Basic Skills Mathematics Year 7

HAESE

&

HARRIS PUBLICATIONS

Core Skills Mathematics

7 Helen Hall Sue Norris Cheryl Ross Mandy Spiers Wendy Stimson Chris Haines Stan Pulgies

CORE SKILLS MATHEMATICS 7 Helen Hall Sue Norris Cheryl Ross Mandy Spiers Wendy Stimson Chris Haines Stan Pulgies

B.Ed., Dip.T. B.Ed., Dip.T. B.Ed., Dip.T. Dip.T. B.Ed. B.Ed., Grad.Cert.Ed., Dip.T. M.Ed., B.Ed., Grad.Dip.T.

Haese & Harris Publications 3 Frank Collopy Court, Adelaide Airport SA 5950 Telephone: (08) 8355 9444, Fax: (08) 8355 9471 email: [email protected] web: www.haeseandharris.com.au National Library of Australia Card Number & ISBN 1 876543 68 X © Haese & Harris Publications 2004 Published by Raksar Nominees Pty Ltd, 3 Frank Collopy Court, Adelaide Airport SA 5950 First Edition

2004

Cartoon artwork by John Martin Artwork by Piotr Poturaj and David Purton Cover design by Piotr Poturaj. Cover photograph: Copyright © Digital Vision® Ltd Computer software by David Purton and Eli Sieradzki Typeset in Australia by Susan Haese (Raksar Nominees). Typeset in Times Roman 11/12\Er_

This book is copyright. Except as permitted by the Copyright Act (any fair dealing for the purposes of private study, research, criticism or review), no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Enquiries to be made to Haese & Harris Publications. 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 the educational institution or the body that administers it has given a remuneration notice to Copyright Agency Limited (CAL). For information, contact the Copyright Agency Limited. Acknowledgements: the Correlation Chart at the end of the book relates to the R-7 SACSA Mathematics Teaching Resource published by the Department of Education and Children’s Services. The Publishers also wish to acknowledge The Royal Agricultural & Historical Society of S.A. Inc. for permission to include the map of the Royal Adelaide Show.

FOREWORD

We have written this book to provide a sound course in mathematics that Year 7 students will find easy to read and understand. Our particular aim was to cover the core skills in a clear and readable way, so that every Year 7 student can be given a sound foundation in mathematics that will stand them in good stead as they begin their secondary-level education. Units are presented in easy-to-follow, double-page spreads. Attention has been paid to sentence length and page layout to ensure the book is easy to read. The content and order of the thirteen chapters parallels the content and order of the thirteen chapters in Mathematics for Year 7 (second edition) also published by Haese & Harris Publications and that book could be used by teachers seeking extension work for students at this level. Throughout this book, as appropriate, the main idea and an example are presented at the top of the left hand page; graded exercises and activities follow, and more challenging questions appear towards the foot of the right-hand page. With the support of the interactive Student CD, there is plenty of explanation, revision and practice. We hope that this book will help to give students a sound foundation in mathematics, but we also caution that no single book should be the sole resource for any classroom teacher. We welcome your feedback. Email: [email protected] Web: www.haeseandharris.com.au HH SN CGR MS WS CAH SP

Active icons – for use with interactive student CD By clicking on the CD-link icon you can access a range of interactive features, including: ! spreadsheets ! video clips ! graphing and geometry software ! computer demonstrations and simulations.

CD LINK

TABLE OF CONTENTS

TABLE OF CONTENTS

Chapter 1 WHOLE NUMBERS

Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7

Our number system Operations with whole numbers Problem solving with whole numbers Rounding and approximation One million and beyond Number opposites Review of chapter 1

8 10 12 14 16 18 20

Chapter 2 NUMBER PROPERTIES

Unit 8 Unit 9 Unit 10 Unit 11 Unit 12 Unit 13 Unit 14

Number operations and their order Factors of natural numbers Multiples and divisibility rules Powers of numbers Square and cube numbers Problem solving methods Review of chapter 2

22 24 26 28 30 32 34

Chapter 3 SHAPES AND SOLIDS

Unit 15 Unit 16 Unit 17 Unit 18 Unit 19 Unit 20 Unit 21 Unit 22 Unit 23 Unit 24

Points and lines Angles Angles of a triangle and quadrilateral Polygons Classifying triangles and quadrilaterals Constructing a triangle and bisecting angles 90° and 60° angles and Circles Polyhedra and nets of solids Drawing solids Review of chapter 3 Review of chapters 1, 2 and 3

36 38 40 42 44 46 48 50 52 54 56

Chapter 4 FRACTIONS

Unit 25 Unit 26 Unit 27 Unit 28 Unit 29 Unit 30 Unit 31 Unit 32

Representation of fractions Equivalent fractions and lowest terms Fractions of quantities Fraction sizes and types Adding and subtracting fractions Multiplying fractions Problem solving with fractions Review of chapter 4

58 60 62 64 66 68 70 72

Chapter 5 DECIMALS

Unit 33 Unit 34 Unit 35 Unit 36 Unit 37 Unit 38 Unit 39

Representing decimals Place value Rounding decimal numbers Ordering decimals Adding and subtracting decimals Multiplying and dividing by powers of 10 Multiplying decimal numbers

74 76 78 80 82 84 86

TEST YOURSELF

5

6

TABLE OF CONTENTS

Unit 40 Unit 41 Unit 42

Dividing decimals by whole numbers Fractions and decimal conversions Review of chapter 5

Unit 43 Unit 44 Unit 45 Unit 46 Unit 47 Unit 48 Unit 49 Unit 50 Unit 51

Percentages and fractions Percentage, decimal and fraction conversions Percentages on display and being used Representing percentages Quantities and percentages Money and problem solving Discount and GST Simple interest and other money problems Review of chapter 6 Review of chapters 4, 5 and 6

94 96 98 100 102 104 106 108 110 112

Chapter 7 MEASUREMENT (LENGTH AND MASS)

Unit 52 Unit 53 Unit 54 Unit 55 Unit 56 Unit 57 Unit 58

Reading scales Units and length conversions Perimeter Scale diagrams Mass Problem solving Review of chapter 7

114 116 118 120 122 124 126

Chapter 8 MEASUREMENT (AREA AND VOLUME)

Unit 59 Unit 60 Unit 61 Unit 62 Unit 63 Unit 64 Unit 65

Area (square units) Area of a rectangle Area of a triangle Units of volume and capacity Volume formulae Problem solving Review of chapter 8

128 130 132 134 136 138 140

Chapter 9 DATA COLLECTION AND REPRESENTATION

Unit 66 Unit 67 Unit 68 Unit 69 Unit 70 Unit 71

Samples and population Collecting and interpreting data Interpreting graphs Mean and median Line graphs Review of chapter 9

142 144 146 148 150 152

Chapter 10 TIME AND TEMPERATURE

Unit 72 Unit 73 Unit 74 Unit 75 Unit 76 Unit 77 Unit 78

Units of time Differences in time Reading clocks and timelines Timetables Time zones Average speed and temperature Review of chapter 10

154 156 158 160 162 164 166

Review of chapters 7, 8, 9 and 10

168

Chapter 6 PERCENTAGES

TEST YOURSELF

TEST YOURSELF

88 90 92

TABLE OF CONTENTS

Chapter 11 ALGEBRA

Unit 79 Unit 80 Unit 81 Unit 82 Unit 83

Geometric and number patterns Formulae and variables Practical problems and linear graphs Solving equations Review of chapter 11

170 172 174 176 178

Chapter 12 TRANSFORMATION AND LOCATION

Unit 84 Unit 85 Unit 86 Unit 87 Unit 88 Unit 89 Unit 90 Unit 91 Unit 92

Number planes Transformations and reflections Rotations and rotational symmetry Translations and tessellations Enlargements and reductions Using ratios Bearings and directions Distance and bearings Review of chapter 12

180 182 184 186 188 190 192 194 196

Chapter 13 CHANCE

Unit 93 Unit 94 Unit 95 Unit 96 Unit 97

Describing chance Defining probability Tree diagrams and probability Expectation Review of chapter 13

198 200 202 204 206

Review of chapters 11, 12 and 13

208

TEST YOURSELF ANSWERS

210

Correlation chart: R-7 SACSA Mathematics Teaching Resource

239

INDEX

243

7

CHAPTER 1

8

WHOLE NUMBERS

Unit 1

Our number system

Numbers less than one million The chart shows the place value of each digit in a number. Thousands Hundreds Tens Units 1 2 0

Units Hundreds Tens 9 9

This digit represents 900.

The number shown is one hundred and twenty thousand, nine hundred and ninety three.

Units 3 This digit represents 90.

Exercise 1 1 What number is represented by the digit 8 in the following? a 38 b 81 c 458 e 1981 f 8247 g 2861 i 60 008 j 84 019 k 78 794

d h l

847 28 902 189 964

2 What is the place value of the digit 7 in the following? a 497 b 37 482 c 856 784

d

755 846

3 Write down the place value of the 3, the 5 and the 8 in each of the following: a 53 486 b 3580 c 50 083 d

805 340

4

a Use the digits 6, 4 and 8 once only to make the largest number you can. b Write the largest number you can using the digits 4, 1, 0, 7, 2 and 9 once only. c What is the largest 6 digit numeral you can write using each of the digits 2, 7 and 9 twice? d How many different numbers can you write using the digits 3, 4 and 5 once only?

5 Put the following numbers in ascending order (smallest first): a c e

57, 8, 75, 16, 54, 19 1080, 1808, 1800, 1008, 1880 236 705, 227 635, 207 653, 265 703

b d f

660, 60, 600, 6, 606 45 061, 46 510, 40 561, 46 051, 46 501 554 922, 594 522, 545 922, 595 242

6 Write the following numbers in descending order (largest first): a c

361, 136, 163, 613, 316, 631 498 231, 428 931, 492 813, 428 391, 498 321

b d

7789, 7987, 9787, 8779, 8977, 7897, 9877 563 074, 576 304, 675 034, 607 543, 673 540

7 Write the numeral for: a 8 £ 10 + 6 £ 1 b 6 £ 100 + 7 £ 10 + 4 £ 1 c 9 £ 1000 + 6 £ 100 + 3 £ 10 + 8 £ 1 d 5 £ 10 000 + 2 £ 100 + 4 £ 10 e 2 £ 10 000 + 7 £ 1000 + 3 £ 1 f 2 £ 100 + 7 £ 10 000 + 3 £ 1000 + 9 £ 10 + 8 £ 1 g 3 £ 100 + 5 £ 100 000 + 7 £ 10 + 5 £ 1 h 8 £ 100 000 + 9 £ 1000 + 3 £ 100 + 2 £ 1

The numbers in question 7 are in expanded form.

DEMO

WHOLE NUMBERS (CHAPTER 1)

8 Write in expanded form: a 975 e 56 742

b f

c g

680 75 007

d h

3874 600 829

9083 354 718

9 Write the following in numeral form: a twenty seven b eighty c six hundred and eight d one thousand and sixteen e eight thousand two hundred f nineteen thousand five hundred and thirty eight g seventy five thousand four hundred and three h six hundred and two thousand eight hundred and eighteen. 10 What number is: a one less than eight b two greater than eleven c four more than seventeen d one less than three hundred e seven greater than four thousand f 3 less than 10 000 g four more than four hundred thousand h 26 greater than two hundred and nine thousand?

= + or ¼ > <

11 In the following replace ¤ by = or + : a 375 + 836 ¤ 1200 c 978 ¡ 463 ¤ 515 e 455 + 544 ¤ 999 g 2000 ¡ 1010 ¤ 990

b d f h

12 In the following replace ¢ by > or < : a 5268 ¡ 3179 ¢ 4169 b c 672 + 762 ¢ 1444 d e 20 £ 80 ¢ 160 f g 5649 + 7205 ¢ 12 844 h

29 £ 30 ¢ 900 720 ¥ 80 ¢ 8 700 £ 80 ¢ 54 000 6060 ¡ 606 ¢ 5444

reads reads reads reads

‘is ‘is ‘is ‘is

equal to’ approximately equal to’ greater than’ less than’

79 £ 8 ¤ 640 7980 ¥ 20 ¤ 400 50 £ 400 ¤ 20 000 3000 ¥ 300 ¤ 10

Number systems The number system we use is called the Hindu-Arabic system. It uses the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. We call them digits. We use them to make up our natural (whole) numbers. Natural numbers are the numbers that we count with (1, 2, 3, 4, 5, 6, .....). Click on the icon to read about other number systems that have been used around the world.

CIV Egyptian

Roman

Mayan PRINTABLE MATERIAL

Chinese Japanese

These all represent the number 104.

9

10

WHOLE NUMBERS (CHAPTER 1)

Unit 2

Operations with whole numbers

Addition and subtraction When we subtract a smaller number from a larger one we find their difference.

When we add numbers we find their sum. For example:

For example:

32 427 + 3274

3 11 9 10

4200 ¡ 326 3874

1 1

3733

sum

difference

Exercise 2

DEMO

1 Do these additions: a

392 + 415

b

601 + 729

c

1917 + 2078

d

913 24 + 707

e

217 106 + 1274

f

9004 216 23 + 3816

b e h

72 + 35 921 + 1234 32 + 627 + 4296

c f i

421 + 327 6214 + 324 + 27 912 + 6 + 427 + 3274

2 Find: a 42 + 37 d 624 + 72 g 90 + 724 3 Do these subtractions: a 97 ¡ 15 d

DEMO

602 ¡ 149

4 Find: a 47 ¡ 13 d 40 ¡ 18 g 503 ¡ 127

b

63 ¡ 19

c

247 ¡ 138

e

713 ¡ 48

f

6005 ¡ 2349

c f i

33 ¡ 27 623 ¡ 147 5939 ¡ 3959

b e h

62 ¡ 14 214 ¡ 32 5003 ¡ 1236

5 The cards have place values as shown. i Find the sum. ii Find the difference.

3

2

ª

5

3

ª

7

ªª ªªª ªª

9 5

2

§ § § § §§§ § §

8

8

§ § § §

§ § § §

4

4 9

§ § § § § § §

9

U

T

§ §

4

7

4 7

H

§ §

4

A

7

A

9 9

A

4

Th

9

7

b

U

T

7

H

8

Th

8

a

An ace has the value 1.

A

WHOLE NUMBERS (CHAPTER 1)

Multiplication and division We multiply numbers to find their product.

23 £ 100 = 2300

53 £ 16

12¡700 ¥ 100 = 127

When we divide one number by another, the result is the quotient. 4 1 7

1

6

318 530

25

product

848 6 Find: (You could do these mentally.) a 50 £ 10 b d 69 £ 100 e g 123 £ 100 h j 49 £ 10 000 k

50 £ 100 69 £ 1000 246 £ 1000 490 £ 100

c f i l

50 £ 1000 69 £ 10 000 960 £ 100 4900 £ 100

7 Find: (You could do these mentally.) a 24¡000 ¥ 10 b e d 45¡000 ¥ 10 h g 72¡000 ¥ 10 j 6000 ¥ 10 k

24¡000 ¥ 100 45¡000 ¥ 100 72¡000 ¥ 100 6000 ¥ 100

c f i l

24¡000 ¥ 1000 45¡000 ¥ 1000 72¡000 ¥ 1000 6000 ¥ 1000

d h l

53 £ 24 642 £ 36 368 £ 73

8 Find: a 24 £ 5 e 27 £ 15 i 274 £ 21

b f j

9 Do these divisions: a 3 42 d

5

c g k

37 £ 4 56 £ 49 958 £ 47 b e

375

4

216

7

6307

62 £ 8 324 £ 45 117 £ 89

c f

10 Find: a 24 ¥ 4 d 240 ¥ 5

b e

125 ¥ 5 624 ¥ 3

c f

11 Find: a d g j m

b e h k n

63 ¥ 4 143 ¥ 2 661 ¥ 8 1201 ¥ 5 8463 ¥ 4

c f i l o

45 ¥ 2 97 ¥ 8 439 ¥ 5 1033 ¥ 4 7349 ¥ 2

quotient

10 42

312 ¥ 6 7353 ¥ 9 81 ¥ 5 275 ¥ 4 955 ¥ 2 4699 ¥ 8 7999 ¥ 5

8 11

168 6809

Sometimes we need to add zeros to complete the division. For example 8 7: 7 5 4

3 5 31 : 3 0 2 0

So, 351 ¥ 4 = 87:75

12 Solve the following problems: a Find the product of 29 and 12. b Find the quotient of 368 and 23. c Find the difference between the product of 7 and 6 and the quotient of 500 and 50. 13 This sum is not correct. By changing only one of the digits, make it correct:

386 + 125 521

DEMO

DEMO

11

12

WHOLE NUMBERS (CHAPTER 1)

Unit 3

Problem solving with whole numbers

Two examples of problem solving are:

² John bought 5 kg of potatoes, 3 kg of carrots, 7 kg of onions and 25 kg of pumpkin. We can find the total weight of John’s vegetables by adding the weight of each vegetable. Total weight = 5 + 3 + 7 + 25 = 40 kg

Exercise 3

² Jason buys 217 baskets of fresh cherries for a supermarket chain at $38 a basket. What will be the total cost? Total cost = 217 £ $38 = $8246

We need a number sentence to answer the question.

Addition and subtraction 1 Jack bought 4 separate lengths of timber. Their lengths were as follows: 5 m, 1 m, 7 m, and 9 m. If all four lengths of timber were put end to end, what would the total length be? 2 Jenny bought a Play Station for $255. She also purchased another controller for $50, a Play Station game for $95 and a bag to store these in for $32. How much did she pay altogether? 3 Kerry needed to lose some weight to be chosen in a light weight rowing team. He weighed 60 kg but needed to weigh 54 kg. How much weight did he need to lose? 4 Stephen made $72 worth of phone calls in one month. His parents said they would only pay $31 of this. How much did Stephen have to pay? 5 Rima went on an overseas trip that required three plane flights. The first flight was 2142 km long, the next one was 732 km long and the third one was 1049 km long. How long were her flights in total? 6 Bill measured out a straight line that was 6010 cm long on the school grounds. He actually went too far. The line should have been 4832 cm long. How much of the line will he need to rub out?

Multiplication and division 7 Carlo lifted five 18 kg bags of potatoes onto a truck. How many kg of potatoes did he lift altogether? 8 My three brothers and I received a gift of $320. If we shared the money equally amongst ourselves, how much did each person receive? 9 A relay team of nine people took 738 minutes to complete a relay race. If each team member took exactly the same time how long did each team member take? 10 A maths textbook is 245 mm long. If I put 10 books end to end how far would they stretch? 11 24 people each travelled 28 km to play sport. How far in total did they travel? 12 If I write 8 words per minute, how long would it take me to write 648 words?

WHOLE NUMBERS (CHAPTER 1)

13

Two step problems This is an example of a problem solved using two steps. How much change from $50 would you receive after buying three bags of potatoes at $14 a bag? Step 1: Total cost of potatoes = $14 £ 3 = $42 Step 2: Change is $50 ¡ $42 = $8 13 Sara bought a shirt costing $29 and a pair of jeans costing $45. How much change did she get from $100? 14 Glen bought three T-shirts costing $42 each and a pair of shoes costing $75. Find the total cost of his purchases. 15 Miki had 65 minutes of time left on her prepaid mobile phone. She made a 10 minute call to Rupesh, a 7¡¡minute call to her mother and a 26 minute call to her boyfriend Michael. How many minutes did she have left after making these calls? 16 Maria bought five 3 kilogram bags of oranges. The numbers of oranges in the bags were: 10, 11, 12, 12 and 10. Find the average number of oranges in a bag. 17 Lachlan had a herd of 183 goats. He put 75 in his largest paddock and divided the rest equally between two smaller paddocks. How many goats were put in each of the smaller paddocks? 18 George had $436 in his bank and was given $30 cash for his birthday. How much money did he have left if he bought a bicycle costing $455? 19 The cost of placing an advertisement in the local paper is $10 plus $4 for each line of type. If my advertisement takes 5 lines, how much will I pay? 20 How much would June pay for 8 iced buns if 3 buns cost her 54 cents? 21 A football team had kicked 12 goals 13 points. They had another kick for goal as the siren sounded. Their final score was 91 points. Did the last kick score a goal or a point? (1 goal = 6 points) 22 Marcia saved $620 during the year and her sister saved twice that amount. How much money did they save in total? 23 Anna had $463 in her savings account and decided to bank $20 a week for 14 weeks. How much was in the account at the end of that time? 24 Tony’s wages for the week were $496. He was also paid for 3 hours overtime at $18 per hour. How much did he earn in total? 25 Alicia ran 6 km each day from Monday to Saturday and 12 km on Sunday. How far did she run during the week? 26 A plastic crate contains 100 boxes of ball point pens. The boxes of pens each weigh 86 grams. If the total mass of the crate and pens is 9200 g, find the mass of the crate.

14

WHOLE NUMBERS (CHAPTER 1)

Unit 4

Rounding and approximation

We round off if we do not need to know the exact number.

Rules for rounding off: ²

If the digit after the one being rounded off is less than 5 (i.e., 0, 1, 2, 3 or 4) we round down.

²

If the digit after the one being rounded off is 5 or more (i.e., 5, 6, 7, 8, 9) we round up.

For example, to round off:

63

to the nearest 10,

63 + 60

fWe round down, as 3 is less than 5:g

275

to the nearest 100,

275 + 300

fWe round up, as 7 is greater than 5:g

8467

to the nearest 1000,

8467 + 8000

fWe round down, as 4 is less than 5:g

+ means “is approximately equal to”

Exercise 4 1 Round off to the nearest 10: a 75 b 78 e 3994 f 1651 i 49 566 j 30 942

c g k

298 9797 999 571

d h l

2379 61 015 128 674

2 Round off to the nearest 100: a 78 b 468 e 25 449 f 14 765

c g

998 130 009

d h

2954 43 951

3 Round off to the nearest 1000: a 748 b 5500 e 65 438 f 123 456

c g

9990 434 576

d h

43 743 570 846

4 Round off to the accuracy given: a $45 387 (to the nearest $1000) b 328 kg (to the nearest ten kg) c a weekly wage of $485 (to the nearest $100) d a distance of 4753 km (to the nearest 100 km) e the annual amount of water used in a household was 362 498 litres (to the nearest 1000 litres) f the profit of a company was $487 374 (to the nearest $10 000) g the population of a town is 37 495 (to the nearest one thousand) h the population of a city is 637 952 (to the nearest hundred thousand) i the number of times the average heart will beat in one year is 35 765 280 times (to nearest million) j a year’s loss by a large mining company was $1 517 493 826 (to nearest billion).

DEMO

DEMO

DEMO

WHOLE NUMBERS (CHAPTER 1)

One figure approximations Rules:

² ²

Leave single digits as they are. For all other numbers, round the left most digit and put zeros in the other places.

For example: 57 £ 8 + 60 £ 8 + 480

15

Estimating with money For example: Estimate the cost of 19 pens at $1:95 each. 19 £ $1:95 When we estimate + 20 £ $2 with money with cents we round to the + $40 nearest whole dollar.

294 ¥ 48 + 300 ¥ 50 +6

We estimate using approximations to get a good idea of what the answer should be. 5 Estimate the cost of: a 195 exercise books at 98 cents each c 18 show bags at $3:45 each e 4 dozen iceblocks at $1:20 each

b d f

27 sweets packets at $2:15 a packet 12 bottles of drink at $2:95 a bottle 3850 football tickets at $6:50 each.

6 Estimate the following products using 1 figure approximations: a 55 £ 3 b 62 £ 7 c 88 £ 6 e 389 £ 7 f 4971 £ 3 g 57 £ 42 i 85 £ 98 j 3079 £ 29 k 40 989 £ 9 7 Estimate the following quotients using 1 figure approximations: a 397 ¥ 4 b 6849 ¥ 7 d 6000 ¥ 19 e 80 000 ¥ 37 g 549 ¥ 49 h 3038 ¥ 28

c f i

d h l

275 £ 5 73 £ 59 880 £ 750

79 095 ¥ 8 18 700 ¥ 97 5899 ¥ 30

8 Multiply the following. Use estimation to check that your answers are reasonable. a 79 b 445 c 3759 £7 £8 £9 9 Divide the following. Use estimation to check that your answers are reasonable. a b c 6 366 8 1080 4 392

In the following questions, round the given data to one figure to find the approximate value asked for: 10 In her bookcase Lynda has 12 shelves. Estimate the number of books in the bookcase if there are approximately 40 books on each shelf. 11 Miki reads 217 words in a minute. Estimate the number of words she can read in one hour. 12 A bricklayer lays 115 bricks each hour. If he works a 37 hour week, approximately how many bricks will he lay in one month? 13 Joe can type at 52 words per minute. Find an approximate time for him to type a document of 3820 words. 14 In a vineyard there are 189 vines in each row. There are 54 rows. Find the approximate number of vines in the vineyard.

PRINTABLE MATERIAL

16

WHOLE NUMBERS (CHAPTER 1)

Unit 5

One million and beyond

One million is 1 000 000. 1 000 000 = 100 £ 100 £ 100

One million $1 coins placed side by side would stretch 25 km. 1 000 000 £ 25 mm = 25 000 000 mm = (25 000 000 ¥ 1 000 000) km = 25 km

1m

A cube made of one million MA unit blocks would measure 1 m long £ 1 m wide £ 1 m high.

Millions hundreds tens 5

25 mm

units 3

Thousands hundreds tens units 4 7 9

Units hundreds tens 6 8

units 2

The number displayed in the place value chart is 53 million, 479 thousand, 682.

Exercise 5 1 In the number shown on the chart above, the digit 9 has the value 9000 and the digit 3 has the value 3 000 000. Give the value of the: EXTRA a 8 b 5 c 6 d 4 e 7 f 2 ACTIVITIES

2 Write the value of each digit in the following numbers: a 3 648 597 b 34 865 271 3 Read the following stories about large numbers. Write each large number using figures. a A heart beating at a rate of 70 beats per minute would beat about thirty seven million times in a year. b Australia’s largest hamburger chain bought two hundred million bread buns and used seventeen million kilograms of beef in one year. c The Jurassic era was about one hundred and fifty million years ago. d One hundred and eleven million, two hundred and forty thousand, four hundred and sixty three dollars and ten cents was won by two people in a Powerball Lottery in Wisconsin USA in 1993. e A total of twenty one million, two hundred and forty thousand, six hundred and fifty seven Volkswagen ‘Beetles’ had been built to the end of 1995. 4 In the following questions, how many times does the given container need to be filled to hold 1 000 000 units? a fuel tank holding 50 litres b packet containing 250 sugar cubes c school hall seating 400 students d rainwater tank holding 2000 litres e case packed with 100 oranges f carriage for 80 passengers g restaurant feeding 125 diners h computer disk cartridges with 40 disks i crates holding 160 cans j stackers storing 8 CDs

WHOLE NUMBERS (CHAPTER 1)

17

5 Arrange these planets in order of their distance from the Sun, starting with the closest. Venus 108 200 000 km Saturn 1 427 000 000 km Earth 149 600 000 km Uranus 2 870 000 000 km Mercury 57 900 000 km Jupiter 778 300 000 km Pluto 5 900 000 000 km Neptune 4 497 000 000 km Mars 227 900 000 km 6 Use the table to answer these. a Which continent has the greatest area? b Name the continents with an area greater than 20 million square kilometres. c Find out which continents are completely in the Southern Hemisphere.

Continent Africa Antarctica Asia Australia Europe North America South America

Area in square km 30 271 000 13 209 000 44 026 000 7 682 000 10 404 000 24 258 000 17 823 000

7 How long would a car, travelling non-stop at 100 kilometres per hour, take to travel a million kilometres? 8 How long would a motor cyclist travelling non-stop at 50 kilometres per hour take to travel one million kilometres? 9 How many hours would a jumbo jet, flying non-stop at 500 kilometres per hour, take to fly 1 million kilometres?

10 A $5 note is 135 mm long. a How far would one million $5 notes laid out end to end in a straight line stretch? b If you walked from one end of the line to the other at a speed of 5 kmph, how long would it take? 11 How long would a satellite orbiting the earth at 8000 kmph take to fly 1 million kms? 12 One million one dollar coins stacked on top of one another would be 2700 metres high. That is about 8 times higher than Auckland’s Sky Tower, 9 times higher than Sydney’s Centrepoint Tower and over 3 times higher than the DIB-200 in Tokyo.

800m 800m

447m

How many coins are needed to build a stack:

400m

a one metre high (to the nearest 10 coins)

300m

b the height of each of the illustrated buildings (to the nearest 1000 coins)?

Comparative sizes of structures 380m 328m 305m

300m

200m

100m

0 DIB-200, Tokyo

NUMBER SEARCH PROBLEMS

Click here for some Number Search Problems.

Sears Tower, Chicago

Empire State Building, New York

Sky Tower, Auckland

Centrepoint Tower, Sydney

Eiffel Tower, Paris

18

WHOLE NUMBERS (CHAPTER 1)

Unit 6

Number opposites

Negative numbers are the opposite of positive numbers.

Some words showing number opposites are:

Zero is our reference point. negative numbers

positive numbers

-4 -3 -2 -1

0

1

2

3

Negative 5o C below zero (¡5o C) a decrease of 2 kg a loss of $1000 10 km south

Positive 5o C above zero an increase of 2 kg a profit of $1000 10 km north

¡3 is the opposite of +3

4

Negative numbers can be shown as ¡3 or ¡3 Positive numbers can be shown as +3 or +3

Opposites are the same distance from 0.

Exercise 6 1 Copy and complete the table:

Statement a b c d e f g h i j

20 m above sea level 45 km south of the city a loss of 2 kg in weight a clock is 2 min fast she arrives 5 min early a profit of $4000 2 floors above ground level 10o C below zero an increase of $400 winning by 34 points

Directed number +20

Opposite to statement 20 m below sea level

2 Write positive or negative numbers for the position of the lift, the car, the parking attendant and the rubbish skip.

(Use the bottom of each object.)

Directed number ¡20

+3 +2 +1 0 -1 -2 -3 -4 -5

ground level

3 If right is positive and left is negative, write the numbers for the positions of A, B, C, D and E using zero as the reference position. B

A -4

E

D

0

C +4

4 Write these temperatures as positive or negative numbers. Zero degrees is the reference point. a 11o above zero b 6o below zero c 8o below zero d 29o above zero e 14o below zero 5 Write these gains or losses as positive or negative numbers: a $30 loss b $200 gain c $431 loss d $751 loss e $809 gain

WHOLE NUMBERS (CHAPTER 1)

19

6 If north is the positive direction, write these directions as positive or negative numbers: a 7 metres north b 15 metres south c 115 metres south d 362 metres north e 19:6 metres south +6 is the same as 6. Usually we simply say “6”. If a number is negative we must use the minus sign.

7 Write the opposite of the following numbers: a +6 b ¡4 c 16 d 0 e ¡2 f ¡40 8 4 + 7 = 11 whereas 4 ¡ 7 = ¡3

DEMO

This can be seen by movement along the number line: -7

+7

0

-5

5

10

4

-3

11

If necessary, use the number line to find: -10

a e i m q

0

-5

b f j n r

5+6 2¡6 ¡2 + 1 4+2 ¡2 ¡ 4

5¡6 2+6 ¡2 + 3 4¡2 7¡3

5

10

c g k o s

2+7 ¡1 + 2 ¡2 ¡ 3 2¡4 3¡7

9 This number line is vertical. As you go up the number line, the numbers increase, and as you go down, the numbers decrease. a Write the directed number for each of the points marked on the number line. b Write i iii v vii 10

True or False for these: B is higher than D ii D is lower than A iv C>E vi B and D are opposites viii °C 40

A 30 D 20 E

10

above freezing

B 0 F C

-10 -20

below freezing

A<E B
d h l p t 5 4 3 2 1 0 -1 -2 -3 -4 -5

2¡7 ¡1 ¡ 2 2¡3 ¡2 + 4 ¡3 + 7 A B

C D

E

The temperatures of cities A, B, C, D, E and F were recorded at 12 noon on a certain day last year. a What was the temperature of each of the cities? b How many o C is city D warmer than city: i E ii B iii F iv C? o c How many C is city C cooler than city: i A ii E iii F iv B? d What is the difference in temperature between: i A and B ii D and E iii E and C iv F and C v B and F vi D and F?

20

WHOLE NUMBERS (CHAPTER 1)

Unit 7

Review of chapter 1

Review set 1A 1 Write T(true) or F(false) to the following statements: a Counting from one to one million in units would take about one million seconds. b 1 000 000 = 1000 £ 1000 c 10 000 000 6= 10 £ 100 £ 1000 d 100 £ 100 > 1000 ¥ 10 2 Use the digits 3, 8, 0, 4, 1, 7 to make the largest number you can. 3 What is the place value of the 8 in the following numbers? a 3894 b 508 415 c 856 042

d

38 475 042

4 Write in ascending order (smallest number first): 673 502, 674 551, 654 662, 765 442, 750 467 5 Round off: a 35 to the nearest 10 6

a c e g

b

4384 to the nearest 1000

Find the difference between 207 and 28: Determine the product of 17 and 26. Determine 205 £ 21. Determine 20 £ 5 £ 35:

b d f h

c

463 994 to one figure.

Find the quotient of 432 and 16. By how much does 431 exceed 258? What number is 642 greater than 179? Estimate $2:95 £ 48:

7 Write the numeral for: 126 350 greater than four million. 8 Find an approximate value for: 650 £ 750: 9 How many $27 concert tickets can be bought for $1026? 10 During 3 days of practice, a golfer hit 24 more balls each day than the previous day. How many golf balls did she hit in the 3 days if she hit 376 on the first day? 11 Students collect and fill 154 crates which each hold two dozen bottles. A recycling depot pays 5 cents for each empty bottle. How much money did the students raise for the school? 12 Damien bought a pair of jeans for $39 and a T-shirt for $32. How much change did he get from $100? 13 -5 -4 -3 -2 -1

0

1

2

3

4

5

6

Write out the operation represented on the number line above. 14

a Write the opposite of a withdrawal of $30 from the bank. b Find i

2¡5

ii

¡2 ¡ 5

WHOLE NUMBERS (CHAPTER 1)

21

Review set 1B 1 Write True or False for the following statements: a 8703 ¡ 6679 = 2124 b The place value of the 6 digit in the number 526 947 857 is a million. c 504 £ 1998 + 1 000 000 d 4 863 663 < 4 863 363 2 Round off a 25 to the nearest 10

3636 to the nearest 1000

b

c

527 382 to one figure.

3 Use the digits 4, 7, 3, 0 and 1 to make the largest number you can. 4 What is the place value of the 4 in the following numbers? a 7846 b 204 320 c 6425 5

a c e g

Find Find Find Find

the the the the

difference between 205 and 197. product of 32 and 15. value of 302 £ 23. value of 30 £ 4 £ 25 £ 53

b d f h

d

24 096 035

Find the quotient of 594 and 9. By how much does 450 exceed 231? What number is 387 greater than 199? Estimate $3:85 £ 52

6 Write these numbers in ascending order (smallest first): 569 207, 96 572, 652 097, 795 602, 79 562 7 Write the numbers that are: a thirty seven greater than one hundred and ninety four thousand two hundred and twenty b the product of 395 and 49

c

fifty seven multiplied by zero.

8 Estimate the following using 1 figure approximations: a 197 £ 234 b 1802 ¥ 387 9 Find the cost of 24 concert tickets at $112 each. 10 Janet has 5 children. There are triplets aged 16 years and two other children aged 13 and 11 years. What is the combined age of Janet’s children? 11 Find the approximate mass in grams of one can of cat food if a carton containing 96 cans weighs 18 kilograms. (1 kilogram = 1000 grams) 12 Would $200 be enough to pay for:

a $69 ‘Cheap Deal’ flight to Melbourne, a $114 return ticket and an $18 ticket to the football? Show your working. 13 -8

-6

-4

-2

0

2

4

6

8

Write out the operation represented on the number line above. 14

a Write the opposite of losing by 2 goals. b On the weather report Philip noticed that the temperature was 25o C in Adelaide, ¡1o C in London and ¡5o C in Berlin. i How much warmer was it in Adelaide than London? ii How much colder was it in Berlin than London?

22

NUMBER PROPERTIES

CHAPTER 2

Unit 8

Number operations and their order

+ means add, plus, more

The sum of 7 and 8 is 7 + 8 = 15.

¡ means take, subtract, less

The difference between 7 and 8 is 8 ¡ 7 = 1.

£ means times, multiply

The product of 7 and 8 is 7 £ 8 = 56.

¥ means divide, share

The quotient of 8 and 2 is 8 ¥ 2 = 4. dividend

divisor

quotient

Exercise 8 1 Solve the following problems: a Find the sum of 23 and 37. b Find the difference between 37 and 56. c Find the product of 14 and 28. d Calculate the quotient of three hundred and seventy four and seventeen. e By how much is the sum of 90 and 50 greater than the difference between 90 and 50? f What is the product of the first 5 even numbers? g Find the sum of the first 10 odd numbers. h How much greater than 273 is 632? i How much smaller than 394 is 209? 2 Mount Cook in New Zealand is 3765 m above sea level, whereas Mount Kosciuszko in Australia is ´ 2231¡¡m high. How much higher is Mount Cook than Mount Kosciuszko? ´ 3 A couple saved $12 654 towards the cost of their first home. How much would they need to borrow if they are to purchase a home for $109 850?

FOR SALE $

109 850

4 What is the difference between the AFL’s prediction of 120 000 for the first three games in July and the actual attendances of 36 287, 27 615 and 32 974? 5 Four friends shared a Lotto payout of $13 828. How much did they each win? 6 The organisers of a concert set out four identical blocks of seating. If each block has 35 rows, each with 40 seats, how many people can be seated? 7 A softdrink company fills and packs 540 dozen cans of drink every hour. How many cans are filled and packed in an 8 hour day? 8 All rooms of a hotel cost $88 per day to rent. The hotel has 5 floors and 37 rooms per floor. What is the total rental received per day if the hotel is fully occupied? 9 An investor bought 5000 shares at $2:15 each on the stock market. He sold half of them at $2:65 each. A few weeks later he sold the rest at $2:85 each. What was his total profit? 10 A basketball team scores the following goals in their season of 10 games: 24, 15, 108, 26, 35, 23, 31, 19, 27, 50. What is their total goal score for the season? 11 Jacob’s nursery sells 145 dozen flower plants each week. How many plants do they sell in 16 weeks? 12 If an aeroplane seats 143 passengers and 19 full aircraft leave Adelaide each day, how many passengers are carried?

NUMBER PROPERTIES (CHAPTER 2)

Order of operations Examples: ² 3+4£5 = 3 + 20 f£ firstg = 23

Rules for doing mathematical operations in the correct order B E D M A S

operate within the brackets first exponents (or powers) next division and multiplication (left to right) addition and subtraction (left to right)

13 Find: a d g j m

²

b e h k n

6+9¡7 18 ¥ 3 + 9 8¥2+7 4+5¡3£2 5£3£2¡1

The letters BEDMAS will help you remember the correct order.

15 ¡ (4 + 6) ¥ 5 = 15 ¡ 10 ¥ 5 fbrackets firstg = 15 ¡ 2 f¥, left to rightg = 13

7£6+4 50 + 6 ¡ 7 5+4£3¡1 7£9¥3 7£9¡3£9

14 Find: (Remember to complete the brackets first.) a (9 + 6) £ 2 b (23 ¡ 7) £ 2 d 4 £ (7 + 5) e 26 ¡ (7 ¡ 5) £ 4 g (27 ¡ 27) £ 9 h (13 ¡ 7 + 5) £ 8 15 Find: (Remember the order for exponents.) a 2 £ 32 ¡ 9 b 22 ¥ 4 + 1

c

c f i l o

18 + 9 £ 3 7+8¥2 24 ¥ 6 + 4 £ 5 4+3£8¥4 8+6¥3£4

c f i

(16 + 4) ¥ 10 (27 + 5) £ 3 32 ¡ (6 £ 3) ¥ 9

(9 + 7) ¥ 22

DEMO

exponent

32

=3£3 =9

16 Make these statements true by putting in brackets where necessary: a 6 + 3 £ 2 = 18 b 21 ¡ 7 £ 3 = 0 c 8+4¡3£2=6 d 50 ¥ 5 + 5 = 5 e 5 £ 3 ¡ 1 + 7 = 17 f 4 + 4 £ 4 ¥ 16 = 2 g 50 ¥ 5 + 5 = 15 h 9 £ 7 + 5 + 2 = 110 i 9 £ 7 + 5 ¡ 2 = 66 17 Put brackets where necessary to make each answer correct: a 96 ¥ 4 + 8 £ 10 ¡ 9 = 71 b 96 ¥ 4 + 8 £ 10 ¡ 9 = 32 c 96 ¥ 4 + 8 £ 10 ¡ 9 = 95 d 96 ¥ 4 + 8 £ 10 ¡ 9 = 8 18 State whether the following equations are true or false: a (3 + 2) £ 6 ¥ 10 = 3 b 18 ¥ (3 £ 2) + 5 = 17 d 3 + 6 £ 4 ¥ 2 = 18 e 5 + 5 £ 5 ¥ 10 = 5 g 18 + 7 ¡ 3 + 4 = 27 h 3 £ (5 + 7) ¥ 12 = 3 19

a Design a worded problem to go with the operations: i

(2 + 3) £ 4

ii

8 ¥ 2 + 5.

b Use diagrams or pictures to represent the problems in a.

c f i

(3 + 6 ¥ 2) ¥ 3 = 2 40 ¥ 10 £ 4 + 4 = 20 54 = 3 + 6 £ 3

For 4 £ 5 ¡ 10 = 10 : Susan saved $5 each week for 4 weeks. Then she bought a book for $10. She had $10 left.

23

24

NUMBER PROPERTIES (CHAPTER 2)

Factors of natural numbers

Unit 9 The factors of a natural number are the natural numbers whose product produces the number. 8 4 2

2 2

The common factors of two or more numbers are the factors shared by the numbers.

8 can be written as 8 = 1 £ 8 or 8 = 2 £ 4

8 has 1, 2, 4 and 8 as factors 12 has 1, 2, 3, 4, 6 and 12 as factors

1, 2, 4 and 8 are the factors of 8.

1, 2 and 4 are the common factors of 8 and 12. The highest common factor (HCF) of 8 and 12 is 4.

They divide exactly into 8.

A prime number has exactly two factors, 1 and itself. For example, 5 has two factors, 1 and 5. It is prime. A composite number has more than two factors. For example, 8 has four factors 1, 2, 4 and 8. It is composite.

Exercise 9 1

a c d

2 List a e i

List all the factors of 15. b List all the factors of 16. Complete this equation: 16 = 2 £ ::::: Write all pairs of factors which multiply to give 16. all the factors of each of the following numbers: 8 b 36 c 48 f 63 g 39 j 35 k

DEMO

d h l

40 30 60

3 Copy and complete the factorisations below: a 33 = 3 £ ::::: b 55 = 5 £ :::::: d 50 = 10 £ ::::: e 27 = 9 £ ::::: g 35 = 5 £ ::::: h 72 = 8 £ ::::: j 49 = 7 £ ::::: k 121 = 11 £ ::::: m 64 = 16 £ ::::: n 108 = 12 £ :::::

c f i l o

4 Write the largest factor (not itself) of each of the following numbers: a 12 b 18 c 27 e 44 f 75 g 90

42 84 81

28 = 4 £ ::::: 42 = 2 £ ::::: 99 = 11 £ ::::: 48 = 6 £ ::::: 88 = 2 £ ::::: d h

48 39

5 List all the prime numbers less than 50. 6 Copy and complete these factor trees: a b 28 2

14

c

80 2

40 2

75 3

20

25

NUMBER PROPERTIES (CHAPTER 2)

To write a number as a product of prime numbers, we divide it by prime numbers, starting with the smallest prime number first.

