 2 2 2 2 3 2

c i

No. mistakes 0 1 2 3 4 5 6 7 8 9

f 4 4 1 0 1 1 2 1 2 4

d i

No. mistakes 0 1 2 3 4 5 6 7 8 9

f 0 3 4 1 0 2 2 1 5 2

100 80 60 40 20 0 1

4

7

10

13

0

EXERCISE 11I 1 a b c d

i i i i

70:9 41:7 128:0 20:2

ii ii ii ii

75 iii 72:5 26, 58 iii 45 108 iii 120:5 26 iii 21

EXERCISE 11J 1 a i 7 ii no mode iii 35 b i yes ii no iii no, not a central value c i Mean = 6:9, Median = 7 are both good measures ii Mean = 5, Median = 4:5 are both good measures iii Mean = 29, Median = 26 are both good measures 2 a i 9:2 ii 29 b i Yes ii No, affected by outlier 110 c i Median = 9 is good measure, Mode = 8 is not a central value. ii Median = 9 is satisfactory, Mode = 8 is not central. 3 a 5 b No - it is not a central value c Mode = 5 is not central, Mean = 4 is satisfactory. 4 a i 6:25 ii 7 iii 6:5 b Mode 5 a $966 000 b $32 200 c i 25 ii 5 d $27 500 e $25 000 f i Mean ii Mode g Median - half employees earn more and half earn less. 6 a Mean = 23:8, Mode = 29, Median = 23:5 b Mean and median are central and typical. Mode is not central but is important because more people had to wait this length of time than any other. 7 a Mean = 4:1, Mode = 6, Median = 4:5 b Mode 8 a Mean = $320 000 b The mean has been affected by the high price $480 000. No mode. Median = $280 000 is the most appropriate. 9 Mode

ii

521

Mean = 4:1 Mode = 3 Median = 3:5

ii Mean = 4:8 Mode = 1, 8 Median = 5

ii Mean = 4:3 Mode = 0, 1, 9 Median = 4:5

ii Mean = 5:1 Mode = 8 Median = 5:5

100

95

75

25

5

0



C

M

Y

K


522 100

ANSWERS

e i

No. mistakes 0 1 2 3 4 5 6 7 8 9

95

75

25

5

0

f

Mean Mode Median

4:1 3 3:5

f 4 0 0 2 1 2 1 2 4 4 4:8 1, 8 5

ii Mean = 5:4 Mode = 8, 9 Median = 6:5

15

2

10

2 3 4 5 6

5:1 8 5:5

40

148

140

145

10

45

153

150

18

10

155 28

15

20

73

70

48 iv 60 v

25

78

80

78

80

30 83

84

82

5

7

9

e 3

2

1

3

4

5

8

4

6

11

7

8

9 19

14

a 0

2

4

6

8

10

21 22

b

24

12

14

27

16

18

20

31 33

35

30

22 15

17 89

19

21 23

91

25

29

27

95

100

103

c 88

90

92

94

18

96

98

22

100

102

27

104 30 31

d 75

15

63


17

19

23

21

4 5

1

25

29

27

7

31

15

e

73

30 iv 59 v 69 43 iv 55 v

76

3

1

35 iv 54 v 65 42 iv 61 v

74

72

0

ii 35 iii vii 25 ii 40 iii vii 20 ii 36 iii vii 28 ii 46 iii vii 13 ii 31 iii vii 24


135

5 71

D1 = 26, D4 = 44, D9 = 74 D1 = 12, D4 = 24, D9 = 35 D1 = 30, D4 = 36, D9 = 51 D1 = 58, D4 = 98, D9 = 174 83 50 75 45 78 45 76 56 74 39

35

g 8

a 7 b 2 c 5 d 6 e 5 f 5 g 19 h 11 i 9 j 21:5 k 25 l 6:5 m 20 n 9 o 3 a 21 ¡ 7 = 14 b 35 ¡ 23 = 12 c 53 ¡ 31 = 22 d 28 ¡ 19 = 9 a 1 b 2 c 1 a ii IQR + 19 b ii IQR + 20 c IQR + 6 The marks in Lara’s class were more closely clustered around the median. a Tom b Matthew c Matthew

a i vi b i vi c i vi d i vi e i vi

30

7 0

EXERCISE 11N 1

25

d

a 8 b 7 c 159 d 4 e 7 f 7 h 5 a 4 b 4 c 5 3 No

a b c d

40

139

130

c

5:4 8, 9 6:5

EXERCISE 11M 1

20

6

4:3 0, 1, 9 4:5

32

141 125

EXERCISE 11L 1

15

b

EXERCISE 11K

2

28

130

g Mode h Mean

1

23

a

100

0

2

4

6

8

10

12

14

16 95

EXERCISE 11O 1 a e 2 a d 3 a c 4 a b

6, 1:4 b 6, 2:8 c 6, 4:2 d 28, 2:6 90, 1:4 f 250, 5:6 g 6, 2:5 h 5, 0 3:56, 1:10 b 16:9, 1:11 c 1:79, 1:16 122:96, 1:37 66:85, 3:35 b 72:83, 12:89 31:67, 13:30 d 165:93, 5:57 4:5, 1:26 i 9:5, 1:26 ii 6:5, 1:26