25

All composite numbers can be written as the product of prime factors.

For example, for 24 and 18: 2 2 2 3

2 18 24 12 3 9 6 3 3 3 1 1 24 = 2 £ 2 £ 2 £ 3 and 18 = 2 £ 3 £ 3 Notice that 2 £ 3 is common to both 24 and 18. So 2 £ 3 = 6 is the highest common factor (HCF) of 24 and 18. 7 Express each of the following numbers as the product of prime factors: a 16 b 36 c 28 d 56 f 75 g 168 h 252 i 305 8 Find a e i m 9

the highest common factor of: 9, 15 b 7, 28 20, 40, 80 f 25, 50, 75 56, 14, 28 j 27, 108, 63 26, 39, 13 n 25, 35, 50, 60

c g k o

18, 27, 24, 96,

e j

30 45 60, 132 32, 48, 72

d h l p

63 392 21, 33, 48, 10,

28 55, 77 8, 4 18, 36, 20

a Beginning with 14, write three consecutive even numbers. b Beginning with 35, write five consecutive odd numbers. c Write two even numbers which are not consecutive and which add to 10.

10 Use the words “even” and “odd” to complete the following sentences correctly: a The sum of two even numbers is always .....

Even numbers are always divisible by 2. They have 2 as a factor.

b The sum of two odd numbers is always ..... c The sum of three even numbers is always ..... d The sum of three odd numbers is always .....

Odd numbers are not divisible by 2.

e The sum of an odd number and an even number is always ..... f When an even number is subtracted from an odd number the result is .....

There is only one even prime number.

g When an odd number is subtracted from an even number the result is .....

How odd!

h The product of two odd numbers is always ..... i The product of an even and an odd number is always .....

Click on the icon to see Eratosthene’s method for finding prime numbers.

DEMO

Challenge: You must place each factor once only on the diagram. These are the rules: Circle A contains factors of 36. Circle B contains factors of 18. Circle C contains factors of 24.

B A C

26

NUMBER PROPERTIES (CHAPTER 2)

Unit 10

Multiples and divisiblity rules

To find multiples of a number, we multiply it by the natural numbers. The multiples of 10 are 1 £ 10, 2 £ 10, 3 £ 10, 4 £ 10, ..... that is, the multiples of 10 are 10, 20, 30, 40, 50, 60, ..... The multiples of 15 are 15, 30, 45, 60, 75, 90, ..... 30 and 60 are common multiples of 10 and 15. 30 is the lowest common multiple (LCM) of 10 and 15.

The lowest common multiple of two or more numbers is the smallest number into which all of them will divide exactly.

Exercise 10 1 List the numbers from 1 to 30. a Put a circle around each multiple of 3. b Put a square around each multiple of 4. c List the common multiples of 3 and 4 which are less than 30. 2 In the question following use the list of multiples of 15 given: 15 30 45 60 75 90 105 120 135 150 State which of these numbers are common multiples of both: a 15 and 10 b 15 and 9 c 20 and 30

d

4 and 30

3 Find the lowest common multiples of these sets: a 3, 6 b 4, 6 e 6, 8 f 2, 4, 6

d h

12, 15 3, 4, 5

c g

5, 8 15, 12

4 A piece of rope is either to be cut exactly into 12 metre lengths or exactly into 18 metre lengths. Find the shortest length of rope satisfying these requirements. 5 Two bells toll at intervals of 6 and 9 seconds respectively. If they start to ring at the same instant, how long will it take before they will again ring together?

6 Two different arcade games cost 4 and 5 tokens respectively. Two brothers each play one of the games and spend the same amount. How many tokens did they each need? 7 Three long distance runners train to drink at 3, 5 and 6 kilometre intervals respectively. At what distance will they all drink? 8 Four students each have a special bar to play. The piano player repeats her piece every 4th bar, the saxophone player every 5th bar, the drummer every 6th bar and the xylophone player every 10th bar. If the piece of music contains 100 bars, at what stages will they all play together?

NUMBER PROPERTIES (CHAPTER 2)

27

A number is divisible by another if we get a whole number when we divide the first number by the second.

Standard divisibility tests Number

Divisibility Test

2

If the last digit is 0 or even, then the original number is divisible by 2.

3

If the sum of the digits is divisible by 3, then the original number is divisible by 3.

4

If the number made from the last two digits of a number is divisible by 4 then the original number is divisible by 4.

5

If the last digit is 0 or 5 then the number is divisible by 5.

6

If the number is even and divisible by 3 then it is divisible by 6.

9 Answer true or false: a 45 is divisible by 5 c 92 is divisible by 3 e 56 235 is divisible by 3 g 1088 is divisible by 6 10 Which of these numbers are divisible by 3? a 75 b 96 e 509 f 816 i 817 203 j 246 642

There is a test for divisibility by 7 but it is too difficult to use.

b d f h

75 is divisible by 2 126 is divisible by 3 1042 is divisible by 4 2124 is divisible by 6

c g k

186 9657 123 456 789

d h l

254 8433 124 124 124

11 Find all the possible values of the missing digit if these numbers are divisible by 3: a 32 b 224 c 1282 d 6234 e 5287 f 4527 g 892 216 h 348 210 12 Decide whether these numbers are divisible by 4: a 3784 b 8804

c

6794

d

32 418

A number of attributes

Activity What to do:

Construct 3 intersecting circles as shown. multiples of 3 and even Use a pencil to write down all the even numbers A B from 1 to 100 inside circle A. Inside circle B write down all the multiples of 3 between 1 and 100. Transfer any even number which is also a multiple of 3 to the area where both circles A and B intersect. even and odd and square square In circle C write down all the square numbers from 1 to 100. C Transfer common numbers to the intersecting areas. From your diagram, what numbers are: a multiples of 3 and square numbers b square numbers and even c in the shaded areas? 1 2

3 4 5 6 7 8

a b

What are the attributes of the numbers in the central shaded area? Using the numbers from 30 to 50, what numbers have the attributes of being not even, not square and not multiples of 3?

28

NUMBER PROPERTIES (CHAPTER 2)

Powers of numbers

Unit 11

A power, index or exponent shows that a number has been multiplied by itself several times.

34 means 3 has been multiplied by itself 4 times. 34 = 3 £ 3 £ 3 £ 3 (= 81) 34 6= 3 £ 4 as 3 £ 4 = 12. 2 £ 2 £ {z 2 £ 3 £ 3} =

|

expanded form

4

3

2| 3 {z £ 32}

power or exponent form

power, index or exponent

base number

32 £ 24 = 3 £ 3 £ 2 £ 2 £ 2 £ 2 = 9 £ 16 = 144

Exercise 11 1 Write each number in exponent form: a 6£6£6£6 b d 5£5£5£3£3 e g 2£2£4£4£4£5£5 h

c f i

13 £ 13 £ 13 £ 13 £ 13 8£8£8£3£3£3 3£3£3£5£5£9£9

2 Convert into a single whole number without using a calculator: a 2£3£5 b 22 £ 3 d 2 £ 32 £ 5 e 22 £ 32 £ 11 g 3 £ 42 £ 10 h 25 £ 32 £ 10

c f i

33 £ 2 23 £ 52 £ 11 12 £ 23 £ 34

3 Write in expanded form and use a calculator to find the value of: a 54 b 73 d 125 e 1003

c f

37 145

4 Use a calculator to write as whole numbers: a 25 + 74 b 35 £ 53 d 82 £ 92 £ 15 e 55 ¡ 44 ¡ 33 ¡ 22

c f

66 + 33 52 £ 53 £ 54

5 Without using a calculator, work out the a 25 and 52 b 2 d 9 £ 2 and 9 e 10 2 g 2 and 10 h

4£4£4£4£4£4 9£9£2£2£2£2 11 £ 11 £ 11 £ 3 £ 3 £ 6

difference between the pairs of values: 34 and 43 c 63 and 36 35 and 3 £ 5 f 84 and 83 53 and 3 £ 5 i 503 and 50 £ 50 £ 50

6 Arrange these power expressions in ascending order: a 38 , 65 , 210 , 84 , 57 , 104 b 7 Work out in simplest form the answers to the following pattern of numbers:

12 112 1112 11112 11 1112 111 1112 1 111 1112

= = = = = = =

58 , 95 , 273 , 1003 , 127 , 10002

What does the resulting pattern of answers have in common with the following words: “eye”, “dad”, “mum”, “radar”, “racecar”, “rotator”?

0 0 0 0 0 0 0 Units

1 0 0 0 0 0 0 Tens

1 0 0 0 0

1 0 0 0 0 0 Hundreds

1 0 0 0 Ten thousands

Hundred thousands

Millions

101 102 = 10 £ 10 = 103 = 10 £ 10 £ 10 = 104 = 10 £ 10 £ 10 £ 10 = 1 105 = 10 £ 10 £ 10 £ 10 £ 10 = 1 0 106 = 10 £ 10 £ 10 £ 10 £ 10 £ 10 = 107 = 10 £ 10 £ 10 £ 10 £ 10 £ 10 £ 10 = 1 0 0 Tens of millions

Powers with Base 10

Thousands

NUMBER PROPERTIES (CHAPTER 2)

We can write the number 5042 as: 5042 = (5 £ 1000) + (4 £ 10) + (2 £ 1) 5042 = (5 £

103 ) +

(4

£ 101 )

expanded notation

+ (2 £ 1)

power notation

8 Write the simplest numerals for each of the following: a (8 £ 100 000) + (6 £ 10 000) + (2 £ 1000) + (9 £ 100) + (5 £ 10) + (3 £ 1) b (6 £ 1 000 000) + (9 £ 100 000) + (8 £ 10 000) + (7 £ 1000) + (9 £ 10) + (6 £ 1) c (3 £ 1 000 000) + (5 £ 10 000) + (7 £ 100) + (9 £ 1) d (4 £ 106 ) + (8 £ 105 ) + (9 £ 104 ) + (2 £ 103 ) + (2 £ 102 ) + (6 £ 101 ) e (2 £ 107 ) + (3 £ 105 ) + (6 £ 104 ) + (9 £ 103 ) + (6 £ 101 ) + (8 £ 1) f (106 ) + (104 ) + (103 ) + (102 ) + (9 £ 101 ) g 9 thousands and 8 hundreds and 3 tens and 6 units h 8 hundred thousands + 9 ten thousands + 6 hundreds + 3 tens + 7 units i 5 ten millions + 8 hundred thousands + 7 ten thousands + 5 thousands 9 Write these numbers using expanded notation: a 9738 b 29 782 e 800 888 f 1 247 091

c g

d h

40 404 49 755 400

10 Expand these numbers using power notation: a 658 b 3874 c 95 636 d e 505 750 f 1 274 947 g 36 600 000 h i four hundred thousand six hundred and eighty seven j twenty three million, six hundred and ninety seven thousand five hundred

657 931 6 777 777 100 100 4 293 375

11 Challenge: z

103

}|

{

z

104

}|

{

£ 10 £ 10 £ 10 £ 10 £ 10 £ 10} 103 £ 104 =10 | {z 107

= 107 a Find:

i

102 £ 105

ii

103 £ 106

iii 104 £ 104

b Without writing 1025 as |10 £ 10 £ ::::: £ 10}, find: {z 25 of them

i

25

10

4

£ 10

ii

1025 £ 1020

iii

1025 £ 1060

29

30

NUMBER PROPERTIES (CHAPTER 2)

Square and cube numbers

Unit 12

A square number is the product of two identical numbers. For example: 4 = 22 , 9 = 32

For 22 we say “two squared”.

and 16 = 42 .

4, 9 and 16 are square numbers. A cube number is the result of cubing a whole number. For example: 8 = 23 , 27 = 33

and 64 = 43 .

8, 27 and 64 are cube numbers.

7Z = 7 ‘seven to the one’

For 23 we say “two cubed”.

7X = 49 ‘seven squared’

7C = 343 ‘seven cubed’

~` is the symbol meaning “the square root of ”.

The square root of a number is the number that when multiplied by itself gives the first number. p 9 = 3, that is, the square root of 9 is 3. p 16 = 4. 42 = 16 so

For example: 32 = 9 so

Exercise 12 1 Find a e i

the value of: 42 322 22 + 42

2 Use a calculator to find: a 1362 3 42 = 16 ends in a 6 and

b f j

52 722 (2 + 4)2

c g

72 52 ¡ 22

d h

102 (5 ¡ 2)2

b

4082

c

11672

d

23052

52 = 25 ends in a 5.

a List all the possible numbers that a square number could end in. b Is 638 254 916 823 620 058 a square number? 4 Find the square root of: a 1 b 16

c

5 Find: p a 49

c

b

p 64

36 p

100

d

81

e

144

d

p 0

e

p 400

6 Find the first 10 cube numbers, beginning with 13 = 1. 7 Find: a 23 ¡ 22

b

53 ¡ 5

c

43 + 23

d

73 ¡ 72

NUMBER PROPERTIES (CHAPTER 2)

8 ,

31

represents the first 3 cube numbers.

,

a Draw a sketch of the representation of 43 . b Explain why these diagrams do represent 13 , 23 , 33 and 43 .

Square number patterns

Activity 1

A

B

C

Diagram A shows that the sum of the first 3 odd numbers is 32 i.e., 1 + 3 + 5 = 9 B shows that the sum of the first 4 odd numbers is 42 i.e., 1 + 3 + 5 + 7 = 16 C shows that the sum of the first 5 odd numbers is 52 i.e., 1 + 3 + 5 + 7 + 9 = 25: a b c d

Draw the next two diagrams in this pattern, D and E. Write down the statements for D and E like those for A, B and C. Add the first and last number in each sum then divide it by 2. What do you find? What numbers when squared are equal to these sums? i 1 + 3 + 5 + 7 + ::::: + 21 = ii 1 + 3 + 5 + 7 + ::::: + 23 = iii 1 + 3 + 5 + 7 + ::::: + 35 = iv 1 + 3 + 5 + 7 + ::::: + 39 =

2 Use graph paper. Construct overlapping squares as shown. On the top right corner of each square, write down the total number of squares enclosed by the larger square. On your piece of paper, what is the largest number of smaller squares that you can enclose with a larger square? List your square numbers in ascending order.

1

4

9

16

25

36

3 Use the following formula to complete the exercises. A square number results from one being added to the product of any four consecutive whole numbers. 1 £ 2 £ 3 £ 4 + 1 = 25 = 52

2 £ 3 £ 4 £ 5 + 1 = 121 = 112

a

3 £ 4 £ 5 £ 6 + 1 = ...... = 192

b

4 £ 5 £ 6 £ 7 + 1 = ...... = 22

c

5 £ 6 £ 7 £ 8 + 1 = ...... = n2

d

6 £ 7 £ 8 £ 9 + 1 = ...... = ¢2

e

7 £ 8 £ 9 £ 10 + 1 = ...... = °2

Use any four consecutive whole numbers to show that the formula works. 4 Find other patterns and formulae using square numbers to share with the class. Click on the icon for an activity on Triangular numbers.

ACTIVITY

32

NUMBER PROPERTIES (CHAPTER 2)

Problem solving methods

Unit 13 Method 1: Trial and error

Method 2: Making a table and looking for a pattern

Bob and Vince are brothers. The product of their ages is 48 and the sum of their ages is 16. How old are they?

How many top and bottom rails would be needed to build a fence with 55 posts?

As 6 £ 8 = 48, we guess their ages are 6 and 8. But 6 + 8 = 14 6= 16, so 6 and 8 are not correct. Try again: guess 12 and 4 as 12 £ 4 = 48. 12 + 4 = 16 X So Bob and Vince are 12 and 4 years old.

We draw a table: Number of posts Number of rails

2 2

3 4

4 6

5 8

The pattern is: We subtract 1 from the number of posts and multiply the result by 2. So for 55 posts we need (55 ¡ 1) £ 2 = 54 £ 2 = 108 rails

Exercise 13 1 Solve these problems using the trial and error method: a Find consecutive whole numbers that add up to 51. (Consecutive numbers are numbers that follow each other, for example, 4 and 5 or 24 and 25.) b In a jar there are some spiders and beetles. If there are 13 creatures in total and the number of legs adds to 86 how many of each creature are in the jar? c How many two digit numbers are there in which the tens digit is less than the ones digit? d Using the digits 2, 3, 4 and 5 in that order and the symbols £, ¡, + in any order, make a mathematical sentence that equals 9. 2 Solve the following using Method 2. a How many different two course meals can I make with eight main courses and 16 desserts? Hint: Try with two mains and one dessert, then two mains and two desserts, etc. Make a table of your findings and look for a pattern. b A club team has three shirt colours (yellow, green and grey) and four shorts colours (black, white, blue and red). How many different uniforms are possible, if they have different colours for shorts and tops? (Remember to start with a smaller number of tops and shorts and look for a pattern.) c If there are 15 people in a room and everyone must shake hands with everyone else how many handshakes will there be in total? Hint: What if there were only two people in the room? Three people in the room? etc. Make a table and look for a pattern. d How many diagonals would a 12-sided polygon have? (What is the name of a 12-sided polygon?) Remember that a diagonal is a straight line that joins two vertices of a polygon and is not a side of the polygon.

For example:

A rectangle has 2 diagonals.

A pentagon has 5 diagonals.

NUMBER PROPERTIES (CHAPTER 2)

33

Method 3: Modelling or drawing a picture

Method 4:

Jan has three different tops and two different skirts. How many different combinations of skirts and tops can she wear?

I think of a number, divide it by 3 then square it. The result is 16. What is the number?

There are six different outfits.

The answer is 12.

You can use symbols instead of drawing if you wish.

Working backwards

We start with the result and work backwards: Start with 16. (the result) Find the square root. It is 4. Multiply by 3 to get 12. (the opposite of dividing by 3) Always check your answer by working the other way: 12 ¥ 3 = 4, 42 = 16 X

3 Do the following by modelling or drawing the picture. Remember to state your answer clearly and show any working you do. a A square table has four seats around it. In how many different ways can four people sit around the table? b This one can be modelled by counting seconds instead of minutes with a person/counter leaving the group at the correct time. Year 7 and 8 students are doing the same orienteering course. A year 7 student leaves every 6 minutes and a year 8 student every 3:5 minutes. If the event begins at 9 am with one student from each year group leaving together, when is the next time a student from each year level will leave together? c Darren numbered the pages of his art folder using a packet of stickers. On each sticker there was one digit from 0 to 9. He started with page 1. When he finished he noticed he had used 77 stickers. How many pages were in his art folder? d When Stephen put 10 counters in a bag it was

When James put 13 in his bag it was Who had the biggest bag?

1 2

1 3

full.

full and when Blair put 7 in his bag it was

1 4

full.

4 Attempt these problems using the working backwards method. a The number of rabbits in my rabbit farm double each month. At the end of last month there were 24 000 rabbits. When were there 1500 rabbits? b I think of a number and multiply it by 2, then add 2 and subtract 10. This result is then divided by 4. I end up with 4. What number did I think of originally? c

Nikora left home at a certain time. He biked for 20 minutes, then walked for a further 15 minutes. He rested there for half an hour before continuing on to Sam’s house which was a further 25 minutes walk away. Nikora played at Sam’s house for 45 minutes before moving on to his grandmother ’s home, which took him another 20 minutes. He arrived at his grandmother’s home at noon, just in time for lunch. What was the time he left his home that morning?

d I am trying to work out David’s age. He told me that if I add 6 to his age, then multiply by 3, add another 17 then divide by 8 I will end up at his sister Susie’s age, which is twice his brother Bill’s age which is 5 years old. How old is David?

34

NUMBER PROPERTIES (CHAPTER 2)

Review of chapter 2

Unit 14 Review set 2A 1

a

Find the difference between 10 and 2.

b

Decrease 10 by 2.

c

Find the quotient of 10 and 2.

d

Find 10 to the power of 2.

e

Share 12 between 3:

f

Increase 12 by 3. p h Find 49.

g Calculate 12 lots of 3. 2 Evaluate the following: a 5+6£3¡9

b

c

(5 + 6) £ 3 ¡ 9

5 + (6 ¥ 3 £ 9)

3 Insert brackets where necessary to make the following statements correct: a 5 + 2 £ 4 = 28 b 6£7¡5¥3=4 4 Copy and complete the factor tree:

30 2

15

5 Write down the 4th prime number after 23. 6

a c

List all the factors of 56. b Find the highest common factor of 12 and 30.

Determine the LCM of 4 and 10.

7 For the set of numbers between 30 and 60, list all the: a multiples of 8 b square numbers

c

prime numbers

8 A motorist has her oil, battery and radiator checked every 5000 km, her tyres rotated every 15 000 km and her engine tuned every 20 000 km. At what stage will all three services happen together? 9 In the following, is the first number divisible by the second number? (Answer yes or no.) a 3675, 10 b 47 368, 4 c 974 580, 5 10 Convert each number into natural form: a 32 £ 22 £ 10 b 24 £ 102 £ 2 11 Simplify: a 5 £ 104 + 7 £ 102 + 3 £ 10 b (4 £ 1 000 000) + (2 £ 10 000) + (3 £ 100) + (5 £ 1) 12 List 4 consecutive powers of 10 after 100. 13 Use trial and error to solve the following problem:

Hera paid for her $69 jeans with coins she had saved. She used only $2 and $1 coins and noticed she was able to pay using the same number of each coin. How many $1 coins did she use to pay for her jeans?

d c

129 408, 3

102 £ 102 £ 102

NUMBER PROPERTIES (CHAPTER 2)

35

Review set 2B 1 Simplify: a 3£5¡4£2 c 12 ¥ 4 £ 2 + 7

b d

3 + 13 £ 2 (3 + 4) £ 2 + 20 ¥ (9 ¡ 4)

2 Insert brackets where necessary to make the following statements correct: a 32 ¡ 4 £ 2 = 10 b 4 + 2 £ 8 ¡ 5 = 18 3 Copy and complete the factor tree:

36 2

4

a b c d e

18

List all the factors of 64. List the prime numbers between 10 and 20. Find the LCM of 15 and 21. Express 120 as the product of prime factors. Find the highest common factor of 18 and 45.

5 Which of the following are prime numbers: 27, 37, 57, 67? 6

a Which of the following are divisible by 3:

216, 732, 1649?

b Which of the following are divisible by 4:

9316, 73 046, 22 000?

7 A long piece of licorice is to be cut exactly into 4, 6 and 9 cm lengths. Find the shortest length of licorice satisfying these requirements. 8 Complete the following: a 49 = 22

b

27 = ¢3

e

d

121 = ¢2 p 400 = ¤

c

p 81 = ¤

9 Write the simplest numerals for these numbers: a 22 £ 5 £ 102 b 32 £ 7 £ 103 c 10 £ 102 £ 103 d 3 £ 104 + 7 £ 103 + 1 £ 10 + 2 £ 1 e (6 £ 1 000 000) + (5 £ 100 000) + (4 £ 1000) + (3 £ 100) + (8 £ 1) 10 Use the method of working backwards from the result to solve the following problem:

Jillian enjoyed gardening and liked to keep a record of the height of her rose bush. After recording the starting height she noted that it grew a further 5 cm before halving in height when she pruned it by mistake. It then continued to grow nicely for another 20 cm before a hedge trimmer accidently cut one third off the height. At this stage Jillian recorded the height of the rose bush to be 24 cm. How high was the rose bush when she started measuring it?

36

SHAPES AND SOLIDS

Unit 15

Points and lines

CHAPTER 3

A point marks a position. It does not have any size. We use a dot to show where it is.

B

A

plural of vertex.

D

B

à ! à ! This is line AB or BA.

B

This is line segment AB or BA.

A

A B A

This figure has 4 sides and four vertices (corner points). Vertices is the

C

Parallel lines never meet. Parallel lines are shown using arrow heads. Two lines in the same plane are either parallel or intersecting.

¡! This is ray AB.

DEMO

point of intersection

Exercise 15 1 Give two examples in the classroom which indicate: a a point b

DEMO

a line

2 In geometry, what is meant by the word(s): a vertex b point of intersection Draw diagrams to illustrate each.

c

parallel lines?

3 Give all ways of naming the straight lines shown: a b D

L

E

C

M

(Hint: In b there are 6 answers.) 4 A

What is the intersection of: Ã ! Ã ! a AB and BC Ã ! Ã ! b CB and CA?

B C

5 What is the intersection of: a AB and BC b AB and AC?

A

6 P

Q

R

S

B

C

What is the intersection of: Ã ! a PQ and PR b PQ and QS c QR and PS d PR and SQ?

SHAPES AND SOLIDS (CHAPTER 3)

7 a b

K

Correctly name the line segments which form the sides of this figure. Name the line segments which intersect at vertex M.

L

N B

A

8

C E

D

37

M

For the given figure: a name the line segments forming its sides b name the line segments which intersect at B c name 4 line segments (not all drawn) which could intersect at C.

Activity

Straight line surprises Part 1: You will need:

a sheet of blank paper, a ruler and a sharp pencil.

What to do: 1 Mark a point somewhere near the centre of the paper. 2 Line up one edge of your ruler so that it passes through the point. Draw a line along the other edge of the ruler across the paper. 3 Change the position of the ruler but still keep one edge passing through the point. Draw a second line so that it intersects with your first line. 4 Change the position of the ruler again, so one edge still passes through the point and the other edge allows you to form a triangle. Draw the third line. 5 Rule lots more lines like you did in 4. 6 Describe what happens to the shape formed by the intersecting lines as more lines are drawn. 7 Why is this shape forming? DEMO

Part 2:

DEMO

You will need: a sheet of 5 mm graph paper, a ruler, a sharp pencil. What to do: 1 On graph paper, draw a horizontal base line and mark the numbers from 0 to 16 on it as shown in the diagram on the next page. 2 Draw a vertical line at O, and mark on it the numbers from 1 to 16 at the intersection of the horizontal lines, as shown. 3 Rule a straight line from the intersection at vertical 1 to the intersection at horizontal 1. 4 Rule a straight line from the intersection at vertical 2 to the intersection at horizontal 2. Repeat this process until all the horizontal points have been joined. 5 Then draw a vertical line at 16 on the base line and repeat the pattern. 6 A real challenge is to turn the page upside down and repeat the pattern so that you have drawn 4 sets of straight lines.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 2 3 4 5 6 7 8 9 10 etc.

38

SHAPES AND SOLIDS (CHAPTER 3)

Unit 16

Angles

Classifying angles The size of an angle is measured in degrees.

Naming Angles (Three Point Notation) A

Name

Figure

Degrees

Revolution

360o one full turn

Straight angle

180o 1 2 turn

Right angle

90o 1 4 turn

Acute angle

between 0o and 90o less than 14 turn

Obtuse angle

between 90o and 180o between 14 and 12 turn

Reflex angle

between 180o and 360o between 12 and 1 turn

B

C

We call this: ² angle ABC (or angle CBA) or ² ]ABC (or ]CBA) Notice that vertex B is in the middle of the three letters. ]BCA is the angle marked ² DEMO

Exercise 16 1 Draw a diagram to illustrate: a a 12 turn b

an obtuse angle

a full turn

c

2 Classify each angle as acute, obtuse, reflex, etc: a 60o b 120o c 180o

d

a reflex angle

d

240o

e

300o

3 Use 3-point notation to name the following angles and state the type of angle in each case: a b c P A L

B

R

C

Q

4 Draw and label an angle with the name: a ]CDE b ]QPT 5

B

A

6 In the figure, name: a a right angle b an obtuse angle c two acute angles

K

angle MTD

c

d

M

reflex ]SNP

This figure contains 8 angles. Name all of them using 3-point notation. Do not list ]BAC as well as ]CAB as these are the same angle.

C

D B

A

C

D

39

SHAPES AND SOLIDS (CHAPTER 3)

7 State the size of angle AOB shown on these protractors: a 50

10 2 0

0

80 90 100 110

120

13

50

0

13

40

A

B

0

30

30

120

O

8 Measure all angles of the following triangles. Use 3-point notation to write down your answers. For example ]PRQ = 38o . A a b

170 180

10 2 0

80 90 100 110

160

170 180

0

70

150

160

O

60

0 14

150

A

O

c

0 14

40

50

0

40

A

B 70

13

base line of protractor

b 60

120

30

170 180 160

O

centre

80 90 100 110

10 2 0

40 30

150

A

70

170 180 160 150

0 14

B

13 0

60

0 14

50

80 90 100 110 70 120

0

60

10 2 0

32°

0

DEMO

B

DEMO

D E

B

F

C

9 Measure all angles of the following figures. Use 3-point notation to write down your answers: R a b B A S

Q T

D MORE EXERCISES

Click here for more practice exercises.

Angles are everywhere around us.

C

P

40

SHAPES AND SOLIDS (CHAPTER 3)

Unit 17

Angles of a triangle and quadrilateral

Angles of a triangle The sum of the angles of a triangle is always 180o . DEMO

a°

a + b + c = 180:

If we tear the angles and rearrange them, we can see that their sum is 180o.

65°

DEMO

40°

Since the angles of a triangle add to 180o , and 65 + 40 = 105 then x = 180 ¡ 105 ) x = 75

line such as the bottom of the chalk board

C

A

and so the third angle measures 75o .

B

A

C

x°

c°

b°

B

Find the third angle of (i.e., find x):

Exercise 17 1 Find the unknown angles. The diagrams have not been drawn accurately: a b c 80°

a° 60°

61°

b°

35°

73°

75°

d

e

c°

f 44°

d° e° 37°

g

11°

132°

f°

h 67.2°

i

i°

27°

h°

g°

48° k°

36°

69°

j°

2 Harder: Write down a rule connecting the unknowns in: a b

c

p°

a° b° r°

q°

a°

3 Challenge: Find the unknown angles: a b

c 3a°

b°

a°

2a° a°

40°

b°

b°

b°

a°

[2a means 2 £ 2]

SHAPES AND SOLIDS (CHAPTER 3)

Angles of a quadrilateral The sum of the angles of a quadrilateral is always 360o . a°

a + b + c + d = 360:

d°

b°

Find the value of x in: x°

DEMO

c°

Mark the angles of a quadrilateral. If we tear and rearrange them at a point, we can see that their sum is 360o . a° d°

d°

b°

c° a°

48°

117°

b°

90 + 117 + 48 = 255 and x + 255 = 360 then x = 360 ¡ 255 so x = 105

DEMO

c°

4 Find the unknown angles. The diagrams have not been drawn accurately: a

b

a°

70°

d

x°

x°

100°

100°

c

115° b°

65°

e

x°

f

x°

h 100°

120°

70°

d°

b°

c°

A quadrilateral can be divided into two triangles by drawing a diagonal.

The sum of the angles of a quadrilateral is 2 £ the sum of the angles of a triangle = 2 £ 180o = 360o .

x°

x°

x°

x°

y°

r° 70°

5 Write down a rule which connects the unknowns in: a b a°

x°

i q°

p°

120°

70°

130°

c°

x°

g p°

120°

m°

n°

c

100°

A pentagon has five sides. It can be divided into three triangles by drawing two diagonals. The sum of the angles of a pentagon is 3 £ the sum of the angles of a triangle = 3 £ 180o = 540o Draw a pentagon and measure its angles to check that this is true.

41

42

SHAPES AND SOLIDS (CHAPTER 3)

Unit 18

Polygons

Polygons are straight-sided closed figures that do not cross themselves and can only be drawn on a plane surface.

polygons

A closed figure has no gaps.

We name polygons according to the number of sides they have.

triangle quadrilateral pentagon

hexagon

octagon

A regular polygon has all its sides the same length and all its angles the same size. These polygons are marked to show that they are regular:

not polygons equilateral triangle

square

regular pentagon

regular hexagon

Exercise 18 1 Give one reason why these are not polygons: a b

d

e

c

f

2 Which of these are regular polygons? Give a reason. a b

c

e

d

f

Equal sides are shown by using the same small markings on them. Equal angles are shown by using the same symbols (for example, ² or a°).

SHAPES AND SOLIDS (CHAPTER 3)

3 Using the given code, name the polygons which follow: tri = 3, quad = 4, penta = 5, hexa = 6, octa = 8, deca = 10 a b c

e

f

i

d

g

j

h

k

4 Draw an example of: a a quadrilateral d a decagon

b e

l

an equilateral triangle a regular pentagon

c f

a hexagon an octagon

5 Draw and name polygons with the following descriptions: a six equal sides and six equal angles b three equal sides c five equal sides, but with unequal angles 6 Using a ruler and protractor, classify the following shapes as regular (R) or irregular (I) polygons: a b c d

e

f

g

h

6 edges

and 3 regions

Euler’s Rule Consider the figure below.

It has 5 vertices, 5 4 3

2

1 1 2

5

6 4

1 3

2

(Outside the figure counts as a region.)

3

In any closed figure, the number of edges is always two less than the sum of the number of vertices and regions, i.e., E = V + R ¡ 2: Click on the icon to investigate Euler’s Rule.

EULER’S RULE

43

44

SHAPES AND SOLIDS (CHAPTER 3)

Unit 19

Classifying triangles and quadrilaterals

Triangles are classified according to the lengths of their sides.

Quadrilaterals are classified according to the number of: ² equal sides ² parallel sides ² right angles

scalene all sides of different length

trapezium one pair of opposite parallel sides

isosceles two sides of the same length

parallelogram both pairs of opposite sides parallel rhombus parallelogram with four equal sides

equilateral all sides of the same length

rectangle right angled with both pairs of opposite sides parallel and equal in length

An equilateral triangle is also isosceles.

Parallel lines are shown using arrow heads or

square right angled with both pairs of opposite sides parallel, all sides equal in length kite two pairs of adjacent sides equal in length

Exercise 19 1 Measure the lengths of the sides of the triangles. Use these measurements to classify each as equilateral, isosceles or scalene: a b c d

2 Sketch these quadrilaterals. Use markings to show parallel and equal sides: a a parallelogram b a rhombus c a kite. 3 Use a ruler to help classify the following: a b

d

e

c

f

SHAPES AND SOLIDS (CHAPTER 3)

4 This is a flowchart. Use it to state whether the statements below are true or false. Quadrilaterals

Trapezia

Parallelograms

Kites

Rhombi

Rectangles

Squares

a Rectangles, squares and rhombi are parallelograms. b A rectangle is a parallelogram with four equal angles of 90o . c A rhombus is a rectangle. d A square is a rhombus with four equal angles of 90o . e A trapezium is a parallelogram.

Parallel and perpendicular lines In the figure AB is parallel to DC. We write this as AB k DC.

A

B

D

C

Perpendicular means at right angles to .

Also AD is perpendicular to DC. We write this as AD ? DC. k reads is parallel to ? reads is perpendicular to

5 Using k and ?, write statements about the following figures: a

b

P

Q

d

R L

K

N

M

A

D

e

B

c

H

C

P

Q

S

R

I

K

f

W

X

Z

Y

6 Draw the figures from these instructions. A freehand labelled sketch is needed in each case. a AB is 4 cm long. BC is 3 cm long. AB ? BC. b PQ is 5 cm long. RS is 4 cm long. RS k PQ and RS is 3 cm from PQ. c ABCD is a quadrilateral in which BC k AD and AB ? AD.

J

45

46

SHAPES AND SOLIDS (CHAPTER 3)

Unit 20

Constructing a triangle and bisecting angles

Constructing a triangle

VIDEO CLIP

To construct a triangle with sides 4 cm, 3 cm and 2 cm long.

Aim:

Step 1:

Step 2:

Step 3:

Step 4:

Draw a line segment the length of one of the sides (say AB). It is often best to choose the longest side. Use this as the base of the triangle. Open your compass to a radius equal to the length of one of the other sides (say AC). Using this radius draw an arc from A as shown. Note: Construction lines should not be erased. Now open the compass to a radius equal to the length of the other side, BC. Draw another arc from B to intersect the first arc.

Join A and B to the point of intersection of the two arcs to form triangle ABC.

A

4 cm

B

4 cm

B

In geometry, construct means that you may only use a drawing compass and straight edge.

2 cm A

3 cm 4 cm

A

B

C 2 cm A

3 cm 4 cm

B

Exercise 20 1 Accurately construct, with ruler and compass, a triangle with sides 4 cm, 5 cm and 6 cm. 2 Draw AB of length 5 cm. Set the compass points 5 cm apart. With centre A, draw an arc of a circle above AB. Likewise with centre B draw an arc to intersect the other one. Let C be the point where these arcs meet. Join AC and BC. a What type of triangle is ABC?

(Give a reason.)

b Measure angles ABC, BCA, CAB using a protractor.

C

5 cm

A

There are 6 parts to a triangle. Can you name them?

5 cm

B

5 cm

c Copy and complete: “All angles of an equilateral triangle measure .......o ” 3 With centre A, draw an arc of a circle of radius 5 cm. Choose any two points B and C on this arc (not too close together). Join AB, BC and AC.

B A

a What type of triangle is ABC? (Give a reason.) b Measure angles ABC and ACB using a protractor.

C

c Copy and complete: “In an isosceles triangle the angles opposite the equal sides are .......”

SHAPES AND SOLIDS (CHAPTER 3)

47

4 Use a ruler and compass to construct: a an equilateral triangle with sides 6 cm long (Use question 2.) b an equilateral triangle with sides 4:8 cm long c an isosceles triangle with equal sides 6 cm long and third side 4 cm long (Use question 3.) d an isosceles triangle with equal sides 4:4 cm long and third side 6 cm long. A

Bisecting an angle Aim:

VIDEO CLIP

To bisect angle ABC. B

Step 1:

With centre at B, draw an arc which cuts BA at P and BC at Q.

C P

B

A

C

Q A

Step 2:

With Q as centre, draw an arc within angle ABC.

P B

Step 3:

Keep the same radius. With centre P draw another arc to intersect the previous one at M.

C

Q A P

B

M C

Q A

Step 4:

Join B to M. BM bisects angle ABC, i.e., ]ABM = ]CBM.

P B

Q

M C

5 Draw 4 angles of your choice of different degree measure (2 acute and 2 obtuse). Bisect each angle using the given Bisecting an Angle method. 6 Draw any triangle ABC with sides greater than 6 cm and bisect each of the angles using the compass method. What do you notice about the 3 angle bisectors? Repeat with another different shaped triangle. Copy and complete: “The angle bisectors of any triangle appear to ..... point.” 7 Use what you learned in questions 2 and 3 to find the unknowns in the following, which are not drawn accurately. a

67°

b

x°

The triangle is isosceles, so the angles opposite the equal sides are equal in size.

x° 48°

c

d y cm (x+20)° 5 cm

x°

y cm

40° x°

) this angle is xo as well.

4 cm 40°

70°

Find the value of the unknown:

x + x + 40 = 180 f180o in a triangleg ) x + x = 140 ) x = 70

48

SHAPES AND SOLIDS (CHAPTER 3)

Unit 21

90° and 60° angles and Circles

Constructing a 90° angle Aim: Step 1:

X

To construct a 90o angle at X on line segment XY.

Y

With centre X, draw an arc which cuts XY at A. X

Step 2:

Step 3:

Keep the same radius throughout. With centre A draw an arc which meets the first arc at B. With centre B draw another arc to meet the first arc at C.

C

B X

With centres at B and C, draw arcs of equal radius above B and C, to meet at Z. Join XZ. Angle YXZ is a right angle.

Y

A

Y

A

VIDEO CLIP

Z C

B X

Y

A

Constructing a 60° angle Aim: Step 1:

To construct a 60o angle at X on line segment XY.

With centre X, draw an arc which cuts XY at Z.

X

Y

centre

X

Z

Y

Z

Y

W

Step 2:

With centre Z, and the same radius, draw an arc to cut the first one at W.

centre

X

W

Step 3:

VIDEO CLIP

Draw the line from X through W. Angle WXY is 60o . X

Exercise 21

Y

Z

To construct a 30° angle, construct a 60° angle and bisect it. For a 45° angle, construct a 90° angle and bisect it.

1 On the end of a line segment, construct an angle of: o a 30o b 45o c 15o d 22 12

All construction lines must be visible.

2 Construct a 90o angle at the end A of a 12 cm long line segment AB. If angle BAC is the right angle, put C so that AC is 5 cm long. Now join BC and measure its length. If you have been very accurate and have used a sharp pencil you should have found that BC is 13 cm long.

C 5 cm

A

? cm

12 cm

B

SHAPES AND SOLIDS (CHAPTER 3)

Parts of a circle: radius arc

sector

Every point on the circle is the same distance from the centre. This distance is the length of the radius of the circle.

²

The perimeter of a circle is called the circumference. It is the distance around the circle.

²

The diameter of a circle has length equal to twice the length of the radius.

circle

centre

diameter

3

²

a Sketch a circle and mark on it: i centre O ii a radius OA

iii

a diameter BC.

b Draw line segment AB. Explain why ABO is isosceles. D

4 Name the following parts of the circle alongside: a AB b C c DC d bold curve DA e CB f the shaded area

B

C A

5

a Sketch a circle. Shade a sector that covers more than half the circle. b Sketch a circle. Mark on it an arc that is less than one half of the circumference.

6 Draw a circle of radius 3 cm and mark its centre at O.

R

Q

Draw a diameter AB through O meeting the circle at A and B. Construct 60o angles at A on either side of AB. Construct

60o

60°

A

60°

60°

angles at B on either side of AB.

B

60°

O

Let these meet the circle at P, Q, R and S. Join PS and QR.

P

S

Explain why AQRBSP is a regular hexagon. 7 Another easier method of drawing a regular hexagon is to draw a circle (of radius 3 cm, say) and select point A on it. Keeping the same radius as you used for the circle, with centre A, draw an arc meeting the circle at B. With centre B and same radius, draw an arc meeting the circle at C and continue this around the circle to obtain D, E and F. If you have done this very accurately, the final arc will pass through A, the starting point. Join AB, BC, CD, DE, EF, FA. Explain why ABCDEF is a regular hexagon.

B

C

D

A

O

E

F

49

50

SHAPES AND SOLIDS (CHAPTER 3)

Unit 22

Polyhedra and nets of solids

A solid is a body that occupies space.

cube

cone

cylinder

A prism is a solid whose cross-section is a polygon.

A polyhedron is a solid that has all flat surfaces.

vertex edge

face

sphere

rectangular prism

triangular prism

Prisms and pyramids are named after the shape of their end or base.

square-based pyramid

triangular-based pyramid

Exercise 22 1 Draw a neat diagram to represent a: a cube b d sphere e

DEMO

cone rectangular prism (cuboid)

c f

cylinder triangular-based pyramid

2 Which of the solids shown above have curved surfaces or a combination of flat and curved surfaces? 3 Name the shape which best resembles: a a basketball b the top part of a funnel d a six-faced die e a cornflakes packet 4 Draw a neat diagram to represent: a a triangular prism b c an octagonal prism d 5

c f

a rectangular-based pyramid a hexagonal-based pyramid

a Name all the vertices of this cube.

C

B

b Name all the faces of this cube.

A

c Name all the edges of this cube.

D E

H

6 What shapes are the side faces of: a a prism b a pyramid? 7 Draw a tetrahedron. How many a faces b vertices

a tennis ball container a broom handle

c

edges

does it have?

F G

SHAPES AND SOLIDS (CHAPTER 3)

51

Nets of solids A net is a two-dimensional shape which may be folded or shaped to form a solid.