C

M

Y

K

75

25

5

0


ANSWERS 100

95

75

iii (4:5 + 10), 1:26 c i 22:5, 6:29 ii 9, 2:52 iii (4:5 £ 10), (1:26 £ 10) 5 a 20, 4 b 25, 4 c 51, 12 d 85, 20 e 12, 4 f 3:4, 0:8 EXERCISE 11P

0

DIAGNOSTIC TEST 1 A 2 D 3 9:3_ 4 C 5 C 6 D 7 C 8 C 9 B 10 D 11 D3 = 61, D7 = 67 12 B 13 ¾n = 1:13, ¾n¡1 = 1:15 REVIEW SET 11A _ Median = 8, Mode = 6, 1 Mean = 7:6, Range = 4 2 a Class Centre c:f: 3 12 8 20 13 38 18 69 23 80 b

1 a 50 - 59 b 57:9 2 a Stem Leaves 4 89 5 000146689 6 02245 7 1349

30 50 58:5

b c d

3 6

7.5

9.5

5

11 12

10

15

4 a 60 Cumulative frequency

5

Mean and median are appropriate measures. c i Mean = 2:5, Mode = 1, Median = 1 Mode and median are not central. Mean is appropriate measure 3 a 69 b 90 c 58 d 32 e 54 4 a Sample S.D. (¾n¡1 ) b 0:15 5 C REVIEW SET 11C

1 1:51 2 2, 1:12 3 29:8, 3:79


25

523

80 60

50 40 30 20 10 0

40

34.5

44.5

54.5

64.5

74.5

84.5

20

b i 47 ii 66 iii 19 iv 57 v 84 5 a 17:7, 4:5 b 63:5, 22:5

0 3

8

13

18

23

REVIEW SET 11D c i Median = 16 ii D4 = 14 3 a b c d

i i i i

1

9, 11, 12 ii 3 9, 11, 12:5 ii 3:5 9, 11:5, 12 ii 3 9:5, 11:5, 13:5 ii 4

4

20

25

30 32

25

30

36

35

42

40

45

5 20:6, 2:5 REVIEW SET 11B 1 Mean = 86:65, Mode = 87, Median = 87, Range = 3 2 a i Mean = 24, Mode = 5, Median = 5 ii Mean has been affected by the outlier 100. Mode and median are appropriate measures. b i Mean = 5:1, Mode = 4, Median = 5 ii Mode is not a central score.



a b c d e f g h i j k l m

Mean 3:9 4:1 3:7 3:9 4 13:5 6:9 8:9 103:9 1:9 7:8 11:7 390

Mode 5 5 2, 5 5 5 5 8 10 105 3 10 15 500

Median 4 4:5 3:5 4 4:5 4:5 7 9 104 2 8 12 400

Range 4 4 4 4 4 98 4 4 4 4 8 12 400

¾n 1:37 1:45 1:42 1:3 1:34 28:86 1:37 1:37 1:37 1:37 2:74 4:11 137

100

95

75

EXERCISE 12B 1 a i ii b i c i

Similar - angles equal, sides in proportion e=4 Similar ii e = 2 Not similar - sides not in proportion

C

M

Y

K

25

5

0


524

ANSWERS

75

2

f g h a

i i i i

5 4

vii 2 viii 25

b i

5

2 3 3 5

vi 3 4

2 5

iii 1 2

vii

2 5

4 3

a 5 b

c

1 3

v

3 4

ix

2 3

x

iv 4 5

viii

4 3 16 5

h 3 i

5 3

ix 1:5 x 1:2

a 1:4 b 0:8 c 0:7 a 2:4 b 13 c 1:2 d g

5

ii

1 2

e i 1 : 20 000 ii

3 4

j d

1 2 3 2

e

1 3

5 6

f

2 3

e 2:5

ii y = 2

c i

ii z = 10:5

d i e i f i h i 2

3

1 3 7 e = 12 e = 25 e = 17 20 e = 94 e = 11 25

4 a 1 : 1000 b 1 : 10 000 c 1 : 4000 d 1 : 500 e 1 : 500 000 5 252:5 m £ 180 m 6 a = 50 m, b = 35 m, c = 62 m, d = 38 m 7 a 1 : 80 b 1:6 m c 0:96 m 8 a 1 : 1000 b 49 m 9 b 123 m 10 b 43 m EXERCISE 12F 1 a 39:6 m £ 20 m b 26:4 m c i 2 m ii 4 m d 6 m e 0:8 m f 32 m2 g 21:2 m h 38 m2 i 45:6 m3 j 45 600 L k 12 2 a

EXERCISE 12C 1 a i e = 1:5 ii y = 7:5, z = 15 b i e=

1 20 000

front elevation

plan

ii w = 5:2 ii 10:2 ii 6:75 g i

e = 0:4 ii 3:2

ii 8:36

a y = 15, z = 14:4 b y = 6, z = 10:5 c y = 10, z = 13:2 d y = 15:75, z = 10:7 e y = 12:8, z = 15 a i 4 ii y = 16 b i 1:5 ii z = 4:5 c i 1:5 ii w = 18 d

i 4 ii y = 16

b

plan

e i 1:5 ii v = 12 f i 2:25 ii y = 33:75 g i 2:4 ii

k = 16:8 h i 0:4 ii z = 4:8

i i 0:8 (or 1:25) ii w = 7:2, t = 7:5 j i

5 6

or (1:2) ii y = 22:5, w = 24

EXERCISE 12D 1 a 8 b 8m 2 5 11

4 3

5 6

a

b 1:5 m 3

front elevation

side elevation

0

e = 23 - not all sides in proportion e=1 iii 2:5 iv 3 v 43 vi

Similar ii Not similar Similar ii 1:5 ii 2

side elevation

95

1 5000

d i 1 : 5000 ii

d i Not similar - sides not in proportion e i Similar ii e = 13

100

6m

6 12 cm 7 16 m 9 12:6 m 10 1:15 m

x = 8, y = 12, z = 16

EXERCISE 12E 1 a i 9 mm ii 18 mm iii 90 mm b i 6:5 mm ii 13 mm iii 65 mm c i 16 mm ii 32 mm iii 160 mm d i 18 mm ii 36 mm iii 180 mm e i 24 mm ii 48 mm iii 240 mm 2 a i 8:5 m ii 17 m iii 340 m b i 17:5 m ii 35 m iii 700 m c i 21 m ii 42 m iii 840 m d i 30 m ii 60 m iii 1200 m e i 13 m ii 26 m iii 520 m 1 3 a i 1 : 1000 ii 1000 b i 1 : 400 ii c i 1 : 5000 ii


1 400

c

plan 100

95

front elevation

25

side elevation

5

1 5000


75

0

C

M

Y

K


ANSWERS 100

j 68:2 m2

d

front elevation

plan

95

75

side elevation 25

k 17 m2

DIAGNOSTIC TEST 2 B 3 a Similar b Not similar 4 B 5 B 6 C 7 A 8 D 9 B 10 A 11 C 12 A REVIEW SET 12A 3 a Similar b Not similar 4 a 1:5 or 23 b 3:5 or 27 5 a i 2:5 ii x = 27:5 b i 2 (or 12 ) ii y = 16, z = 6:5

5

e

plan

0

525

side elevation

REVIEW SET 12B 2 a Similar b Not similar 3 a i 2:7 ii x = 10:8 b i 0:6 (or 53 ) ii x = 6, y = 18 13 c i 3 ii x = 9 4 3:5 m 5 360 m