A solid is formed when the net is folded along the dotted lines

becomes cube

becomes

square-based pramid

There is always more than one net for any solid.

and

both become

triangular-based pyramid

8 Match the net given 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) Square-based pyramid

d

D

(4) Cylinder

9 Make models of the solids in 8, with gluing tabs, and construct each solid at a reasonable size.

Activity

NETS

Making cones Click on the icon for instructions on how to make cones.

MAKING CONES

52

SHAPES AND SOLIDS (CHAPTER 3)

Unit 23

Drawing solids

Oblique projections

Isometric Projections

Drawing a cube:

isometric graph paper Sides go back at 45°

front face

1

shortened sides

dotted lines indicate hidden edges

3

B

2

4

sides go back at 30° on both sides

A side lengths are not altered

Edge AB appears closest to us, and this is often the starting edge.

Exercise 23 1 Draw an oblique projection diagram of a box which has sides 3 units by 1 unit by 2 units. (Start with a 3 unit by 2 unit front rectangle.) 2 Redraw the following figures on isometric graph paper. Use the darkened edge as the starting edge. a b c

3 Copy these objects (oblique projections) and then draw each of them as isometric projections. a b c

4 Harder: On isometric paper draw two different shapes which can be made from four cubes of the same size and which have at least one face in full contact with one of the other cubes. The diagram alongside shows one possible shape.

SHAPES AND SOLIDS (CHAPTER 3)

Constructing block solids Draw top, front, back, left and right views of:

The views are:

BLOCK BUSTER

top or plan left back

2

1

2

1

top front

front

back

left

right

right

5 Draw top, front, back, left and right views of: a b

c You may find it helpful to use centicubes to plan your drawings.

d

e

f

6

top view

3

2

2

1

1

1

a Explain what the numbers in the top view indicate. b Add up the numbers. What information does this give you? 7 Draw the 3-dimensional object whose views are: a 2

b

2

8 Harder:

1 1 top front

back

left

right

front

back

left

right

1 2 top

Draw three objects made from five cubes whose view from the top is

.

53

54

SHAPES AND SOLIDS (CHAPTER 3)

Unit 24

Review of chapter 3

Review set 3A 1 Name the intersection of: a AC and BD Ã ! b BC and AD

B

A

D

C

In the diagram shown alongside, what is represented by: a the double arrows b the dots c the mark ?

2

3 What angle is formed inside the hands of a clock showing: a

3 o’clock

b

6 o’clock?

4 Find the value of the missing angles in each of the following which are not drawn accurately: a

b 130° 23°

a°

5 Draw the following: a an isosceles triangle

b

b°

a circle showing a sector

6 What is the aim of this construction? P N

Q

M

O

P N

M

Q

P O

N

Q

M

P

R O

N

Q

M R O

7 Draw and name the solid that corresponds to the following net:

8 Draw a net for a square-based pyramid. 9 For this isometric drawing, draw the: a top b left c right d front e back view.

front

SHAPES AND SOLIDS (CHAPTER 3)

Review set 3B 1 In the diagram alongside, what is represented by: a the arrows b AB Ã ! c CD d X?

A

D X

C

B

2 Name the following: a

b

c

3 Classify the following triangles by measurement of their sides: a b

c

4 Find the value of the missing angles in each of the following which are not drawn accurately: a

b 68° c° 43°

a°

5 For the given figure, name:

E

A

a all vertices b all edges

F B

c all faces. 6 Draw the following solids: a a triangular prism b

D

C

a cylinder

7 Draw a net for a triangular-based pyramid. 8 For the following views, draw an isometric diagram:

top

left end

right end

front

9 Draw an oblique projection for a rectangular prism of 2 £ 1 £ 2 units.

back

55

56

REVIEW OF CHAPTERS 1, 2 AND 3

TEST YOURSELF: Review of chapters 1, 2 and 3 1 Replace ¤ by > or <: a 4607 ¤ 4670

b

50 £ 1000 ¤ 50 021

2 Simplify: a 13 ¡ 3 £ 2

b

5+3£2

c

3 Name these triangles according to their side lengths. a b

4 Use the digits 5, 7, 8, 9, 0, 1

2 £ (5 + 3) ¡ 7

c

once only to make the largest number you can.

5 Over the three nights of the netball championships the attendance was 3028, 3347 and 3517. a What was the total attendance? b The total attendance was estimated at 10 000. How short of this estimate was the actual total attendance? 6 Find the value of a in these inaccurately drawn figures: a b

c 96°

150° a° 54°

124° a°

7 What is the place value of the 6 in these numbers? a 362 b 36 071 8

a c

List the factors of 24: Find the LCM of 3 and 8:

9 Round off: a 426 to the nearest 10

c b d

b

a°

46 070 832

Find the prime numbers between 20 and 30. Find the HCF of 20 and 25.

426 to the nearest 100.

10 A length of rope can be cut exactly into 6 m or 8 m lengths. Find the shortest length of rope for which this is possible. 11 Draw a net for making a triangular-based pyramid. 12 Find: a the product of 12 and 17

b

the sum of 37, 48 and 116.

13 Copy and complete: a 64 = ¤2

b

p 100 = ¢

14 Draw a sketch of: a a cone

b

a rectangular box

c

8 = ¤3

15 80 cans of fruit weigh 40 kg. Find the weight of one can in grams (1 kg = 1000 grams). 16 Draw a net of a square-based pyramid.

REVIEW OF CHAPTERS 1, 2 AND 3

17 Write in simplest form: a 2£3£6 c 22 £ 23 £ 24

b d

57

22 £ 32 5 £ 104 + 3 £ 102 + 5 £ 10 + 6

18 Find the smallest number which can be made up from the digits 7, 1, 0, 2 and 8, using the digits once only. 19 Pam spends $38 on a T-shirt, $69 on a pair of jeans and $117 on a pair of shoes. How much change from $250 will she receive? 20 Find the total cost of 2 dozen jars of jam at $2:70 each. 21 Write these numbers in ascending order (smallest first). 36 183, 31 368, 33 681, 38 631, 31 836 22 Find x in these inaccurately drawn figures: a b

c 50°

67° 120°

x°

x°

x°

23 For this diagram, draw views from: a the top b the left c the right d the front front

e the back 24

a Write the opposite of winning by six goals. b The weather report showed that while Adelaide was 21o C, Tokyo was ¡3o C and Moscow was ¡17o C. By how many o C was: i Adelaide warmer than Tokyo ii Moscow colder than Adelaide?

25 True or False? a 3 674 763 > 3 647 483 b 5607 ¡ 2489 = 3118 c The difference between 369 and 963 is 396 26 Find the approximate value of 293 £ 39: 27 How many times is the first 5 larger than the second 5 in 28 Find the value of: a 6+9£2¡2 29 In the diagram alongside, what is represented by: a the arrows b PQ Ã ! c RS d T?

b

35 659? c

2 £ (6 ¡ 4) ¥ 2

P

T

R

2£6¡4¥2

S

Q

58

FRACTIONS (CHAPTER 4)

Unit 25

Representation of fractions

The fraction three eighths can be represented in a number of ways.

as a shaded region

Diagram

Number line

CHAPTER 4

Ei_\ means that one whole is divided into 8 equal parts and we are considering 3 of these parts.

three eighths

Words

Symbol

0

Ei_

or

as pieces of a pie

three eighths

1

numerator bar denominator

Exercise 25 1 Copy and complete the following table:

Symbol

Numerator

one half

a

b

3 4

c

2 3

d

Words

Denominator

Meaning

2

One whole divided into two equal parts and one is being considered.

One whole divided into four equal parts and three are being considered.

three quarters

2

two sevenths

g

0 one half

1

0 1 three quarters

0 two thirds

3

1

7

One whole divided into nine equal parts and seven are being considered.

e

f

Number Line

5

8

0

1

FRACTIONS (CHAPTER 4)

2 What fraction of each of these shapes is shaded? a b

d

e

c

f

3 Carefully copy 3 identical sets of each of the following shapes, then answer each question.

a In the first set divide each whole shape into two equal parts. Each part is one half of the whole shape. b In the second set divide each whole shape into three equal parts. Each part is ..... c In the third set divide each whole shape into four equal parts. Each part is ..... d Which shapes were the most difficult to divide equally? 4 Using identical square pieces of paper, make 2 copies of this tangram. Number the pieces on both sheets. Cut one of the sheets into its seven pieces. Use the pieces to help you work out the following: a How many triangles like piece 1 would fit into the largest square?

1 3 4

2

5

b What fraction of the largest square is piece 1? c What fraction of piece 1 is piece 3?

6

d What fraction of the largest square is: i triangle 1 iii square 4

7

ii triangle 3 iv parallelogram 6?

5 Copy the given shape exactly. a If each rectangle is half of the one before it, how much of the shape is unshaded if the whole square is 1? b Check your answer to a by drawing a grid within the large square. Use the boundaries of the shaded square as the dimensions of the smallest squares in your grid.

Qr_

Qw_

c How many of the smallest squares fit into your large square?

Complete the shading of part b to make a chessboard and then answer the following: d What fraction of the whole chessboard is the unshaded area? e What fraction of the total chessboard is the first row of squares? f What fraction of the total chessboard are the unshaded squares in the first row?

59

60

FRACTIONS (CHAPTER 4)

Unit 26

Equivalent fractions and lowest terms Equivalent (equal) fractions represent the same amount.

Multiply to write

1 4

1 4

with denominator 16: 1 4

4 16

and

are equivalent fractions.

=

1£4 4£4

=

4 16

DEMO

9 12

Divide to find equivalent fractions: 9 12

3 4

and

=

9¥3 12¥3

=

3 4

are also equivalent.

Exercise 26 1 Multiply to find equivalent fractions: 5 5 5£2 10 5£2 25 a b = = = = 6 6£2 2 7 7£5 2 2 Divide to find equivalent fractions: 8¥2 2 10 ¥ 5 2 8 10 = = = = a b 10 10 ¥ 2 5 15 15 ¥ 2 3

c

4 4£2 40 = = 5 5£2 2

c

2¥2 2 18 = = 20 20 ¥ 2 10

3 Express with denominator 8: a

1 4

1 2

b

c

3 4

d

1

To obtain equivalent fractions we multiply (or divide) both the numerator and denominator by the same number.

4 Express with denominator 30: a

1 2

b

4 5

c

5 6

d

3 10

b

3 4

c

1

d

0

b

1 4

c

4 5

d

9 10

5 Express in sixteenths: a

1 8

6 Express in hundredths: a

1 2

7 Find 2 if: 2 7 = a 3 21 e

4 16 = 5 2

b

4 2 = 5 15

c

2 9 = 13 39

d

27 2 = 63 7

f

6 3 = 2 4

g

7 35 = 8 2

h

28 14 = 2 25

FRACTIONS (CHAPTER 4)

61

Lowest terms are the simplest form of a fraction. We write a fraction in lowest terms by dividing the numerator and denominator by their highest common factor. Examples: ² = =

12 16 12¥4 16¥4 3 4

²

fas 4 is the HCF of 12 and 16g

= =

80 100 80¥20 100¥20 4 5

fas 20 is the HCF of 80 and 100g

8 Reduce to lowest terms: a e

8 10 24 42

b f

9 36 55 77

c

18 20 39 52

c

45 80 3 51

c

g

21 28 48 84

d

72 96 60 80

d

12 20 24 81

d

h

15 35 6 30

9 Reduce to lowest terms: a e

12 15 49 91

b f

g

h

We divide the numerator and denominator by their highest common factor (HCF).

35 49 15 55

10 Simplify: a e

56 77 250 1000

b f

g

h

15 45 45 180

11 Which of these fractions are in lowest terms? a

15 20

b

1 3

c

13 24

d

64 72

e

6 9

f

21 28

g

22 24

h

5 6

i

75 100

j

14 15

k

9 100

l

39 52

b

175 200

c

32 80

d

875 1000

12 Reduce to lowest terms: 132 144

a

Activity

Representations of equal (equivalent) fractions What to do: 1 Construct 5 identical 4 cm sided squares. Draw lines to create your halves and quarters. Name the equal fractions created in each square. Qw_

2

Wr_

a Write the numbers 1 to 10 on one ice block stick, one centimetre apart. b Write the multiplies of 3, one centimetre apart on another stick.

1

2

3

4

5

6

7

8

9

10

3

6

9

12 15 18 21 24 27

30

c Place one stick directly above the other to show that

1 2 3 4 3 , 6 , 9 , 12 ,

...., ...., ...., ...., ....,

10 30

are equal.

d On a third stick mark off the multiples of 5 from 5 to 50. Place the multiples of 3 stick directly above the multiples of 5 stick and complete these equivalent fractions. 3 6 5 , 10 ,

...., ...., ...., ...., ....,

30 50 .

62

FRACTIONS (CHAPTER 4)

Unit 27

Fractions of quantities

Matthew was given a box of chocolates. 5 had red wrappers, 4 had blue, 4 had gold and 2 had green. There were 15 chocolates in total.

What fraction of 1 metre is 37 cm? 37 cm as a fraction of 1 metre

DEMO

Notice that ² ²

5 had red wrappers so the fraction with red wrappers =

5 15

=

11 did not have gold wrappers so the fraction without gold wrappers =

1 3

=

37 cm 1 metre

=

37 cm 100 cm

=

37 100

11 15

The numerator and denominator must have the same units.

Exercise 27 1 What fraction of each of these different quantities has been circled? a b

c

2 Use a full pack of 52 playing cards for these questions. Calculate which fraction of the full pack are: a

all the red cards, e.g.,

c

all the aces, e.g.,

e

7

3

5

b

all the spades, e.g.,

d

all the picture cards, e.g.,

all the odd numbered cards

f

all the even numbered black cards.

3 What fraction of one hour is: a 30 minutes b 10 minutes

c

45 minutes

d

12 minutes ?

4 What fraction of one day is: a 1 hour b

c

30 minutes

d

1 minute ?

d

1 minute ?

¨

©

A

§

4 hours

5 Use your calculator for this question. What fraction of one week is: a 5 days b 12 hours c 2 hours As the fraction of the week got smaller, what happened to the denominator? 6 In lowest terms, state what fraction of: a 1 metre is 20 cm b 2 metres is 78 cm d 3 kg is 750 g e 1 day is 5 hours g November is two days h a decade is one year

c f i

ª

1 kg is 500 g 1 hour is 23 minutes 2 dollars is 27 cents

7 Jenny scored 27 correct answers in her test of 40 questions. What fraction of her answers were incorrect? 8 James was travelling a journey of 420 km. His car broke down after 280 km. What fraction of his journey did he still have to travel?

J

¨

Q

ª

K

©

1 kg = 1000 g A decade is 10 years.

FRACTIONS (CHAPTER 4)

9 Since

6 ¥ 2 = 3,

1 2

63

of 6 is 3. What number is:

a

1 2

of 10

b

1 2

of 36

c

1 3

of 12

d

1 3

of 45

e

1 4

of 20

f

1 4

of 44

g

1 5

of 30

h

1 5

of 120

i

1 10

j

1 2

of $1:20

k

1 4

of 1 hour (in min)

l

1 12

of 650 g

of 600?

10 Fill in the missing fraction:

is ..... of

a

is ..... of

b

11 Damien only won one third of the games of tennis that he played for his school team. If he played 15 games, how many did he win? 12 One fifth of the students at a school were absent because of colds. If there were 245 students in the school, how many were away? 13 One sixth of the cars from an assembly line were painted white. If 222 cars came from the assembly line, how many were painted white? 14 Draw sketches of the amounts of money which represent Write their numerical value beneath your sketches. a b

d

1 4

of each of the following.

e

15 There are 360o in 1 revolution (one full turn). one quarter turn

a half turn

c

f

a Find the number of degrees in: i one quarter turn ii a half turn iii three quarters of a turn. b What fraction of a revolution is: i 30o ii 60o iii 240o ?

three quarters of a turn

16 One morning two fifths of the passengers on my bus were school children. If there were 45 passengers, how many were school children? 17 Richard spent two thirds of his working day installing computers, and the remainder of the time travelling between jobs. If his working day was 8 hours, how much time did he spend travelling?

Finding the whole amount 2 5

of Freddy’s money was $5260.

So

1 5

was $2630.

)

5 5

was $2630 £ 5 = $13 150

Click on the icon for more problem solving questions.

EXTRA QUESTIONS

64

FRACTIONS (CHAPTER 4)

Unit 28

Fraction sizes and types 3 4

Which fraction is larger,

or 23 ?

The lowest common denominator (LCD) of two or more fractions is the lowest common multiple of their denominators.

If they have the same denominators, we can compare them. The LCM of 4 and 3 is 12. So the LCD of 3 4

Now

3£3 4£3

=

9 12

and

3 4

>

and

=

8 12

2 3

9 12

and 3 4

so

is 12:

>

2 3

=

2£4 3£4

=

> means “is greater than” < means “is less than”

8 12

2 3

Exercise 28 1 Find the LCM of: a 7, 3 e 6, 8, 9

b f

5, 3 10, 5, 6

c g

3, 6 5, 6, 11

d h

12, 18 12, 4, 9

c

5 9

d

4 7

2 Find the lowest common denominator of: 1 4

a

and

5 8

b

2 3

and

3 4

3 4

and

3 Write each set of fractions with the lowest common denominator. Then write the original fractions in ascending order (smallest to largest): a

1 1 2, 4

b

2 3 3, 4

c

1 4 2, 7

d

5 3 8, 4

e

7 5 10 , 6

f

7 3 9, 4

g

5 8 8 , 10

h

9 7 1 25 , 20 , 4

d

1 9 3 2 , 20 , 5

and

5 9

Ascending means going up. Descending means going down.

4 Write each set of fractions with a common denominator. Then arrange them in descending order: 1 2 7 2 , 5 , 10

a

b

1 5 3 2, 8, 4

c

1 7 4 2 , 12 , 6

5 Sanjay scored 16 out of 20 in a test. Robert scored 25 out of 30 in a different test. a Write each of the students’ scores as a fraction. b Write the two fractions with a lowest common denominator. c Which of the two students scored higher as a fraction of the total possible score in their test?

Activity

Fraction strips What to do: 1 Use a sheet of paper 24 cm long and a ruler to copy the fraction chart shown.

2 To compare

left for three 9 10

so

3 4 1 4

and

9 10 ,

1

count across from the

pieces and for nine

1 10

Qw_

pieces. Qt_

Qt_

<

5 12

b

5 6

>

13 16

Qr_

Qr_

Qt_

Qt_

Qt_

6 ths 8 ths 10 ths 12 ths

> 34 .

2 5

Qe_

Qr_

Use the fraction strips to answer these True or False questions: a

Qe_

Qr_

is further to the right than 34 , 9 10

Qw_

Qe_

c

3 5

<

10 16

d

3 4

>

12 16

FRACTIONS (CHAPTER 4)

Improper fractions and mixed numbers

Examples: Improper fraction to whole number

2 3

15 5

represents 23 .

is a proper fraction.

= 15 ¥ 5 =3

Its numerator is less than its denominator.

Improper fraction to mixed number 5 4

21 5

is an improper fraction. =

Its numerator is greater than its denominator.

20 5

1 5

+ 1 5

=4+ = 4 15

represents 54 .

Mixed number to improper fraction 5 4

2 45

can be written as a whole number and a fraction. 5 4

1 14

4 5

=2+

= 1 14

is a mixed number.

=

10 5

=

14 5

+

4 5

6 Write as a whole number: a

16 4

b

20 5

c

18 6

d

40 8

e

30 6

f

30 3

g

30 10

h

30 1

i

30 30

j

64 8

k

125 25

l

63 7

7 Write as a mixed number: a

5 4

b

7 6

c

18 4

d

19 6

e

15 2

f

17 3

g

16 7

h

23 8

i

22 7

j

35 9

k

41 4

l

109 12

8 Draw diagrams to show: a

7 6

b 3 12

9 Write as an improper fraction: a

3 12

b

4 23

c

2 34

d

1 23

e

1 12

f

3 34

g

1 45

h

6 12

i

4 59

j

5 78

k

6 67

l

1 11 12

10 Use 2 dice. Use one to roll the numerator and the other to roll the denominator. Find: a the smallest fraction it is possible to roll b the largest proper fraction it is possible to roll c the largest improper fraction (not a whole number) it is possible to roll d the number of different fractions it is possible to roll. e List the different combinations that can be simplified to a whole number.

numerator is the upper face bar denominator is the upper face

65

66

FRACTIONS (CHAPTER 4)

Unit 29

Adding and subtracting fractions

Adding fractions For mixed numbers:

If fractions have the same denominator, add the numerators. + 4 6

1 12 + 2 16

= 1 6

+

5 6

=

If fractions have different denominators: 2 3

+

=

2£4 3£4

=

8 12

=

11 12

1 4

+

+

1£3 4£3

fLCD is 12g

3 12

fadd numeratorsg

=

3 2

+

13 6

=

3£3 2£3

+

=

9 6

13 6

=

22 6

fadd numeratorsg

=

11 3

freduce to lowest termsg

= 3 23

fwrite as mixed numberg

+

fwrite as improper fractionsg 13 6

fLCD is 6g

² If necessary, convert mixed numbers to improper fractions.

Rules

² If necessary, change the fractions to fractions with the lowest common denominator. ² Add the numerators while the denominators stay the same.

Exercise 29 1 Without showing any working, add the following: a

1 4

+

2 4

b

3 10

d

4 7

+

2 7

e

4 9

g

3+

2 3

h

2+

a

1 5

+

3 10

b

3 5

e

3 4

+

1 3

f

7 10

i

3 4

+

1 6

j

5 9

+

5 6

1 4

+

1 3

b

3 5

+

7 10

+

2 3

+

+

3 10

10 9 5 8

+

7 8

c

1 6

+

4 6

f

3 5

+

4 5

i

1+

7 10

+

6 10

2 Find:

+

7 10

c

1 2

+

1 4

d

1 2

+

1 10

1 3

g

2 3

+

1 2

h

5 6

+

5 8

k

3 7

+

3 14

l

4 9

+

2 5

c

5 9

+

5 6

d

3 4

+

7 8

d

2 14 +

h

3 4

+

3 Find: a

+

1 2

+

5 20

+

1 3

+

2 3

4 Find:

5

a

2 + 1 13

b

3 12 + 2 12

c

2 23 + 1 13

e

3 34 +

f

2 23 + 1 12

g

2+

1 3

1 3

+ 1 12

1 8

+ 1 + 1 13

a Find the sum of 1 12 and 2 15 . b Find the average of 3 18 , 2 12 and 3 38 . c Frank has 2 12 m, 3 14 m and 1 13 m of water pipe. He has two pipe joiners. What length of pipe can he make?

FRACTIONS (CHAPTER 4)

Subtracting fractions We use the same rules to subtract fractions as we do to add them, except we subtract the numerators. If fractions have the same denominator: 7 9

5 9

¡

=

7¡5 9

=

2 9

For mixed numbers: 2 ¡ 1 13

fsubtract numeratorsg

If fractions have different denominators: 4 5

3 4

¡

=

4£4 5£4

=

16 20

=

1 20

¡

¡

3£5 4£5

=

2 1

=

2£3 1£3

¡

=

6 3

4 3

=

2 3

¡

¡

4 3

fwrite as improper fractionsg 4 3

fwrite with LCDg

fsubtract numeratorsg

fLCD is 20g

15 20

6 Find without showing any working: a

3 4

1 4

b

7 9

e

1¡

11 13

f

19 20

i

3¡

7 10

j

4¡

a

1 3

¡

1 4

b

5 6

e

3 4

¡

3 8

f

5 6

i

3 8

¡

1 4

j

7 15

1+

2 5

b

1 6

9 Find: a 3 12 ¡ 2 12

b

3 78 ¡ 1 12

f

¡

c

7 8

¡

5 8

d

1¡

5 6

g

5¡

1 2

h

2¡

3 5

6 7

k

1¡

1 7

l

1 ¡ ( 17 + 27 )

¡

1 3

c

3 4

¡

1 5

d

1 2

¡

3 10

¡

1 2

g

2 3

¡

1 6

h

4 5

¡

1 3

k

11 12

¡

3 4

l

7 10

c

5 12

+

5 6

d

3 4

2 23 ¡ 1 13

c

2 35 ¡ 1 25

d

2 23 ¡ 1 12

3 12 ¡ 1 56

g

3¡

h

3 34 ¡ 2 13

¡

4 9 13 20

¡

¡

2 7

7 Find:

1 3

¡

¡

3 15

1 8

¡

8 Find: a

e 10

¡

3 10

+

1 4

¡

1 8

7 10

¡

2 3

+

1 6

a Find the difference between 3 14 and 2 38 . b By how much does 2 12 exceed 78 ? c How much larger than 2 12 is 3 18 ?

11 Complete the magic squares where each row, column and diagonal must have the same sum.

3 Qw_

1 We_

1 Er_ 3 2 Er_

3

0

1 Qw_

6 Qw_

2 Qe_

Qe_ 5

Test your friends by making up two of your own.

67

68

FRACTIONS (CHAPTER 4)

Unit 30

Multiplying fractions

During the basketball season, a player drinks of a litre of milk five days a week.

3 4

For multiplying fractions:

How much milk does the player drink each week?

=

+

3 4

+

3 4

+

3 4

=

3 4

5 1

=

£

3 4

=

3+3+3+3+3 4

=

15 4

15 4

=

litres

litres

3 5

£

10 2 1

f cancel common factors g f multiply numerators, multiply denominators g

5 6

=

3 2

£

5 6

f write as fractions g

=

13

£

5 62

f cancel common factors g

=

5 4

2

= 1 14

c a£c a £ = b d b£d

The rule is:

f write as fractions g

1 12 £

DEMO

5£3 1£4

10 1

=6

or by multiplication: 5£

£

6 1

We can get the answer by addition: 3 4

3 5 1

=

=

+

£ 10

=

We can show this using diagrams:

3 4

3 5

f write as a mixed number g

Exercise 30 1 Find the missing number: a

5£

¤ 2 = 3 3

b

6£

¤ 3 = 5 5

c

3£

¤ 7 = 8 8

2 Write as a mixed number: a

3£

3 5

b

6£

4 7

c

5£

2 3

d

9£

3 4

e

6£

1 4

f

3£

7 8

g

8£

1 3

h

4£

5 6

i

9£

1 2

j

7£

4 10

k

2£

11 12

l

4£

4 5

m

3£

5 7

n

5£

4 9

o

9£

4 5

p

10 £

3 Simplify the following by cancelling common factors then multiplying: a

2 3

£

3 2

b

3 10

1 3

c

3 4

£

4 5

d

7 6

£

2 5

e

3 5

£

25 6

f

8 3

£

15 4

g

1 2

£

2 3

h

1 2

£

2 3

i

2 3

£

6 5

a

2 3

£

4 5

b

3 8

£

4 5

c

3 4

£

5 9

d

4 7

£

7 9

e

2 23 £

f

2 23 £

g

1 1 14 £ 1 15

h

4 3

i

2 25 £ 2 12

j

6 34 £

k

1 14 £

l

3 7

£

3 1

£

£

3 4

£

1 3

Cancelling common factors keeps the numbers smaller and easier to handle.

15 2

4 Find:

8 9

£

1 7

6 7 9 10

£

2 3

6 7

£ 3 12 £ 1 13

Change a mixed number to an improper fraction before you multiply.

FRACTIONS (CHAPTER 4)

69

5 Find: a

2 3

of 12

b

3 5

d

2 7

of 21

e

3 10

g

4 5

of 60

h

j

3 4

of

3 4

k

To find

2 3

=

c

3 4

of 4

f

3 8

of 16

1 13 of 9

i

3 7

of 49

1 4

l

1 2

of 17 12

of 10 of 20

of 6

“of ” means that we multiply. Qw_\ of 5 = Qw_ £ 5

of an hour, we write

2 3

of 60 minutes

2 3

£ 60

f1 hour = 60 minutesg

= 40 minutes 6 Find: a

3 4

of a metre

b

2 3

of one day

c

3 5

d

5 6

of an hour

e

7 10

of a litre

f

3 20

of a century of a kilogram

7 The whole value of each of the following groups of shapes appears beneath them. What is the value of the coloured shapes in each group? a b c d 90

30

70

200

8 Lisa had $117. She spent one third of her money on new jeans. How much did the jeans cost? 9 While Evan was on holidays, one eighth of the tomato plants in his greenhouse died. If he had 96 plants alive when he went away, how many were still alive when he came home? 10 A business hired a truck to transport boxes of equipment. The total weight of the equipment was 3 tonnes, but the truck could only carry 58 of the boxes in one load. a What weight did the truck carry in the first load? Remember 1 tonne = 1000 kg. b If there were 80 boxes, how many did the truck carry in the first load?

Multiplying fractions using diagrams 11

Use a rectangle to find

1 2

of 35 .

Shade

3 5

of a rectangle.

12

Divide the rectangle into halves and shade 12 of the 35 . 3 10

is shaded.

Checking our answer:

1 2

£

3 5

=

1£3 2£5

=

3 10

13

1 3

of 23 .

a

Use a rectangle diagram to find

b

Check your answer using the rule for multiplying.

a

Use a rectangle diagram to find

b

Check your answer using the rule for multiplying.

1 4

of 13 .

Write down the fraction multiplication and answer for the following shaded rectangles: a b

70

FRACTIONS (CHAPTER 4)

Unit 31

Problem solving with fractions

Donna trains three times a week. On Monday she ran 2 12 km, on Wednesday she ran 1 78 km and on Friday she ran 1 38 km. How many km did she run altogether?

After a party, three eighths of the birthday cake was left over. Usman ate half of this. What fraction of the cake did he eat?

2 12 + 1 78 + 1 38

Donna ran =

5 2

+

15 8

+

11 8

=

5£4 2£4

+

15 8

+

=

20 8

=

46 8

=

23 4

+

15 8

+

1 2

of

1 2

of

3 8

=

1 2

£

=

3 16

Usman ate and

11 8

fLCD is 8g

11 8

3 8

3 8

i.e., Usman ate

3 16

of the cake.

flowest termsg

= 5 34 i.e., Donna ran 5 34 km.

Exercise 31 1

a

Find the sum of 2 13 and 45 .

c

Find the product of

5 8

and

2 Balance the following scales: a

c

e

3 4

+

5 8

1 4

+

5 6

7 10

+

8

12

7 25

100

+

5 8

+ 12

+ 100

4 10 .

b

d

f

b

Find the difference between 2 13 and 45 .

d

How much larger than

2 9

+

5 6

4 18

+ 18

2 3

+

3 8

16

+

4 7

+

15 42

42

15 + 42

3 In a class of 28, four students were late handing in their projects. What fraction of the class were late? 4 Tom paid $2800 deposit on a car. He borrowed a further $8400 to pay for the car. What fraction of the car’s total cost was Tom’s deposit? 5 When Susan drove her car out of the yard the fuel tank was 12 full.

She used 13 of a tank to take her friends for a drive. How much fuel remained in the tank?

9

3 10

is 37 ?

FRACTIONS (CHAPTER 4)

71

6 Alice has 42 birds in an aviary; 26 are canaries and the rest are budgerigars. a What fraction of the birds are budgerigars? b If half the budgerigars are female, what fraction of all the birds are male budgerigars? 7 Tony plays his computer games for an hour and a quarter each week night. On Saturday he plays for three and a half hours and he plays for four and three quarter hours on Sunday. At this rate how much time does Tony spend playing computer games during one year? 8 A swimmer swims 37 of the way in the first hour and the second hour. What fraction has the swimmer left to swim?

2 5

in

9 To make a 20 kg blend of 5 different nuts, a wholesaler mixes 6 kg of peanuts, 4 kg of almonds, 3 kg of walnuts and 2 kg of cashews. The rest are macadamias. What fraction of the blend are the: a macadamias b peanuts c walnuts? 10 Wi filled one aquarium 34 full of water. He filled an identical aquarium 11 16 full of water. If the volume of one aquarium was 48 litres, how much water did he use altogether? 11 Zoe’s development company plans to subdivide 60 hectares of land into a housing development. One tenth of the land must be used for parks and gardens and 14 will be required for roads and walkways. How many blocks with an area of 15 hectare will she be able to create? 12 Which is the better score in a mathematics test: a 17 out of 20 b 21 out of 25? 13 An orchardist picked

1 4

of his orange crop in July and

2 3

of his crop in August.

a How much of his crop remained to be picked in September? b If he picked 600 cases in September, how many cases did he pick that season? 14 A snail crawls 3 35 metres in the first 14 hour, 2 23 m in the second 14 hour and 1 12 metres in the third How much further did it crawl if at the end of one hour it had reached 10 metres? 15

1 4

hour.

Joe’s Burger Shop makes 16 meat patties with every kilogram of minced beef. In his Double Pattie Delight, Joe uses 2 meat patties. His other varieties use only one pattie. If Joe sells 600 burgers in one week and 13 of them are his Double Pattie Delights, how much beef mince does Joe use in one week?

16 What fraction would 4 different pizzas need to be cut into if: a 12 people were to have one piece of each of the pizzas b each person was to have 2 pieces from each pizza? 17 A road crew repainting lines completed 3 23 km on day 1, 2 78 km on day 2 and 3 34 km on day 3. How many kilometres did they complete in total? 18 It takes 7 23 hours to fly from Adelaide to Singapore. The plane flies over Darwin 3 12 hours after leaving Adelaide. How long will it be before the plane lands in Singapore?

72

FRACTIONS (CHAPTER 4)

Unit 32

Review of chapter 4

Review set 4A 1 What fraction is represented by the shading in these diagrams? a b c

2 Find the fractions represented by the points on the number lines: a b 2

3

1

2

3 In a class of 24, three students were late handing up their projects. What fraction of the class was this? 4 What number must

1 2

be multiplied by to get an answer of 4?

5 Write the fractions in lowest terms: 21 28

a

b

6 Express 56 ,

2 3

and

7 9

15 24

c

120 300

with lowest common denominator.

Then write the original fractions in order of size, beginning with the smallest.

7 Write T for true or F for false. 3 9

a

=

15 40

b

3 47 =

24 7

c

76 8

= 9 12

d

375 1000

=

3 8

8 Find: a

2 12 + 3 45

b

6 14 ¡ 3 23

c

2 3

£ 2 12

3 4

b

5 6

c

5 8

of 1 kg

9 Find: a

of $28

of 1 hour (in minutes)

10 In lowest terms, state what fraction of: a

one week is 3 days

b

one metre is 35 cm

11 Solve the following problems: a There were 2728 paying spectators at a match. If three quarters supported the home team, how many supported the visiting team? b Three fifths of the students in a school order their lunch from the canteen. 142 do not. How many students are there in the school? c

3 7

of the students of a school attended a film night. If there were 840 students in the school, how many attended the film night?

FRACTIONS (CHAPTER 4)

Review set 4B 1 Divide and shade each of these shapes to show the fractions written underneath: a b c

7 12

3 8

5 6

2 Find the fractions represented by the points on the number lines: a b 1

2

4

5

3 Write these fractions in lowest terms: 24 27

a 4

b

a Convert

39 8

20 32

c

120 260

c

5 67 =

to a mixed number.

b What fraction of $9:00 is $1:80? c What fraction of 1 km is 800 m? 5 Express 25 , 34 and 13 20 with lowest common denominator. Then write the original fractions in order of size, the largest being first. 6

a If

3 4

of a number is 21, find the number.

b Find the values of 2 and 4 given that

3 4

=

2 20

=

27 4.

7 Write T for true or F for false: 3 7

a 8 Find:

=

6 14

a

=

15 35

2 19 ¡

b 5 6

675 1000

=

5 8

b 3 12 + 2 25

9 Solve the following problems: a A man who weighed 90 kg went on a diet and lost 10 kg. What fraction of his original weight did he lose? b

2 5

of a flock of sheep numbered 240. Find the size of the whole flock.

c Melissa works 2 nights a week after school. On the first night she works 2 23 hours and on the second 3 12 hours. What is her total time worked for the week? 10 Anne can type a 1 hour

2 3

b

of a page in 1 12

hours?

1 4

of an hour. How many pages can she type in

41 6

73

74

DECIMALS

Representing decimals

Unit 33

Australia’s currency (money) is called decimal, because it uses base 10.

CHAPTER 5

one cent

The decimal point separates whole numbers from fractionals.

is a tenth of

i.e.,

1 10

or 0:1 of 10 cents = 1 cent

is a tenth of

i.e.,

1 10

or 0:1 of $1 or 100 cents = 10 cents

is a tenth of

i.e.,

1 10

or 0:1 of $10:00 = $1:00

is a tenth of

i.e.,

1 10

or 0:1 of $100:00 = $10:00

Exercise 33 1 Using the example, change these currency values to decimals of one dollar: a

is $61.10 b

c

2 If seven dollars 45 cents is $7:45, what is: a 4 dollars 47 cents b 15 dollars 97 cents d 36 dollars e 150 dollars g 85 dollars 5 cents h 30 dollars 3 cents?

c f

seven dollars fifty five cents thirty two dollars eighty cents

3 Change these amounts to decimals using the dollar as the unit: a 35 cents b 5 cents d 3000 cents e 487 cents h 638 475 cents g 3875 cents

c f

405 cents 295 cents

4 Express these amounts as a fraction of a dollar. a e i

$1:75 $31:13 105 cents

b f j

$3:25 $243:08 $0:07

79 (e.g., $6:79 is 6 100 dollars)

c g k

$52:40 649 cents 3755 cents

d h l

$0:87 428 cents 100 010 cents

DECIMALS (CHAPTER 5)

Multi Attribute (MA) blocks

75

Decimal Grids

Tenths Hundredths Thousandths

Units

1 • 3 4 7 10 lots of 0.1

represents 1.347

100 lots of 0.01

represents 4¡´¡0.1¡=¡0.4

represents 27¡´¡0.01¡=¡0.27

5 Write the decimal value represented by the MA blocks if the largest block represents one: a b

c

d

6 In question 5, if

represents one cent and

represents one dollar:

a what is the decimal currency value of each example b what is the total value? 7 Write the decimal that represents the shaded area: a b

8

i ii

c

How many rectangles are shaded in these diagrams? What decimals are represented? a b

d

76

DECIMALS (CHAPTER 5)

Place value

Unit 34

Number 7 hundredths

Ten s Un it s De c. Ten Poin t t Hu hs ndr Th edt ous hs and t hs

The decimal point separates the whole numbers from the decimal part.

0 "

23\+\qF_p_\+\q_pL_p_p_ 2 3 "

If a number is less than 1, we write a zero in front of the decimal point.

0 7

Written numeral 0"07

Word form zero point zero seven

4 0 9

23"409

twenty three point four zero nine

the 4 stands for qF_p_

Exercise 34 1 Express the following in 2 different written forms: a 0:6 b 0:45 c 0:908

d

e

8:3

56:864

3 Draw up a place value table in your exercise book using the headings: Number

Place the following into the table: a 8 tenths b 3 thousandths c 7 tens and 8 tenths e 2 hundreds, 9 units and 4 hundredths g 5 thousands, 20 units and 3 tenths

Th ou Hu sand nd s Ten reds s Un it s De c. Ten Poin t t Hu hs ndr Th edt ous hs and t hs

2 Write as decimal numbers: a seventeen and four hundred and sixty five thousandths b two point nine eight three c thirty two point seven five two d twelve and ninety six thousandths e three and six hundred and ninety four thousandths f four and twenty two hundredths

Written numeral

"

d f h

9 thousands and 2 thousandths 8 thousands, 4 tenths and 2 thousandths 36 units and 42 hundredths

4 State the value of the digit 3 in the following: a 4325:9 b 6:374 e 43:4444 f 82:7384

c g

32:098 24:8403

d h

150:953 3874:941

5 State the value of the digit 5 in the following: a 18:945 b 596:08 e 75 948:264 f 275:183

c g

4:5972 358 946:843

d h

94:8573 0:0005

6 A drawing pin has been placed to show the decimal place on these abacuses. i

ii

.

iii

.

a What is the value represented in i, ii and iii? b What is the sum of all 3 amounts? c What is the difference in value between i and ii + iii?

.

DECIMALS (CHAPTER 5)

²

² Write

Write 5:706 in expanded fraction form (as a whole number and fractions). 5:706 7 + = 5 + 10 7 10

=5+

0 100

39 1000

6 1000

+

in decimal form. 39 1000

6 1000

+

77

=

30 1000

=

3 100

+

9 1000

9 1000

+

= 0:03 + 0:009 = 0:039 7 Express in expanded fraction form (as a whole number and fractions): a 5:4 b 14:9 d 32:86 e 2:264 g 3:002 h 0:952 j 2:973 k 20:816 m 9:008 n 154:451

c f i l o

2:03 1:308 4:024 7:777 808:808

8 Write in decimal form: a

6 10

d

8 10

+

9 1000

g

5 10

+

6 100

j

1 10

+

1 1000

+

8 1000

b

19 100

c

4 10

e

52 1000

f

5 100

+

2 1000

h

2 1000

+

3 10 000

i

9 100

+

4 1000

k

4+

3 10

+

l

3 100

+

8 10 000

8 100

+

7 1000

3 100

+

9 State the value of the digit 2: 324 4 1000

b

47 62 100

c

42 946 100

d

695 24 1000

e

3652 1 10 000

f

8 254 10

g

2 57 10

h

652 5 1000

i

1027 59 10 000

7 10 000

§

in expanded fractional form.

Units

What do these hands represent in: i oral and decimal form

5

.

5

§ § § § §

5

5

6

8

ª ª ª ª

b ª ª ª ª

6

Thousandths

.

ªª ªª ªª

4

5

9

§ § § § §§§ § §

d A

2

5 A

2

3

3

ª ª ª

3

.

ªª ª ªª

5

7

.

§ § § § § § §

3

8

8 8

§

7

A

a Which of the hands in question 10 has the highest value in the: i thousandths place ii tenths place iii ten thousandths place iv hundredths place? b Order the hands from highest to lowest value.

§ § § §

§ § § §

8

A

11

Hundredths

4

§ § § §

4

c

6

A

8

A

Tenths

7

expanded fractional form?

A

a

ii

Dec. Point

6

+

4

4 1000

4

5

+

ª ª ª

9

3 100

3

2

+

9

•

9

9 10

§

3

2+

2

4

10 In the decimal place value card game, this hand represents the number 2:9347 or

7

a

Ten Thousandths

78

DECIMALS (CHAPTER 5)

Rounding decimal numbers

Unit 35

Sometimes completely accurate answers are not required and so we round off to the required accuracy. (to 3 decimal places) (to 2 decimal places) (to 1 decimal place)

g

g

+ 6000

(to the nearest 10) 1 zero (to the nearest 100) 2 zeros (to the nearest 1000) 3 zeros g

g

+ 5700

0:5864 + 0:586 + 0:59 + 0:6

g

g

5716 + 5720

Rules for rounding off decimal numbers If, for example, an answer correct to 3 decimal places is required, we look at the fourth decimal place. ²

If the number in the fourth decimal place is 0, 1, 2, 3 or 4, leave the first 3 digits after the decimal point unchanged.

²

If the number in the fourth decimal place is 5, 6, 7, 8 or 9, increase the third digit after the decimal point by one.