front elevation

REVIEW SET 12C 1 a i 43 (or 34 ) ii x = 13 13 , y = 10:5 b i

3 a i 10 ii 9 iii 4 b P - pantry, W.O. - wall oven, F - fridge, W.C. - water closet, ENS - ensuite, WIR - walk in robe, LIN - linen cupboard c i 5:6 m £ 4:6 m ii 3:6 m £ 4:2 m iii 4:0 m £ 3:4 m iv 4:2 m £ 3:6 m v 3:8 m £ 3:4 m vi 5:4 m £ 4:0 m vii 7:0 m £ 2:7 m d i 262 m2 ii 28 squares e 1:89 m3 f i 7:2 m ii $51:20 g a i 100 mm ii 250 mm h i 2600 £ 1800 mm ii 88 iii 6 i i 4:0 m ii 6:8 m iii A j Bedroom 1 i 4:2 m ii 7:2 m Bedroom 3 i 4:2 m ii 7:2 m Bedroom 4 i 3:8 m ii 6:8 m 4 b i 18 ii 12 c 14 d 4 e 20:89 £ 10:59 m f 63 m g 221 m2 h $675/m2 i 55:3 m3 j 65:24 m k 12 m l i 8:4 m ii 8:6 m m $1938 n 12 m o i 8:4 m ii 8:6 m p $1938 5 a 1 : 100 b 6340 £ 5486 mm c 250 mm d 7 e 3900 £ 5200 mm f 6840 mm g 3400 mm h 2740 mm i W1 1800 £ 2000, W3 2700 £ 2000, W4 1800 £ 600 j 1800 £ 2100 k 34:8 m2 l 19:5 m2 6 a i 9 ii 3 iii 1 b i 2 ii 2 iii 3 iv 1 c 1 : 200 d i 4200 £ 1800 mm ii 4200 £ 3000 mm iii 5400 £ 3400 mm e W2 1800 £ 1200 mm, W5 1100 £ 1200 mm, W6 1800 £ 1200 mm Door 1300 £ 2200 mm f 2700 mm g 5600 mm h 600 mm

2 4

9 4

(or

4 ) 9

b 314 m 3 a 4£3 m b 42 m2

ii y = 11:25, x = 6 23

1 : 250 c

d 441:6 m2 e west

6m

10 m 6m 11 m

5

plan

side elevation

front elevation

REVIEW SET 12D 1 a i 1:6 (or 58 ) ii z = 11:2, y = 6:25 b i 0:25 (or 4) ii x = 3, y = 3:75 2 b 448 m2 c 156 m2 d 34:8% 3 a 1500 £ 2000 mm b 1200 £ 2400 mm c 5:1 m d 9:6 m e 500 mm 4 a 12:1 m b 4 m c 1900 £ 1300 mm and 1600 £ 1100 mm d 2:5 m 5 1:83 m

100

95

75

25

5

0



C

M

Y

K


526

1

a

75

100

Cost ($)

95

b

EXERCISE 13A 25 Cost ($)

100

ANSWERS

20 15

80 60 40 20

10

0

5 0 0

1

2

3

4

5

6

7

7 a

b + $8:80 100 80 60

Cost ($)

0

1

2

3

4

5

6

7

Cost ($)

b + $68 a 140 120 100 80 60 40 20 0

8 a Number of kgs 1

2

3

4

5

6

7

4 Cost ($)

50

b

2

Minutes of calls 0 0

time 0 1 2 3 4

10

20

40

+ 320 km + 1:2 hours

c d

distance 0 90 180 270 360

30

1

2

3

4

5

6

400 300 200 100

7

Time (h)

b + $1:70 c + 4:7 minutes

0 0

1

2

3

4

200 150

9 a

100 50 Number of kgs 0 0

1

2

3

4

5

6


cost $3:00 $27:00 $51:00 $75:00 $99:00

b c d

150 km

c

distance 300 200 100 0

100

95

7

b + $116 c + 3:6 kg kilometres 0 10 20 30 40

time 0 1 2 3

+ $90 + 7 km


75

400

Kilometres

a

a

km

3

1

6

+ $55 + 22 km

100

0

b + $86 c + 2:4 kg

5

c d

40

0

0

a

cost $5:00 $37:00 $69:00 $101:00 $133:00

30

150

b

Number of kgs

0

4

20

40 20

3

kilometres 0 10 20 30 40

Kilometres

a Cost ($)

2

Cost ($)

0

10

Number of litres

25

5

km

0

300 200

25

100

Time (h)

0 0

1

2

C

3

M

Y

4

K

5

0


527

ANSWERS 100

US $

95

75

80 60 40 20 0

11

8 b 2 c ¡6 d ¡ 1000

1

2 1

20

40

60

80

100

x -1

Yen

8000 6000 4000 2000 0

2

y 2

Aust $ 20

40

60

1

80 100

x -2

x-run 6 7 7 6 3 1 0

slope 0

y-rise 0 1 2 3 5 7 5

1 7 2 7 1 2 5 3

1 5

4 5

c

Line segment BC DE AC BE AE AF

d

7 6

x-run 2 4 4 6 8 9

e

7 3

y-rise 1 2 2 3 4 5

-1

-1

3

x -1

1

c ¡ 23

-2

4

y

9 2

3 2 1

gradient 1 2 1 2 1 2 1 2 1 2 1 2

-3 -2 -1

4 5

8 a gradient = ¡4, cuts axis at 5 b gradient = ¡2, cuts axis at 3 c gradient =

3 , 2

cuts axis at ¡1

d gradient = ¡ 74 , cuts axis at 3 e gradient = f gradient =


5 , 2 1 , 3

cuts axis at ¡5

-1

x 1

2

2

4

-2

5 a

y 8 6 4 2

d ¡1 e ¡2 f ¡9

6 a OP, PQ, RS, TU b QR, ST, UV c TU d ST e VW f PQ 7 a 5 b ¡2 c ¡4 d 32 e ¡ 34 f 6 h