Exercise 35 1 Write these numbers correct to 1 decimal place: a 2:43 b 3:57 c 4:92

d

6:38

e

4:275

2 Write these numbers correct to 2 decimal places: a 4:236 b 2:731 c 5:625

d

4:377

e

6:5237

3 Write 0:486 correct to: a 1 decimal place

b

2 decimal places

4 Write 3:789 correct to: a 1 decimal place

b

2 decimal places

b

2 decimal places

5 Write 0:183 75 correct to: a 1 decimal place d 4 decimal places 6 Find a c e

decimal approximations for: 3:87 to the nearest tenth 6:09 to one decimal place 2:946 to 2 decimal places

b d f

c

DEMO

3 decimal places

4:3 to the nearest whole number 0:4617 to 3 decimal places 0:175 61 to 3 decimal places

7 Evaluate correct to the number of decimal places shown in the square brackets: a

17 4

[1]

b

73 8

[2]

c

4:3 £ 2:6

[1]

d

0:12 £ 0:4

[1]

e

8 11

[2]

f

0:08 £ 0:31

[3]

g

(0:7)2

[1]

h

37 6

[2]

i

17 7

[3]

To find 27 correct to 3 decimal places we first write 27 as a decimal to 4 decimal places, then round. 7

0:2 8 5 7 2 : 0 60 40 50

)

2 7

+ 0:286

DECIMALS (CHAPTER 5)

79

We often shorten very large numbers using letters and decimals to represent them. Salaries, real estate prices and profits or losses of large companies use this form. K represents thousands

mill represents millions

bn represents billions

$27:5 K = $27 500 $19 829 = $19:829 K + $19:83 K rounded to 2 dec. places

$2 378 425

37 425 679 420 37 425 679 420 bn 1 000 000 000 = 37:425 679 420 bn + 37:43 bn

2 378 425 mill 1 000 000 = $2:378 425 mill + $2:38 mill =$

=

rounded to 2 dec. places

8 State the value in whole numbers of the following: a $38:7 K b $43:2 K

c

rounded to 2 dec. places

$98:9 K

9 Round these figures to 1 decimal place of a thousand dollars: a $56 345 b $32 475 c $23 159 10 Convert these salary ranges to 1 decimal place of a thousand dollars: a $70 839 - $73 195 b $158 650 - $165 749

c

$327 890 - $348 359

11 Round these figures to 2 decimal places of a million: a 3 179 486 b 91 734 598 d 1 489 701 e 30 081 896

c f

23 456 654 9 475 962

12 Expand these to whole numbers: a 21:65 mill b 1:93 mill

c

16:03 mill

13 Round these figures to 2 decimals of a billion: a 3 867 900 000 b 2 713 964 784 d 2 019 438 421 e 4 209 473 864 000 14 Expand the following to whole numbers: a 3:86 bn b 375:09 bn d 4:13 bn

d

212:45 mill

c f

97 055 843 899 549 000 000 000

c

21:95 bn

e

0:97 mill

Remember one million = 1¡000¡000 one billion = 1¡000¡000¡000.

15 Leo resigned from his job where his salary was $58:5 K. He accepted a new position with a salary of $82:7 K. Write these salaries in whole numbers and find Leo’s increase in salary. 16 A real estate agent sold properties valued at $170:2 K, $295:8 K and $672:1 K. Write the values in whole numbers and find their total. 17 The value of the grape harvest in the Barossa Valley was $67:4 mill in 2003. In the same year the value of the grape harvest in McLaren Vale was $69:9 mill. a Write as whole numbers: i the Barossa value b

i ii

ii

the McLaren Vale value

Which area had the better value? By how much was it better?

18 McGyver and Sons Engineering made a record profit of $1:2 bn in 2003. In 2002 the profit was $86:7 mill. i $1:2 bn ii $86:7 mill a Write as whole numbers: b Find the increase in profit from 2002 to 2003.

80

DECIMALS (CHAPTER 5)

Ordering decimals

Unit 36 We can mark decimal numbers on a number line. 1.50

1.51

1.52

1.53

1.54

1.55

1.56

1.57

1.58

1.59

You can write zeros at the end of decimal numbers as this does not affect the place value of the other digits. e.g., 1.6 = 1.60

1.60

1:58 < 1:60 because 1:58 is to the left of 1:60: 1:56 > 1:51 because 1:56 is to the right of 1:51. Arrange 0:1, 0:12 and 0:102 in order from smallest to largest. 0:1 = 0:100 =

100 1000

²

0:12 = 0:120 =

120 1000

²

0:102 = 0:102 =

102 1000

Write all numbers with the same number of decimal places by adding zeros. Compare the numbers.

) the numbers are 0:1, 0:102, 0:12 from smallest to largest.

Exercise 36 1 Write down the value of the number at A on these number lines: a b A 6

7

c

48

e

f

3.7

0.09

A

0.1

2.41

2.42

2.43

2.44

2.45

2.46

0.2

A

l

0.1

2.40

4.20

j 0.41

A

2.39

A

4.19

A

2.38

2.1

h 1.96

0.40

2

2.0

A

1.95

k

152

A

3.8

i

A

151

A

g

14

d

A

47

A

13

0.04

2.47

2.48

0.05

2.49

2.50

2.51

2.52

2.53

2.54

Use the number line to compare the pairs of numbers. Write which is the greater: a 2:42, 2:5 b 2:48, 2:38 c 2:45, 2:54 3 Use a number line to show these numbers and then write them in order from smallest to largest. a 1:73, 1:70, 1:69 b 0:79, 0:77, 0:76 c 5:431, 5:427, 5:425 4 Insert the correct signs f>, < or =g to make the statements true: a 0:7 2 0:8 b 0:06 2 0:05 d 4:01 2 4:1 e 0:81 2 0:803 g 0:304 2 0:34 h 0:03 2 0:2 j 0:29 2 0:290 k 5:01 2 5:016 m 21:021 2 21:210 n 8:09 2 8:090

c f i l o

0:2 2 0:19 2:5 2 2:50 6:05 2 60:50 1:15 2 1:035 0:904 2 0:94

DECIMALS (CHAPTER 5)

5 A

81

Maha went to the greengrocer and bought some apples, bananas, grapes and pears.

B

a Which fruit was the most expensive per kg? Price/kg: $4.15 weight: 2.68kg Cost: $11.12

C

Price/kg: $3.65 weight: 2.80kg Cost: $10.22

D

b Which fruit did she spend the least amount of money on? c Which fruit did she buy the largest amount of (in kg)? d Which scales show the highest total price? e Which scales show the lowest weight (in kg) of fruit?

Price/kg: $4.35 weight: 2.58kg Cost: $11.22

Price/kg: $3.95 weight: 2.56kg Cost: $10.11

6 Arrange in ascending order (lowest to highest): a 0:8, 0:4, 0:6 b 0:4, 0:1, 0:9 c 0:14, 0:09, 0:06 d 0:46, 0:5, 0:51 e 1:06, 1:59, 1:61 f 2:6, 2:06, 0:206 g 0:095, 0:905, 0:0905 h 15:5, 15:05, 15:55

7 Arrange in descending order (highest to a 0:9, 0:4, 0:3, 0:8 b c 0:6, 0:596, 0:61, 0:609 d e 6:27, 6:271, 6:027, 6:277 f g 8:088, 8:008, 8:080, 8:880 h

lowest): 0:51, 0:49, 0:5, 0:47 0:02, 0:04, 0:42, 0:24 0:31, 0:031, 0:301, 0:311 7:61, 7:061, 7:01, 7:06

8 At the athletics meet, 5 competitors recorded these times for an event: Matthew - 10:05 seconds, Sam - 10:015 seconds, Jason - 10:5 seconds, Saxby - 10:55 seconds Eli - 10:055 seconds. a Show their times on a number line. b Write their times in order with the winning time first. c Write their names in order from first to last. 9 Continue the number patterns by writing the next three terms: a 0:1, 0:2, 0:3, .... b 0:9, 0:8, 0:7, .... c 0:2, 0:4, 0:6, .... d 0:05, 0:07, 0:09, .... e 0:7, 0:65, 0:6, .... f 2:17, 2:13, 2:09, .... g 7:2, 6:4, 5:6, .... h 3:456, 3:567, 3:678, ....

Ascending means lowest to highest.

Descending means highest to lowest.

82

DECIMALS (CHAPTER 5)

Adding and subtracting decimals

Unit 37 Examples: Addition

Subtraction

Find 3:84 + 0:372

Find 6:7 ¡ 0:637

3:8 4 0 + 0:3 7 2 1

1

6 9 10

4:2 1 2

6:0 6 3

Notice that the decimal points are vertically underneath each other.

Exercise 37

add zeros so that the place values line up.

6:7 0 0 ¡ 0:6 3 7

add a zero so that the place values line up.

DEMO

1 Find: a 0:4 + 0:5 d 0:17 + 0:96 g 0:4 + 0:8 + 4 j 30 + 0:007 + 2:948

b e h k

0:6 + 2:7 23:04 + 4:78 0:009 + 0:435 0:0036 + 0:697

c f i l

0:9 + 0:23 15:79 + 2:64 0:95 + 1:23 + 8:74 0:071 + 0:677 + 4

2 Find: a 1:7 ¡ 0:9 d 2 ¡ 0:6 g 4:5 ¡ 1:83 j 5:6 ¡ 0:007

b e h k

2:3 ¡ 0:8 4 ¡ 1:7 1 ¡ 0:99 1 ¡ 0:999

c f i l

4:2 ¡ 3:8 3 ¡ 0:74 10 ¡ 0:98 0:18 + 0:072 ¡ 0:251

3

DEMO

a Add 2:094 to the following: i

36:918

ii

0:04

iii

0:982

iv

5:906

iii

13:06

iv

24

b Subtract 1:306 from the following: i

2:407

ii

1:405

4 Add: a 31:704, 8:097, 24:2 and 0:891 c 1:001, 0:101, 0:011, 10:101 and 1 e 4, 4:004, 0:044 and 400:44

b d f

3:56, 4:575, 18:109 and 1:249 3:0975, 1:904, 0:003 and 16:2874 0:76, 10:4, 198:4352 and 0:149

5 Use two step operations to find: a 0:18 + 0:072 ¡ 0:251 d 5 + 0:444 ¡ 3:222

b e

4:234 ¡ 3:26 + 1:4 5:26 ¡ 3:111 + 6

c f

2:11 + 0:621 ¡ 0:01 15 ¡ 3:29 + 10:2

6 Subtract: a 29:712 from 35:693 d 3:7 from 171:048 g 3:333 from 22:2 j $109:75 from $115:05

b e h k

6:089 from 7:1 9:674 from 68:3 38:018 + 17:2 from 63 $24:13 from $30:10

c f i l

19 from 23:481 8:0096 from 11:11 (47:64 ¡ 18:79) from 33:108 $38:45 and $16:95 from $60

7

a What length is 1:6 cm less than 4:22 cm? b What distance is 4:2 km more than 3:55 km?

8

a Add three point seven nine four two, eleven point zero five zero nine, thirty six point eight five nine four and three point four one three eight.

DECIMALS (CHAPTER 5)

83

b Find the sum of seventeen and four hundred and twenty five thousandths, twelve and eighty five hundredths, three and nine hundred and seven thousandths and eight and eighty four thousandths. c Add thirteen hundredths and twenty thousandths and one and four hundredths. d Find the sum of fourteen dollars seventy eight, three dollars forty, six dollars eighty seven and ninety three dollars and five cents. 9

a By how much is forty three point nine five four greater than twenty eight point zero eight seven? b How much less than five and thirty eight hundredths is two and six hundred and forty nine thousandths? c What is the difference between nine and seventy two hundredths and nine and thirty nine thousandths? d How much have I got remaining from my sixty four dollars seventy five if I spend fifty seven dollars ninety?

10 Helena is 1:75 m tall and Fred is 1:38 m tall. How much taller is Helena than Fred? 11 On the first day of school the morning minimum temperature was 18:6o C and the maximum afternoon temperature was 35:9o C. What was the range of temperatures on this day? 12 John gets $5:40 pocket money, Pat gets $3:85 and Jill $7:85. How much pocket money do they get altogether? 13 What is the total length of these three pieces of timber: 2:755 m, 3:084 m and 7:240 m? 14 Our class went trout fishing and caught five fish weighing the following amounts: 10:6 kg, 3:45 kg, 6:23 kg, 1:83 kg and 5:84 kg. What was the total weight of all five fish? 15 In a fish shop, four large fish weigh 4:72, 3:96, 3:09 and 4:85 kg. What must the minimum mass of a fifth fish be if the customer wants a minimum of 20 kg of fish? 16 How much change from $100 is left after I buy items for $10:85, $37:65, $19:05 and $24:35? 17 Shin needed to save $62:50 for a computer game. He had $16:40 in his bank to start with and earned the following amounts doing odd jobs: $2:45, $6:35, $19:50, $14:35. Does he have enough money? If he does not, how much more does he need to earn? 18 At a golf tournament two players hit the same ball, one after the other. First Jeff hit the ball 132:6 m. Janet then hit the ball a further 204:8 m. How far did the ball travel altogether? 19 Out of interest I weighed myself weekly. In the first week I put on 1:2 kg while in the second week I lost 1:6 kg. Unfortunately I put on another 1:4 kg in the third week. If at the beginning I weighed 68:4 kg, how much did I weigh after the three weeks?

84

DECIMALS (CHAPTER 5)

Unit 38

Multiplying and dividing by powers of 10

Multiplication Remember 101 = 10 102 = 100 103 = 1000 104 = 10 000 etc.

Examples: 8:3 £ 10 = 8:3 £ 101 = 83 0:0932 £ 100 = 0:0932 £ 102 = 9:32 4:32 £ 1000 = 4:3200 £ 103 = 4320

When multiplying by 10 = 101 , shift the decimal point 1 place right.

The index or power number indicates the number of zeros.

When multiplying by 100 = 102 , shift the decimal point 2 places right. When multiplying by 1000 = 103 , shift the decimal point 3 places right.

Exercise 38 1 Multiply the numbers to complete the table:

Number

2 Find: a e i m q

43 £ 10 4:6 £ 10 0:8 £ 100 0:24 £ 1000 0:0094 £ 101

a

0:0943

b

4:0837

c

0:0008

d

24:6801

e

$57:85

b f j n r

£10

£100

8 £ 1000 0:58 £ 100 3:24 £ 100 2:085 £ 102 0:718 £ 100 000

£1000

c g k o

£104

£106

5 £ 106 3:09 £ 100 0:9 £ 1000 8:94 £ 103

3 Write the multiplier to complete the equation: a 5:3 £ 2 = 530 b 0:89 £ 2 = 890 d 38:094 £ 2 = 3809:4 e 70:4 £ 2 = 704 2 g 65:871 £ 2 = 6587:1 h 0:0006 £ 2 = 600

d h l p

c f i

0:6 £ 10 2:5 £ 100 0:845 £ 1000 0:053 £ 1000

0:89 £ 2 = 8900 38:69 £ 2 = 386:9 0:003 934 £ 23 = 3:934

4 A cinema ticket costs $13:50. If ten friends went to see a film together, what would be the total cost? 5 1 km = 1000 m. So 4:75 km = 4:750 £ 1000 m = 4750 m. Convert to the smaller units by multiplying by a power of 10: a $4:75 to cents b 12:56 kL to litres d 13:86 tonnes to kg e 9:847 m to mm

c f

3:86 cm to mm 2:08 kg to g

DECIMALS (CHAPTER 5)

Division

When dividing by 10n shift the decimal point n places to the left.

Examples: 0:6 ¥ 10 = 0:6 ¥ 101 = 0:06

85

fWhen dividing by 10 = 101 , shift the decimal point 1 place to the left.g

0:37 ¥ 1000 = 000:37 ¥ 103 = 0:000 37

6 Divide the numbers to complete the table:

fWhen dividing by 1000 = 103 , shift the decimal point 3 places to the left.g

a

647:352

b

93 082:6

c

42 870

d

10:94

7 Find: a 2:3 ¥ 10 d 3 ¥ 10 g 394 ¥ 10 j m p

¥10

Number

8:007 ¥ 10 579 ¥ 100 0:03 ¥ 10

¥100

¥1000

DEMO

¥105

b e h

3:6 ¥ 100 58 ¥ 10 7 ¥ 100

c f i

42:6 ¥ 100 58 ¥ 100 45:8 ¥ 100

k n q

24:05 ¥ 1000 579 ¥ 1000 0:03 ¥ 100

l o r

632 ¥ 10 000 579 ¥ 10 000 0:046 ¥ 1000

8 Write the divisor to complete the equation: a 9:6 ¥ 2 = 0:96 b 38:96 ¥ 2 = 0:3896 d 5:8 ¥ 2 = 0:0058 e 15:95 ¥ 2 = 1:595 g 3016:4 ¥ 2 = 30:164 h 874:86 ¥ 2 = 0:874 86

c f

6:3 ¥ 2 = 0:063 386 ¥ 2 = 0:0386

9 When a group of 100 employees won second prize of $13 352 in a lottery, they divided the money equally between them. How much did each person receive? 10 1 L = 1000 mL. So, 987 mL = 987: 0 ¥ 1000 L = 0:987 L Convert to the units given by dividing by a power of 10: a 4975 m to km b 5685 g to kg c 3095 mm to cm d 75 400 cents to dollars e 47 850 litres to kL f 2348 kg to tonnes g 26 cm to m h 5655 mm to m i 500 m to km

1000 m 100 cm 10 mm 1000 mm 1000 kg 1000 g 1000 L

= = = = = = =

1 1 1 1 1 1 1

km m cm m t kg kL

11 How many cents are there in $96:55? 12 Jess was 1:65 m tall and Tom measured 149:5 cm. How much taller was Jess? 13 Emma needed 1:5 kg of sugar, but discovered that she was 300 g short of that amount. How much sugar did Emma have?

86

DECIMALS (CHAPTER 5)

Multiplying decimal numbers

Unit 39 Examples: ²

Step 1: Remove the decimal point, i.e., £ by 101 .

3 £ 0:6 = 18: = 1:8

²

0:4 £ 0:03 = 0012: = 0:012

Step 2: Find 3 £ 6 = 18 Step 3: Replace decimal point,

i.e., ¥ by 101 .

Step 1: Remove decimal points,

i.e., £ by 103 .

Step 2: Find 4 £ 3 = 12 Step 3: Replace decimal point,

i.e., ¥ by 103 .

The number of decimal places in the question equals the number of decimal places in the answer. INVESTIGATION

DEMO

Exercise 39 1 State the number of decimal places in the following products. (Do not calculate the answer.) a 8 £ 5:7 b 12:98 £ 7:6 c 1:2 £ 5:3 d 11:296 £ 11:34 e 1:076 £ 5:2 f 0:0006 £ 0:13 2 Find a d g j m p

the value of: 0:8 £ 7 2:4 £ 3 0:3 £ 0:02 1:2 £ 0:12 30 £ 0:003 700 £ 1:2

3 Given that 34 £ 28 = 952, a 34 £ 2:8 d 0:34 £ 2:8 g 0:034 £ 2:8

b e h k n q

c f i l o r

9 £ 0:04 6:5 £ 4 0:04 £ 0:004 0:12 £ 11 (0:6)2 (0:09)2

find the value of the following: b 3:4 £ 2:8 e 0:034 £ 28 h 0:034 £ 0:028

0:4 £ 0:6 2:7 £ 5 7 £ 0:005 5:05 £ 0:09 0:08 £ 80 0:4 £ 0:3 £ 0:2

c f i

34 £ 0:028 0:34 £ 0:28 340 £ 0:0028

4 Given that 57 £ 235 = 13 395, find the value of the following: a 5:7 £ 235 b 5:7 £ 23:5 d 5:7 £ 0:235 e 57 £ 0:235 g 0:57 £ 0:235 h 5:7 £ 0:000 235

c f i

5:7 £ 2:35 0:57 £ 2:35 570 £ 0:235

5 Find a d g j m

c f i l o

0:5 £ 5:0 3:8 £ 4 0:04 £ 0:04 (0:03)2 2:5 £ 0:004

the value of: 0:4 £ 6 0:03 £ 9 0:9 £ 0:8 0:16 £ 0:5 1:2 £ 0:06

b e h k n

0:11 £ 8 0:03 £ 90 0:007 £ 0:9 (0:2)2 (1:1)2

6 Find the perimeter of these regular polygons: a b 4.09 m

c 30.75 cm

6.045 km

DECIMALS (CHAPTER 5)

d

e

87

f

36.5 mm

2.56 m 3.75 cm

g

h

i 3.68 m

1.25 mm

8.51 cm

7 A stone weighed 5:6 kg. If Duncan was able to lift 8 stones of this weight, how much weight could he lift? 8 I need 4:5 m of hose to water my garden. If hose costs me $3:40 per metre, how much will it cost me to buy my hose? 9 Fred needed at least 25 metres of timber. He found 6 pieces of timber in a shed, each 3:9 m long. Did he have enough altogether? How much timber did he have over or did he still need to find? 10

a Find the cost of 45 litres of petrol at 87:8 cents per litre. b Find the cost of 9:6 metres of pipe at $3:85 a metre. c Find the capacity of 6 dozen 1:25 litre bottles.

11 A caterer orders 5700 pies and 3600 pasties to sell at a football match. The pies and pasties each have a mass of 0:16 kg. What is the total mass of the: a pies b pasties c pies and pasties? 1 d How many heated vans ( 2 tonne capacity) are needed to deliver the pies and pasties? e If the caterer has a profit margin of 29:7 cents on each pie or pasty, what is her total profit if she sells the lot? 12

HAZEL’S PIZZA SHOP MENU Pizza Supreme Mexican Hawaiian Pasta Bolognese Napoletana Chips Drinks Cola Juice

Small $13:50 $11:80 $9:90

Large $15:50 $13:60 $11:70

$7:50 $6:50 $2:50

$13:80 $12:00 $4:10

$2:50 $3:00

$3:50 $3:80

Family $19:80 $17:50 $15:80

Find the cost of: a 4 large Hawaiian pizzas and 4 small chips b 1 family Mexican pizza, 3 large chips and 4 large juices c 5 large colas and 5 small chips d 6 large Napoletanas and 6 large juices e 2 small Supreme pizzas, 2 small Bolognese and 4 large colas. f 3 small Hawaiian pizzas, 2 small Napoletanas, 4 small chips, 1 large chips and 5 small colas.

88

DECIMALS (CHAPTER 5)

Dividing decimals by whole numbers

Unit 40 Examples: ²

4 )

²

4

)

1:1 6 4 : 6 24

Put a decimal point directly above the decimal point of the number to be divided (the dividend).

4:64 ¥ 4 = 1:16

So 4 divides into 4:64 exactly 1:16 times.

0:8 7 5 3 : 5 30 20

Add zeros if necessary to complete the division.

3:5 ¥ 4 = 0:875

Exercise 40 1 Find: a e i m

3:2 ¥ 4 24:16 ¥ 8 5:004 ¥ 9 0:354 ¥ 6

2 Find: a $0:84 ¥ 4 e $5:20 ¥ 8 i $114:75 ¥ 9

b f j n

7:5 ¥ 5 2:46 ¥ 6 52:5 ¥ 5 3:44 ¥ 8

c g k o

b f j

$0:57 ¥ 3 $50:65 ¥ 5 $787:50 ¥ 7

c g k

1:26 ¥ 3 0:72 ¥ 9 8:004 ¥ 6 0:045 ¥ 3 $2:68 ¥ 4 $82:56 ¥ 3 $1040:00 ¥ 8

d h l p d h l

3:57 ¥ 7 81:6 ¥ 4 0:042 ¥ 6 4:25 ¥ 5 $3:90 ¥ 5 $5:22 ¥ 9 $189:96 ¥ 6

3 Solve these problems: a If 5 pens cost $7:75, find the cost of 1 pen. b How much money would each person get if $76:50 is divided equally among 9 people? c One 3:5 m length of timber is cut into five equal pieces. How long is each piece? d How many 7 kg bags of potatoes can be filled from a bag of potatoes weighing 88:2 kg? e If $96:48 is divided equally among six people, how much does each person get? f The football club spent $189:20 on 8 trophies for their best players. How much did each trophy cost? g Paul worked at the local supermarket for 9 hours and was paid $69:30. How much did he earn per hour? 4 The perimeter of each of the following regular polygons is given. Find the length of one side to the nearest 2 decimal places. a b c d

P = 48:88 metres

P = 30:72 km

P = 138:72 cm

P = 34:2 millimetres

DECIMALS (CHAPTER 5)

Calculator practice with decimals 5 Choose the correct answer and then check using your calculator: a 4:387 £ 6 i 263:22 ii 26:322 iii 2:6322 b 59:48 £ 9 i 5:3532 ii 5353:2 iii 535:32 c 18:71 £ 19 i 355:49 ii 35:549 iii 35 549 d 0:028 £ 11 i 3:080 ii 0:0308 iii 0:308 6 Estimate the following using 1 figure approximations: a 8:6 £ 5:1 b 9:8 £ 13:2 d 1:96 £ 3:09 e 15:39 £ 8:109 g 0:976 £ 92:8 h 109:4 £ 21:84

iv iv iv iv c f i

2632:2 53:532 3554:9 30:800 12:2 £ 11:9 39:04 £ 2:08 1446 £ 49:2

Find the actual answers using your calculator. Solve these problems using your calculator. 7 How many $3:60 hamburgers can be bought for $104:40? 8 Thirteen people share a $47 446:75 lottery jackpot. How much do they each collect? 9 21 DVDs cost $389:55. How much does one DVD cost? 10 A square has a perimeter of 12:66 metres. Find the length of each side of the square. 11 How many 2:4 metre lengths of piping are needed to make a drain 360 metres long? 12 The heights of the girls in the Primary School Basketball team were measured in metres and the results were:

1:56, 1:43, 1:51, 1:36, 1:32, 1:45, 1:39, 1:38 Find the mean height. 13 Janine’s weekly earnings for 6 weeks were: $272:25, $301:50, $260:40, $278:85, $284:70 and $288:30. Find the average amount Janine earned per week. 14 A piece of wood is 6:4 m long and must be cut into short lengths of 0:36 m. a How many full lengths can be cut? b What length is left over? 15 When Emily’s family travelled from Adelaide to Eston they used 1:5 tanks of petrol. The tank held 62 litres of petrol.

a How many litres of petrol did they use travelling to Eston? b If petrol cost $0:90 per litre, what was the cost of fuel to travel from Adelaide to Eston? c If the car used 10 litres of petrol per 100 km, how far is it from Adelaide to Eston and back? d If they travelled a total of 2040 km while they were away, how many kilometres did they travel while in Eston?

89

90

DECIMALS (CHAPTER 5)

Fractions and decimals conversion

Unit 41 To convert fractions to decimals we can:

Use Division

Use Multiplication Examples: 4 5

²

=

4£2 5£2

=

8 10

9 25

²

= 0:8

²

=

9£4 25£4

=

36 100

=

7£125 8£125

=

875 1000

0: 4 2: 0

2 5

= 0:4

²

0: 4 4 4 4 4: 0 4 0 4 0 4 0

9

4 9

= 0:4444 :::: = 0:4

= 0:36 Fractions can be written as terminating decimals or recurring decimals.

7 8

²

5

Terminating decimals end. 0:4 is a terminating decimal.

DEMO

Recurring decimals repeat themselves without end. 0:4 is a recurring decimal. The bar above the 4 indicates this.

= 0:875

0:37 is a recurring decimal also. 0:37 = 0:373737:::: without end.

Exercise 41 1 Write as decimals using multiplication: a

7 10

b

1 2

c

2 5

d

3 10

e

4 5

f

1 4

g

4 25

h

3 4

i

1 8

j

5 8

k

7 20

l

6 25

m

13 20

n

11 125

o

4 14

p

2 15

q

5 35

r

9 2 20

s

7 1 25

t

358 500

c

3 8

d

9 8

g

4 78

h

5 38

2 Use division to write as a decimal: a 35 b 95 e

2 34

5 45

f

3 Convert the following fractions to decimals. Use a bar to show the repeating pattern of digits. a

1 3

b

2 3

c

1 6

d

1 7

e

2 7

f

1 12

g

2 9

h

5 6

i

3 11

j

7 12

4 Copy and complete the following pattern:

Fraction:

1 9

2 9

Decimal:

0:1

0:2

3 9

4 9

5 9

6 9

7 9

8 9

9 9

5 Write as decimals: a

23 32

b

11 16

c

17 80

d

11 25

e

3 1 16

f

3 14

g

2 15

h

9 11

i

7 2 30

j

97 50

k

6 13

l

49 160

m

5 3 12

n

31 123

o

23 45

DECIMALS (CHAPTER 5)

91

To convert decimals to fractions we write the decimal with a power of 10 in the denominator then simplify if possible. Examples:

0:6 =

6 10

6:44 44 = 6 + 100

=

3 5

= 6 11 25

²

²

0:625

²

=

625 1000

=

5 8

6 Write as fractions in simplest form: a 0:1 b 0:7 e 0:9 f 0:6 i 0:18 j 0:65 m 0:75 n 0:025

c g k o

0:5 0:19 0:05 0:04

7 Write these as fractions in simplest form: a 0:8 b 0:88 e 0:49 f 0:06 i 0:085 j 0:702

c g k

0:888 0:064 0:3

d h l

0:551 0:096 0:6

8 Write as mixed numbers in simplest form: a 2:8 b 4:5 e 22:32 f 46:19 i 3:260 j 4:014

c g k

3:6 28:42 13:025

d h l

7:2 5:002 12:001

d h l p

0:2 0:25 0:07 0:375

For you to remember This table contains commonly used fractions. Copy and complete the table by calculating the decimal form. 1 2

=

1 3

=

1 4

=

1 5

=

1 6

=

1 8

=

1 9

2 2

=

2 3

=

2 4

=

2 5

=

2 6

=

2 8

=

1 11

=

3 3

=

3 4

=

3 5

=

3 6

=

3 8

=

1 20

=

4 4

=

4 5

=

4 6

=

4 8

=

1 25

=

5 5

=

5 6

=

5 8

=

1 40

=

6 6

=

6 8

=

1 99

=

7 8

=

8 8

=

=

Challenge Convert each of the following mixed numbers into a decimal number and then fit the decimals into the grid alongside. One of the numbers has been inserted to get you started. 315 15

5

.

7

7 27 25

9 23 40

2 78

54 11 20

7 2 10

7 5 10

9 392 10

Notice that the decimal point occupies one square on the grid.

92

DECIMALS (CHAPTER 5)

Review of chapter 5

Unit 42 Review set 5A

1 Given that the boundary of the square represents one unit, what decimals are represented in the following grids? a

2

b

a Express 2:049 in expanded rational form (whole number and fractions). b State the value of the digit 2 in 51:932

3 Round off correct to 1 decimal place: a 0:465 b

$35 650 to $K

4 Given that 58 £ 47 = 2726, evaluate: a 5:8 £ 47 b 5:8 £ 0:47

c

c

5:8 £ 4:7

c

5:6 ¥ 10

8 094 387 to mill

5 Find: a

6:2 £ 1000

b

2:158 £ 100

d

4:2 ¥ 100

6 Convert a 7

$352:76 to cents

b

8:94 L to mL

a Find the difference between 246 and 239:84 b Find 0:03 £ 0:5 c A square has sides of length 3:7 m. What is its perimeter? d How much would each person get if $82:40 was divided equally between four people?

8 Write the following decimal numbers in ascending order:

0:216, 0:621, 0:062, 0:206, 0:026 9 In 3 seasons a vineyard produces the following tonnage of grapes: 638:17, 582:35 and 717:36. a What was the total tonnage for the 3 years? b Find the average tonnage for the 3 years. 10 A marathon runner stops for a drink 13 of the way at the 14:1 km mark. How far has he still to run? 11

a Write in decimal form: i 12 ii

3 8

iii

2 3

b Convert these decimals to fractions in lowest terms: i 0:6 ii 0:85 iii 0:2

DECIMALS (CHAPTER 5)

Review set 5B 1 If

represents one thousandth, what are the decimal values of the following?

a

b

c What is the sum of a and b? 2

a Convert 8 +

7 10

+

9 1000

to decimal form.

b State the value of the digit 6 in 9:016 3 Round off correct to 2 decimal places: a

b

9:4357

4 Given that 26 £ 53 = 1378, a

2:6 £ 5:3

b

$29 762 to $K

c

3 472 613 250 to bn

c

2:17 ¥ 100

evaluate: 2:6 £ 0:053

5 Find: a

1:89 ¥ 10

b

1:114 £ 1000

b

97:82 kg to mg

6 Convert to decimal form: a 7

7408 cm to metres

a Find the product of 4:2 and 1:2 b Evaluate 3:018 + 20:9 + 4:836 c Find the difference between 423:54 and 276:49 d Determine the total cost of 14 show bags costing $7:85 each.

8 Write these decimal numbers in descending order:

0:444, 4:04, 4:44, 4:044, 4:404 9 The first horse in a 1000 metre sprint finished in 56:98 seconds. The second and third horses were 0:07 seconds and 0:23 seconds behind the winner. What were the times of the: a second horse b third horse?

10

a Write as fractions in lowest terms: i 0:46 ii 0:375

iii

0:05

b Write as decimals: i 34 ii

iii

7 9

4 25

93

94

PERCENTAGES (CHAPTER 6)

Percentages and fractions

Unit 43

Percent means ‘out of one hundred’. 100 100

1 100

or

or 100%

1%

5 out of 100 =

5 100

= 5% = 5 percent

50 out of 100 =

50 100

= 50% = 50 percent

Most common fractions can be changed into percentages by first converting into fractions with a denominator of 100. For example: ²

²

²

=

CHAPTER 6

1 5

=

20 100

=

1 4

= 20%

=

25 100

=

7 25

= 25%

=

7£4 25£4

=

28 100

= 28%

Exercise 43 1 In each of these patterns there are 100 tiles.

a

b

Write the number of coloured tiles as a fraction of 100.

2 In this circle there are 100 symbols. Count each type then write the number of each type of symbol as a fraction of 100. a M= b C= c L= d X= e V=

3

i ii a

X M V M X V C X X C L X X C L V C X C X X X V M X V M C X C V X V X V V X M V LM X C M X V X V CX V L X C L C X X M C V X V L CL VV VM X C X XC X X V L V X LV X V X V L M X V C X C XM V X V V

What percentage is shaded in these diagrams? What percentage is unshaded? b c

4 Estimate the percentage shaded: a b

0 10 20 30 40 50 60 70 80 90 100

Check to see that your numerators total 100.

d

c

0 10 20 30 40 50 60 70 80 90 100

d

0 10 20 30 40 50 60 70 80 90 100

PERCENTAGES (CHAPTER 6)

In a class of 25 students, 6 have black hair.

To change a fraction to a percentage, we write the fraction with 100 in the denominator.

The fraction with black hair =

Examples:

= 13 25

²

=

13£4 25£4

=

52 100

²

= =

95

557 1000 557¥10 1000¥10 55:7 100

=

6 25 6£4 25£4 24 100

So the percentage with black hair is 24%.

= 55:7%

= 52%

5 Write the these fractions as percentages: a

19 100

b

3 100

c

37 100

d

54 100

e

79 100

f

50 100

g

100 100

h

85 100

i

6:6 100

j

34:5 100

k

75 1000

l

356 1000

6 Write as fractions with denominator 100, and then convert to percentages: a

7 10

b

1 10

c

9 10

d

1 2

e

1 4

f

3 4

g

2 5

h

4 5

i

7 20

j

11 20

k

7 25

l

19 25

m

23 50

n

47 50

o

1

7 Copy and complete these statements: a Fourteen percent means fourteen out of every ....... b If 53% of the students in a school are girls, 53% means the fraction c 39 out of one hundred is ......%. 8 In a class of 25 students, 13 have blue eyes.

::::::: : :::::::

Remember to write the fraction with 100 in the denominator.

a What percentage of the class have blue eyes? b What percentage of the class do not have blue eyes? 9 There are 35 netballers. 14 of them are boys. What percentage are girls?

10

A pack of 52 playing cards has been shuffled and 25 cards have been dealt as shown. a What percentage of the cards shown are: i hearts ii black iii picture cards iv spades? b If an ace is 1 and picture cards are higher than 10, what percentage of the cards shown are: i 10 or higher ii 5 or lower iii higher than 5 and less than 10? c From a full pack, what percentage are: i red ii picture cards iii diamonds iv spades or clubs? (J, Q and K are picture cards.)

96

PERCENTAGES (CHAPTER 6)

Percentage, decimal and fraction conversions

Unit 44 Percentages ²

85% 85 100 85¥5 100¥5 17 20

= = =

Fractions ²

Percentages ²

2:5% = = = = =

2:5 100 2:5£10 100£10 25 1000 25¥25 1000¥25 1 40

²

= 21 ¥ 100 = 21:0 ¥ 100 = 0:21

fto remove the decimalg fto simplifyg

21%

Decimals

²

First convert to a fraction with denominator 100, then write in simplest form.

12 12 % = 12:5% = 12:5 ¥ 100 = 12:5 ¥ 100 = 0:125

140% = 140 ¥ 100 = 140 ¥ 100 = 1:4

Divide the percentage by 100 to obtain the decimal.

Exercise 44 1 Write as a fraction in lowest terms: a 43% b 37% e 90% f 20% i 75% j 95% m 25% n 60% q 5% r 44%

c g k o s

50% 40% 100% 80% 200%

2 Write as a fraction in lowest terms: a 12:5% b 7:5% e 97:5% f 0:2%

c g

0:5% 0:05%

d h

17:3% 0:02%

3 Write as a decimal: a 50% e 85% i 15%

b f j

30% 5% 100%

c g k

25% 45% 67%

d h l

60% 42% 125%

4 Write as a decimal: a 7:5%

b

18:3%

c

17:2%

d

106:7%

h

6 12 %

l

4 14 %

e

0:15%

f

8:63%

g

37 12 %

i

1 2%

j

1 12 %

k

3 4%

d h l p t

70% 25% 3% 300% 350%

97

PERCENTAGES (CHAPTER 6)

Fractions

(2 methods)

Percentages

Decimals

3 4

0:27 = 0:27 £ 100% = 27%

Write as a decimal. Multiply by 100%.

= 0:75 = 0:75 £ 100% = 75% or 3 4

=

3£25 4£25

=

75 100

Percentages

Multiply the decimal by 100% to obtain the percentage.

Write as a fraction with denominator 100.

= 75% 5 Change to percentages by writing as a decimal first: a

1 10

b

8 10

c

4 10

d

3 5

e

2 5

f

1 2

g

3 20

h

1 4

i

19 20

j

3 50

k

39 50

l

17 25

m

3 8

n

1

o

11 100

p

3 8

q

1 3

r

2 3

6 Copy and complete these patterns: a 1 is 100% b 15 = 20%

c

1 3

is 33 13 %

d

1 4

is ......

1 2

is 50%

2 5

= ::::::

2 3

is ......

2 4

=

1 4

is ......

3 5

= ::::::

3 3

is ......

3 4

= ::::::

1 8

is ......

4 5

= ::::::

4 4

= ::::::

5 5

= ::::::

1 16

is ......

1 2

7 Change the following into percentage form by multiplying by 100%: a 0:37 b 0:89 c 0:15 e 0:73 f 0:05 g 1:02

d h

0:49 1:17

8 Change the following into percentage form by multiplying by 100%: a 0:2 b 0:7 c 0:9 e 0:074 f 0:739 g 0:0067

d h

0:4 0:0018

9 Copy and complete the table below:

Percent a

20%

b

40%

Fraction

0:2 2 5

c d

3 4

Fraction

g

k l

Decimal 0:35

12:5% 5 8

i j

0:85 2 25

Percent

h

0:5

e f

Decimal

100% 3 20

0:375

is .......

98

PERCENTAGES (CHAPTER 6)

Percentages on display and being used

Unit 45 We can convert

1 4,

0:42, 33% to percentages and plot them on a number line.

²

1 4

²

0:42 = 0:42 £ 100% = 42%

²

33%

=

1 4

£ 100%

= 25% = 33%

Using the percentages we can arrange the numbers in order from lowest to highest. Qr_ 0%

10

20

33% 30

0.42 40

50

60

70

80

90

100%

Exercise 45 1 Convert each set of numbers to percentages and plot them on a number line: a

f 35 , 70%, 0:65g

b

f55%,

9 20 ,

0:83g

c

f0:93, 79%,

17 20 g

d

f0:85, 34 , 92%g

e

f 27 50 , 67%, 0:59g

f

f47%, 0:74,

18 30 g

g

f 34 , 0:65, 42%g

h

f0:39, 58%,

i

f 58 , 73%,

7 2 20 , 5 g

13 20 ,

0:47g

2 Write each of the following number line positions in fraction notation with 100 as the denominator, as decimals and using % notation: a 0%

20

40

60

80

100%

0%

20

40

60

80

100%

0%

20

40

60

80

100%

Which is bigger, 24% or Qw_ ?

24%

b

1 2–

c

This is a table of conversions between fractions, decimals and percentages that are frequently used. Try to learn them. Percentage

Fraction

Decimal

Percentage

Fraction

Decimal 0:05

100%

1

1:0

5%

1 20

75%

3 4

0:75

33 31 %

1 3

0:3

50%

1 2

0:5

66 32 %

2 3

0:6

25%

1 4

0:25

12 21 %

1 8

0:125

20%

1 5

0:2

6 41 %

1 16

0:0625

10%

1 10

0:1

1 2%

1 200

0:005

PERCENTAGES (CHAPTER 6)

99

3 Refer to the illustration given and then complete the table which follows:

Students a b c d e f g

4

a

b

c

Number

Fraction

Fraction with denominator of 100

Percentage

wearing shorts with a ball wearing skirts and dresses wearing shorts and with a ball wearing track pants, baseball cap and striped top wearing shorts or track pants every student in the picture

Column A represents the students of room 16 who are driven to school. i What percentage are driven to school? ii What percentage find some other way to get to school? Column B represents the students of room 16 who play a musical instrument. i What percentage play a musical instrument? ii In lowest terms, what fraction play a musical instrument?

Percentage

Room 16 of Greenfields School 100 90 80 70 60 50 40 30

20 Column C represents the students of room 16 who 10 play sport for the school teams. 0 i What percentage play sport for the school A B teams? ii In lowest terms what fraction does not play sport for the school teams?

C

D

E

F

d

Column D represents the students who regularly use the internet or CD-Roms. i What percentage regularly use the internet or CD-Roms? ii If there are 30 students in this class, how many do not use CD-Roms or the internet regularly?

e

Column E represents the students who have been overseas. i What percentage have not been overseas? ii What fraction of the students is still to go overseas? (lowest terms)

f

Column F represents the students who can type more than 20 words a minute. If there are an equal number of boys and girls in this class of 30 and 3 more girls than boys can type more than 20 words a minute, what percentage of the girls can type over 20 words a minute?

100

PERCENTAGES (CHAPTER 6)

Representing percentages

Unit 46 Graphical representation

Passenger 20.4% Motor Cyclist 7.9% Pedestrian 21.7% Cyclist 5.3% Driver 44.7%

We often see percentages marked on pie charts and other statistical graphs. On pie charts the sector angle must accurately show the actual percentage. This pie chart shows the percentages of different road users who were killed in road accidents.

FATALITIES BY ROAD USERS

The sum of the percentages should be 100%. Can you explain why it may not be exactly 100% for a graph like this?

Exercise 46 1 The sectors of this pie chart of percentages represent 3 age groups of people living in Australia in 1996. Match your prediction with the graph and give reasons for your choice. a Under 15 b 15 - 64 c 65 and over.

21%

14%

2 Look at these diagrams. Find the unknown percentages: a b

15%

lemon squash¡/ lemonade

8%

mineral water

17%

other

Steel garbage is 5% of the total. The depth of steel garbage measures 2:5 mm on this graph. i

Use your ruler to find the percentage of plastic garbage.

ii

Then find the percentage of food garbage.

food ?%

paper 21%

?%

65%

cola brands

glass 16% plastic ?% garden 7% steel 5% other 4%

Sales of all carbonated softdrinks

aluminium 1%

Contents of a garbage can

2.97%

3 Name the states and territories whose percentage of Australia’s total area is represented by the figures shown on the graph. You may find it helpful to study a map of Australia to compare the areas with the percentages shown on the graph.