2

-1

b No matter which pair of points on a line are used, the gradient is the same. 5 a 0 b ¡ 27

2

y

7 1

f

1

1

b i 0 ii 1, undefined iii increase 150 3 2 2 a 1000 = 20 b 10 = 15 c 34

g ¡ 12

2

3

Line segment AB CD EF GH IJ KL MN

3 a 0 b g 1

1 -1

EXERCISE 13B

4 a

6 25

Aust $

0

1 a

f

y

5000 yen is about $70

0

11 40

e

EXERCISE 13C

0

25

5

9 a 13 12

10 $50 US is about $78 AUS

x

-4 -6

b

-2

4

-2

100

95

y

2

75

x

-6 -4 -2 -2

2

4

6

-4 25

-6 -8

5

cuts axis at ¡1 0


C

M

Y

K


528 100

ANSWERS

c

y

i

y

2

2 95

1

1

x

75

-2

-1

1

x

1

-4 -3 -2 -1 -1

2

2

-2

-1

-3

-2

-4

25

y

j

5

2

y

d

1

1

0

x -2

-1

1

2 -2

2

-1

x

1

-1

-1

-2

-2

-3

6 a 7 b ¡6 c ¡1 d ¡1 e h ¡2 i ¡1 j 1 e

y

The y-intercept is the number in the equation of the line. i.e., not the number that is the co-efficient of x.

2 1

x -2

-1

1

-1

EXERCISE 13D 1 a

2

Christopher’s run

4

Distance (m)

-2

f

1 f 3 g 2

y

3

100 80 60 40 20 0

Time (s) 0

5

10

15

20

25

2 1

x 1

-1

3

2

b about 8 seconds c gradient = 4, speed = 4 m/s 2 a

4

Craig’s drive

g

Distance (km)

-1

y

4 3 2

150 100 50

Time (h)

0

1 -3 -2 -1

200

0

x 1

-1

2

1

2

3

4

3

b about 80 km c gradient = 50 d 3 a

-2

km/hr 100

Conversion 95

2 -1

80

y

1

-3 -2

x -1

1

2

3

-2 -3 -4



Euro

h

75

60 40 20

Aust $

0 0

20

40

60

25

80 100

5

b about 66 c gradient = 0:78 d conversion rate

C

M

Y

K

0


ANSWERS

e Yes 4 a Yes, the line has the same gradient as it is straight.

95

b

75

The runner would get tired and so the speed would decrease. A longer race would have a slower speed so the distance could be covered.

5 a Yes, the line is straight. b

25

+ $68 b + 4 km gradient = 2:5, cost per km = $2:50 intercept = 5, flag fall = $5 + $260 b + 42 gradient = 6, $6 per person intercept = 50, set up cost = $50 + $420 b + 2 12 hours gradient + 70, cost/hour = $70 intercept = $80, call out charge = $80

EXERCISE 13F 1 a about 55 minutes b Plan 30 c Plan 30, lower minimum up to about 10 minutes of calls 2 a about 90 minutes b Plan 55 c Plan 60, cheaper d about $20 3 a books cost

EXERCISE 13E 1

a b c 2 a d 3 a b

4

c a b c d e f

i $105 ii $65 iii $125 i 15 mins ii 30 mins iii 60 mins $20 d i $145 ii $165 iii $185 $42 b $20 extra c i $42 ii $52 iii $62 $240 i $172 ii $182 iii $191 i $9000 up to $10 300 ii $12 300 up to $14 100 iii $4000 up to $7400 No, no point on the graph. i $2:40 ii <, 6 iii 125 g iv 125 g, 250 g, $8 v 500 g i $2:40 ii $4:80 iiii $14:80 i 50 g ii 125 g iii 500 g separate = $12:80, together = $14:80, save $2 sending separately cost $4:80 each, $8 together, save $3:20 sending together Next level for > 500 g would be at least $6:80 but more likely $12 or more ) send separately.

0 1000 2000 3000

Book production

20000

3

4

5

600 400 200

Time (h)

e

1

2

3

c about 2350 d i about $4000 loss ii about $1000 profit iii about $3500 loss e i about 1300 books ii about 3000 books number 0 100 300 500 700

produce 3000 3800 5400 7000 8600

income 0 1500 4500 7500 10 500

b

100

Cost ($000’s)

6

om

Number of books (’000s)

Time (h) 2

in c 0

5

1

cost

10000

0

10

0

15000

5000

15

0

8000 11 000 14 000 17 000

b

4 a

5 Cost ($)

0

6 a c d 7 a c d 8 a c d

Cost ($)

5

Not necessarily as he may need petrol stops, there may be traffic, he may need a rest etc.