Click on the icon for an activity.

ACTIVITY

22.48%

17.52%

32.88%

0.089% 12.80% 10.43%

0.026%

PERCENTAGES (CHAPTER 6)

101

Geometric representation 20 50

There are 50 squares. 20 are shaded. What percentage is shaded?

= 0:4 = 0:4 £ 100% = 40%

so 40% is shaded and 60% is unshaded f100% ¡ 40%g 4 Copy and complete the following table, filling in the shading where necessary:

Figure

Fraction Percentage Percentage shaded shaded unshaded

a

b

Figure

Fraction Percentage Percentage shaded shaded unshaded

e

3 4

37:5%

f

30%

PRINTABLE WORKSHEET

c

d

2 3

g

1 6

h

30%

Click on the icon for the activity that matches percentages with geometric representations. 5 Construct a square with 10 cm sides. Divide it into 1 cm squares. a How many squares must you shade to leave 65% unshaded?

ACTIVITY

EXTRA PRACTICE

b In lowest terms, what fraction of the overall square is unshaded? 6 Construct a rectangle 10 cm by 5 cm. Divide it into 1 cm squares. Shade 7 squares blue, 9 squares red and 20 squares yellow. What percentage of the rectangle is: a red b blue c not shaded d either red or blue? 7 Use a compass to draw a circle. Colour 50% of your circle red, 25% blue, 10% orange, 5% green, 5% purple, 5% yellow.

[Hint: 100% of a circle = 360o, so 1% of a circle = What fraction of the whole circle is: a blue b d orange or blue e

360o 100

= 3:6o and 20% of a circle =

red red or purple or yellow

c f

£ 20 = 72o :]

orange coloured?

8 Divide a circle into 5 equal sectors of 20%. Colour 15 of the circle red, 40% yellow, If you drew 4 such identical circles: a what percentage of all the circles would be i blue ii red? b What fraction of the 4 circles would be yellow?

360o 100

1 5

blue and 20% green.

102

PERCENTAGES (CHAPTER 6)

Quantities and percentages

Unit 47

One quantity as a percentage of another We can only compare quantities with the same units. 3 apples as a percentage of 10 apples is possible.

3 apples as a percentage of 7 bananas is not possible.

To express 800 m as a percentage of 2 km, 800 m 2 km 800 m = 2000 m =

800¥20 2000¥20

=

40 100

Write the quantities as a fraction. Convert to the same units.

Out of 56 cakes baked, a shop sold 49. We express this as a percentage as 49 cakes 56 cakes =

49¥7 56¥7

=

7 8

= 0:875 = 87:5%

= 40%

Exercise 47 1 Choose a common name (denominator) which could be sensibly used to express one quantity as a percentage of the whole in each case. a coffee, tea b hamburgers, pizza c Virgin Blue, Qantas d fins, wetsuit, goggles e train, bus, tram f e-mail, letters, fax, telephone g saxophone, clarinet, recorder, trumpet h Holden, Ford, Mitsubishi, Toyota 2 Express the first quantity as a percentage of the second: a 10 km, 50 km b $2, $8 o o d 120 , 360 e 60 cents, $2 g 400 mL, 2 L h 6 months, 4 years j 48 kg, 1 tonne k 36 cents, $2 m 25 cm, 0:5 m n 48 min, 2 hours p 90 cents, $45 q 5 mg, 2 g

c f i l o r

3 m, 4 m 90o , 360o 50 g, 1 kg 5 mm, 8 cm 180 cm, 3 m 6 hours, 2 days

3 Express as a percentage: a 13 marks out of a possible 25 b 72 marks out of a possible 80 c 427 books sold out of a total 500 printed d 650 square metres of lawn in a 2000 square metre garden e 27 400 spectators in a 40 000 seat stadium f An archer scores 95 points out of a possible 125 points. 4 What percentage is: a 42 of 60 d 3 minutes of one hour g 420 kg of 1 tonne

b e h

34 of 40 175 g of 1 kg 16 hours of 1 day

c f i

48 seconds of 2 min 440 mL of 2 L 174 cm of 1 m?

PERCENTAGES (CHAPTER 6)

103

Finding percentages of quantities We can find a percentage of a quantity using these steps: 10% of 7 m = 0:1 of 700 cm = 0:1 £ 700 cm = 70 cm

35% of 5000 people = 0:35 of 5000 = 0:35 £ 5000 = 1750 people

Write the % as a decimal. ‘of’ means multiply.

Remember that the word ‘of’ indicates that we multiply.

5 Find: a 20% of 360 hectares

b

25% of 4200 square metres

c

5% of 9 m (in cm)

d

40% of 400 tonnes

e

10% of 3 hours (in min)

f

8% of 80 metres (in cm)

g

30% of 2 tonnes (in kg)

h

4% of 12 m (in mm)

i

15% of 12 hours (in min)

j

75% of 250 litres (in mL)

6 A school with 485 students enrolled takes 20% of them for an excursion to the museum. How many are left at school? 7 An orchardist picks 2400 kg of apricots for drying. If 85% of the weight is lost in the drying process, how many kilograms of dried apricots are produced?

8 A council collects 4500 tonnes of rubbish each year from its ratepayers. If 27% is recycled, how many tonnes is that? 9 A marathon runner improves her best time of 3 hours by 5%. What is her new best time?

10 Damian was 1:5 metres tall at the beginning of the school year. At the end of the year his height had increased by 5:6%. What was his new height? 11 The fruit drink made at a packaging plant consists of water (65%), blended with pure juice. If the plant produced 25:5¡¡kL of fruit drink last season, how many litres of this was pure juice? (Remember: 1 kL = 1000 L) 12 Dan played 30 games of baseball in a summer season. If the team lost 27% of those games, how many games did they win? 13 Which is the larger amount? a 40% of a litre or 13 of a litre c 8% of $100 or 85 cents e 33% of 1000 or 13 of 1000

b d f

20% of one metre or 14 of a metre 5% of a kilolitre or 5000 millilitres 30% of a kg or 315 g

104

PERCENTAGES (CHAPTER 6)

Money and problem solving

Unit 48 Finding a percentage of an amount

20 cents = 100 cents

20% of one dollar could look like

or

We can find

20 100

Finding one amount as a percentage of another To find 25 cents as a percentage of $1,

$16 as a percentage of $80

first write $1 as 100 cents.

=

16 80

Then 25 cents as a percentage of $1

=

16¥8 80¥8

=

2 10

=

25 100

£ 100%

= 25%

20% of $3500 = 0:2 of $3500 = 0:2 £ $3500 = $700

A as a percentage of B A is £ 100% B

£ 100% £ 100%

£ 100%

= 0:2 £ 100% = 20%

Exercise 48 1 Copy and complete:

This fraction represents ...... cents out of every ............ cents

This fraction represents ...... dollars out of every ............ dollars.

2 Find: a d g j m

10% of $47 11% of $20 83% of $720 12% of $2950 17:5% of $4000

3 Express: a $5 as % of $20 d $20 as % of $80 g $1:50 as % of $30 j $40 as % of $60

b e h k n

30% of $180 37% of $700 36% of $4:50 45% of $9700 6:8% of $40

c f i l o

b e h k

$15 as % of $150 $25 as % of $125 35 cents as % of $1:40 $334 as % of $33 400

c f i l

=

:::::: ::::::

=

= ...... %

:::::: 100

= ...... %

70% of $21 27% of $150 8% of $48:50 54% of $2500 10:9% of $50 000 $3 as % of $20 $6 as % of $120 $8 as % of $24 $9:95 as % of $99:50

PERCENTAGES (CHAPTER 6)

105

4 Write these scores as percentages. Arrange them in descending order. a Jan threw 18 goals from 25 shots, Jill 30 from 40, Jessie 38 from 50 and Jenny 20 from 32. b Jeff threw 21 from 30, Jake 40 from 60, Joel 34 from 50 and Juan 50 from 80. 5 In a series of three matches, Kim scored 5 goals from 9 shots, 7 from 11 and 4 from 5, and Kathy scored 15 from 20, 7 from 14 and 9 from 16. a Who was the more accurate scorer overall? b By what percentage was one girl better than the other? 6 Each of the following students saved a percentage of their allowance. Arrange the names of the students and their percentage saved in descending order. a Tom saved $6 from a $10 per week allowance, Tina $35 from $70 per month, Tao $13 from $25 per fortnight and Toni $11 from $20 per week. If each student was promised an extra 10% on the amount they saved over one year, how much more would be received by: b Tao

c Toni?

7 Nicky pitched 9 strikeouts and 4 walks against the 36 batters who faced her. What was Nicky’s percentage of: b walks? a strikeouts 8 A goal kicker had 80 kicks for goal during the football season. He kicked 56 goals. What percentage of his scoring attempts were: a goals b not goals? 9 Out of $1200, Sarah gets paid 30% and Jack gets paid 45%. Peter is paid the remainder. How much does each person receive?

Complete the crossnumber by writing all the clues as percentages. 1

2

3

4

5

6

7 10

8 11

14 18

16 19

21 25

13

15

22 26

20 24 27

2 5

16

4 5 6 7

17

23

15

3

9

12

Across 1 1 4 0:07 6 1 2 1 100

8 10

0:67

12 13 14

1 0:09 0:87

11 20

18 19 20 21 22

1 20 326 100

0:03 1 100 2 25 4 5 4 25

24 25

0:05

26 27

0:21 0:85

37 50

Down 2 0:56 3 0:4 3 4 4 7 8 9 11 12 13 17

1:58 3 5 703 100

0:57 3 20 24 25 14 50

18

3 10

19 21

1:11

23

0:06

24 26 27

21 25 11 20 1 50

0:08

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106

PERCENTAGES (CHAPTER 6)

Discount and GST

Unit 49

If the marked price of a wetsuit is $200 and 15% discount is offered, the discount is The selling price is found by:

15% of marked price = 15% of $200 = 15% £ $200 = 0:15 £ $200 = $30 discount

Normal price Less 15% discount Selling price

$200 ¡ $30

Discount % is a percentage decrease.

$170

Exercise 49 1 If there is a 10% discount on a pair of shoes, originally priced at $90, find the amount of discount. 2

take

a If the marked price of a DVD player is $320 and 15% discount is offered, find the actual selling price. b A camera’s normal price is $460. By buying it duty free, it is 25% less. What is the selling price duty free? c A supermarket is offering 2% discount on the total of your shopping docket. How much will you pay if your docket is $130? d If the marked price of a computer is $2600 and 12% discount is offered, what is the new selling price?

3 Find the selling price after these discounts have been made: a b c

d

e

f

g

h

i

j

k

l

4 A television set is priced at $456 with 10% discount in store A. In store B a similar television set is priced at $525 with 20% discount. Which television set would cost less?

Remember to round off to the nearest cent.

PERCENTAGES (CHAPTER 6)

Goods and services tax (GST) Many goods and services sold in Australia include a goods and services tax (GST).

107

The rate of GST is 10%. This is a percentage increase.

For example, in order to make a profit a shop must sell an item for $80, and the GST must be added to this price. The GST is

10% of $80 = 0:1 £ $80 = $8

The shop sells the item for $80 + $8 = $88

5 What is the GST which must be added to the following items? a b

Selling price = $20 + GST

Selling price = $800 + GST

6 What is the selling price of these items? a b

Selling price = $2200 + GST

c

Selling price = $56 + GST

c

Selling price = $9.50 + GST

Selling price = $135 + GST

7 What is the GST on an item which would otherwise sell for: a $100 b $10 c $16

d

$320?

8 What is the price, GST included, on a service which would otherwise cost: a $100 b $48 c $2000

d

$640?

9 A shopkeeper needs to sell a pair of shoes for $160 to make the profit she wants. a What is the GST she must add on? b What must she sell the shoes for? 10 A bicycle shop sells bicycles for $250 and GST is to be added to this price. a What is the GST amount? b How much will the customers have to pay for a bicycle? 11 Challenge a Rachael pays her hairdresser $44 for a cut and colour. How much GST was included in the bill? b Martin receives a bill for $528 from the plumber. How much GST was included in the bill?

The bill is: hairdressers charge + 10% GST

108

PERCENTAGES (CHAPTER 6)

Unit 50

Simple interest and other money problems

When a person borrows money from a bank or a finance company, the borrower must repay the loan in full, and pay an additional charge which is called interest.

Simple interest is often called flat rate interest and is not used as often as compound interest.

The total amount to repay on a loan of $5000 for 3 12 years at 8% per annum simple interest can be calculated in this way: The simple interest charge for 1 year = 8% of $5000 = 0:08 £ $5000 = $400 ) simple interest for 3 12 years = $400 £ 3:5 = $1400 The borrower must repay

$5000 + $1400 fprincipal + interestg = $6400

DEMO

Exercise 50 1 Copy and complete the following table, following the example above.

Principal

Interest rate (p.a.)

Time (years)

Interest for one year

Total interest

$2000

10%

2

0:1 £ $2000 = $200

2 £ $200 = $400

$1000

15%

1

$5000

8%

4

$20 000

12%

1 12 p.a. is short for per annum.

2 Find the simple interest when: a $2000 is borrowed for 1 year at 15% per annum simple interest b $3500 is borrowed for 2 years at 10% per annum simple interest c $5000 is borrowed for 4 years at 8% per annum simple interest d $20 000 is borrowed for 1 12 years at 12% per annum simple interest e $140 000 is borrowed for

1 2

year at 20% per annum simple interest.

3 Find the total amount to repay on a loan of: a $2000 for 5 years at 8% p.a. simple interest b $6500 for 3 years at 10% p.a. simple interest

DEMO

c $8000 for 4 12 years at 12% p.a. simple interest d $10 000 for 10 years at 10% p.a. simple interest. 4 Alex borrowed $4000 for 2 years at 5% simple interest. How much did Alex repay at the end of 2 years? 5 Margot invested $6000 for 3 years at 8% simple interest. How much interest did she earn in that time?

PERCENTAGES (CHAPTER 6)

109

Who uses percentages? Extension: These are some of the words that are used regularly where money is being considered: discount, interest, rates, commission, taxation, rebates, deposit, profit, loss, increase, decrease, gross, nett, deduction. If a shopkeeper buys a mountain bike for $500 and sells it a few days later for $400 he has made a loss of $100 or 20%. 6 Kirsten bought a house for $267 000. Soon afterwards she had to move interstate and sold the house for $253 000. Find her: ¶ µ loss £ 100% a loss b loss as a percentage of her cost find cost

A car dealer buys a second hand car for $3800, spends a further $400 fixing the engine and putting better tyres on it and then sells it for $5040. He makes a profit of $840 on his costs. This is a 20% profit as a percentage of his costs. 7 A store keeper buys a set of golf clubs for $300 and sells them for $420. Find the: µ a profit b profit as a percentage of the cost find

¶

profit £ 100% cost

To buy a block of land for $60 000 you may be asked to put down a deposit of 20%. By paying the $12 000 deposit you can then arrange a loan to borrow the remaining 80% balance. Most lenders will not provide a loan for the full 100% value of the land. 8 Martin is looking for a loan to buy a car costing $25 500. He is asked to pay a deposit of 20%. a How much is the deposit? b What percentage does Martin have to borrow? c How much does Martin have to borrow?

When you start working you may be on a salary of $500 per week. This is your gross salary and represents 100% of what your employer pays you. However, you will be expected to pay about 17% in taxation and you may choose to make a 5% contribution to superannuation. This would mean that 22% is taken off in deductions leaving you with a nett salary of 78% or $390:00. 9 Maria’s gross salary was $800 a week. a If she paid 25% in taxation, what was her nett salary? b If she also paid 2% of her gross salary in superannuation, what was her nett salary?

When a real estate agent sells your home for $150 000 and charges you 4 12 % commission he receives $6750 and you receive $143 250. You receive less than the house was sold for. 10 A real estate agent sold my home for $300 000 and charged me 2% commission. How much commission did she receive?

110

PERCENTAGES (CHAPTER 6)

Review of chapter 6

Unit 51 Review set 6A 1

7 25

a

Write

with a denominator of 100:

b

Change

c d e f

Convert 0:45 to a percentage. Express 6 minutes as a percentage of one hour. Find 30% of $600. Find 140% of 2 kilometres (in metres).

1 3

to a percentage.

2 What percentage of the diagram is shaded?

3 Convert to percentages and plot on a number line: f 18 , 52%, 0:8g 4 Samantha had a budget of $200 to spend on clothes. She paid $58 for jeans and $82 for shoes. She spent 24% of her budget on a jacket and the balance on a baseball cap. a How much was the baseball cap? b What percentage of her budget did she spend on the baseball cap and shoes? 5 Write the first quantity as a percentage of the second: a 45o , 360o b 2 mm, 5 cm 6 A small country town had 280 households. 45% used a wood fire to warm their homes, 30% used electricity, 15% gas and the rest used oil or kerosene. How many households used gas, oil or kerosene?

7 A variety store is having a “20% off the ticket price” sale. If I bought a $38:90 toaster, a $79:90 sleeping bag, 2 bath towels at $12:90 each and a $5:40 blank video tape, how much would I save? 8 A plumber charges $940 for supplying and installing a new hot water service. a How much GST must be added? b What amount is the customer charged? 9 Joshua bought $690 of goods at the hardware store. He was allowed 5% discount for paying cash. How much did he pay to the hardware store? 10 Find the simple interest when $2400 is borrowed for 2 years at 12% p.a.

PERCENTAGES (CHAPTER 6)

Review set 6B 1

a Write 40% as a fraction in lowest terms. b Convert 0:45 to a percentage. c Write

7 25

as a percentage.

d Find 85% of $1200. e Find 16% of 4 m (in cm). 2 What percentage of the diagram is unshaded?

3 Write f 34 , 0:78, 72%g

as percentages and then plot them on a number line.

4 Express the first quantity as a percentage of the second. a 13 goals from 25 shots b 58 cm from 2 m

c

500 mL from 5 L

5 Anthony lost 6 marks in a test out of 25. What percentage did he score for the test? 6 What percentage is 650 kilometres of a 2000 km journey? 7 One hundred students agree to come to a fund raising school disco. What price should the committee charge each student if the DJ costs $180, balloons and streamers cost a further $20 and they want to make a 50% profit on their costs? 8 A fridge has a selling price of $840 but a discount of 15% is given. a Find the discount. b What is the actual price paid for the fridge? 9 A dentist charges $270 for dental treatment and GST must be added to this amount. a What is the GST amount? b How much will the customer have to pay? 10 Maryanne received 12% p.a. simple interest on her $3500 investment. a How much interest did she earn after 2 years? b What was her new balance? 11 The deposit on a new car was 20%. If the car cost $16 800, how much was the deposit? 12 A house was bought for $145 000 and sold for a 10% profit. How much was it sold for?

111

112

REVIEW OF CHAPTERS 4, 5 AND 6

TEST YOURSELF: Review of chapters 4, 5 and 6 1 What fraction of the diagram is shaded? a

b

2 What percentage of the diagram is shaded? 3 Find the fractions represented by the points on the number lines: a b 2

A

B

4 Draw a rectangle and shade

7 10

3

1

C

2

D

of it.

5 If the dollar represents the unit, what is the decimal value of:

6 Write in decimal form: a 26 045 cents as dollars 7

a Write

3 20

b

4500 metres as kilometres.

with denominator 100.

b Write 0:24 as a percentage. c Write

4 9

as a decimal number.

8 Write 6:095 in expanded fraction form. 9 Copy and complete the pattern in lowest terms: 2, 1 56 , 10

a Write

47 3

......, ......, 1 13 , 1 16 , 1, ......

as a mixed number.

b State the value of the digit 5 in the number 41:452 c Write 0:64 as a fraction in simplest form. 11

a Write 750 metres as a percentage of 1 kilometre. b Find 20% of 300 g.

12 Convert to percentages and plot on a number line: f 25 , 55%, 0:63g 13 What fraction of 2 dollars is 45 cents? Answer in lowest terms. 14

a Find the lowest common multiple of 9 and 12. b Write

7 12

and

c True or false, 15 Write 16 If

3 8

4 25

5 9

with lowest common denominator. 7 12

> 59 ?

as a percentage.

of a number is 21, find the number.

REVIEW OF CHAPTERS 4, 5 AND 6

17 Find a 18 Find

2 9

1 12 +

4 9

b 3 23 ¡ 1 34

c 3£

113

5 8

of $36:

19 Find the value of: a 1:27 + 5:063

b

5:063 ¡ 1:27

c

d

6:3 £ 0:9

6:3 ¥ 0:9

20 An electrician charges $615 for parts and labour. a How much GST must he add to his bill? b What amount does the customer pay? 21 Continue the number pattern by writing the next 3 terms: 4:51,

4:42, 4:33, ......

22 Jan received 10% discount when she bought a coat marked at $210. How much discount did she receive? 23 Find the simple interest payable when $1500 is borrowed for 2 years at 7% p.a. 24 Find the values of ¤ and ¢ if

¤ 25 5 = = : 8 16 ¢

25 Round off correct to 1 decimal place: a 0:947 b

$87 500 to $K

c

8 705 059 to mill

26 Jon had two $50 notes, a $5 note, and three 20 cent coins. Write this amount as a decimal of 1 dollar. 27 How many litres of drink are in 7 bottles which each hold 1:25 L? 28

3 4

of a box of apples were eaten. If 7 apples remained, how many had been eaten?

29 How much change would you receive from $50 if you bought five postage stamps costing $1:45 each and an express post bag costing $8:20? 30 6 students ran equal distances in a 4:8 km relay. How many metres did each student run? 31 When Reiko ordered new carpet she was asked to pay 20% deposit. If the carpet cost $3200, how much deposit did she pay? 32 A piece of steak weighing 4:35 kg is cut into 3 equal slices. What is the weight of each piece? 33 The number of shoppers in the mall on Saturday was 20% more than on Friday. If there were 1850 shoppers on Friday, how many were there on Saturday? 34 Find the total cost of 15 bus tickets costing $2:25 each. 35 How many 1:2 kg books would weigh 2:4 tonnes? 36 In a class of 30 students, two fifths of the students play sport after school. How many students do not play sport after school? 37 Sue cut 15 metres of wire mesh from a 45 metre roll. What percentage of the roll was left? 38 Arkie had $50 to spend on food. He spent $12 on fruit and vegetables, 18% on meat and $23 on groceries. a How much did he spend on meat? b What percentage of his money was left? 39 A variety store is having a “15% off everything” sale. How much in total would you pay for 2 CDs normally costing $28:50 each and a shirt normally priced at $33?

114

MEASUREMENT (LENGTH AND MASS)

Unit 52

Reading scales

In everyday life we measure many things. Some common measuring instruments are shown here. 17

18

19

35

cm

37

38

F

100

1 000 10 000 9 0 1

E

8 7

FUEL

40

41

42

°C

6 5 4

2 3

2 3

1 0 9 4 5 6

9 0 1 8 7

8 7

6 5 4

10 2 3

2 3

1 0 9 4 5 6

KWH

8 7

9 0 1 8 7

KILOWATT HOURS

6 5 4

2 3

The electricity meter shows 26 593 kWh.

There are 8 main divisions from empty to full on this fuel gauge. 5 8

39

Each small division is 0:1o C. The thermometer shows 36:8o C

Each small division is 0:1 cm (1 mm). The ruler shows 17:4 cm.

The fuel gauge shows

36

full.

CHAPTER 7

Exercise 52 1 Read these ruler measurements (in cm): a b 20 30 d

16

e

17

10

20

25

26

c

f

10

11

18

19

2 Read the temperature (in o C) for these thermometers: a b 33

34

35

36

37

38

39

40

°C

35

36

37

38

39

40

41

42

°C

c

3 Read these fuel gauges: a F

35

36

37

38

39

40

41

42

°C

35

36

37

38

39

40

41

42

°C

d

b

c

F

E

F

E FUEL

E FUEL

FUEL

4 Read as accurately as possible the speeds shown on these speedometers: a b c 60 40 20 0

80 100 120 140 160 KM/H 180 200

60 40 20 0

80 100 120 140 160 KM/H 180 200

60 40 20 0

80 100 120 140 160 KM/H 180 200

MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)

5 Find the weights, in kilograms, shown by these bathroom scales: a b

71

72

45

c

46

6 Find the quantity of electricity used as shown by these meters: a b 100 1 000 10 10 000 9 0 1 8 7

6 5 4

2 3

2 3

1 0 9 4 5 6

9 0 1 8 7

8 7

6 5 4

2 3

2 3

KILOWATT HOURS

1 0 9 4 5 6

8 7 8 7

10 000

9 0 1

9 0 1

6 5 4

8 7

7 Find the mass (in grams) on these scales: a b

6 5 4

2 3

2 3

1 0 9 4 5 6

9 0 1 8 7

8 7

6 5 4

64

10 2 3

2 3

1 0 9 4 5 6

9 0 1 8 7

KILOWATT HOURS

500 g 1 kg

KWH

8 7

6 5 4

2 3

c

500 g 0

100

1 000

KWH 2 3

63

0

500 g 1 kg

8 Find the quantity of fluid (in mL) in these jugs: a b

0

1 kg

c

1000 mL

1000 mL

1000 mL

800

800

800

600

600

600

400

400

400

200 100

200 100

200 100

9 For the following lines: i estimate the length ii using a ruler, measure the length to the nearest mm. iii What was the error in your estimation? a b c

d

e

f

g

h

115

116

MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)

Unit 53

Units and length conversions

The earliest units of measurement used were lengths related to parts of the body. All of these measurements were inaccurate because people are different sizes.

We use the metric system of measurement, which is more accurate.

cubit span

The units of length are millimetres (mm), centimetres (cm), metres (m) and kilometres (km).

pace

The units of mass are milligrams (mg), grams (g), kilograms (kg) and tonnes (t).

Exercise 53 1 State what units you would use to measure the following: a the mass of a person b the distance between two towns c the length of a sporting field d the mass of a tablet

e f g h

the the the the

length of a bus mass of a car width of this book mass of a truck

Activity

Measuring instruments What do these instruments measure? Match the instrument to its name:

pocket watch builders square

thermometer fuel gauge

electricity metre sphygmomanometer

micrometre sextant

a

b

c

d

e

f

g

h

ACTIVITY

Click on the icon to find the activity on ‘Measures and who uses them’.

MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)

117

Conversion diagram for length To convert smaller units to larger ones we divide. ¥10 mm

¥100

¥1000

cm £10

Learn these conversions

m

1 cm = 10 mm 1 m = 100 cm 1 m = 1000 mm 1 km = 1000 m

km

£100

£1000

To convert larger units to smaller ones we multiply. Convert

²

640 cm to m smaller unit to larger ) divide

²

3:8 km to m larger unit to smaller ) multiply

²

7560 mm to m smaller unit to larger ) divide

640 cm = 640 ¥ 100 m = 6:4 m

3:8 km = 3:8 £ 1000 m = 3800 m

7560 mm = 7560 ¥ 1000 m = 7:56 m

2 Write the following in metres: a 900 cm b 643 cm e 9000 mm f 13 500 mm i 2 km j 6:8 km

c g k

4753 cm 620 mm 0:5 km

d h l

35 cm 58 mm 0:826 km

3 Write the following in centimetres: a 7m b 13:8 m e 85 mm f 1328 mm i 1 km j 0:5 km

c g k

0:34 m 402 mm 0:02 km

d h l

0:02 m 0:4 mm 0:003 km

4 Write the following in millimetres: a 7m b 3:4 cm

c

78 cm

d

0:46 m

e

0:26 cm

5 Write the following in kilometres: a 4500 m b 17 458 m

c

200 m

d

16 400 cm

e

653 000 cm

6 If the distance from your home to school is 750 metres, how far in kilometres do you travel to and from school in a week? 7 Zoe is a triathlete. She has to swim 200 m, ride her bicycle for 7:5 km and run 2500 m. What is the total distance Zoe has to travel in a metres b kilometres? 8 Convert all lengths to metres and then add: a 3 km + 110 m + 32 cm c 153 m + 217 cm + 48 mm

b d

72 km + 43 m + 47 cm + 16 mm 15 km + 348 m + 63 cm + 97 mm

9 Write these in the same units and then put in order from longest to shortest: a 37 mm, 4 cm b 750 cm, 8 m, 7800 mm c 1250 m, 1:3 km d 0:005 km, 485 cm, 5:2 m e 3500 mm, 347 cm, 3:6 m f 0:134 km, 128 m, 13 000 cm g 4:82 m, 512 cm, 4900 mm h 72 m, 7150 cm, 71 800 mm 10 Calculate your answer and write it in the units given in brackets: a 6 m ¡ 23 cm (cm) b 9 cm ¡ 25 mm (cm) c 3:8 km ¡ 850 m (m) d 17 m ¡ 8 m 49 cm (m)

118

MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)

Unit 54

Perimeter

The perimeter of a figure is a measurement of the distance around its boundary. For:

Triangle

Square

In figures, sides having the same markings show equal lengths.

Rectangle

b

a

s

w

c

l

the formulae for finding the perimeters of these figures are: P =a+b+c

P =4£s

P = (l + w) £ 2

[P = 4 £ side length]

[P = (length + width) £ 2]

7 cm

²

8 cm

3 cm 9 cm

P

Always give the units of measurement, for example, cm.

²

17 cm

= 3 + 7 + 9 cm = 19 cm

P

DEMO

= (8 + 17) £ 2 cm = 25 £ 2 cm = 50 cm

Exercise 54 1

i ii a

Estimate the perimeter of each figure. Check your estimate with a ruler. b

c

2 Find the perimeter of each of the following triangles: a b

c

19 cm

4.2 km

15 m

13 cm 27 cm 11 m

3 Find the perimeter of: a

b

c 4.5 cm

12 cm

10.2 km

9.8 cm 3.1 km

MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)

4 Find the perimeter of: a

b

c 6 cm

10 cm

9 km

7 km

5 cm

d

4 km

13 km

5 km

e

f 11 cm

8 cm

10 cm 7 cm

g

h

18 cm 11.3 cm

i

10 cm

7.2 cm 7.2 cm

10 cm

9.6 cm

5 Use a piece of string to find the perimeter of the following: a b c

d

6 Solve the following problems: a A rectangular paddock 120 m by 260 m is to be fenced. Find the length of the fence. b How far will a runner travel if he runs 5 times around a triangular block with sides 320 m, 480 m and 610 m?

Draw a diagram to help solve these problems.

c Find the cost of fencing a square block of land with side length 75 m if the fence costs $14:50 per metre. 7

a What is the perimeter of an equilateral triangle with 35:5 mm sides? b If the perimeter of a regular pentagon is 1:35 metres, what is the length of one side? c One half of the perimeter of a regular hexagon is 57 metres. What is the length of one of its sides? d One third of the sum of the lengths of sides of a regular dodecagon is 39 cm. What is its perimeter? e The perimeter of 2 identical regular octagons joined along one side is 98 cm. What is their combined perimeter when they are separated?

8

a Find the length of the sides of a square with perimeter 56 cm. b Find the length of the sides of a rhombus which has a perimeter of 72 metres.

A dodecahedron has 12 sides.

119

120

MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)

Unit 55

Scale diagrams A scale diagram is a drawing or plan which has the same proportions as the original object. The scale is the ratio scale length : actual length.

If the scale is 1 : 20, we can find the: a actual length if the scale length is 3:4 cm b scale length if the actual length is 1:2 m. a

actual length b = 20 £ scale length = 20 £ 3:4 cm = 68 cm

km 0

10

20

30

40

On this map scale, 5 cm (the drawn length) represents 50 km (the actual length).

scale length = actual length ¥ 20 = 1:2 m ¥ 20 = 120 cm ¥ 20 = 6 cm

Scale = 5 cm : 50 km = 5 cm : 50 £ 1000 £ 100 cm = 5 : 5 000 000 = 1 : 1 000 000

Exercise 55 1 For the following scales, state if the drawing or the actual object is larger than the original: a 1 : 500 b 3:1 c 2:5 d 1:4 e 1 : 10 000 2 Find the scale if: a an aeroplane has wingspan 50 m and its scale length is 50 cm b a truck is 15 m long and the diagram has its scale length 12 cm

wingspan

c a bacterium has body length 0:005 mm and its scale length is 10 cm.

3 Find the actual length for a scale length of 5 cm if the scale is: a 1 : 50 b 1 : 2000 c 1 : 10 000

d

1 : 5 000 000

4 If the scale is 1 : 5000, find: a the actual length if the scale length is i

4 cm

ii

5:8 cm

iii

2:4 cm

iv

12:6 cm

iii

20 m

iv

108 m

iii

8:2 cm

iv

0:8 cm

iii

5:6 m

iv

12:2 m

b the scale length if the actual length is i

500 m

ii

175 m

5 If the scale is 1 : 200, find: a the actual length if the scale length is i

3 cm

ii

4:5 cm

b the scale length if the actual length is i

200 m

ii

18 m

6 The drawing of a gate alongside has a scale of 1 : 100. Find: a the width of the gate b the height of the gate c the length of the diagonal support.

(Note: The posts are not part of the gate.)

width of the gate

onal

diag

50

MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)

7 Using the scale on the map of Australia, find the distance from a Sydney to Perth b Adelaide to Darwin c Melbourne to Alice Springs d Canberra to Brisbane.

N

Darwin Cairns

Mt. Isa

Alice Springs

Rockhampton Brisbane

Kalgoorlie

Perth

Adelaide Canberra Sydney Melbourne scale: 1 cm represents 600 km Hobart

8 If the plan of a house wall alongside has been drawn with a scale of 1 : 200, find: a the length of the wall b the height of the wall c the measurements of the door d the measurements of the windows. 9 For the truck alongside, find: a the actual length of the truck b the maximum height of the truck.

(Scale:

10 a

1 : 100)

Measure the length of the body of the dragonfly and find the scale for the diagram. Using the scale in a, find: i the length of the head ii the wingspan iii the greatest width of the rear wing.

length of body

b

121

(Real length = 50 mm)

11 Using the scale shown on the map, find: a the actual distance shown by 1 cm b the map distance required for an actual distance of 200 km c the distance from i A to B ii D to E iii C to F.

E

A

F

D Scale: 1¡:¡500¡000

C

B

122

MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)

Unit 56

Mass The mass of an object is the amount of matter it contains.

Units of mass are milligrams (mg), grams (g), kilograms (kg) and tonnes (t).

Conversion diagram To convert smaller units to larger we divide. ¥1000 mg

¥1000

¥1000

g

Learn these conversions

kg

£1000

1 g = 1000 mg 1 kg = 1000 g 1 t = 1000 kg

t

£1000

£1000

To convert larger units to smaller units we multiply.

Convert:

²

350 g to kg smaller unit to larger ) divide

²

350 g = 350 ¥ 1000 kg = 0:35 kg

7 500 000 mg to kg smaller unit to larger ) divide

²

7 500 000 mg = 7 500 000 ¥ 1000 ¥ 1000 kg = 7:5 kg

8:5 t to kg larger unit to smaller ) multiply 8:5 t = 8:5 £ 1000 kg = 8500 kg

Exercise 56 1 Give a c e g i k m o q s

the units you would use to measure: a person’s mass the mass of an egg the mass of an orange the mass of a raindrop the mass of your school lunch the mass of a refrigerator the mass of a school ruler the mass of a bulldozer the mass of a calculator the mass of an ant

b d f h j l n p r t

the the the the the the the the the the

mass mass mass mass mass mass mass mass mass mass

of of of of of of of of of of

a a a a a a a a a a

ship book lounge suite boulder cricket bat dinner plate slab of concrete leaf computer horse

2 Which of these devices could be used to measure the items in question 1? A

B

spring balance

C

kitchen scales

3 Convert these grams into milligrams: a 2 b 34

c

350

D

bathroom scale

d

4:5

weigh bridge

e

0:3

MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)

4 Convert these tonnes into kilograms: a 4 b 25

c

3:6

d

294

e

0:4

5 Convert these kilograms to grams: a 6 b 34

c

2:5

d

256

e

0:6

6 Convert these milligrams to grams: a 3000 b 2500

c

45 000

d

67:5

e

9:5

7 Convert these kilograms to tonnes: a 4000 b 95 000

c

4534

d

45:6

e

0:8

8 Write the following in grams: a 8 kg b 3:2 kg e 4250 mg f 75 420 mg

c g

14:2 kg 6:8 t

d h

380 mg 0:56 t

9 Convert the following to kilograms: a 13 870 g b 3:4 t

c

786 g

d

3496 mg

10 Calculate your answers in kilograms: a 520 g + 2:1 kg + 16 kg c 1:5 kg ¡ 750 g e 4:2 t ¡ 3 t + 300 kg

b d f

700 g + 1600 g + 63 g 2 t ¡ 763 kg 15 kg ¥ 2

11 Solve the following problems: a Find the total mass, in kilograms, of 200 blocks of chocolate, each 120 grams. b If a nail has mass 25 g, find the number of nails in a 5¡¡kg packet. c Find the mass in tonnes of 15 000 bricks if each brick has a mass of 2:2 kg. d A box of 150 tins of dog food weighs 205 kg. If the empty box weighs 25 kg, find the mass of each tin. e A carton with a mass of 350 g holds 12 boxes of cereal. Each box of cereal has a mass of 850 g. Find the total mass of the carton full of boxes of cereal. 12 Write in the same units. Then list in ascending order (smallest to largest). a 2400 mg, 2 g b 6700 g, 7 kg c d 0:004 t, 3:6 kg, 3800 g e 1900 mg, 1:5 g, 0:002 kg

1420 kg, 1:4 t

13 Write both masses as kg. Find the cost per kg for each of them. Which is the better buy? a b

Soap Powder 3 kg $6.60

Soap Powder 2 kg $4.50

Muesli 1 kg

$16.75

Activity Sheet

Click on the icon for the Measurement Message Activity Sheet.

Muesli 500 g $9.10

123

124

MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)

Unit 57

Problem solving

A rectangular housing block 40 m by 22 m is to be fenced. The fencing costs $75 per metre. The length of fencing needed = (40 + 22) £ 2 m = 62 £ 2 m = 124 m

40 m 22 m

The cost of the fence = 124 £ $75 = $9300

²

Think about what the question is asking and the units you will work in.

²

A labelled diagram is often helpful.

²

Set out your answer in a clear and logical way.

²

You may need to write your final answer in a sentence.

Exercise 57 1 A farmer fences a 250 m by 400 m rectangular paddock with a 3 strand wire fence. a Find the total length of wire needed. b Find the cost of the wire if wire costs $2:40 per metre. 2

a A house owner has a block of land 30 m by 75 m (30 m across the back). If he wishes to fence two sides and the back of the block, what is the total length of the fence needed? b If the fence is to be made of “Good Neighbour” panelling which comes in sheets 2 m wide costing $18:50, what will be the cost of the fence?

3 A carpenter has to make a window frame with the dimensions shown. What is the total length of timber he requires?

5 cm 120 cm

150 cm

4 6m

a Henry edges his garden with railway sleepers. If his garden has two plots as shown, find the total length of sleepers required.

Plot 1 2m

Paths 6m

Plot 2

2m

b If each sleeper is 2 m long and weighs 40 kg, find:

20 m

5 A supermarket buys cartons of canned peaches. Each carton contains 12 cans and each can weighs 825 g. Find the mass in kilograms of a carton of peaches.

i the total number of sleepers needed ii the total mass of sleepers.

MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)

6

125

a A couple wish to build a brick fence along the 60¡¡m front of their block of land. If they want 12 rows of bricks and each brick is 20 cm long, find the number of bricks required. b If each brick weighs 2:5 kg, find the total mass in tonnes of the bricks needed.

7 3.5 m

7.5 m 10 m

A builder needs to construct a pergola with the dimensions as shown. The support posts cost $15 per metre and the timber for the top costs $4:50 per metre. a Find the total length of timber for the top and hence the cost of this timber. b Find the cost of the posts. c Find the total cost of building the frame for the pergola if nails and other extras cost $27.

8 A grazier has a property with the dimensions illustrated. One of the farmhands is asked to check the fence on his motorbike. If he can travel at 15 km/h, how long will it take him to check the whole fence?

5.5 km 12.5 km

6 km

a Using the scale diagram alongside, find the total length of timber required to make the gate frame shown.

9

Scale 1 : 60

b If the timber costs $4:50 per metre, find the total cost of the timber used.

10 A 30 m picket fence is to be built as shown. There is a 2 m post every 2 m, to which the rails are attached. If the timber for the pickets costs $1:80 per metre, for the rails costs $2:50 per metre and for the posts costs $4:50 per metre, find: a the number of posts and hence the total length of timber required for the posts

rail

1.2 m

picket 10 cm

10 cm

b the total length of rails needed c the number of pickets needed and the length of timber needed to make these pickets d the total cost of the fence.

Click on the icon for a worksheet with more problem solving questions.

PRINTABLE WORKSHEET

126

MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)

Unit 58

Review of chapter 7

Review set 7A 1 Read the scales: a

b

c

d

F

5L

80 100 120 60 140 40 160 KM/H 180 20

E

0

FUEL

2 1

200

2 Convert: a 356 cm to mm d 83 000 kg to t

b e

3

3200 g to kg 7:63 m to mm

3 Find the perimeter of: a

4

kg 5 0 6

c f

450 m to km 630 cm to m

b 12 cm

3.8 m

17 cm

4 If the scale is 1 : 500 000 find: a the actual length if the scale length is i

3:8 cm

ii

6:4 cm

iii

12:2 cm

iii

130 km

b the scale length if the actual length is i

50 km

ii

22 km

5 Look at the scale diagram. Use your ruler to find the actual dimensions given that the scale is 1¡¡:¡¡2000. Which of the following could it represent? A C

a bathroom a swimming pool

B D

a beach towel a sports field

6 Kym competes in the 200 metre, 400 metre, 800 metre, 1500 metre and 5000 metre running events on sports day. How many kilometres does she run? 7

a Find the total mass in kg of 1500 oranges if the average mass of an orange is 180 g. b If a truck can carry 1400 kg of soil, how many truckloads will be needed to remove 42 tonnes of soil?

8 A rectangular farming block with dimensions as shown is to be fenced with a 3-strand wire fence. a Determine the perimeter of the block. 180 m

b Determine the total length of wire required. c If the wire costs $1:75 per metre, find the total cost of the wire.

320 m

MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)

127

Review set 7B 1 Read the gauge for the amount of electricity used: 10 000 9 0 1 8 7

2 Convert: a 3480 g to kg d 5:4 m to cm

b e

3 Find the perimeter of: a

b

100

1 000

6 5 4

8623 mm to m 13:2 t to kg

2 3

2 3

1 0 9 4 5 6

9 0 1 8 7

8 7

6 5 4

10 2 3

2 3

1 0 9 4 5 6

KILOWATT HOURS

c f

KWH

8 7

9 0 1 8 7

6 5 4

2 3

4:6 g to mg 13:3 km to m

13.2 km 10 cm 6 cm

8.3 km

4 If the scale is 1 : 2 500 000 determine: a the actual length if the scale length is

i

4:8 cm

ii

0:7 cm

b the scale length if the actual length is

i

120 km

ii

98 km

5 This is a scale diagram of a toy cat. Use your ruler and the given scale to find: a the length of each whisker b the distance between the tips of the ears.

Scale: 1 : 10

c If the length of the cat’s tail is 2:5 cm on the diagram, how long is the cat’s actual tail?

6 At the hardware store, Max bought 4 offcuts of timber, measuring 500 mm, 750 mm, 400 mm and 800¡¡mm long. How many metres did he buy in total? 7 How many 25 cm rulers placed end to end are needed to measure to a length of 3:5 m? 8 If a bag of nails contains 50 nails and each nail weighs 45 g, find the total weight of 100 bags of nails. 9 How many 1:8 kg bricks can be carried by a truck which has a maximum allowable carrying mass of 3:6 tonnes? 10 Find the total length of edging required to surround the lawn and two garden beds shown.

5m

8m

8m 5m 20 m

16 m

128

MEASUREMENT (AREA AND VOLUME)

Unit 59

Area (square units)

The area of a figure, no matter what shape, is the number of square units (unit2 or u2 ) it encloses.

1 mm 2

1 square millimetre (mm2 ) is the area enclosed by a square of side length 1 mm.

1 cm 2

1 square centimetre (cm2 ) is the area enclosed by a square of side length 1 cm. 1 square metre (m2 ) is the area enclosed by a square of side length 1 m. 1 hectare (ha) is the area enclosed by a square of side length 100 m. 1 square kilometre

(km2 )

This area is 100 mm 2.

10 mm 10 mm

is the area enclosed by a square of side length 1 km.

1 cmX =100 mmX

Exercise 59 1 Find the area in square units of each of the following shapes: a b c

2

a Check to see that the following shapes all have the same area. b What is the perimeter of each? i ii

CHAPTER 8

d

iii

iv

v vi

vii

vii

c What does this exercise tell you about the area and the perimeter of a shape? 3

a In the given sketch, how many tiles have been used for i the floor

ii the walls?

(Do not forget tiles behind and under the sink cabinet and in the shower.) b These tiles are only sold in square metre lots. There are 25 tiles for each square metre. How many square metres need to be bought? c The tiles cost $36:90 per square metre and the tiler charges $18:00 per square metre to glue them. What is the total cost of tiling?

MEASUREMENT (AREA AND VOLUME) (CHAPTER 8)

129

Conversion diagram ¥100

¥10 000

mm2

cm2 £100

For example, to convert:

¥10 000 m2

£10 000

£10 000

6

b e h k n q t

£100

² 350 000 m2 to ha smaller unit to larger ) divide

0:56 m2 = 0:56 £ 10 000 cm2 = 5600 cm2 4 What units of area would most sensibly of the following? a the floor space in a house b c wheat grown on a farm d e a freckle on your skin f g microchip for a computer h i postage stamp j k sheep station l m fingernail n

km2

ha

² 0:56 m2 to cm2 larger unit to smaller unit ) multiply

5 Convert: a 452 mm2 to cm2 d 3579 cm2 to m2 g 550 000 mm2 to m2 j 4400 mm2 to cm2 m 0:7 cm2 to mm2 p 0:8 m2 to cm2 s 0:5 km2 to ha

¥100

350 000 m2 = 350 000 ¥ 10 000 ha = 35 ha

be used to measure the areas

Remember to change larger units to smaller units we multiply, while to change smaller units to larger units we divide.

a dog’s paw carpet for a doll’s house Tasmania bathroom mirror your school grounds suburban railway station pupil of your eye 7:5 m2 to cm2 6:3 km2 to ha 5:2 cm2 to mm2 0:6 ha to m2 480 ha to km2 8800 mm2 to cm2 550 ha to km2

c f i l o r u

5:8 ha to m2 36:5 m2 to mm2 6800 m2 to ha 200 ha to km2 25 cm2 to mm2 6600 cm2 to m2 10 cm2 to m2

a In the given picture, how many pavers were used for: i the driveway ii the patio? b The pavers in the patio are the same as the pavers in the driveway. If there are 50 pavers for every square metre, how many square metres of paving were laid? c If the cost of the pavers is $16:90 per m2 , and the cost of laying them is $14 per m2 , what is the total cost of the paving? d One paver is 20 cm long and 10 cm wide. How far would all the pavers used in this example stretch if they were placed: i end to end in a straight line ii side by side in a straight line? e What do you notice about the answers to d i and d ii?

10 rows of 28 bricks

30 rows of 18 bricks

130

MEASUREMENT (AREA AND VOLUME) (CHAPTER 8)

Unit 60

Area of a rectangle Since a square is a rectangle with equal length and width.

Area of rectangle = length £ width A=l £ w

A = length £ length A= l£l

)

) A=l2 Examples:

²

² 4.2 m

DEMO

1

16.3 m

6m

Area = length £ width = 16:3 £ 4:2 m2 = 68:46 m2

2

12 m

Area = Area 1 + Area 2 = 6 £ 6 + 12 £ 6 m2 = 36 + 72 m2 = 108 m2

Exercise 60 1 Find the area of the following rectangles: a b

c 12 km

28 mm

18 cm 40 cm 18 mm

4 km

2 Find the area of the following squares: a b

c 200 m 200 m

15 m

8.4 cm

3 Find the shaded areas: a

b

in hectares

c

2m

5 cm

12 cm

8 cm

6 cm 4 cm

20 cm

d

6m

15 m

e

f 3 m 10 m 4m 3m

12 m 2m

5 m 3m

3m 3m

20 cm 10 cm 30 cm

20 cm

MEASUREMENT (AREA AND VOLUME) (CHAPTER 8)

131

A rectangle has area 20 units2 and its sides are a whole number of units. We can find all the possible lengths L, widths W and perimeters P for it, in this way: The factors of 20 are: 1, 2, 4, 5, 10, and 20. Now,

area = length £ width, so the possible rectangles are: 20 £ 1, 10 £ 2, 5 £ 4. 1 unit

20 units

Perimeter = (L + W ) £ 2

2 units 10 units 4 units

P = (20 + 1) £ 2 = 42 units

(or 42 u)

P = (10 + 2) £ 2 = 24 units

(or 24 u)

P = (5 + 4) £ 2 = 18 units

(or 18 u)

5 units

4 Using only whole units, write all the possible lengths, widths and perimeters of the following rectangular areas. For each question, use a scale drawing to represent one of the answers: a 12 m2 b 36 cm2 c 64 km2 d 48 mm2 e 64 u2 f 144 mm2 5 Using only whole numbers for sides, write all possible areas which can be found from rectangles or squares with perimeters of: a 12 m b 20 m c 36 km Illustrate the possible answers for a. 6 A rectangular garden bed 3 m by 5 m is cut out of a lawn 10 m by 8 m. Find the area of lawn remaining. 7 A rectangular wheat field is 450 m by 600 m. a Find the area of the field in hectares. b Find the cost of planting the field if planting costs $180 per hectare. 8 A floor 3:5 m by 5 m is to be covered with floor tiles 25 cm by 25 cm square. a Find the number of tiles required. b Find the total cost of the tiles if each tile costs $3:50.

Investigation

Estimating areas of irregular shapes How can we find the area of shapes that are not regular? At best we can only estimate the answer. One method of doing this is to draw grid lines across the figure. Then we count all the full squares and, as we do so, cross them out. Then count squares which are more than half a square unit as 1 (²), and those less than half a square unit as 0. So our estimate for the total area is 26 square units.

Estimate the areas of the shapes. Is b true or false?

a

b

<

132

MEASUREMENT (AREA AND VOLUME) (CHAPTER 8)

Unit 61

Area of a triangle Area of triangle =

height

1 2

£ base £ height DEMO

base

base

Examples: ²

base

20 m

²

² 7 cm

8 cm

10 m

6m

15cm

12 cm

8m

Area of triangle = 12 £ base £ height

Area of triangle = 12 £ base £ height

Shaded area = area rectangle ¡ area triangle

=

=

= 20 £ 10 ¡

1 2

£ 12 £ 8 cm2

= 48 cm2

1 2

£ 15 £ 7 cm2

= 52:5 cm2

= 200 ¡ 24 = 176 m2

1 2 m2

£ 8 £ 6 m2

Exercise 61 1 Find the area of the following triangles: a b

c

8 cm

7m

7 cm

5 cm 12 m

d

11 cm

e

10.6 m

f

7m

5.2 m

4m

4.8 m

3m

2 Find the shaded area: a 12 cm

b

c

4m

6m 2m

8 cm

10 m

6 cm 2 cm

6m 7m

4 cm

5m

10 cm

3 Find the shaded area: 6 cm

8 cm 4 cm 20 cm

12 cm

5m

MEASUREMENT (AREA AND VOLUME) (CHAPTER 8)

4

a Find the area in hectares of the triangular paddock shown.

1200 m

b How much would it cost to plant the paddock with wheat if planting costs $150 per hectare?

800 m

5 Above the bricks on both ends of this house, the shaded area has been filled with weatherboard.

3m

a How much weatherboard was used?

8m

b What was the total cost if weatherboard costs $13:90 per m2 ?

6 Find the shaded area a 100 cm

133

b

50 cm 70 cm

18 cm

6 cm 10 cm

35 cm

30 cm

Investigation

Areas with a spreadsheet

The spreadsheet below is used to calculate the area of a rectangle and a triangle given the dimensions of each figure.

SPREADSHEET

What to do:

1 Open a new spreadsheet and type into the cells the labels and formulae as shown. 2 Change the dimensions of the shapes and check that your spreadsheet gives you the correct answers in each case.

A B C 1 Rectangle Length Width 10 6 2 Base Height 3 Triangle 12 7 4

D

E Area =B2*C2 Area =(B4*C4)/2

Areas of parallelograms, trapezia, circles This extension material is found by clicking on the icon and printing off the pages required. AREA OF PARALLELOGRAM

AREA OF TRAPEZIUM

AREA OF CIRCLE

DEMO PARALLELOGRAM AREA

DEMO TRAPEZIUM AREA

DEMO CIRCLE AREA

134

MEASUREMENT (AREA AND VOLUME) (CHAPTER 8)

Unit 62

Units of volume and capacity

Volume The volume of a solid is the amount of space it occupies. This space is measured in cubic units.

1 mmD All sides have a length of 1 mm.

1 cm

1 cm 3 1 cm

The little 3 in mm3 , cm3 , and m3 indicates that the shape has 3 dimensions L £ W £H

1 cubic millimetre (mm3 ) is the volume of a cube with a side of length 1 mm. 1 cubic centimetre (cm3 ) is the volume of a cube with a side of length 1 cm. 1 cubic metre (m3 ) is the volume of a cube with a side of length 1 m.

Conversion diagram ¥1000 mm3

¥1 000 000 cm3

£1000

m3 To change larger units to smaller units we multiply.

£1 000 000

To change smaller units to larger units we divide.

For example, to convert ²

0:163 m3 to cm3 larger unit to smaller unit ) multiply 0:163 m3 = 0:163 £ 1 000 000 cm3 = 163 000 cm3

²

7953 mm3 to cm3 smaller unit to larger unit ) divide 7953 mm3 = 7953 ¥ 1000 cm3 = 7:953 cm3

Exercise 62 1 Choose the most suitable units to measure the volumes of the following: a a block of cheese b a coin c a margarine container d the space inside a room e a truckload of bark chips f a die g a brick h a shipping container 2 Perform these conversions: a 8 mm3 to cm3 d 0:64 cm3 to mm3 3 Perform these conversions: a 500 mm3 to cm3 c 5 000 000 cm3 to m3 e 2 000 000 mm3 to m3

b e

0:06 m3 to cm3 3 m3 to mm3 b d f

1 cm

c f

11:8 cm3 to mm3 0:0075 m3 to mm3

7000 mm3 to cm3 450 000 cm3 to m3 5 400 000 000 mm3 to m3

MEASUREMENT (AREA AND VOLUME) (CHAPTER 8)

135

Capacity The capacity of a container is a measure of the amount of fluid it can contain. The units of capacity are the

² ² ² ²

millilitre (mL) litre (L) kilolitre (kL) megalitre (ML)

The relationship between capacity units and volume units is:

1 L = 1000 mL 1 kL = 1000 L 1 ML = 1000 kL 1 mL = 1 cm3 1 L = 1000 cm3 1 kL = 1 000 000 cm3 = 1 m3

Millilitres and litres are used for small capacities while kilolitres and megalitres are used for large capacities such as swimming pools and reservoirs. For example, to convert: ²

8 L to mL larger unit to smaller ) multiply

²

²

3400 cm3 to L smaller unit to larger ) divide 3400 cm3 to L 3400 cm3 = 3400 ¥ 1000 L = 3:4 L

12:4 kL to L 12:4 kL = 12:4 £ 1000 L = 12 400 L

8L = 8 £ 1000 mL = 8000 mL

4 What a c e g i k m o q

12:4 kL to L larger unit to smaller ) multiply

units of capacity are most suitable to measure the following? perfume bottle b thermos flask Olympic pool d 6 cylinder car engine drinking glass f teardrop household water use h roll-on deodorant service station petrol tank j model aeroplane engine oil refinery tanks l ocean tanker reservoirs n domestic gas use ocean p pipeline baby’s bottle r beads of perspiration

5 Convert: a 5:6 kL to L d 7200 cm3 to L g 0:0625 L to mL

b e h

3540 mL to L 6:3 kL to m3 400 cm3 to mL

c f i

760 000 L to ML 12:4 kL to mL 3:5 ML to kL

6 A jug contains 350 mL of cordial concentrate. How much water needs to be added to make 2 L of cordial? 7 Jane has a bottle containing 1:25 L of water. If she pours 600 mL into her school lunch bottle, how much water remains? 8 A kiosk sells 18 cans of soft drink each containing 375 mL. How much is this in litres? 9 A bottle of wine contains 750 mL. Find the number of bottles that could be filled from a vat containing 1:2 kL.

136

MEASUREMENT (AREA AND VOLUME) (CHAPTER 8)

Unit 63

Volume formulae

Rectangular prisms A rectangular prism is a 3-dimensional solid which has the same size (cross-section) for its whole length. h

Volume of rectangular prism = length £ width £ height = area of base £ height

4£2£3

w

l

Solids of uniform cross-section

area of base

height

area of base

Volume of solid of uniform cross-section = area of base £ height Examples: ²

²

²

8 cm

8 cm

6 cm

10 cm

Volume = area of base £ height = 10 cm £ 6 cm £ 8 cm = 480 cm3

12 cm

7 cm 5 cm

20 cmS

Volume = area of base £ height = 20 cm2 £ 7 cm = 140 cm3

Volume = area of base £ height = 12 £ 8 £ 5 cm2 £ 12 cm

= 20 £ 12 cm3 = 240 cm3

Exercise 63

DEMO

1 Find the number of cubic units in each of these solids: a b c

d

2 Find the volume of the following rectangular prisms: a b

5 cm

5m 4 cm

3 cm

12 m

c

7m

10 cm

11 cm 1 cm

MEASUREMENT (AREA AND VOLUME) (CHAPTER 8)

3 Find the volume of the following solids of uniform cross-section: a b area 17.3 cmX area 25 mX

2m

c

area 38 mX 8.3 m

2.8 cm

d

e

f

area 3.4 mX

area 74 cmX

137

area 5.1 mX

21.6 m

21.3 m

18 cm

4 Find the volume of the prisms shown: a b 3 cm

c 6 cm 5m

4 cm 6 cm

8m

7m

5 List all the possible dimensions (side lengths) for all the rectangular prisms which can be made using the same number of cubic units as these shown. Use only whole units. a

b 2m

6

8.4 cm

For example: 36 m3 = 36 m £ 1 m £ 1 m = 18 m £ 2 m £ 1 m =9m£4m£1m =9m£2m£2m =6m£6m£1m =6m£3m£2m =4m£3m£3m

3m 4m

9 cm

3m

3 cm 6 cm

4m

6m

3.2 cm

a Using only whole units, list all possible dimensions for rectangular prisms with these volumes: i

24 cm3

ii

40 m3

iii 64 mm3

b Draw scale diagrams for all answers for a i . 7 Find the capacity in mL of: a

b

c

6 cm

5 cm

7 cm

Area 75 cmX

8 cm

6 cm 12 cm

6 cm

8 What is the capacity (in litres) of a rectangular fuel tank 80 cm by 60 cm by 15 cm? 9 How many times could a water container 15 cm by 8 cm by 5 cm be filled from a 40 L container? 10 Find the amount of water (in kL) required to fill the swimming pool shown alongside.

Hint: Draw a line on the trapezium side of the swimming pool to divide it into a rectangle and a triangle.

20 m 2.5 m 1m

6m

138

MEASUREMENT (AREA AND VOLUME) (CHAPTER 8)

Unit 64

Problem solving

A concrete path 1:5 m wide is to be laid around a 20 m by 8 m swimming pool. Concrete of the required depth costs $41 per square metre. The area of the path = area of large rectangle ¡ area of small rectangle = (20 + 1:5 + 1:5) £ (8 + 1:5 + 1:5) ¡ 20 £ 8 m2 = 23 £ 11 ¡ 20 £ 8 = 253 ¡ 160 m2 = 93 m2

path 1.5 m

The cost of concrete = 93 £ $41 = $3813

pool

8m

20 m

A neat, labelled diagram is often useful.

Exercise 64 1 A room 5 m by 6 m by 3 m high is to have its walls timber panelled. a Find the area of timber panelling required. b If the timber panelling costs $12:50 per square metre, find the total cost of the panelling. 2 A rectangular playing field 120 m by 80 m is to be surrounded by a 10 m wide strip of bitumen. a Find the area of bitumen.

120 m 80 m

10 m

b If each truckload of bitumen covers 50 m2 , how many truckloads of bitumen will be required? 3 If each page of a book is 25 cm by 15 cm, find the total area (in m2 ) of paper used in a book of 420 pages. 4 A roll of toilet tissue contains 1000, individually perforated, 110 mm £ 100 mm sheets. a Find the area of tissue in each roll. b How many such rolls would be needed to cover a 50 metre by 66 metre paved area? 5 The area of a rectangle is 1 hectare. Find the width of the rectangle if it has a length of: a 100 m b 1 kilometre c 250 metres d 2000 metres e 800 metres f 1:25 km g 500 metres h 12:5 metres 6 Vertical blinds are to be fitted to a 1430 mm wide by 1200 mm high window. Each slat in the blind is 13 cm wide and is cut to the full height of the window. When they are hanging and the blinds are closed, each slat overlaps another by 2 cm. a How many slats are needed for this window? b What area (in m2¡) of reflective sheeting is needed to cover the glass?

MEASUREMENT (AREA AND VOLUME) (CHAPTER 8)

139

A gardener orders 30 cubic metres of top soil to be spread over a triangular garden with dimensions shown. volume We can find the depth of soil required using depth = : area Area of garden bed = 12 £ base £ height = 12 £ 40 £ 30 = 600 m2 volume area 30 m3 = 600 m2 = 0:05 m = 5 cm

Depth =

‘Depth’ here would mean ‘average depth’.

30 m

40 m

) top soil can be spread to a depth of 5 cm. 7 To celebrate her 3 years in business a baker bakes a large cake with the dimensions shown. How much icing must she make if she covers the top of the cake to a depth of 5 mm with icing?

view from above 60 cm 90 cm

How much canvas is needed for a tent which has three identical sides like this?

8 4m 3m

9 Engineers dug a 150 metre £ 80 metre £ 17 metre deep hole to dump the town’s rubbish. How much compacted rubbish can be dumped if the engineers need a depth of 2 metres of soil on top once the hole is full of rubbish? 10 How many 300 mL spring water bottles can be filled from a rectangular container 3 m £ 2 m £ 1:5 m? 11

a How much water is in this rainwater tank if it is

3 4

140 cm

full?

60 cm

b How many 8 litre buckets could be filled from it?

100 cm

How many 30 cm by 20 cm by 90 cm fuel tanks can a car manufacturer fill from its 27:54 kL storage tank?

12 90 cm

30 cm

20 cm

13 Draw a sketch to show the best way to pack the maximum number of the smaller prism into the larger box. How many can be packed?

3 cm

3 cm

9 cm

6 cm 18 cm

12 cm

140

MEASUREMENT (AREA AND VOLUME) (CHAPTER 8)

Unit 65

Review of chapter 8

Review set 8A 1 Convert: a 3:56 ha to m2

b

2 Find the shaded areas: a

b

357 000 mm2 to m2

8 cm

c

7:2 cm3 to mm3

5 cm

14 cm

3 Find the shaded areas: a

b 6m

6.2 m 3m

8m 5m

4 Find the volume of: a

b

c

area 23.2 cm 2

3m 4m

5 Convert: a 380 mL to L

b

9m

5:4 kL to m3

6 The outside of a shed with the dimensions shown is to be painted. a Find the total area to be painted (including the roof). b If a litre of paint covers 15 m 2 and two coats of paint are needed, what quantity of paint will be required? 7

c

8.6 cm

7528 cm3 to L

2m

3m

5m

a How many posters 120 cm long by 90 cm high can Lotus stick on her 3:6 m by 3 m high bedroom wall? b She wants the space between the posters equal. What is the area of each space if the top row of posters is level with the ceiling and the bottom row is level with the floor?

8

a How many 2 cm by 3 cm stamps can fit on a sheet 200 mm by 300 mm? b If each stamp costs 50 cents, what is the cost of half a sheet?

9 Find the number of kilolitres that will be held in a rectangular rainwater tank 5 m by 3 m by 4:5 m.

MEASUREMENT (AREA AND VOLUME) (CHAPTER 8)

141

Review set 8B 1 Convert: a 3400 m2 to ha

b

2 Find the shaded areas: a 4.2 cm

3:2 cm2 to mm2 b

c

7:2 m2 to mm2

6 cm

5 cm 6 cm

3 Find the shaded area: a

7m

4m

b 8m

12 m

17 m

16 m 25 m

15 m

4 Convert: a 45 000 L to kL

b

5 Find the volume of: a

b

8900 mm3 to cm3

c

4:6 kL to L

c 4m

4m

30 cm

5m

10 m

30 cm

6 How many 10 cm £ 5 cm £ 10 cm containers can be filled from a container with dimensions 1 m £ 1 m £ 12 m? 7 Using only whole units, how many different rectangular prisms can be made from 63 cm3 ? 8 How many kilolitres of sea water are needed to fill this seal’s enclosure at the zoo? 5m area of base 346.2 m 2

9 A wall of a house has two windows and a door with the dimensions illustrated. If the wall is wallpapered and the wallpaper costs $3:75 per square metre, find the cost of papering the wall.

1m

1m

0.8 m

3m 2m 5m

10 Find the area of floorboards showing if a 5 m by 3 m carpet is placed on the floor of a 6:5 m by 8 m room.

142

DATA COLLECTION AND REPRESENTATION

Unit 66

Samples and population

Statistics is about collection, organisation, display, analysis and interpretation of data.

Some words we need to understand Data

Data is the information we collect. It could be opinions or measurements.

Population

The whole group of objects or people about whom we want to make truthful statements.

Sample

The group chosen to take part in a survey or to be measured or tested in some way. A sample should be as large as reasonably possible.

Random sample

A sample selected in such a way that any person or object has as much chance as any other of being selected. The sample must represent the whole population.

Bias

If the sample is not randomly selected the results of an investigation could be biased. They may not represent the whole population.

Inferences

Conclusions you make based on your survey or investigation. For example, after completing a survey on the chocolate eating habits of students, you might infer (conclude) that most year 7 students eat chocolate once a week.

Exercise 66 Explain why the following surveys may provide biased data:

1

a asking farmers if the government needs to give financial assistance to farmers in times of drought b asking 16 year olds if it should be easier for 16 year olds to get a licence to drive a car c asking girls if they spend too much money on clothes d asking boys if girls spend too much money on clothes

CHAPTER 9

e asking a football team if they like ballet f asking people in an expensive restaurant whether taxes are too high. 2 Suggest how to select a random sample of: a 400 adults b bottles of soft drink at a factory c 30 students at a school d words from the English language Mention the advantages and disadvantages of the method you suggested. 3 How would you randomly select: a one ticket out of 5 tickets c one of the numbers 1, 2, 3, 4, 5 or 6

b d

one of the letters A or B a card from a pack of 52 playing cards?

4 Discuss how you would gather data in these situations: a Your fundraising group wants to try a new activity. You need to know if the community will support it. b You have a new design in hair ties for sale by hairdressers. You want to know how many of each colour to make. c You own a chain of fresh fruit juice shops. You are thinking of opening another shop in a new area. You need to know if it is likely to be successful.

DATA COLLECTION AND REPRESENTATION (CHAPTER 9)

143

From a school of 400 students, a random sample of 60 students was selected. 13 were found to have blue eyes. From this information we can say that: ²

There are 400 students in the population.

²

There are 60 students in the sample.

²

13 out of 60 students in the sample have blue eyes ) the fraction of the sample with blue eyes is

²

13 60 :

We can estimate how many in the population have blue eyes to be: =

13 60

of 400

13 60

£ 400

f 13 60 of the sample have blue eyesg

+ 87

Calculator: 13

÷

60

×

400

=

) approximately 87 students in the school have blue eyes. 5 From a colony of 10 000 ants, 300 are collected to examine for red eye colour. 36 were found to have red eyes. a How many ants form the population? b How large was the sample taken? c What percentage of the sample had red eyes? d Estimate the total number of red-eyed ants. 6 50 people are randomly selected from the 750 who attended the opening night of a new play. Of the 50 people, 33 said that they liked the play. a How many people attended the play (the population)? b How large was the sample? c What percentage of the sample did not like the play? d Estimate the total number of people who did not like the play. 7 A local council wants to know the opinions of its ratepayers about a proposed new shopping centre. It conducts a survey of 85 randomly selected ratepayers. 61 of them are in favour of the shopping centre. There are 5231 ratepayers in the council area. a How large was the sample? b How large was the population? c What percentage of the sample were in favour of the shopping centre? d Estimate the number of ratepayers in the council area who were not in favour of the shopping centre.

Activity

Ratings 1 Use the internet, encyclopaedias or library to find out: a what a ‘ratings survey’ is b why radio and television stations want to know the results of ratings surveys c how ratings surveys are conducted.

2 Contact a radio or television station or the agency which conducts surveys in your area and ask for a copy of their survey questions. 3 Find out what other methods radio and television stations use to gather information about their audiences.

144

DATA COLLECTION AND REPRESENTATION (CHAPTER 9)

Unit 67

Collecting and interpreting data

Tally/Frequency tables The frequency of a category is the number of items in that category. This data shows how students in a class travel to school on a particular day. The code is W = walk, Bi = bicycle, Bu = bus, C = car, T = train The data is:

W Bi Bu T C Bi C W Bi Bu Bi C C Bi Bu W Bu Bu T C Bi Bi Bu T C C Bi C C C W W Bu T C

We organise the data in a tally / frequency table. Method of Travelling Walk Bicycle Bus Car Train

Tally © jjjj © © jjj jjjj © © jj jjjj © © jjjj © j jjjj © © jjjj Total

Frequency 5 8 7 11 4 35

The mode is ‘Car’.

The tally is a way of recording using strokes.

It is the category that occurs most frequently.

Exercise 67 1 A survey of eye colour in a class of 28 year 7 students was carried out and the results were: Br Bl Gn Bl Gn Br Br Bl Gn Gr Br Gr Br Br Bl Br Bl Br Gr Gn Br Bl Br Gn Gr Br Bl Gn where Br = brown, Bl = blue, Gn = green, Gr = grey a Complete a tally / frequency table for the data.

b What is the mode of the data?

2 Students in a science class obtained the following levels of achievement: DCCAACCDCBCCCDBCCCCEBACCBCBC a Complete a frequency table for the data above. b Use your table to find the: i

number of students who obtained a C

ii

fraction of students who obtained a B.

c What is the mode of the data? 3 Tourists staying in a city hotel were surveyed to find out what they thought about the service by the hotel staff. They were asked to choose E = excellent, G = good, S = satisfactory or U = unsatisfactory. The results were: EGGSE USSGG SGUGG ESGUG SSEGG

a Complete a tally / frequency table for the data. b What is the mode of the data? c Suggest a reason why this survey would be carried out.

Graphing with technology Click on the icon to load an easy to use statistical graphing package. It can be used to draw a variety of graphs including pie graphs and bar graphs. Make sure that you read the help file. Click on the spreadsheet icon to use a spreadsheet for statistical graphing.

STATISTICS PACKAGE

SPREADSHEET

DATA COLLECTION AND REPRESENTATION (CHAPTER 9)

145

Working with data The number of goals scored in 25 soccer matches is: 3 0 2 1 2 3 1 1 3 2 2 0 2 1 3 1 2 0 3 2 4 1 2 2 0 Goals 0 1 2 3 4

Tally jjjj © j jjjj © © jjjj jjjj © © jjjj © j

Frequency 4 6 9 5 1

We organise the data in a tally / frequency table and display it on a column graph.

We can see, for example, that 2 or more goals were scored

Number of goals 10 frequency 8 6 4 2 0 0 1 2 3

4

goals

Zero goals were scored 4 times. The percentage of matches when 0 goals were scored

9 + 5 + 1 = 15 times.

=

4 25

£ 100%

= 0:16 £ 100% = 16% i.e., on 16% of occasions. 4

a Complete a tally / frequency table for the number of children in 30 families: 04621 32402 12502 31421 24330 45224 b Use your table to find the: i number of families with two children

ii

fraction of families with three children.

5 These are the ages of children at a party: 12, 11, 17, 12, 14, 13, 11, 12, 15, 13, 12, 14, 11, 14, 12, 10, 12, 11, 13, 14 a Organise the data in a tally / frequency table. b How many attended the party? c How many were aged 12 or 13? d What percentage were 13 or more years old? e Display the data on a column graph. 6

An exceptional hockey player scored the following number of goals each match for 25 matches: 4 3 6 1 5 8 4 2 2 4 6 0 5 1 9 3 7 2 6 6 8 3 6 2 10 a Organise the data in a tally / frequency table. b Graph the data on a column graph. c On how many occasions did the player score 5 or more goals in a match? d On what percentage of occasions did the player score 4 or more goals in a match?

7 The number of goals kicked by a soccer player each match for the 2002 season was: 3 0 4 2 0 3 31 2 1 1 2 3 3 2 2 5 0 2 1 4 3 a Complete a frequency table of the given data. b Use the table to find the number of games where the player kicked: i exactly 3 goals ii at least 3 goals.

146

DATA COLLECTION AND REPRESENTATION (CHAPTER 9)

Unit 68

Interpreting graphs

Column graphs

² Iced coffee is the least popular drink. fshortest columng ² ‘Soft drink’ is the mode. ftallest columng

Recess time drinks

frequency

The given graph shows the type of drink purchased by students at recess time. We can obtain information from the graph, such as:

10

The total number of students purchasing drinks = 27 + 35 + 18 + 10 = 90 ) % drinking chocolate milk =

18 90

30 20

² 27 students drink orange juice. ²

40

0 e ang

juic

e

Or

£ 100% = 20%

ft So

DEMO

Exercise 68 1 A survey of eye colour in a class of 28 year 7 students was taken and the results were: a Illustrate these results using a hand drawn column graph.

type

k k fee mil drin cof e t d a l Ice oco Ch

Eye colour Frequency

Brown 11

Blue 7

Green 6

Grey 4

20

c What percentage of the surveyed people drive Fords? 3 Yearly profit and loss figures for a business can be easily illustrated on a column graph as shown. a In what years was a profit made? b What happened in 1996? c What was the overall profit (or loss) over the 6-year period?

5 4 3 2 1 -1 -2

1993 1994 1995 1996

SHOP A

1997

1998 year

SHOP B

Jun May

a In what month were the most profits for each shop?

Apr

b What were the profits for each shop during May?

Mar

c What feature(s) of the graph indicate the effectiveness of the advertising? d Find the total profit for each shop over the 6 month period.

type of car

Profits for XNP Nominees millions of dollars

4 Back-to-back bar graphs are often used to compare two sets of data.

The graph alongside compares the profits of two pizza shops from the same chain of stores over several months. Shop A undertook extensive advertising during this period.

Other

b Which make of car is most popular?

Toyota

0

Holden

10

month

a Use the graph to estimate the frequency of each type of car.

30

Ford

2 Given is a column graph of the type of vehicle driven by 120 randomly selected people.

Type of car driven

40

Mitsubishi

c What percentage of the students have blue eyes?

number of cars

b What is the most frequently occurring eye colour (the mode)?

Feb Jan

8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 profit in 1000s of dollars

DATA COLLECTION AND REPRESENTATION (CHAPTER 9)

Pie graphs A pie graph (or pie chart) is another way to display data.

Favourite sport

The pie graph shows the results of a survey of 120 Year 8 students. From the graph we can see that:

Cricket 15%

² The most popular sport is tennis. flargest sector angleg ² Netball is the least popular sport. fsmallest sector angleg ² 25% of students gave basketball as their favourite sport.

Rugby 13%

Tennis 37%

Netball 10%

25% of 120 = 0:25 £ 120 = 30 ) basketball is the favourite sport of 30 students.

Basketball 25%

² The number of students whose favourite sport is cricket is 15% of 120 = 0:15 £ 120 = 18 DEMO

Click here to see how a pie graph is constructed.

Household water use

5 The pie graph alongside illustrates the proportion of water for various household uses. a For what purpose is the most water used?

Laundry 12%

Cleaning 5%

b For what purpose is the least amount of water used?

Garden 44%

c If the household used 400 kilolitres of water during a particular period, estimate the quantity of water used in: i showering

ii

Showering 39%

cleaning.

Clothing sizes

6 The pie graph alongside shows the percentages of women who wear certain sizes of clothing. a Find what size is most commonly worn.

Other 10% Size 10 15% Size 16 10%

b A group of 200 women attends a fashion parade. Estimate how many would wear size 14 clothing. 7 This pie graph shows both the percentages and the actual amounts the council spent for each sector. a Briefly describe what the graph is about. b Comment on the usefulness of having both percentages and amounts shown. c What percentage of total funding is spent on: i Recreation and Health d How much money is spent on: i Environment

Size 12 30%

CITY COUNCIL FUNDING FOR SERVICES

Community & Cultural Development $46 mill Environment City Development Solid Waste 4% $26 mill $53 mill $69 mill Water & Sewerage 2% 5% 6% $268 mill 25% Recreation & Health $93 mill 8% Transport $121 mill 11%

City Management $184 mill 17%

Roads and Stormwater Drainage $235 mill 22%

ii

Community and Cultural Development?

ii

City Development?

e On what service is the largest amount spent? f How much is spent in total?

Size 14 35%

147

148

DATA COLLECTION AND REPRESENTATION (CHAPTER 9)

Unit 69

Mean and median

The mean and median are measures of the ‘middle’ of a set of numerical data.

The mean The mean (or average) is the total of all scores divided by the number of scores. To find the mean of 7, 11, 15, 6, 11, 19, 23, 0 and 7, we find their sum and divide by 9, because we have 9 numbers.

Remember that the mean is a measure of the middle of a set of scores.

7 + 11 + 15 + 6 + 11 + 19 + 23 + 0 + 7 9 99 x= 9 x = 11 x=

) )

Exercise 69

x is the symbol used to represent the mean.

1 Find the mean of 1, 2, 3, 4, 5, 6 and 7. 2 Calculate the mean of the scores 7, 8, 0, 3, 0, 6, 0, 11 and 1. 3 In Michael’s last 12 games of the basketball season he scored 23, 18, 36, 29, 38, 44, 18, 52, 47, 20, 50 and 42 points. What was his mean point score over this period? 4 Compare the performance of two groups of students in the same mental arithmetic test out of 10 marks.

Group X: 7, 6, 6, 8, 6, 9, 7, 5, 4, 7

Group Y: 9, 6, 7, 6, 8, 10, 3, 9, 9, 8, 9

a Calculate the mean of each group. b There are 10 students in group X and 11 in group Y. Because of unequal numbers in each group it is unfair to compare their means. True or false? c Which group performed better at the test? 5 A cricketer has scored 23, 34, 2, 17, 83, 0, 19 and 28 in previous innings. In the next innings, at 23, he is given not out when the TV replay clearly shows he was caught behind. He then continues and is eventually out at 131. Find: a his actual mean score b what his mean score would have been if he was given out at 23. 6 The given data shows the goals scored by girls in the local netball association. a Find the mean number of goals for each goal shooter. b Which goal shooter has the best average performance?

Name Sally Brown Jan Simmons Jane Haren Peta Piper Lee Wong Polly Lynch Sam Crawley

Goals 238 235 228 219 207 199 197

Games 9 10 9 7 8 7 6

7 Oscar worked for 6 weeks during the school Christmas holidays. He earned $268, $310, $297, $143, $198 and $166. Find his mean weekly wage. 8 A farmer weighed 10 of his 15-week old piglets. Their masses (in kilograms) were: 18:2, 20:1, 30:0, 30:1, 25:7, 19:9, 30:2, 28:6, 24:3 and 26:4. Find the mean mass.

DATA COLLECTION AND REPRESENTATION (CHAPTER 9)

149

The median The median of a set of scores is found by placing the scores in order of size and then choosing the middle score (or average of the two middle scores). Finding the median from raw data

DEMO

4 3 2 1 0

To find the median of these sets of scores: a 7, 9, 8, 6, 7, 10, 8, 7, 9 b 2, 1, 3, 4, 4, 2, 3, 2 First we write the scores in order of size. In order of size the scores are: 6, 7, 7, 7, |{z} 8, 8, 9, 9, 10

a

middle score

) the median is 8.

two middle scores

fthe average of 2 and 3 is 2 12 g

9 Find the median of: a 3, 2, 2, 5, 4, 4, 3, 2, 6, 4, 5, 4, 1

14

15

16

17

18 age

The median is 15.

|{z}

) the median is 2 12 .

12 13 mode

We mark off pairs from either end.

The scores are: 1, 2, 2, 2, 3 , 3, 4, 4

b

Ages of band members frequency the median X is 15 X X X X X X

b

The mode is the category that occurs most frequently. It is 14 (the tallest column).

7, 11, 4, 8, 6, 9, 8, 8, 1, 3

c

2, 5, 3, 3, 6, 3, 5, 4, 5, 1, 7

10 Which is the better measure of the ‘middle’ for the following data; the mean or median? 1, 2, 1, 1, 3, 1, 4, 1, 2, 1, 9, 11 (Find each of these measures first.) 11 A cricketer has scores of 3, 8, 42, 11, 0, 0, 12, 113, 7 and 17. a Find the cricketer’s median score. b Find the cricketer’s mean score. c Which of the measures (median or mean) best describes the cricketers batting performance? 12 From the data about shoe sizes, determine: a Find the shoe size occuring most frequently (the mode)

6 4

b the median size.

2 5

13 For each of the following graphs, find a

10 8

frequency

6

i

7

8

b

Goals scored

0

1

2

3

4

9

10 11 12 13 size

the median

6

4 2 0

The mode is size 8, because it occurs most often.

Shoe size number of students

goals

ii

the mode.

Ages of choir members 6 frequency 5 4 3 2 1 0 9 10 11 12 13

age

14 Two year 7 mathematics classes sat for the same mathematics test out of 20 marks. Their results were: Class 7P: 19 20 11 15 16 17 17 14 16 17 20 18 17 16 15 15 16 16 17 16 Class 7Q: 14 13 16 17 20 13 16 15 18 12 13 14 17 14 12 13 13 14 10 a State the highest and lowest marks for each class. b Find the mean and median of the results for each class. c Which class performed better at the test?

150

DATA COLLECTION AND REPRESENTATION (CHAPTER 9)

Unit 70

Line graphs

The graph shows the weight of a girl as recorded on her birthday each year. It is her birthday today.

Girl’s weight

From the graph we can estimate that: ² ² ² ² ²

80

her weight at birth was 3 kg 60 her weight at age 10 was 45 kg her weight today is 60 kg 40 her weight increase from age 8 to 13 was 20 60 ¡ 40 = 20 kg The horizontal graph between years 6 and 8 indicates the same weight on these birthdays (even though there would be changes in weight during this period).

kg

Every graph must have: ² a title ² x and y axes labelled

2

4

6

age (years) 10 12 14

8

Exercise 70 1 Kiri rides her bicycle to the shop. She has hooked up a device which measures her pulse rate. The data is later graphed over 1 minute time intervals. a Find her pulse rate after 2 minutes.

Pulse rate pulse rate (heart beats/min)

140 120 100

b Find her pulse rate after 7 minutes.

80

c During what time intervals did her pulse rate increase?

60

d Find the change in her pulse rate during the interval from 4 to 9 minutes.

40 20

e What was her highest recorded pulse rate?

0

f Is this a time series graph?

time (min) 1 2 3 4 5 6 7 8 9 10

0

2 The rainfall for various months of the year is given in the following table:

Month Rainfall (mm)

Ja 50

Fe 80

Ma 50

Ap 100

Ma 200

Ju 350

Ju 270

Au 160

Se 100

Oc 80

No 40

De 70

a Plot the data using a line graph with months on the horizontal axis. b During which period of 4 months did most rain fall? c What was the driest month? d What percentage of the year’s rainfall fell in winter (June to August)?

DEC

NOV

SEP

OCT

AUG

JUL

e Find the average of the monthly temperatures.

months JUN

d Why is the graph decreasing from February to August?

MAY

c Which 4 month period was coldest?

APR

b Which month was the hottest?

FEB

on the temperature axis?

35 30 25 20 15 10

MAR

a What is the meaning of

Temperatures average temp (°C)

JAN

3 The following graph gives the average maximum daily temperature for all months of the year for a country town.

DATA COLLECTION AND REPRESENTATION (CHAPTER 9)

4 The following graph shows the income and costs of a small business over a 13 week period. a What were the income and cost figures for week 5?

Income and costs thousands of dollars 10

b What was the profit during week 5?

8

c During which period did the weekly income fall?

6

d Which week had the greatest profit: i in the first 6 weeks ii in the entire 13 weeks?

4

co

sts

2 1 200 sales 180 160 140 120 100 80 60 40 20 0

3

5

7

9

11

week 13

Sales figures, Feel Rite Airconditioning

quarters

Jun -0 Se 0 p-0 De 0 cMa 00 rJun 01 -0 Se 1 p-0 De 1 cMa 01 rJun 02 Se 02 p-0 De 2 cMa 02 rJun 03 -0 Se 3 p-0 De 3 c-0 3

5 This graph shows quarterly (three monthly) sales figures for Feel Rite Airconditioning since the business started in 2000. a i Can you see any repeating pattern in the data? If so, what might explain the pattern? ii One year the management decided to have a major advertising campaign during a time of year that is normally quiet. When do you think this was? b Can you see any long-term trend in the data?

e

m

co

in

151

c For practice using computer software you could enter the Feel Rite data into a spreadsheet and produce your own graph. Try entering labels, adjusting scales, altering the size of the graph and so on.

Activity

A survey What to do: 1 Brainstorm a list of ways information is gathered. 2 Organise the list into categories such as questionnaires, face to face interviews, opinion polls, telephone surveys, etc.

3 Find a way of recording the data including details about the person who provided it, e.g., adult male/female, child male/female. 4 Prepare a set of questions to ask your families which every student in the class will use. Your aim is to find out: a which members of the households have been involved in providing data in the last 12 months b what methods were used to collect this data. 5 Have every student in your class use the prepared questions to interview every member of their household. 6 Decide on a time you want all the information collected by. 7 Collect all the information gathered by each student. 8 Organise and present the information in tables and graphs. 9 For one set of data, draw a column graph and a pie chart. 10 Put titles and keys on all graphs and tables. 11 Discuss: a the results b how you could have improved the accuracy of the data gathering.