7 a + $12 000 b + $59 000 c gradient + 0:36, cents per dollar of tax rate in this bracket. d gradient + 0:42 i.e., 42 cents per dollar tax in this bracket 8 a + $10 000 b + $58 000 c gradient = 0:3, 30 cents per dollar tax rate in this bracket d 30 cents in $ e gradient = 0:42, cents per dollar tax rate in this bracket f 42 cents in the dollar

Cost ($)

100

12 10 8 6 4 2 0

95

75

ce produ me inco 0

200

400

600

800

Number of items

0 0

2

4

6

8

529

10

c about 440 items

25

5

0



C

M

Y

K


530 100

6

ANSWERS

a (4, 2)

EXERCISE 13G

y 8

95

1 a

6 4

75

2 -2

x 2

-2

4

6

8

25

5

b (2, ¡1)

30 25 20 15 10 5 0

x 0

y

2

4

6

8

b

2

0

y

40

-2

2

4

y

30

x

-2

20

-4

10

x

0

c (¡1, ¡2)

2

4

6

2

4

6

8

-10

c

2

10 -4

-2

y

8

2

6 -2

4

-3

2

-4

0

x 0

8

d

d (1, 3)

8

5

y 6

x

2 4

-5

4

6

8

-10

2 -6 -4 -2

y

0

x 2

-2

4

-15

6

-20

-4

-25

e

e (4, 2) 10

y

4

8

2

x

6

0

4

x 2

-2

4

5

100

95

y

75

10

3 2

5

1

-1 -2

8

15

4

-4 -3 -2 -1

6

-6 20

y

4

-4

6

f f ( 35 , 1 35 )

2

-2

2 -6 -4 -2

y

1

2

3

x

0

4

x

-3

-5

2

4

6

8

-10

25

5

0



C

M

Y

K


ANSWERS

95

75

25

a 5:5 b ¡4 c y = 5:5x ¡ 4

C

a 0:5 b 5 c y = 0:5x + 5

D

a ¡3 b 2 c y = ¡3x + 2

E

a ¡1 b 4 c y = ¡x + 4

F

a 3 b ¡5 c y = 3x ¡ 5

5

Conversion 100 80 60 40 20

Aust ($)

0 0

20

40

60

80 100 120

DIAGNOSTIC TEST

b

about 73 c gradient = 68

1 B 2 C 3 D 4 A 5 D 6 A 7 C 8 C 9 B 10 C 11 A 12 C 13 B

d

conversion of dollars to euros e yes

REVIEW SET 13A

6 a Laundry liquid

1 a Cost ($)

20

Number of litres

0 0

1

2

3

4

5

6

7

b about $16 time 0 1 2 3 4

b

distance 400 300 200 100 0

4

6

8

¡2 c 4 d y = ¡2x + 4

REVIEW SET 13B 1 a

Distance to Sydney

b

2

-2 -4 -6 -8 -10 -12

10

2 a

Scatter plot 6 4 2

30

Red frogs

80

Cost ($)

0

B

Distance (km)

5

a gradient + 3 b y-intercept + 2 c equation is y = 3x + 2

Euro

2 A

60 40

500

20

Number of kg

400

0 0

300 200

b

100

2 a

Time (h)

0 0

1

2

3

4

5

c about 250 km 3 gradient =

1 , 2

intercept = 3

4

1

2

3

4

5

6

7

$45 time 0 1 2 3 4

distance 0 70 140 210 280

y

100

6

b

5 4 3 2 1 -4 -3

-2 -1

-1

x 1

2

3

4

Distance (km)

100

531

Distance to Sydney

300

95

200

75

100

Time (h) 0

1

2

3

4

5 5

-2

c 250 d about 1:4 hours



25

0

C

0

M

Y

K


100

ANSWERS

4

5 a

y

4

100

Euro

532

3

95

2 75

60 40

1 -2 -1

80

x 1

-1

2

20

3

Aust ($)

0 0

20

40

60

80

100 120

25

5

0

6

a c d a c

b about 68

about $250 b about 42 gradient = 6, this is the cost per person 50, this is the fixed cost of $50 about 35 minutes b about 75 minutes after about 45 minutes, the graphs cut

1

1

Cost ($'s)