152

DATA COLLECTION AND REPRESENTATION (CHAPTER 9)

Unit 71

Review of chapter 9

Review set 9A 1 Write T(true) or F(false) for the following statements: a Investigating what Australian students like to eat by asking every student in a school is a survey. b Asking every tenth person at a netball match whether they liked sport would be a random survey. c Asking 100 people in a shopping centre who they would vote for in a state election is too small a sample for a poll. 2 A medal is awarded to the best and fairest player in a national sporting competition. Umpires award 3 votes to the player they feel was the best and fairest in each game. 2 votes are awarded for second best and 1 vote for the third best.

Listed below are the votes awarded to a recent winner. The first vote from the left was for the first game, the second vote was for the second game, etc. 0 2 0 3 1 0 3 2 3 1 1 3 2 0 3 2 1 0 3 2 a Use the tally method to prepare a frequency chart to show the votes awarded to the winner for each game. b Draw a column graph to show the frequency of the votes. c Draw a line graph to show the progressive vote total at the end of each game. Your horizontal axis should show each week in the season. d In how many games did the winner not receive votes? e What was the winner’s total number of votes? f In what percentage of games did the winner receive votes? g What was the i

mean

ii

median of the winner’s votes?

3 Use the line graph to answer the following questions: a On how many days was the minimum temperature i below 15o C

ii

above 20o C?

b On what day was the: i highest maximum ii lowest maximum iii greatest difference between maximum and minimum iv smallest difference between maximum and minimum? c How many times did a daily minimum temperature exceed the lowest maximum for the month? 4 Use the circle graph alongside to answer the following questions:

Temperature (to nearest °C) 40 35 30

25 20 15 10

max min

5 0

1

4

7 10 13 16 19 22 25 28 31

DEATHS FROM SMOKING-RELATED DISEASES

a What was the major smoking-related disease resulting in death? b What 2 groups of diseases made up 50% of all smokingrelated deaths? c If 20 000 people died in one year as a result of smoking, how many died from: i heart disease ii lung cancer iii other cancers?

Day

Heart disease 24%

Chronic bronchitis and emphysema 23% Other circulatory diseases 15%

Lung cancer 27%

Other cancers 8% Other 3%

DATA COLLECTION AND REPRESENTATION (CHAPTER 9)

153

Review set 9B 1

a In a pie chart, a full circle is divided into ........ to show each type or category. b The number of times a particular result occurs in a table of information is called the ........ c In a column graph the difference in ........ represents the difference in the number of objects in each group.

2 The data below represents birth months in a year 7 class. January is represented by the number 1, February by the number 2 and so on up to December which is 12. Boys are shown in black and girls in blue. 6 7 3 9 5 5 9 12 10 4 1 12 6 3 5

7

7

4

10

3

7

1

9

5

9

4

8

7

11

4

52 42

21 39

54 46

a Prepare a tally / frequency table to show this data. b Answer these questions: i ii iii iv v vi vii

How many students were in the class? How many girls were in the class? In which month were the most boys born? What fraction of the class was born in April? In which month were the least number of students born? What percentage of the class was born in March? In which months were the least number of girls born?

3 The heights (in centimetres) of 10 of the tallest hockey players are: 175, 183, 176, 179, 177, 188, 176, 177, 178, 181.

What is the

a

median height

b mean height?

4 The following are the ages of teachers in a school:

56 43

37 48

44 53

51 45

49 51

23 23

40 58

47 50

43 63

38 45

61 31

37 51

a From this data make a column graph. ii

mean age

5 The column graph represents the value of one month’s sales at Stan’s Super Savings Store. a

i ii

What goods represent the highest value of electrical items sold? Give two reasons why this may have happened.

b What was the total value of entertainment items? c What was the total value of goods sold? d If 200 small kitchen appliances like kettles, toasters, irons etc. were sold, what was their average price?

70

Electrical items sold Sales (thousands of $s)

60 50 40 30 20 10

cro wa ve ov all ens kit che na Re frig ppl Wa era shi tor ng s ma chi Ai nes rc on dit ion So ers un ds y s tem Tel evi s sio ns DV ets Dp lay ers

median age

Sm

i

Mi

b Find the

items

154

TIME AND TEMPERATURE

Units of time

Unit 72 1 week = 7 days 1 millennium = 1000 years 1 day = 24 hours 1 century = 100 years 1 hour = 60 minutes 1 decade = 10 years 1 minute = 60 seconds 1 year = 12 months = 52 weeks = 365 days (or 366 in a leap year)

Thirty days have September, April, June and November. All the rest have thirty one except February, which has twenty eight and twenty nine in a leap year.

The number of days in the month varies: January February March April May June

31 28 (29 in a leap year) 31 30 31 30

July August September October November December

31 31 30 31 30 31

Exercise 72 1 Do these conversions: a 120 minutes to hours d 28 days to fortnights g 208 weeks to years j 3000 years to millennia

b e h k

72 hours to days 77 days to weeks 90 years to decades 1095 days to years

CHAPTER 10

2 Copy and complete: a 150 sec = ...... min ...... sec c 200 min = ...... hours ...... min e 56 months = ...... years ...... months g 60 hours = ...... days ...... hours i 7300 years = ...... millennia ...... centuries

c f i l b d f h j

240 seconds to minutes 144 months to years 800 years to centuries 3 hours to seconds

90 min = ...... hour ...... min 53 days = ...... weeks ...... days 73 years = ...... decades ...... years 500 sec = ...... min ...... sec 160 months = ...... years ...... months

3 Find the number of minutes in: a one day

b

one week

c

one 365 day year

4 Find the number of seconds in: a one day

b

one fortnight

c

one 365 day year

5 Consider a four year period which includes a leap year. Find the number of: a days b hours c minutes 6 Find a b c d e f

by adding or subtracting in columns: 3 h 7 min + 5 h 23 min 5 h 17 min + 3 h 25 min + 4 h 35 min 11 h 43 min + 2 h 24 min + 5 h 16 min 7 h 53 min ¡ 3 h 36 min 17 h 42 min ¡ 12 h 53 min 10 h 32 min + 5 h 47 min ¡ 7 h 57 min

DEMO

Addition

Subtraction 4

83

5 h 23 min + 2 h 47 min

5 h 23 min ¡ 2 h 47 min

7 h 70 min = 8 h 10 min

2 h 36 min

TIME AND TEMPERATURE (CHAPTER 10)

²

To convert 3 days, 9 hours and 42 minutes to minutes, using this method: ) 3 days 9 hours + 42 mins Total

9 hours = 9 £ 60 min = 540 min

3 days = 3 £ 24 hours = 3 £ 24 £ 60 min = 4320 min ²

155

4320 540 42 4902

min

To convert 1635 hours to days and hours using your calculator: 1635 = 68:125 24

fDivide by 24 since there are 24 hours in a dayg

) there are 68 days and 0:125 of a day remaining i.e., 68 days, 3 hours

f0:125 £ 24 = 3 hoursg

7 Convert to minutes: a 7 hours 24 min c 12 days 15 hours 36 min

b d

3 days 5 hours 43 min 2 weeks 3 days 8 hours 17 min

8 Convert to seconds: a 40 min 38 sec c 14 hours 12 min 43 sec

b d

3 hours 35 min 27 sec 22 hours 52 min 11 sec

9 Convert: a 124 hours to days c 873 hours to days

b d

552 hours to days 2167 hours to days

10 Convert: a 67 680 minutes to days

b

31 717 minutes to days, hours and minutes

Research

It takes time The following words are all linked with time: chronometer punctuality millennium bug solstice real time longitude rhythm synchronise pendulum geological time

metronome curfew duration equinox meridian

astronomy chronological time signature sesquicentenary almanac

What to do: 1 Find out what they mean. 2 Add others to the list. 3 Prepare a detailed talk or demonstration on any two of the listed words.

Research

A date with a calendar What to do: 1 Research the development of the current Western calendar. 2 Find out what the Gregorian calendar is. 3 Compare the calendars of Christian, Orthodox, Muslim, Jewish, Chinese and Aboriginal people. 4 List the days of great importance to these people and place these dates on your own calendar.

156

TIME AND TEMPERATURE (CHAPTER 10)

Differences in time

Unit 73

13 / 6 / 02 Day of the month

Month of the year

How many days is it from April 24th to July 17th? We can work it out like this: April has 30 days ) 6 days remain (30 ¡ 24 = 6)

Last two digits of the year

the 13 th of June 2002

So, April May June July Total

6 31 30 17 84 days

Exercise 73 1 Write out in sentence form the meaning of: a Wei joined the club on 17=12=99 c Piri is departing for Malaysia on 30=7=04

b d

Jon arrived on 13=3=00 Sam will turn 21 on 28=5=09

2 What is the date: a one week after 12th August c 5 days after 3rd February e 1 week before 22nd March g 11 days before 25th May i 13 days after 18th June

b d f h j

one week before November 19th two weeks after 7th September 2 weeks before July 1st 3 weeks after 3rd July 1 week before October 5th?

3 Find a c e g

b d f h

May 11th to June 23rd September 19th to January 8th February 6th to August 3rd in a leap year 7=2=03 to 17=5=03

the number of days from: March 11th to April 7th July 12th to November 6th January 7th to March 16th in a non-leap year 6=7=02 to 2=11=02

4 Lou Wong needs to save money to buy a bicycle costing $279. Today is the 23rd of March and the shop will hold the bicycle until May 7th at this price. a How many days does Lou have to save for the bicycle? b How much needs to be saved each day to reach the $279 target? 5 Sean can save $18 a day. Today is May 9th and on November 20th he wishes to travel to Fiji on a package deal costing $5449. If he does not reach the target of $5449 he will have to borrow the remainder from a bank. a How many days has he for saving? b What is the total he will save? c Does he need to borrow money? If so, how much? 6 On 17 th July 1999 Sung Kim proudly announced that he had been in Australia for 1000 days. On what day did he arrive in Australia?

$

5449

p.p.

FR. EXSYDNEY BUSINESS CLASS FIJI • Return business class airfares • 7 days FREE car hire and 5 nights luxury Hilton accommodation. TRAVELAND HOLIDAYS

7 If Christmas day is a Sunday, what day of the week will New Years Day be (following Christmas day)? 8 The 1st of January 2003 was a Wednesday. 2003 was not a leap year. What day of the week was the 1st of January 2004?

TIME AND TEMPERATURE (CHAPTER 10)

157

Example:

If it is now 10:15 am and the plane departs at 3:20 pm, how long is it before departure?

10:15 am

12 noon 3 hours 20 min

1 hour 45 min

Use a time line to help you.

3:20 pm

The time before departure = time before noon + time after noon = 1 hour 45 min + 3 hours 20 min = 4 hours 65 min = 5 hours 5 min 9 What is the time: a 4 hours after 3:00 am c 34 minutes after 6:15 am e 2 hours 13 min after 8:19 pm g 2 hours 55 min before 2 pm i 1 hour 47 mins before 1:30 pm 10 What is the time difference from: a 3:24 am to 11:43 am c 8:29 am to 3:46 pm e 3:18 pm to 11:27 am next day g 2:23 pm Sunday to 5:11 pm Monday

b d f h j

5 hours after 8:00 pm 45 minutes after 7:21 pm 3 hours 27 min after 12:42 pm 5 hours 18 minutes before noon 3 hours 16 minutes before 2 am Monday?

b d f h

7:36 5:32 4:29 3:42

pm to 10:55 pm am to 6:24 pm pm to 2:06 am next day am Tuesday to 7:36 pm Friday?

11 If a courier travelling between two cities takes 2 14 hrs for a one way trip, how many trips can she do in an 8 hour working day? 12 Herbert was born in 1895. How old was he when he had his birthday in 1920? 13 It takes Jill 10 seconds to put each can into her supermarket display. If there are 120 cans to be displayed how long will it take Jill to complete the job? 14 Complete the following table: a b c d e f g h i

Bus departs 9:15 am 6:25 pm 7:15 am 12:25 pm 1:50 pm 8:25 pm 11:18 pm 10:13 am

Bus arrives 12:30 pm 9:50 pm 9:05 am 2:50 pm 3:20 pm 12:15 am 1:50 am

Time taken

2 h 15 min 3 h 15 min 2 h 25 min

15 Mary’s watch loses 3 seconds every hour. If it is correct at 8 am on Wednesday, how slow will it be when the real time is 5 pm on Friday of the same week? 16 If a high tide happens every 6 hours and 20 minutes after the last, and the next is at 1:25 am on Monday, list the times for the next 8 high tides after that one. 17 How many times will a second hand of a clock pass 12 in a 24 hour period starting at 11 pm?

158

TIME AND TEMPERATURE (CHAPTER 10)

Reading clocks and time lines

Unit 74

Time can be measured using a 12-hour (analogue) clock or a 24-hour (digital) clock. 12-hour time

Digital display

24-hour time

midnight

0:00

0000 hours

7:42 am

7:42

0742 hours

midday (noon)

12:00

1200 hours

11:29 pm

23: 29

2329 hours

Analogue display

Remember that 24-hour time always uses 4 digits.

2329 hours = 12:00 + 11:29 hours

am stands for ante meridiem which means ‘before the middle of the day’. pm stands for post meridiem which means ‘after the middle of the day’.

Exercise 74 1 Write as 24-hour time: a 3:13 am e 5:41 pm i 2:15 am m 6:30 pm

c g k o

midnight 8:19 pm 10:52 pm 8:06 am

2 Write these 24-hour times as 12-hour times: a 0300 hours b 0630 hours e 0615 hours f 1545 hours i 1450 hours j 0030 hours m 1145 hours n 1435 hours

c g k o

1800 2017 0720 0200

3 Write as 24-hour time: a half past ten in the morning c twenty past twelve noon e twenty to ten at night

b d f

twenty five past six in the evening a quarter to eleven in the morning twelve minutes past midnight

b f j n

11:17 am noon 9:25 pm 12:55 pm

4 Write these analogue times as 24-hour times: a b

morning

5 What, if anything, is wrong with these 24-hour times? a 0862 hours b 0713 hours

hours hours hours hours

d h l p

12:47 pm 11:59 pm 12:15 am 2 am

d h l p

1200 2348 2330 1215

hours hours hours hours

c

afternoon

evening

c

2541 hours

Click on the icon for the activities ‘How long does it take?’ and ‘Time estimation’.

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TIME AND TEMPERATURE (CHAPTER 10)

159

Time lines Time lines are simple graphs which show times, often dates, below the line, and important events above the line. This time line shows some of the important dates in Sarah’s life: started school

born

1960

started playing netball

finished school

1970

1965

1975

injured knee

1980

first child

married

1985

1990

stopped playing netball

second child

1995

2000

From the time line we can say that ² Sarah was born in 1962. ² She was married in 1987. ² She finished school in 1978. ² She injured her knee in 1983. ² Sarah started playing netball in 1970 and finished playing in 1995, so she played for 25 years. ² Sarah’s first child was born in 1992 and her second child was born in 1998, so there are 6 years’ difference in their ages. AD stands for Anno Domini and means ‘in the year of Our Lord’.

6 This time line shows when various methods of writing first appeared: Chinese characters

Egyptian hieroglyphics

3000

2000

Runic Japanese Alphabet Script

Greek Mayan

1000

BC

0

1000

AD

2000

a What does BC stand for on the time line above? b Estimate when the Runic Alphabet first appeared. c How long was there between the appearance of Egyptian hieroglyphics and Chinese characters? 7 This time line shows the monarchs of England during the 20th century: Edward VII

Edward VIII George VI

George V

1900

1910

1920

1940

1930

Elizabeth II

1950

1960

1970

1980

1990

a Which monarch reigned longest in the 20th century? b How long did the reign of George VI last? c How much longer was the reign of Edward VII than Edward VIII?

2000

Some people use CE (meaning Common Era) instead of AD and BCE (meaning Before the Common Era) instead of BC.

8 This time line shows the periods of various Chinese civilisations: Longshan

Yang Shao

4000

3000

Xia

Shong

2000

Zhou Warlords

1000

BC

0

1000

AD

Use the time line and a ruler to: a find the Chinese civilisation which lasted longest b determine the period for which the Xia dynasty lasted c find how much longer the Longshan civilisation lasted compared with the Shong civilisation. 9 The time line below shows the admission of new teams into the AFL: Sydney

1980

West Coast/Brisbane

1985

Adelaide

1990

Fremantle

Port Adelaide

1995

2000

a When were the Adelaide Crows admitted to the AFL? b How much earlier were the West Coast Eagles admitted than the Fremantle Dockers? c In what year will Port Power celebrate its 10th year in the AFL?

160

TIME AND TEMPERATURE (CHAPTER 10)

Timetables

Unit 75 Timetables are tables of information which tell us when events are to occur. The timetable alongside provides information on phases of the moon and the rising and setting of the planets of our solar system. We can ² ² ²

The Moon First

Sep 21

Sep 29

Qr_

Full

Last

Oct 6

Oct 12

The Sun and Planets Tomorrow

Rise

6 : 15 2 : 29 5 : 59 5 : 51 4 : 43 6 : 01 9 : 10

Sun Moon Mercury Venus Mars Jupiter Saturn

see that: the next full moon is on October the 6th Mercury rises at 5:59 am tomorrow Saturn sets at 8:23 pm tomorrow.

Qr_

New

Set

am am am am am am am

6 : 07 1 : 00 5 : 20 5 : 09 3 : 11 6 : 34 8 : 23

pm pm pm pm pm pm pm

Exercise 75 1 The following arrivals appear on a TV monitor at Sydney International Airport: a Convert each 24-hour arrival time to 12-hour time. b At what time is the Singapore Airlines flight from Bali arriving? c At what time is the Qantas flight from Brisbane arriving? d If fog delays all arrivals by 7 hours, what time will: i the Qantas flight from Brisbane arrive ii the Singapore Airlines flight from London arrive? QF is Qantas

SQ is Singapore Airlines

NZ is Air New Zealand

ARRIVALS From Arr. Time

Flight QF62 QF67 SQ34 SQ42 SQ71 SQ82 NZ97 QF83

Adelaide Melbourne Singapore London Bali Japan USA Brisbane

10:35 11:45 12:50 13:50 14:25 14:45 15:15 16:10

2 Below is a timetable for a tourist bus service in Adelaide for the summer season:

a How many bus services are available? b What is the latest departure time? c What is the earliest arrival time back at the depot? d How long does it take between arrivals at: i the Adelaide Zoo and the Stonyfell Winery ii the Murray Mouth and Victor Harbor?

Departure Times

Bus A

Bus B

Bus C

Bus D

Bus E

Bus F

City depot Adelaide Oval Adelaide Zoo Stonyfell Winery Hahndorf Murray Mouth Victor Harbor Port Adelaide Museum Arrive at City depot

7:30 7:40 8:20 10:15 11:20 1:00 1:40 3:15 4:00 5:00

7:45 7:55 8:35 10:30 11:35 1:15 1:55 3:30 4:15 5:15

8:00 8:10 8:50 10:45 11:50 1:30 2:10 3:45 4:30 5:30

8:15 8:25 9:05 11:00 12:05 1:45 2:25 4:00 4:45 5:45

8:30 8:40 9:20 11:15 12:20 2:00 2:40 4:15 5:00 6:00

8:45 8:55 9:35 11:30 12:35 2:15 2:55 4:30 5:15 6:15

TIME AND TEMPERATURE (CHAPTER 10)

161

e How long does a complete trip last? f If you wanted to be at Victor Harbor no later than 2:15, what bus should you take? g If a friend is meeting the bus at Port Adelaide at 3:30, what bus is it best to travel on?

3 I am visiting Sydney and have obtained this Carlingford to Wynyard train timetable: a What do the following mean: i arr ii dep? b If I catch the 4:17 pm train at Rydalmere, what time will I arrive at Central? c At what time will I have to catch the train from Dundas in order to arrive at Lidcombe by 6:00 pm?

CARLINGFORD-WYNYARD TRAIN TIMETABLE Carlingford Telopea Dundas Rydalmere Camellia Rosehill UA Clyde ……...arr dep Lidcombe…..arr dep Strathfield…. arr dep Central……..arr dep Townhall Wynyard

d If I miss the 5:00 pm train from Clyde, what would be the earliest time that I could arrive at Wynyard? e i If I come out of the cinema at 3:45 pm at Carlingford, what is the time of the first train that I can catch to Strathfield? ii At what time will this train reach Strathfield?

p.m. 3.32 3.34 3.36 3.38 3.40 3.42 3.45X 3.51

p.m. 4.11 4.13 4.15 4.17 4.19 4.21 4.24X 4.26

p.m. 4.45 4.47 4.49 4.51 4.53 4.55 4.58X 5.00

p.m. 5.23 5.25 5.27 5.29 5.31 5.33 5.36X 5.48

p.m. 5.55 5.57 5.59 6.01 6.03 6.05 6.08X 6.18

3.57 4.02 4.03 4.17 4.18 4.21 4.24

4.31 4.36 4.37 4.50 4.51 4.54 4.57

5.06 5.11 5.12 5.26 5.27 5.30 5.33

5.54 5.59 6.00 6.14 6.15 6.18 6.20

6.24 6.29 6.30 6.44 6.45 6.48 6.50

p.m. 6.26 6.28 6.30 6.32 6.34 6.36 6.39X 6.48

p.m. 6.52 6.54 6.56 6.58 7.00 7.02 7.05 7.06

6.54 7.12 6.59 7.18X 7.00 7.23 7.14 7.36 7.15 7.37 7.18 7.40 7.20 7.42

iii If I have an errand at Strathfield that will take half an hour, what is the shortest time that I will have to wait for the next train that I can catch to Townhall? f Calculate the time it takes for the i 3:32 pm ii 5:23 pm iii 6:52 pm trains from Carlingford to reach Central. Can you suggest a reason for the differences in times? 4 The tide timetable below is for a particular day in 2003. Tide times Port Xenon Port Dowell Windcok Joseph’s Bay Paradise Point Sunny Inlet

12:55 1:56 5:15 1:46 2:47 5:12 3:20 5:24 3:22 12:19 12:29 11:29

AM PM AM PM AM PM AM PM AM PM AM PM

0.8 m 7:21 AM 2.5 m 1.2 m 7:13 PM 1.8 m 1.4 m 11:45 AM 1.1 m 1.2 m 9:22 PM 0.6 m 0.9 m 9:53 AM 2.4 m 1.2 m 8:41 PM 1.3 m 0.9 m 10:22 AM 2.6 m 1.3 m 9:13 PM 1.5 m 1.4 m 7:57 AM 0.9 m 1.3 m 9:08 PM 0.6 m 0.5 m 9:03 AM 1.3 m 0.4 m

a When is the tide highest in the morning at Port Xenon? b When is the tide lowest in the afternoon at Paradise Point? c What is the lowest tide at Joseph’s Bay in the morning and at what time does it occur? d What is the highest tide at Port Dowell in the afternoon and at what time does it occur?

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TIME AND TEMPERATURE (CHAPTER 10)

Time zones

Unit 76

The Earth rotates from West to East about its axis. This rotation causes day and night. Every day the earth rotates through 360o .

Prime Meridian

This map shows the Standard Time Zones.

-9 -8

-2

-7

-5

Moscow

-1 London

+13

+10 +11 +9

+7 +5

+3 Paris

Beijing -6

San Francisco

New York

+8

Houston

+1

Cairo

Tokyo +9

+5.5

+3

Bombay

+2

-5 -4

Cairns +8 Johannesburg

Santiago

-10

-9

-8

-7

-6

-5

International Date Line

-3

+10

+9.5

Sydney

Perth

-4

-3

-2

-1 GMT

+1

+2

+3

+4

+5

+6

+7

Brisbane

+8

+9

+10

Time along the Prime Meridian is called Greenwich Mean Time (GMT). Places to the East of the prime meridian are ahead of GMT. At 12 midnight GMT, it is 10 am in Sydney (+10). Places to the West of the prime meridian are behind GMT. At 12 noon, GMT, it is 7 pm in New York (¡5). The numbers in the zones show how many hours have to be added or subtracted from Greenwich Mean Time to work out the standard time for that zone.

Exercise 76 Use the Standard Time Zone map to answer these questions: 1 If it is 12 noon in Greenwich, what is the standard time in: a Moscow b Beijing c Sydney

d

Santiago?

2 If it is 12 midnight (at the end of Monday) in Greenwich, what is the standard time in: a Cairo b Bombay c Tokyo d London? 3 If it is 10 pm on Tuesday in Greenwich, what is the standard time in: a New York b San Francisco c Brisbane

d

Johannesburg?

4 If it is 2:45 am on Sunday in Greenwich, what is the standard time in: a Adelaide b Bombay c Santiago

d

Houston?

5 If it is 3 pm in Moscow on Friday, what is the standard time in: a Adelaide b Beijing c London

d

San Francisco?

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163

As the sun rises in Adelaide, Perth is still in darkness. So it is earlier in the day in Perth than in Adelaide. As the sun rises in Perth, Sydney has already experienced about two hours of daylight. So it is later in the day in Sydney than in Perth.

sun’s rays

AUSTRALIA HAS THREE TIME ZONES

Darwin Cairns

Alice Springs

Mt. Isa Rockhampton Brisbane

Kalgoorlie

Perth Adelaide

Canberra Sydney

Melbourne Hobart WESTERN STANDARD TIME 10:00 am

CENTRAL STANDARD TIME 11:30 am

EASTERN STANDARD TIME 12:00 noon

During summer, Australia uses Daylight Saving Time in the ACT, NSW, SA, Tasmania and Victoria. This is done by putting forward (advancing) the time by 1 hour in these states and the ACT, and results in Australia having 5 different time zones in summer.

6 It is 3:30 pm normal time in Adelaide. What is the time in: a Darwin b Canberra c Brisbane

d

Hobart?

7 If it is 4:20 am normal time in Brisbane, what is the time in: a Mt Gambier b Alice Springs c Kalgoorlie

d

Sydney?

8 It is summer and daylight saving is in operation. Draw a sketch of Australia and mark on it the five different time zones. If it is 3 pm in SA, what is the time in: a WA b NT c QLD d NSW e Tasmania? 9 If it is 7:40 am normal time in SA, what will be the normal time in: a Melbourne b Perth c Darwin

d

Canberra?

10 Determine the arrival times for the following: a a 4 12 hour flight from Melbourne to Perth departing Melbourne at 6:00 am normal time b a 10 12 hour drive from Sydney to Brisbane during daylight saving, leaving Sydney at 8:20 am c a 2 12 hour flight from Adelaide to Alice Springs leaving Adelaide at 4 pm central summer time d a 2 hour flight from Adelaide to Sydney, followed by a 2 12 hour flight from Sydney to Townsville, departing Adelaide at 2:15 pm normal time with a delay of 45 minutes in Sydney. 11 If it is 5 am on Monday in Sydney, what is the standard time in: a New York b Bombay c Tokyo

d

San Francisco?

164

TIME AND TEMPERATURE (CHAPTER 10)

Average speed and temperature

Unit 77 Average speed

If a car travels 720 km in 9 hours,

Speeds are measured in km km/h or h distance which is time

distance its average speed = time

Average speed is the distance travelled in a unit of time.

720 km 9h = 80 km/h =

The rule is:

speed =

distance time

If a car travels for 5 hours at an average speed of 90 km/h, the distance travelled = speed £ time = 90 £ 5 = 450 km

To find distances: distance = average speed £ time

If a car travels 160 km at 80 km/h, the time taken = time =

distance speed

160 80 = 2 hours

distance average speed

=

If you find it difficult to remember this formula, think of a simple example like the one given alongside.

Exercise 77 1 Find the average speed travelled by a vehicle if it covers: a 540 km in 6 hours b 840 km in 12 hours c 664 km in 8 hours d 846 km in 9 hours 2 If a vehicle is travelling at 90 km/h, find how far it will travel in: a 7 hours b 5 hours c 10 hours d 3:5 hours e 11 hours 24 min (Hint: 24 min = 3 How far would you travel in: a 3 hours at an average speed of 85 km/h c

4 12 hours at an average speed of 98 km/h

4 Find how long it will take to travel: a 90 km at 30 km/h b d 750 km at 90 km/h e

24 60

hours)

b

8 hours at an average speed of 110 km/h

d

2 hours 15 mins at an average speed of 76 km/h?

720 km at 120 km/h 208 km at 64 km/h

c

440 km at 80 km/h

5 Mark covers a distance of 18 km on his skateboard in 2 hours. What is his average speed? 6 The distance from Adelaide to Melbourne is 744 km. If a car travels at an average speed of 93 km/h, how long would it take to reach Melbourne? 7 Zoe rode her bike 55 km in 3 hours, while Charlotte rode her bike for 2 hours, covering 39 km. Who travelled at the greater speed and by how much? 8 It takes 1 hour 50 mins flying time from Canberra to Adelaide. If a plane travels at an average speed of 540 km/h, what distance is it from Canberra to Adelaide?

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165

Temperature In most of the world, temperatures are measured in degrees Celsius (o C). 0o C is the temperature at which pure water freezes. 100o C is the temperature at which pure water boils.

DEMO

In a few countries, including the USA, the old Fahrenheit scale (o F) is still used. 32o F is the temperature at which pure water freezes. 212o F is the temperature at which pure water boils. This graph allows us to convert from o C to o F or from o F to o C. Celsius (°C) 100

°C

°F

100

212

80 60 40 20

0

32

Fahrenheit (°F)

0

40

80

120

160

200

Conversion formulae Celsius to Fahrenheit

Fahrenheit to Celsius

C = 5 £ (F ¡ 32) ¥ 9

F = 1:8 £ C + 32 Examples:

To convert 20o C to o F.

To convert 65o F to o C.

F = 1:8 £ 20 + 32 = 36 + 32 = 68 )

20o C = 68o F

C = 5 £ (65 ¡ 32) ¥ 9 = 5 £ 33 ¥ 9 + 18:3 )

65o F + 18:3o C

9 Use the graph to convert these o C temperatures into o F temperatures: a 50o C b 80o C c 20o C 10 Use the graph to convert these o F temperatures into o C temperatures: a 100o F b 50o F c 80o F

DEMO

11 Use the formula for converting o C into o F temperatures to check your answers to question 9. 12 Use the formula for converting o F into o C temperatures to check your answers to question 10. 13

a The highest temperature recorded in Australia is 53o C (at Cloncurry, Queensland in 1889). Use the formula to find this temperature in o F. b The highest temperature recorded in the United States is 134o F (at Death Valley, California in 1913). Use the formula to find this temperature in o C. Give your answer correct to 1 decimal place.

166

TIME AND TEMPERATURE (CHAPTER 10)

Review of chapter 10

Unit 78 Review set 10A

1 What is the best unit of time measurement for the following? a a plane flight from Adelaide to New York b the length of an average lesson c the time between heartbeats d the life cycle of a human being e the age of buildings in ancient Rome f the time for an Olympic sprinter running 100 metres 2 Copy and complete: a c

7 weeks = 2 days 9 14 hours = 2 minutes

3 Write the following in a c

b d

i 12-hour time

ii

quarter to seven in the morning half past nine at night

12 minutes = 2 seconds 1 millennium = 2 years

24-hour time: b

quarter past midnight

4 Find the following: a 9 hours 38 mins + 6 hours 45 mins + 4 hours 18 mins b 7 hours 27 min ¡ 3 hours 49 mins 5

a Find the time difference from 10:15 pm to 6:35 am the next day. b What is the time 40 minutes after 11:33 am?

6 If a bricklayer can lay a brick in 40 seconds, how many can he lay in 4 hours? 7 Josh began saving 15 dollars a day from the 4th April. He needs $3000 to have his teeth straightened on September 27th. a How many days does he have to save? b What is the total he will save? c How much will he still owe the orthodontist? 8 To reach her goal of running 1000 km before the season starts, a netballer plans to run 10 km each day. If the season starts on the 8th September, when should she start her running? 9 Heath drives his car 160 km from Adelaide to Moonta in 2 hours. He then drives a further 36 km to Maitland at a speed of 100 km/h. a What was his speed driving from Adelaide to Moonta? b How long did it take Heath to drive from Moonta to Maitland? c How long did the trip from Adelaide to Maitland take? d How far is it from Adelaide to Maitland? e What was his average speed travelling from Adelaide to Maitland? 10 Use the Standard Time Zone map for this question. If it is 11 am on Saturday in Greenwich, what is the Standard Time in: a Moscow

b

Adelaide?

TIME AND TEMPERATURE (CHAPTER 10)

167

Review set 10B 1 How many: a days from 7th July to 22nd October b hours from 11 pm Monday to noon the following Thursday c minutes from 9:47 am to 11:08 am d seconds from 11:59 pm to 12:04 am? 2 In what a decade b

century c millennium was the year 1786?

3 How many years are there from the start of the first year to the end of the second? a 4

1908 to 1974

b

1545 to 1695

c

14BC to 25AD

a Write the twenty first of November, nineteen sixty three in numerals. b Zoe was born on the seventeenth of October, nineteen eighty nine. In numerals, write the date for Zoe’s 21st birthday. afternoon

5 Write this pm time as: a

an analogue time in words

b

12-hour digital time

c

24-hour time

6 Find the following: a 4 hour 8 min + 5 hour 35 min + 3 hour 47 min b 2 days 9 hours 18 min + 3 days 15 hours 45 min c 3 hours 15 min ¡ 1 hour 57 min 7 Find the average speed of a car which travels 752 km in 8 hours. 8 How long would it take you to travel 440 km at 80 km/h? 9

Spacecraft Vostok 1 Vostok 2 Merc. - Atlas 6 Vostok 6 Voskhod 1 Gemini 8 Gemini 11 Soyuz 1 Apollo 8 Soyuz 4 Soyuz 5 Apollo 11 Apollo 12

A DECADE OF MANNED SPACE FLIGHTS Launch date Duration (h:min) Remarks 12=04=61 1:48 first manned flight 06=08=61 25:18 first flight exceeding 24 hours 20=02=62 4:55 first American to orbit 06=06=63 70:50 first woman in space 12=10=64 24:17 first 3-man crew 16=03=66 10:41 first docking of 2 orbiting spacecraft 12=09=66 71:17 highest Earth orbit altitude (1372 km) 23=04=67 26:37 cosmonaut killed in re-entry accident 21=12=68 147:01 first manned orbit(s) of Moon 14=01=69 71:23 Soyoz 4 and 5 docked and transferred 15=01=69 72:56 2 cosmonauts from Soyuz 5 to Soyuz 4 16=07=69 195:19 first landing of human on the Moon 14=11=69 244:36 second manned lunar landing

a For how many days did Apollo 11 fly? b On what space craft did the first woman in space travel? c How much longer was Apollo 12’s flight than Apollo 8’s? d How soon after the first manned orbit of the moon was the first human on the moon?

168

REVIEW OF CHAPTERS 7, 8, 9 AND 10

TEST YOURSELF: Review of chapters 7, 8, 9 and 10 1 Convert:

a

3620 m to km

2 Convert:

a

mm2

3 In what

a decade

250

to

b

cm2

b

3:55 kg to g 5 ha to

b century

c

56 cm to m

c

4:2 m2 to cm2

m2

c millennium

was the year 1913?

4 Give the missing word in each sentence: a In a column graph, the ...... of a column represents the number of objects in a group. b The number of times a result occurs is shown in the ...... or ...... column of a table of information. 5 Convert:

a

640 mL to L

b

8:2 mL to cm3

3:8 kL to m3

c

6 How many years are there from 1879 to 1942? 7 Find the perimeter of: a

b

c

7m 3m

16.8 cm

4m

2.8 km

1m 5m

4.2 km

8 Find the shaded area: a 9m

b

c

9 Convert:

a

2:6 tonnes to kg

b

0:5 km to m

10 Read the following scales: a b F

FUEL

2 cm

8460 cm2 to m2

c

c 80 100 120 60 140 40 160 KM/H 180 20

E

5m

7m

4.8 m

0

d 2 1

200

5 cm

3

5L

4

kg 5 0 6

11 Write true or false for these statements: a If a politician asked only his friends if they were going to vote for him at the next election, his survey about his election results would be biased. b To find out which brand of dishwashing detergent is most popular you would conduct a census. 12 Find how many: a days from March 3 to May 14 c minutes from 9:15 am until 1:40 pm 13 Find the volume of:

b d

hours from 4 pm on Tuesday to 9 am on Wednesday seconds from 11:59 am until 1:01 pm.

REVIEW OF CHAPTERS 7, 8, 9 AND 10

169

14 Each month a farmer recorded the rainfall on his property. The rainfall in mm for 2002 was: 18:5, 0, 6, 11:5, 19:5, 16:5, 16:5, 21, 12, 9:5, 22, 19 Find a the median b the mean rainfall. 15 If a scale is 1 : 5000, find: a the actual length if the scale length is b the scale length if the actual length is 16 Find the volume of: a

i i

2 cm 50 m

b

ii 0:5 cm ii 125 m 38.5 cm 2

4m

9 cm

5m

6m

c 12 cm 5 cm 26 cm

Write this afternoon time as: a an analogue time in words b 12-hour digital time c 24-hour time

17

18 Australia’s population reached 20 million on the fourth of December in the year two thousand and three. Write this date in numerals. 19 Maria has a square lawn and garden with sides 12 metres. In the middle is a rectangular rose bed with sides 6 m and 4 m. Find the area of lawn.

20 Cricket 15% Football 13%

Tennis 37%

Netball 10% Basketball 25%

6m 4m

The pie chart shows the results of a survey of 120 Year 7 students. All students were asked the question: “What is your favourite sport?” Use the chart to find: a the most popular sport b the number of students whose favourite sport is netball c the number of students whose favourite sport is tennis or football.

21 A plumber often receives emergency calls to repair broken pipes etc. Draw a column graph for these numbers of emergency calls taken over 2 weeks: 2, 1, 3, 3, 2, 0, 1, 2, 4, 1, 1, 2, 0, 3 22 How many 90 cm steps are needed to walk 1:8 km? 23 Find the following: a 3 hours 22 minutes + 2 hours 58 minutes b 4 hours 22 minutes ¡ 2 hours 39 minutes 24

12 m

a Find the speed of a car which travels 426 km in 6 hours. b How long would it take you to walk 22 km if your average speed was 4 km/h?

170

ALGEBRA

Geometric and number patterns

Unit 79 and

.....

.....

are examples of geometric patterns.

For the matchstick pattern ,

,

For the number pattern 1 £ 3 + 2, 2 £ 3 + 2, 3 £ 3 + 2, 4 £ 3 + 2,

, .....

The next two members are: ,

the first 4 members are: 1£3 + 2 = 5 2£3 + 2 = 8 3 £ 3 + 2 = 11 4 £ 3 + 2 = 14

, .....

If we count the number of matches and draw a table: Unit number Matches needed

1 3

2 6 +3

3 9 +3

4 12 +3

5 15

The 100th member is

+3

We can see that the number of matchsticks is increasing by 3 each time. So we can predict the next two members of the pattern will have: Unit number Matches needed

6 18

One matchstick pattern that fits the number pattern is

VIDEO CLIP

7 21

100 £ 3 + 2 = 300 + 2 = 302

etc. 3 + 2

3 + 3 + 2

3 + 3 + 3 + 2

Exercise 79 , 1 unit

1 Consider the pattern

2 units

,

, ....

3 units

a Draw the next 4 members of the pattern. b Copy and complete the table showing the number of matches required to make each member.

Unit number Matches needed

1 3

2 6

3

4

5

6

7

c Without drawing them, write down the number of matches needed to make the members with unit numbers 8, 9, 10 and 11. d Predict the number of matchsticks needed to make the member with: i 30 units ii 50 units iii 200 units

iv

1 000 000 units.

CHAPTER 11

2 For each of the matchstick patterns below: i draw the next two members ii construct a table of ‘matches needed’ for the first 5 members iii without drawing them, predict the number of matchsticks needed to make members 6 and 7. a

,

c

,

3 Look at i ii iii a b c d

, ,

, ..... , .....

b d

,

,

,

,

the number patterns a, b, c and d. For each pattern: find the values of its first 4 members find the value of its 100th member draw a matchstick pattern which fits the number pattern.

1 £ 3 + 1, 1 £ 3 ¡ 1, 1 £ 4 + 2, 1 £ 4 ¡ 3,

2 £ 3 + 1, 2 £ 3 ¡ 1, 2 £ 4 + 2, 2 £ 4 ¡ 3,

3 £ 3 + 1, 3 £ 3 ¡ 1, 3 £ 4 + 2, 3 £ 4 ¡ 3,

4 £ 3 + 1, 4 £ 3 ¡ 1, 4 £ 4 + 2, 4 £ 4 ¡ 3,

.... .... .... ....

, ..... , .....

ALGEBRA (CHAPTER 11)

171

With number patterns such as 3, 7, 11, 15, 19, .... we should be able to continue the number sequence as far as we like. To get the next member from the previous one we simply add 4. Look at these patterns: ²

We look for a pattern we can use for getting from one member to the next.

To get the next member of the pattern 1, 7, 13, 19, 25, .... we add 6, ) the next 3 members are: 31, 37 and 43. Rule: ‘The next member is equal to the previous one, plus 6.’

²

To get the next member of the pattern 50, 47, 44, 41, 38, .... we take 3, ) the next 3 members are: 35, 32 and 29. Rule: ‘The next member is equal to the previous one, minus 3.’

²

To get the next member of the pattern 2, 10, 50, 250, .... we multiply by 5, ) the next 3 members are: 1250, 6250, 31 250. Rule: ‘The next member is equal to the previous one multiplied by 5.’

²

To get the next member of the pattern 128, 64, 32, 16, .... we divide by 2, ) the next 3 members are: 8, 4, 2 Rule: ‘The next member is equal to the previous one divided by 2.’

4 Find the next 3 members of the following number patterns and in each case write down the rule for finding the next member: a d

1, 4, 7, 10, 13, .... 6, 12, 18, 24, ....

b e

11, 15, 19, 23, 27, .... 13, 22, 31, 40, ....

c f

2, 9, 16, 23, 30, .... 7, 20, 33, 46, ....

5 Find the next 3 members of the following number patterns and in each case write down the rule for finding the next member: a d g j m

38, 36, 34, 32, 30, .... 100, 97, 94, 91, .... 1, 2, 4, 8, 16, .... 64, 32, 16, 8, 4, .... 250, 25, 2:5, 0:25, ....

b e h k n

29, 26, 23, 20, .... 250, 242, 234, 226, .... 2, 6, 18, 54, .... 80, 40, 20, 10, .... 2, 3, 5, 8, 12, 17, ....

c f i l o

57, 51, 45, 39, .... 65, 61, 57, 53, 49, .... 2, 8, 32, 128, .... 243, 81, 27, 9, .... 1, 1, 2, 3, 5, 8, 13, ....

6 Using the first number and the rule given, write down the next three numbers in each pattern: a 7; ‘add 6’ b 3; ‘add 9’ c

4;

e g i

150; 4; 97;

‘add 1 12 ’

‘subtract 25’ ‘multiply by 2 and then add 3’ ‘add one then divide by two’

d

56;

‘take 11’

f h j

3:8; 3; 2;

‘reduce by 0:5’ ‘times by 10 and then subtract 4’ ‘multiply number by itself”

7 Write down the missing number from each pattern: a 3, 9, 2 , 21, 27 b 12, 2 , 36, 48, 60 d 3, 6, 2 , 24, 48 e 6, 10, 2 , 21, 28, 36 g 0:08, 0:8, 2 , 80, 800 h 10, 2 , 32, 43 j 100, 50, 2 , 12:5 k 2, 5, 11, 2 , 47

c f i l

75, 60, 2 , 30, 15 3, 9, 27, 2 , 243 2, 6, 24, 2 , 720 96, 2 , 6, 1:5

172

ALGEBRA (CHAPTER 11)

Formulae and variables

Unit 80

M = 2 £ n ¡ 1 is a rule or formula. M and n are variables. If M is the total number of matchsticks needed to build the n-unit figure, we can write this rule in words as: of matchsticks} |The number {z

is |{z}

two times the {z number of units} |

minus | {z }

one : |{z}

M

=

2n

¡

1

When n = 1, n = 2, n = 3, n = 4,

M M M M

=2£1¡1 =1 =2£2¡1 =3 =2£3¡1 =5 =2£4¡1 =7

n M

1 1

2 3

3 5

4 7

We put the results in table form. DEMO

One pattern that fits this rule is:

,

,

,

, .....