100 80

2

-1

60

0

1

2

3

4

5

6

b 0:5 c ¡2 d y = 0:5x ¡ 2

REVIEW SET 13D

cost $7:00 $45:00 $83:00 $121:00 $159:00

1 a

6

Cost ($'s)


4 2

Minutes of calls

b

0

200

Cost ($'s)

8

7

b about $56 c about 3:4 kg

0

150

1

2

3

4

5

6

7

b about $2:20 c about 3:5 minutes

100 50

Distance (km)

2 a

0 0

10

20

30

40

50

c about $65 d about 18 kilometres gradient =

6

-3

Number of kg

0

3

4

-2

40 20

a

Scatter plot

3 2

2

gradient = 87 d the exchange rate

6 a

REVIEW SET 13C a

c

e yes

5 , 2

intercept = 5

4


cost $10:00 $44:00 $78:00 $112:00 $146:00

y 2

b

1

x -2

-1

1

2

3

4

-1

Cost ($'s)

5

150

100

100

95

50

75

Distance (km)

-2

0

-3

0

10

20

30

40

50 25

-4 -5

c about $130 d

about 5 km

5

3 gradient = ¡3:5, intercept = 6 0



C

M

Y

K


ANSWERS 100

4

EXERCISE 14D

y

2

1 a opposite = 35 mm, adjacent = 60 mm, hypotenuse = 70 mm

x

95

-2

2

4

b 2 c 4 a c e

-2

75

-4 25

5

0

5 a c d 6 a c

about $33 b about 85 kilometres about 0:4, the rate of charge per km 8, the flag fall is $8 about 35 minutes b about 80 minutes about 60 minutes, the graphs cross

EXERCISE 14A 1 a e 2 a e 3 a

7:8 cm b 97:7 mm 8:9 cm b 70:5 mm i Yes ii

13:9 cm c f 182:1 cm 15:3 cm c f 14:8 km

23:4 m d 7:4 km 2:7 m d

37:8 m

b i No

6 cm

sin 30o = 0:5, cos 30o = 0:86, tan 30o = 0:58 0:9, 0:4, 2:1 3 c 0:4, 0:9, 0:5 0:8, 0:6, 1:33 b 0:38, 0:92, 0:42 0:47, 0:88, 0:53 d 0:98, 0:22, 4:44 0:69, 0:72, 0:95 f 0:28, 0:96, 0:29

EXERCISE 14E 1 a 0:602 b 0:743 c 1:483 d 0:999 e 0:224 f 0:378 g 2:078 h 0:467 i 0:989 j 0:739 2 a 34:4 b 49:2 c 76:3 d 51:7 e 67:9 f 14:35 g 80:65 h 44:95 i 18:3_ j 36:28 3 a 0:679 b 0:270 c 0:443 d 0:995 e 1:857 f 0:773 g 1:021 h 0:938 i 0:970 j 8:732 EXERCISE 14F 1 a 44:4o b 66:4o c 26:6o f 51:3o g 73:7o h 24:7o

10 cm c i No d i Yes ii

24 m 10 m 26 m

3 a 44o 260 b 72o 330 c 41o 590 d 16o 160 e 29o 320 f 55o 240 g 82o 320 h 34o 130 i 15o 00 j 49o 90 4 a 48o 350 b 51o 190 c 23o 120 d 53o 80 e 69o 270 f 56o 270 g 28o 370 h 63o 370 i 25o 360 j 48o 180 5 a 43o 590 e 59o 380

17 cm 8 cm

15 cm

d 42:8o e 16:3o i 26:7o j 30:7o

2 a 24o 360 b 87o 180 c 47o 120 d 80o 540 e 16o 480 f 35o 510 g 54o 150 h 17o 450 i 36o 360 j 63o 180 k 21o 260 l 9o 410

8 cm

e i Yes ii

533

b 43o 120

c 40o 190

d 46o 80

EXERCISE 14G 1 a A b D c C d B

f i Yes ii

4.5 m

2 a D b C c B d A 3 a D b A c C d B

2.4 m

4 a B b D c D d C

5.1 m 4 7:2 cm 5 8 2:99 m 9 11 a 379 m e 252 m 12 a 165 m

11:3 cm 6 1:06 m 7 Yes 21:4 m 10 97:1 km b 263 m c 178 m d 218 m f 451 m g 218 m h 319 m b 389 m c 169 m

EXERCISE 14B 1 a b c d e f g h

i i i i i i i i

AC MK PQ DF UW SR TV LK


ii ii ii ii ii ii ii ii

BC iii AB ML iii LK PR iii RQ DE iii EF UV iii WV SQ iii QR WT iii WV JK iii LJ


5 a D b C c A d B 6 a g m r

49o b 63o c 53o d 34o e 50o f 35o 69o h 60o i 65o j 48o k 39o l 58o 44o n 50o o 47o p 38o q 55o 41o

7 a 51o 90 b 62o 110 c 48o 350 d 43o 570 e 39o 550 f 73o 440 g 38o 70 h 50o 00 EXERCISE 14H 1 a f k p

21:0 b 15:8 g 240:7 l 20:8 q

100

95

75

18:5 56:5 12:3 90:6

c 65:7 d 62:2 e 16:0 h 107:8 i 36:0 j 47:3 m 68:3 n 23:6 o 93:6 r 5:0

2 a 41:0 b 15:2 c 8:7 d 89:0 e 10:2 f 25:8 g 173:1 h 9:4 i 157:44

25

5

0

C

M

Y

K


534

1

b i

a

b

class 1 to 5 6 to 10 11 to 15 16 to 20 21 to 25

45° 36 m

75

65° 60 m

c

25

d

frequency 16 8 18 31 9

cf 16 24 42 73 82

ii

54°

100

5

e

80 60 40 20

1 to 5

15 m 2 6

17:1 m 3 64:3 m 4 18:0 m 5 233:9 m 147:8 m 7 11 196 m 8 a 16:2 m b 17:8 m

EXERCISE 14J 1 4 9

a 5:7 m b 2:0 m 2 11:7 m 3 70o 49o 5 38:2 m 6 26o 7 32o 8 14 370 m 5:0 m 10 2:8 m

DIAGNOSTIC TEST 1 7

a B b B 2 C 3 D 4 A 8 B 9 C 10 A

B 5 C 6 B

1

a 9:5 m b 6:2 m 2 a XZ b ZY

3

a

4 5 7

a 0:836 b 0:949 c 3:967 a 18o b 65o c 48o 6 a 35o b 34o a 14:5 cm b 48:9 mm 8 a 28:9 b 9:7

5 13

b

12 13

c

6

a 5:9 b 3:3 2 C 3 a 47o 180 b 16:68o a 66o 560 b 81o 400 c 26o 340 5 C a 59:1 b 19:8 7 a 4:0 b 113:2 8 80:0 m a D b B 2 B 3 B 4 a 68o 500 b 46:45o a 0:8119 b 0:9489 c 4:0815 a 36o 520 b 28o 570 c 22o 370 7 5:1 m 8 65o 506 m

REVIEW SET 14D

f

i ii g i iii

10

a 31o b 39o c 54o 3217 m 8 40 m

9 cm

6 cm

3 cm 2 cm

100

b 2:8 m c 2:3 km d 3 a

b

15 17

c

8 15

1 : 125 e

30 cm

4 D 5 A 6 12o

CUMULATIVE REVIEW CHAPTERS 11 - 14 a mean = 7:4, mode = 8, median = 8


95

75

4 3 2

25

1


15

x = 16:3, ¾n = 2:9 x = 56:5, ¾n = 14:5 x = 3 79 ii median = 3 mode = 3 and 5 iv ¾n = 1:397

2 a

Cost ($)

169:7 m 2

3 7

1

12

5

8 17

1

a

9.5 10

7

5

REVIEW SET 14C 1 5 6 9

12 ii 2 Leaves 889 01466789 02455 1348

i Range = 30 ii 48, 56, 65 iii 58:5

5 12

REVIEW SET 14B 1 4 6

iii A median = 15, B + 12 c i 10, 11, d Stem 4 5 6 7 e

REVIEW SET 14A

21 to 25

0

67°

16 to 20

0


36 m

32° 20 m

11 to 15

95

EXERCISE 14I

6 to 10

100

ANSWERS

Number of mins

0 0

1

2

3

4

C

5

6

7

M

Y

K

5

0


ANSWERS

i time ii + $1:70 iii about 3 13 minutes

100

95

75

535

b gradient = ¡3, intercept = 6 c y 1

x -3

-2

-1

-1

25

1

2

3

-2 5

-3 -4

0

-5

d i iii iv 4 a i c i d e f g

i i i i

about $36 ii + 108 km gradient + 2:6, cost per km. intercept + $8, flag fall 9:2 ii 9:8 b i PR ii RQ 5 13

ii

0:9573 16o ii 18:2 ii 50o 280

12 13

iii

5 12

ii 0:3214 iii 0:3285 67o iii 55o 12:6 ii 66o 160

100

95

75

25

5

0



C

M

Y

K


536

ANSWERS

100

95

75

25

5

0

100

95

75

25

5

0



C

M

Y

K

General Maths Preliminary Course Textbook

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