Exercise 80 1 You are given the rule: “the number of matchsticks is three times the unit number”. a Rewrite the rule using a formula. b Make up a matchstick pattern which shows the rule. 2 You are given the rule: “the number of matchsticks is five times the unit number”. a Rewrite the rule using a formula. b Make up a matchstick pattern which shows the rule. 3

i ii

Rewrite the rules below using a formula. Make up a matchstick pattern which fits each rule.

a The number of matchsticks is two times the unit number plus one. b The number of matchsticks is three times the unit number plus two. c The number of matchsticks is three times the unit number minus one.

A rule or formula can be written in words or in symbols. We must be able to convert from one form to the other.

d The number of matchsticks is four times the unit number plus three.

4 For the following rules: i Write down the value of M for n = 1, 2, 3 and 4. Put your answers in table form. ii If M represents the number of matchsticks and n the unit number, write out the formula in words. iii Draw the first four diagrams of a matchstick pattern that fits the rule. a

M =2£n

b

M =n+3

c

M =3£n¡2

d

M =4£n¡3

ALGEBRA (CHAPTER 11)

173

Finding the formula For the pattern

,

,

, .....

the next two members are We draw a table:

If the increases were by +4 instead of +3, the form of the rule would be M = 4 £ n + ¤ or M=4£n¡¤

,

1 2

Unit number (n) Matchsticks needed (M)

2 5 +3

As the n values increase by 1, the M values increase by 3.

3 8 +3

4 11 +3

5 14 +3

Rewrite the table with a line for 3 £ n. n 3£n M

1 3 2

2 6 5

3 9 8

4 12 11

Notice that each M value is 1 less than the value of 3 £ n. So the rule is M = 3 £ n ¡ 1.

5 15 14

We can use our rule to predict the other members. For example, when n = 17, M = 3 £ 17 ¡ 1 = 50 so, 50 matchsticks are needed. 5 Examine the matchstick pattern:

,

,

, .....

a Copy the pattern and add to it the next 3 members. b Copy and complete:

Unit number (n) Matchsticks needed (M)

1

2

3

4

5

6

3

4

5

6

3

4

5

c Find the rule connecting M and n. d Find the number of matchsticks needed to make the 23rd member. 6 Examine the matchstick pattern:

,

,

, .....

a Copy the pattern and add to it the next 3 members. b Copy and complete:

Unit number (n) Matchsticks needed (M)

1

2

c Find the rule connecting M and n. d Find the number of matchsticks needed to make the 43rd member. 7 Look at the following pattern:

,

,

, ......

a Draw the next two members of the pattern. b Copy and complete:

Unit number (n) Matchsticks needed (M)

1

2

c Find the rule connecting M and n. d Find the number of matchsticks needed to make the 57th member. 8 For the following tables, find the formula connecting M and n and so find M when n = 218. a b n 1 2 3 4 5 n 1 2 3 4 5 M 5 9 13 17 21 M 8 15 22 29 36

Click on the icon for more practice exercises.

PRACTICE

174

ALGEBRA (CHAPTER 11)

Practical problems and linear graphs

Unit 81 Example:

A taxi company charges $3 ‘flag fall’ and $2 for each kilometre travelled. If the total charge is $C for travelling n kilometres, you can find the formula connecting C and n and the total charge for travelling say 21 km. First, complete a table of values.

0 3

n C

1 5 +2

2 7 +2

3 9 +2

As the n values increase by 1, the C values increase by 2.

4 11 +2

Rewrite the table with a line for 2 £ n. n 2£n C

0 0 3

1 2 5

For travelling 21 km, n = 21

2 4 7

3 6 9

Notice that each C value is 3 more than the value of 2 £ n. So, the formula is C = 2 £ n + 3.

4 8 11

) C = 2 £ 21 + 3 = 42 + 3 = 45

The charge for travelling 21 km would be $45.

Exercise 81 1 A mechanic charges $40 callout plus $20 for every hour attending a weekend breakdown. So, for a breakdown taking two hours to fix, the total cost would be $40 + 2 £ $20 = $80. The formula the mechanic uses is C = $40 + h £ $20 where C is the total charge for a callout of h hours. Find the total charge for callouts of: a 1 hour b 3 hours c 5 hours 2 A bean plant grows 3 cm each hour during a day. There are 10 hours of sunlight today and the plant starts at 8 cm tall. a Copy and complete the chart showing the height of the bean plant for each hour during the day.

Time (hours) Height (cm)

0 8

1

2

3

4

5

6

7

8

9

10

b Write a rule that connects the hours of sunlight to the height of the bean plant. 3 A satellite TV company charges $75 installation and $42:00 per month from then on. What would it cost to install the service and then use it for: a 7 months b 1 year c 18 months d 5 years? Show how you get your answer. 4 On a rainy day the flow of a river increases by 2 cumecs each hour. When the rain starts the river flow is 8 cumecs. a Construct a table to show the flow each hour for 9 hours. b Write a formula to calculate the flow after h hours. c Calculate the flow after 9 hours of rain.

ALGEBRA (CHAPTER 11)

175

Linear graphs Consider again the taxi company that charges $3 ‘flag fall’ and $2 for each kilometre travelled. The total charge is $C for travelling n km, where the formula C = 2 £ n + 3 was found from a table of values:

n C

0 3

1 5

2 7

3 9

4 11

Rather than use a formula, sometimes a graph is easier to use and find information from. This graph is a linear graph as all points lie on a straight line. It can easily be read from the graph that when n = 12, C = 27 and when n = 18, C = 39.

cost ($C)

40

30

20

10

0

5

Taxi charges

50

0

5

10

a Use only the graph above to find the cost of a taxi ride which is: i 7 km ii 14 km b Why is it easier to use the formula C = 2 £ n + 3

15 20 number of kilometres (n)

iii

17 km

for trips of more than 20 km?

6 On grid paper draw the graph of P against n from the table alongside.

0 7

n P

Make sure that your graph can be extended to n = 25. P is the fee for boarding a pet for n hours. a Find the boarding fee for a pet staying for i 10 hours ii 18 hours

1 10

iii

2 13

3 16

4 19

5 22

6 25

22 hours

b Tania found the formula for calculating the fee. Her formula was P = 3 £ n + 7: i Check that this formula fits the tabled values. ii Check your answers to a. iii Find the boarding fee for a pet staying 35 hours. 7 The following graph shows the growth of a seedling over a period of weeks. a Find the height of the seedling: i when planted ii after 4 weeks iii after 16 weeks

Seedling growth

50

height (cm)

40 30

b If the linear trend continues, how long will it take 20 for the seedling to reach a height of: i 20 cm ii 26 cm 10 iii 33:5 cm?

weeks (n)

0 0

5

10

15

20

176

ALGEBRA (CHAPTER 11)

Solving equations

Unit 82 We can solve simple equations such as x + 2 = 5. We know that 3 + 2 = 5, so x = 3. Similarly if 42 ¥ x = 6, then x = 7 because 42 ¥ 7 = 6. Sometimes we may need to use two steps. To solve the equation 2 £ r + 4 = 10, Alternatively we can use inverse operations.

we can say that so

2£r =6 r=3

as as

6 + 4 = 10 2£3=6

10 + 7 ¡ 7 = 10 and 15 ¡ 7 + 7 = 15 show that +7 and ¡7 are inverse operations. 6 £ 3 ¥ 3 = 6 and 6 ¥ 3 £ 3 = 6 show that £3 and ¥3 are inverse operations. The operations are called inverse as one undoes what the other does. For example, if we have to solve 3 £ x + 11 = 50, we first take 11 from both sides of the equation. The equation stays balanced because you are doing the same operation on both sides.

So 3 £ x + 11 ¡ 11 = 50 ¡ 11 ) 3 £ x = 39 We divide both sides by 3 to leave x = 13

Exercise 82 1 Solve the equations: a a+6=9 d a¡6=9 g a £ 9 = 45 j a¥5=5

b e h k

12 + b = 17 12 ¡ b = 3 6 £ b = 54 b¥6=4

c f i l

x + 11 = 32 x ¡ 20 = 1 3 £ x = 60 49 ¥ x = 7

2 Solve the equations: a 2 £ a + 3 = 27 d 2 £ x + 10 = 30 g 9 £ y + 2 = 29

b e h

3 £ a ¡ 1 = 14 5 £ x ¡ 11 = 9 6 £ y ¡ 5 = 67

c f i

4 £ a + 5 = 13 7 £ x ¡ 12 = 23 3£y+6=6

3 Solve for x: a 2 £ x + 5 = 19 d 4 £ x + 7 = 15 g 6 £ x + 1 = 55

b e h

2 £ x + 3 = 27 3 £ x + 9 = 30 7£x+8=8

c f i

5 £ x + 2 = 22 3 £ x + 11 = 35 7 £ x + 21 = 42

4 Solve for x: a 2 £ x ¡ 1 = 11 d 7 £ x ¡ 8 = 41 g 4 £ x ¡ 3 = 45

b e h

2 £ x ¡ 7 = 15 6 £ x ¡ 2 = 40 5 £ x ¡ 4 = 41

c f i

3 £ x ¡ 5 = 25 9 £ x ¡ 7 = 83 2 £ x ¡ 8 = 18

Checking Solutions In the example above, the equation 3 £ x + 11 = 50 has the solution x = 13. To check that the solution x = 13 is correct, we replace x by 13 in the left hand side of the equation. 3 £ x + 11 = 3 £ 13 + 11 = 39 + 11 = 50 which is the right hand side of the equation, so x = 13 is the correct solution.

ALGEBRA (CHAPTER 11)

177

Problem solving with equations We can use an equation to write: “I multiply a number by 3 and then add 5, to give a result of 23”. Suppose we represent the number with the letter n (for number). Start with a number multiply it by 3 add 5 the result is 23

n 3£n 3£n + 5 3 £ n + 5 = 23

So, the algebraic equation which represents the problem is 3 £ n + 5 = 23. 5 Starting with the number n, rewrite each of these sentences as an equation in algebra. a I think of a number and subtract 5 from it, getting an answer of 16. b I think of a number and subtract 8 from it, giving an answer of 30. c I think of a number and add 3 to it, giving an answer of 31. d The sum of a number and 8 is 11. e The product of a number and 7 is 35. f A number is doubled, and the result is 26. g I think of a number and divide it by 5, getting an answer of 7. h A number is divided by 3 and the result is 9. i I think of a number, multiply it by 7, and subtract 6, giving a result of 64. j A number is multiplied by 8 then 5 is subtracted to give a result of 27. 6 Construct an equation from the sentence(s) given and then use it to solve the problem. a I think of a number and subtract 10 from it, giving an answer of 11. b I think of a number and add 3 to it. The answer is 17. c When I multiply a number by 9, the result is 63. d When I divide a number by 9, the result is 6. e I think of a number, multiply it by 8, then subtract 6. The result is 50. f A number is multiplied by 4, then 3 is added. The result is 23. g A number is trebled, then 1 is added. The answer is 64. h A number is multiplied by 5, then 11 is subtracted. The result is 54. 7 A plant was 6 cm high when I bought it from the nursery. It grew 3 cm a day. After how many days was it 21 cm high? a Set up an equation where d is the number of days after the plant was bought. b Solve the problem. 8 An electrician charges $45 an hour plus a callout fee of $60. A large job costs $465. For how many hours did the electrician work? a Set up an equation where x is the number of hours worked. b Solve the problem. 9 At the start of the school year, Craig has $800 in his bank account. Each week while Craig is studying, he withdraws $15 for pocket money. At the mid year break Craig has $500 left in his account. How many weeks has he been studying? a Set up an equation where w is the number of weeks that Craig has been studying. b Solve the problem.

178

ALGEBRA (CHAPTER 11)

Review of chapter 11

Unit 83 Review set 11A 1 Give the next three members of these number patterns: a 2, 11, 20, ...., ...., .... b 20, 40, 80, ...., ...., .... c d

1 12 , 3 12 , 5 12 , ...., ...., .... 40, 20, 10, ...., ...., ....

2 The following pattern is built out of matchsticks:

,

,

, .....

a Draw the next 2 members of the pattern. b Copy and complete:

Unit number (n) Matchsticks needed (M )

1

2

3

4

5

c Find the rule connecting M and n. d How many matchsticks are needed to build: i 7 squares ii 101 squares? 3 ACME Carpet Cleaning Company charges $35 for the first room it cleans and $15 for every other room. If C dollars is the charge for cleaning n rooms, complete the table of values:

n C

1

2

3

4

5

6

a What is TLC’s fee formula? b Find how much TLC would charge for cleaning a mansion with 27 rooms. 4 VideoOz charges $3 for the first video it hires out and $2 for each video hired out thereafter. The table of charges for video hire is:

Number of videos (n) Charge ($C)

1 3

2 5

3 7

4 5 9 11

a On grid paper, plot the graph of C against n from the table of values. b Determine the VideoOz hire formula for hiring out n videos. c Calculate the hire charge for these numbers of videos: i 9 ii 17 5 I think of a number, multiply it by 6 and subtract 1. The answer is 47. Construct an equation and use it to find the number. 6 Solve the following equations: a f ¡ 12 = 23

b

n¥7= 6

c

8 £ b ¡ 7 = 49

ALGEBRA (CHAPTER 11)

179

Review set 11B 1 The rule is “the next number is equal to the previous one multiplied by 2 then minus 3”. What are the values for the next 3 numbers in these sets: a 5, 7, ...., ...., .... b 10, 17, 31, ...., ...., .... ? 2 The following pattern is built out of matchsticks: ,

,

, .....

a Copy and complete the table of values:

Number of units (n) Number of matchsticks (M)

1

2

3

4

5

6

b Find the rule connecting M and n. c Write down the rule in sentence form. d Use the rule to write the number of matchsticks needed for 80 units. 3 A disc jockey charges $150 to set up the equipment and $40 for each hour to provide entertainment for school discos. a Complete the table of values:

Number of hours (n)

0

Charge ( $C)

150

1

2

3

4

b What is the disc jockey’s fee formula? c How much would the disc jockey charge for: i

a lunch hour disco

ii

3 hours

4 A babysitter charges $12 for the first hour and $8 for each hour after that.

70

cost ($C)

60

a Use the graph to find how much the babysitter would charge for: i 3 hours ii 6 hours.

50

b Write down a formula for the babysitter’s fee.

20

c Use the formula to find how much the babysitter would charge if she worked from 8 am until 10 pm.

5 hours?

iii

40 30 10 0

number of hours 0

2

4

6

8

5 Kathy earns $532 a week plus $21 an hour for working overtime. Last week she earned $637 in total. How many hours overtime did she work? a Set up an equation where x is the number of hours of overtime worked. b Solve the problem. 6 Solve the following equations: a k ¡ 11 = 24

b

m £ 8 = 24

c

4 £ d + 5 = 65

180

TRANSFORMATION AND LOCATION

Number planes

Unit 84 We can use a grid on a number plane to find the exact position of any point.

7 6 5 4 3 2 1

The number plane Vertical axis The two axes join at a right angle

0 Horizontal axis

vertical

The horizontal coordinate is always named and located first.

A

B

A is at (2, 5) B is at (6, 2)

horizontal 1 2 3 4 5 6 7

We say that (2, 5) are the coordinates of A. (2, 5) is an ordered pair. The horizontal coordinate is always named first. DEMO

Exercise 84 1 Copy and draw the given grid. Locate the following points:

A(1, 2) F(4, 3)

B(3, 6) G(1, 6)

C(3, 1) H(5, 1)

6

D(5, 5) I(1, 1)

E(2, 4) J(5, 6)

a When the points GIHJ are joined, is a rectangle or a square formed? b When points ACFE are joined, is a rectangle or a square formed?

5 4 3 2 1

c Is triangle AGJ right angled?

1

2

3

4

5

2 Use the ordered pairs of numbers to locate the letters on the number plane: a (6, 6) b (1, 8) c (7, 4) 9 e (5, 5) f (6, 8) g (1, 1) W C T 8 i (3, 4) j (6, 3) k (0, 0) P Z 7 m (7, 6) n (0, 5) o (2, 2) E B S F 6 p (8, 2) q (4, 6) r (4, 2) Y A 5 L s (7, 1) t (1, 4) u (4, 4) Q N G X 4 v (3, 8) w (5, 3) x (3, 1) J O

8

9

PRINTABLE WORKSHEET

K

V

M U 1 2 3 4 5 6 7 8 9 H

R

D

10

9 8

3 Using this map, find: a the grid coordinates for i Gnometown ii Magic Cave iii Ferry Landing iv where the roads cross Dawson’s River b the places located at i (9:2, 3:7) ii iii (1:6, 3:5) iv

7

Haunted Forest Elftown Cemetery

6

5 4

3

Oasis

Ferry Landing

Gnometown

Da

Mt. Ogre

on ws

Magic Cave

1 0

Treasure Trove

Lion’s Den

2

(5:3, 2:7) (1:4, 5)

Mt. Dragon

iver

CHAPTER 12

0

I

7

(2, 7) (8, 5) (4, 0)

d h l

’s R

3 2 1

6

1

2

3

4

5

6

7

8

9

10

TRANSFORMATION AND LOCATION (CHAPTER 12)

4

12

a Use the map of Australia to find the grid coordinates for: i iii

Perth Cairns

N

11

Darwin

Hobart 10 Adelaide

ii iv

181

Cairns

9

b Find the city/town located at: 8 i (3, 4:6) ii (6:8, 7:4) iii (8:7, 8:7) iv (13, 5:7) 7

Mt. Isa

Alice Springs

Rockhampton

6

Brisbane Kalgoorlie

5 4

Perth

3

Canberra Sydney

Adelaide

2

Melbourne

1

scale: 1 cm represents 800 km

0

1

2

3

4

5

6

7

8

9

Hobart 11 12

10

13

14

Showgrounds map

8 Cookery, Art & Craft

7

TEN CARNIVAL

FAMILY STATE FAIR

Fashion Parades

(upper level)

MOTOR PAVILION (lower level)

Child& Youth Health

$

Agriculture Golden Learning Centre Grains Bar Historical Blacksmith Shop Display

THE OLD RAM SHED

Alpaca & Goat Judging Lawn

Sheep Judging

Rural Services Office

ARD

Welcome Desk

Beef Cattle

Beef Cattle Judging Lawn

Wool Fleece Display

6 What could I find at: a D2 b G9

Parcel Minding Stroller/ Wheelchair Hire

Disabled Access Viewing Platform

B

c

7 Which entrance is closest to:

$

Bar

H5

d a

F8

Entry

Main Arena Spectacular

WAYVILLE PAVILION

RIDLEY CENTRE

Showbags

Be Active Showtime SA

Horses Arnotts Biscuits

Dairy Farmers Fireworks Spectacular

Roundhouse Horse Warm-up Area

Horses

Bar Harness Horses

Exhibition Dairy Cattle Milking Dairy

Dairy Cattle Judging Lawn

D

E2

E

e b

Reserved Car Park

D6

F

f

G3?

Wayville Pavilion?

To Tram

Members Entry

G

H

Baby Change Western Entertainment Arena

Channel 7

Dairy Cattle

C

First Aid

Market Bazaar

Photography/ Art Studio

SOUTH BOULEVARD

Reserved Car Park

Telephone

Pet Centre

Lockers

Horses

Entry

Disabled Access Toilets

Atrium Plaza Food Court

Horse Events

Pigs & People Sows / Piglets

Reserved Car Park

A

$

Vintage Machinery

Beef Cattle

1

SA Volunteers

DUNCAN HALL

Automatic Teller/Bank Toilets

Horticulture Exhibition

BankSA Farmyard Nursery

LEADER PAVILION

Seated Dining

$

Woolworths Good Earth Display

Show Shop

MAIN ARENA

Woodcutting Members

Sheep & Ram Sales

2

Disabled Access Viewing Platform

Members Grandstand

WEST CRESCENT

3

ANZ Seminars

Grandstand

Alpacas & Goats

JUBILEE PAVILION

Lifestyle, Kidz Zone The Shed

Royal Show Office

Triple M Information Booth

ABC

Stock Journal

Taste ROYAL South FARM Australia EXPO The Meating Place

Pig Racing/ Viewing

$

Entry

Garden Cafe Orchids

CENTENNIAL HALL

Jubilee Cafe

Police/ LostChildren

KingswayFoodCourt CWA Cafe

$

Blockbuster Stage

Poultry/Pigeons/Eggs

The Pub

Entry

SHEEP & WOOL PAVILION

The Advertiser

HAMIL TON BOULEVARD

The Milk Bar

4

Channel 10

Boulevard Food Court

Balfours Kitchen

DAIRY PAVILION

5

Investigator Science & Technology Centre

Ferris Wheel RedDoveCafe

2003

To City

TEN CARNIVAL

MICHELL WoolW orks WOOL HALL

6

Reserved Car Park

Bus Entry Stop

GOODWOOD ROAD

Dogs

WEST BOULEV ARD

9

Main Entrance

Reserved Car Park

Car Park

Members Entry

ENT EAST CRESC d Grandstan

Write the closest coordinates for each of the following: a dogs b beef cattle c showbags d Duncan Hall e woodcutting f Ferris Wheel g Blockbuster Stage h poultry / pigeons i Jubilee Pavilion j Pet Centre

WEST BOULEV

5

182

TRANSFORMATION AND LOCATION (CHAPTER 12)

Transformations and reflections

Unit 85 Transformations

Translations, reflections, rotations and enlargements are all examples of transformations. Here are some examples of the four transformations: a translation (or shift)

a reflection

a rotation about O

an enlargement

O

mirror line DEMO

Exercise 85 1 Figure A moves. It is translated, reflected, rotated or enlarged. Give the transformation in each case. a

b

c

d

e

f

A

Lines of symmetry

Activity

Recall that a line of symmetry is a line along which a shape may be folded so that both parts of the shape will match. For example: fold line

DEMO

the line of symmetry is also called a ‘mirror line’

What to do:

1 Copy the following figures and draw on them any lines of symmetry: a

b

c

d

e

2 Check 1 using a mirror. 3 From a magazine or newspaper cut out a picture which seems to have line symmetry. Fold along the line of symmetry and then cut along this line with scissors. Glue half the picture onto a sheet of paper and then draw the other half to make it symmetrical. Display your work on the wall of your room. 4 Click on the icon which enables you to check the symmetry of people’s faces. Write down your observations.

DEMO

TRANSFORMATION AND LOCATION (CHAPTER 12)

Reflection

183

The object has been reflected in the mirror line.

object

The reflection is called the image. The reflection in the mirror is exactly the same as the half of the figure “behind” the mirror. image

2 Place your mirror on the mirror line, shown using dashes, and observe the mirror image. Then draw the object and its mirror image in your work book. a b c

MAHS

3

a Predict and draw the image of these objects if a mirror was placed on the mirror line: i ii iii PRINTABLE DIAGRAMS

b Check your answers to a using a mirror. 4 On grid paper reflect the geometrical shape in the mirror line shown: a b

m

d

Activity

c

m

e

m

f

Making symmetrical shapes Try making symmetrical patterns by folding a strip of paper a number of times and cutting out a shape. For example, how many folds would you need and what shape would you cut out to get the result shown?

184

TRANSFORMATION AND LOCATION (CHAPTER 12)

Rotations and rotational symmetry

Unit 86

We rotate an object by turning it around a fixed point through a certain number of degrees.

90°

180°

O

90°

O

O

90o

90o

anticlockwise This shows a rotation about O.

Every point turns through the same angle relative to O.

180o anticlockwise rotation about O.

O is the fixed point and is called the centre of rotation.

The hands of a clock turn in a clockwise direction. The opposite direction is anticlockwise.

Exercise 86 1 We could rotate the object

clockwise rotation about O.

anticlockwise to obtain: O

A

O

B

C

O

D

O

O

Which of A, B , C , or D is a rotation of the object through: a 180o b 360o c 90o

d

270o ?

2 Copy and rotate anticlockwise each of the following shapes about the centre of rotation O, for the number of degrees shown. Tracing paper can be used. a b c d

PRINTABLE WORKSHEET

O

90°

e

f

O

180°

g

O

270°

h

O

360°

O O

O

90°

180°

270°

90° DEMO

3 Rotate anticlockwise about O through the angle given: a 90o b 180o

c

O

270o

O O

d

180o

e

O

90o

f O

270o O

TRANSFORMATION AND LOCATION (CHAPTER 12)

Order of rotation The order of rotational symmetry is the number of times a shape looks exactly the same and is in the same position during one complete turn about the centre.

185

DEMO

For example: A

B

D

C

180° rotation

C

D

B

A

180° rotation

A

B

D

C

centre of rotation

The rectangle has order of rotational symmetry 2 since it moves back to its original position under rotations of 180o and 360o : 4 For each of the following shapes find the order of rotational symmetry (tracing paper may help): a b c d

e

f

g

h

5 Design your own shape which has order of symmetry of: a 2 b 3 c 1

d

e

4

6

Rotational symmetry

Activity You will need:

a dressmakers pin, scissors.

PRINTABLE WORKSHEET

What to do: 1 Click on the icon and print the page containing the regular polygons. Each regular polygon has its centre of symmetry marked. 2 Cut out each shape with scissors. 3 Starting with the equilateral triangle, place the pin through its centre and hold it to your page by putting your finger on top of it.

Draw around the shape to make your starting figure. Now rotate the shape anticlockwise and count the number of times in the first revolution that the cut out fits onto the original figure. Record your results. 4 Repeat for the square, pentagon, hexagon, heptagon and octagon. 5 Copy and complete the table given:

The number of symmetries is the number of positions where a ‘match’ occurs. 360o is to be included, 0o is not included.

Regular polygon

pin

centre of symmetry

Number of sides

triangle square pentagon hexagon heptagon octagon

6 What do you observe from 5? Write your answer in words.

Number of symmetries

186

TRANSFORMATION AND LOCATION (CHAPTER 12)

Translations and tessellations

Unit 87 Translations object

DEMO

The object A has been translated 4 units right and 3 units down to the image A 0 (“A dashed”).

4 A

Every point is moved the same distance in the same direction. A translation is a sliding movement.

3 A' image

Exercise 87 1 For the given figures, describe the translation from: a A to B b B to A c B to C d C to B e A to C f C to A

B

A

C

2 Copy onto grid paper and translate in the directions shown: a 2 left, b 2 right, 3 down 1 up

c

3 left, 2 down

DEMO

d

3 right, 4 down

e

6 left, 4 up

f

2 right, 5 up

3 The object A has been translated to give the image A0 in each diagram. Give the translation from A to A0 . a b c A

A

A'

A'

A'

d

A

e

f A'

A

A

A'

4 Which of these diagrams shows a translation?

A

A A'

B

C

TRANSFORMATION AND LOCATION (CHAPTER 12)

187

Tessellations A tessellation is a pattern made from identical shapes that fit together without gaps.

These diagrams show tessellations of the triangle and the cross.

5 Copy the shapes onto grid paper and then add 5 or 6 more shapes to show tessellation: a b

6

A

B

C

E

D

7

Give the translation that moves: a A to B b A to C c A to D d C to E

DEMO

a Using a card, make a tile which you can use to create a tessellation using translations only. b Construct your own tessellation, colour it and display it on the wall. c Give instructions on how you created your tile pattern.

Activity

Creating tessellations By following these steps you can create your own tessellating tile. What to do:

Step 1:

Draw a square.

Step 3:

Rub out any unwanted lines and add features.

Step 2:

Step 4:

Cut a piece from one side and ‘glue’ it onto the opposite side.

Photocopy this several times and cut out each face. Combine them.

Now it is your turn. Make your own tessellation pattern and produce a full page pattern with 3 cm by 3 cm tiles. Be creative and colourful. You could use a computer drawing package to do this activity.

188

TRANSFORMATION AND LOCATION (CHAPTER 12)

Enlargements and reductions

Unit 88

These diagrams show the enlargement of an object: a with scale factor 2 b with scale factor object

image

object

Every length on the object is multiplied by the scale factor to produce the image.

1 2

In b the scale factor is less than 1. This causes a reduction.

image

1 2

DEMO

2 4

PRINTABLE WORKSHEET

Exercise 88 1 Enlarge each of these objects by the scale factor given: a b

scale factor 2

c

scale factor 3

d

e

scale factor Qw_

f

scale factor 4

scale factor Qe_

2 Find the scale factor when A is transformed to B: a b A

A

B

scale factor 2

c

B

A

B

3 These diagrams are not drawn accurately. Find the scale factor if A is transformed to B: a b

3 mm

9 mm

A 4 mm

12 mm

c

d A 1.25 m

5m B

15 m A

B

B

24 mm A

6 mm 6 mm

2.50 m

4 For the diagrams in question 3, find the scale factor if B is transformed to A.

B

1.5 mm

TRANSFORMATION AND LOCATION (CHAPTER 12)

Activity

189

Enlargement by grids You will need:

Paper, pencil, ruler

What to do: 1 Copy the picture alongside:

2 Draw a grid 5 mm by 5 mm over the top of the dog alongside as shown: 3 Draw a grid 10 mm by 10 mm alongside the grid already drawn: 4 Copy the dog from the smaller grid onto the larger grid. Start by transferring points where the drawing crosses the existing grid lines to the corresponding points on the new grid. Join these points and finish the picture.

PRINTABLE GRID

5 Use this method to change the size of 3 or 4 other pictures of your choosing. You may like to try making the picture smaller as well as larger, by making your new grid smaller than the original.

Activity

Distorting and transformations In each of the four transformations considered in this chapter, the shape of the image is the same as the object. In this investigation we copy pictures onto unusual graph paper to produce distortions of the original diagram. For example,

PRINTABLE GRIDS

on

becomes

What to do: 1 On ordinary squared paper draw a picture of your own choosing. 2 Redraw your picture on different shaped graph paper. For example,

190

TRANSFORMATION AND LOCATION (CHAPTER 12)

Using ratios

Unit 89 A ratio is a way of comparing quantities of the same kind. Examples:

² If Damian has $3 and Donna has $7, then the ratio of Damian’s money to Donna’s money is 3 : 7.

²

20 cents is to $2 is 20 cents to 200 cents which is 20 : 200

We read this as “3 is to 7.”

= 20 ¥ 20 : 200 ¥ 20 = 1 : 10 in simplest form.

The order is important. 3 : 7 is not the same as 7 : 3

² If the sides of a rectangular field are 1:2 km and 500 m, the ratio of sides is 1:2 km to 500 m = 1200 m to 500 m = 1200 : 500 = 1200 ¥ 100 : 500 ¥ 100 500 m = 12 : 5 1.2 km

Two terms must be expressed in the same units before they can be written as a ratio.

Exercise 89 1 Express as a ratio without simplifying a $9 is to $7 d 8 km is to 7 km g 95 cents is to $1 j 500 metres is to 2 km 2

your b e h k

answer: 5 kg is to 3 kg 2 tonnes is to 11 tonnes 7 months is to 2 years 300 kg is to 1 tonne

c f i l

14 cm is to 5 cm 3 years is to 4 years $5 is to 78 cents 1 m is to 20 cm

From this set of shapes, find the ratio of: a i blue to grey ii blue to white iii blue to grey to white b i triangles to squares ii circles to triangles iii squares to triangles to circles c i blue triangles to white triangles ii grey circles to white circles iii white squares to white triangles to white circles d If this set of shapes was multiplied in exactly the same ratio so that there were now 6 blue squares, how many of each of the following would there be? i white triangles ii blue circles iii grey squares

3 Write the ratio in simplest form: a Tom caught 8 fish and Tamara caught 13. b At a disco there are 3 girls to every boy. c In my class there are three blue-eyed students for every five hazel-eyed students. d There are 30 cars on the road for every 2 motor bikes. e A library spent $15 on books for every $5 on computer software. f There are 35 Power supporters for every 40 Crows supporters. g William weighs 60 kg and Thomas weighs 75 kg.

TRANSFORMATION AND LOCATION (CHAPTER 12)

191

Examples: ²

A bag contains 24 lollies. To share them in the ratio 1 : 5, we must divide them into 1 + 5 = 6 parts. Now 24 ¥ 6 = 4, so each part is 4 lollies. So 1 : 5 = 4 £ 1 : 4 £ 5 = 4 : 20. The shares are 4 lollies and 20 lollies.

²

The ratio of sales of lemonade to cola in a supermarket is 2 : 5. If 60 litres of lemonade are sold, then we can find how much cola is sold. With a ratio of 2 : 5, for every 2 litres of lemonade sold, 5 litres of cola are sold. £30 £30 So for

60 litres of lemonade,

150 litres of cola are sold.

4 Share 24 lollies in these ratios: a 1:1 b 1:2 f 2:6 g 3:9 5

c h

d i

3:1 1:2:3

4:2 1:2:5

e j

3:5 1:3:4

a The ratio of sales of chocolate biscuits to plain biscuits is 4 : 7. If 120 packets of chocolate biscuits are sold, how many packets of plain biscuits are sold? b A sports store sells 9 pairs of Ekin shoes for every 5 pairs of Amup shoes. If 300 pairs of Amup were sold, how many pairs of Ekin were sold? c The ratio of girls to boys passing a maths test was 5 : 4. Given that 100 boys passed, how many girls passed? d In a school, the ratio of students who bring lunch from home to those why buy it at the canteen is 5 : 2. If 70 buy their lunch, how many bring it from home? e The ratio of the sides of a rectangular field is 7 : 3. Given that the shorter side is 75 metres, find:

i

the length of the longer side

ii

the perimeter of the field.

6 The recommended ratio of concentrate to water for a sports drink is 1 : 4. Calculate how many mL of concentrate are needed to: a make 600 mL of sports drink b be added to 16 L of water. 7 Andrew, Brendan and Jill share ownership of a business in the ratio 8 : 3 : 1. They also share the costs and profit of the business in the same ratio. a What fraction of the business does Brendan own, expressed in simplest terms? i Last year, Andrew’s share of the profit was $14 000. b How much profit did the business as a whole make? ii What was Brendan’s share of the profit? c Andrew and Brendan contributed a total of $352 to the quarterly power budget. How much did Jill contribute?

Ratios can involve more than 2 quantities.

192

TRANSFORMATION AND LOCATION (CHAPTER 12)

Bearings and directions

Unit 90 Compass points North, South, East and West are the cardinal points. They are 90o apart.

N W

E

NW

N

45°

W SW

S

S

NE, SE, SW and NW are intermediate points.

NE E

22\Qw_\°

SE SSE

Compass bearings Using the cardinal points we measure acute angles (i.e., between 0o and 90o ). The angles are measured from either North or South to either West or East. N

B

Remember that you must start facing North or South. Then turn towards the East or West.

The observer at A, facing North, needs to turn 35o towards the West to face B.

35°

) the bearing of B from A is N35o W. N

A

A

W

65°

25°

E B

S

The observer at A, facing South, needs to turn (90o ¡ 25o ) towards the east to face B, i.e., needs to turn 65o. ) the bearing of B from A is S65o E

Exercise 90 1 Give the compass bearing of B from A in each of the following: N N a b B

70° A

W

W

E

B

S N

d W

E

E

W

W

4

f

A

E

W

3 2 1 0

S T A B C D E F G H

a b

A

B 15° E

B

S

R

N

E

S

65°

5

A

N

2 Use a protractor to draw fully labelled diagrams to show the compass bearing if: a B from A is N40o E b A from B is S50o W c o o d P from Q is N65 W e X from Y is S80 E f 3

50°

S

B S

B

30° N

e

A

30°

A

N

c

S

C from D is S45o E M from O is N84o E

Give the grid references of: i point R ii point S iii point T Find the bearing of: i R from S ii T from S

TRANSFORMATION AND LOCATION (CHAPTER 12)

193

True bearings True bearings use clockwise rotations from the true north direction. Angles between 0o and 360o are used. N

Examples:

B

35°

For

o 035 | {z }.

the true bearing of B from A is 35o T or

3 digit representation

A N 180°+35° =215°

N B

To find the true bearing of A from B, we need to draw a north line through B.

35°

So the bearing of A from B is 215o T (or 215o ).

A N

A

B

180 ¡ 50 = 130 This diagram shows that the bearing of B from A is 130o T (or 130o ).

130°

50°

B

N

40° A

320°

360 ¡ 40 = 320 This diagram shows that the bearing of B from A is 320o T (or 320o ).

4 Give true bearings for B from A for each of the diagrams in 1. 5 Use a protractor to draw fully labelled diagrams showing true bearings of: a 070o T b 160o T c 213o d 312o (Note: The T is not necessary if we use the 3 digit representation.) 6 Describe the bearings:

a

270o

b

000o

7 Find the true bearing of the eight main compass bearings: a North b North-East d South-East e South g West h North-West 8

10

P

9 8 7

N

q

Start

6

Q

5 4 3 2

Finish

R

1 0

A B C

D E

F

G H

I

J

K

L M

c f

e

096o

A diagram could be useful here.

East South-West

A person who is orienteering must travel from the Start to P, then to Q, then to R and finally to the Finish point. a The Start is given by the grid reference C7. Find the grid references for: i P ii Q iii R iv Finish. b Use a protractor to find the true bearing of: i P from the Start (Hint: Put the centre of the protractor on Start, then measure the angle µ.) PRINTABLE ii R from Q PAGE iii Q from R iv the Start from the Finish.

194

TRANSFORMATION AND LOCATION (CHAPTER 12)

Distance and bearings

Unit 91 We can use this scale diagram to find the distance and true bearing of A from B.

N B

To find the distance:

q

Using a ruler, AB = 4:5 cm So, the actual AB = 4:5 £ 10 km = 45 km To find the bearing of A from B, draw a north line through B.

PHANTOM ISLAND scale: 1 cm represents 10 km

o

Using a protractor, µ = 203 ) A is 203o T from B.

A

Exercise 91 1 P, Q and N are landmarks on the given map and S is a ship at sea. The position of S is given by (7, 10). a What is at the point given by (3, 1)? b What is the compass bearing of N from P? c What is the true compass bearing of: i the ship from P ii the ship from Q PRINTABLE iii Q from P? MAPS

12 11 10 9 8 7 6 5 4 3 2 1

S

P Q N 1 2 3 4 5 6 7 8 9 10 11 12

0

2 X and Y are radar stations on the coastline. Z represents a yacht. a What is the true bearing of the yacht: i from radar station X ii from radar station Y? b

If the map scale is 1 cm represents 20 km, find the distance from: i X to Z ii Y to Z

N

Z

N

X

N

Y

3

a Use the map of Australia and the scale on the next page to estimate the distances from: i Adelaide to Brisbane ii Sydney to Perth iii Melbourne to Darwin b From the map, find the bearing of: i Cairns from Adelaide ii

Kalgoorlie from Hobart

iii

Canberra from Alice Springs

TRANSFORMATION AND LOCATION (CHAPTER 12)

195

Darwin

N

Cairns Mt. Isa Alice Springs Rockhampton

Brisbane Kalgoorlie Perth Adelaide

Sydney

Canberra

Melbourne scale: 1 cm represents 350 km

Hobart

Activity

Scale diagrams – distances in Australia You will need:

some string, a ruler marked in mm, and an atlas or map showing major highways of Australia.

MAP

What to do: 1 Hold one end of your string on Perth and carefully lay your string along the main highway to Adelaide.

Mark the string at this point. Now pull the string straight and measure with your ruler. Use the scale to find the distance by car from Perth to Adelaide. 2 Use this method to find the shortest distance by car (along the major highways marked on the map) from: a

Adelaide to Melbourne

b

Melbourne to Sydney

Activity

c

Perth to Darwin

Orienteering in the schoolyard You will need: What to do:

a direction compass, a trundle wheel (or tape measure).

1 Find 4 or 5 major objects in the school grounds that are easily accessible and where you have a clear line of sight from one to the next. 2 Draw a rough sketch of the situation showing the objects you have selected. For example, flag pole, goal post, corner of building, etc. 3 From a starting point measure distances and directions on a selected pathway from one point to the next. Record all distances and bearings on your rough sketch. 4 Accurately draw your pathway on clean paper, showing all distances and bearings. 5 Give the detailed map to another student and see if he/she can follow your directions accurately.

196

TRANSFORMATION AND LOCATION (CHAPTER 12)

Review of chapter 12

Unit 92 Review set 12A

1 Plot the points with coordinates A(1, 1), B(2, 4), C(6, 6) and D(5, 3). What type of quadrilateral is ABCD? 2

a

Draw the image if a mirror was placed on the mirror line.

b

Draw the axes of symmetry for:

b

Rotate the given figure 180o anticlockwise about O.

m

3

a

Find the order of rotational symmetry for:

O

4

a

Translate the figure three units left and one unit up.

b

Enlarge the figure with scale factor 2.

5 At a barbeque, the ratio of chops to sausages eaten was 4 : 7. If 56 sausages were eaten, how many pieces of meat in total were eaten? 6 Express 40 cm is to 1:2 m as a ratio in simplest form. 7 In an office, the ratio of staff who supported football to those who supported soccer was 5 : 2. If 21 staff gave their opinions, how many supported soccer? 8 X and Y are radar stations on the coastline. Z represents a yacht.

N N

a What is the direction of the radar station X:

Z

i from the yacht Z ii from radar station Y?

X

b If the map scale is 1 cm represents 20 km, find the distance from X to Y.

N

Y

9 For a scale of 1 : 500, find the length drawn on a map if the actual distance is 26:5 m.

TRANSFORMATION AND LOCATION (CHAPTER 12)

Review set 12B

a

P

b

Q

c

R

d

S

e

S

6 5 4 3 Q 2 1

1 Write the coordinates or ordered pairs for the following points:

T

197

P T

1 2 3 4

R

5 6 7

2 Plot 3 vertices A(4, 2), B(7, 2) and C(7, 5) of a rectangle on grid paper and find the coordinates of D, the fourth vertex. 3 Draw the mirror images of:

a

b

m

m

4

a Draw the axes of symmetry of:

b What is the order of rotational symmetry for this shape? 5

a

Rotate the figure shown through 90o anticlockwise about O.

b

Enlarge the figure with scale factor 13 .

O

6 Express 450 mL is to 1:8 L as a ratio in simplest form. 7 The ratio of pies to pasties purchased from the local bakery was 9 : 5. If 63 pies were bought, how many pasties were bought? 8 The map alongside has a scale of 1 cm represents 10 km. Find the distance and direction of:

Lighthouse

Ship

a the Fishing Centre from the Surf Shop

Hilltop

b the ship from the Hilltop

Jetty Surf Shop

c the Lighthouse from the Hilltop.

N

Fishing Centre

9 If a scale is written as 1 : 5000, explain what 1 cm would represent.

Waterfall

198

CHANCE

Describing chance

Unit 93

When we talk about chance, we are considering the likelihood or probability of events occurring (or happening). Some words we use to describe chance are: possible, impossible, likely, good chance, highly probable, probable,

unlikely, improbable,

maybe, doubtful,

certain, often,

uncertain, little chance,

no chance, rarely.

Exercise 93 1 Use some of the words or phrases listed above to describe the chance of the following happening: a A person will live to the age of 140 years. b There will be a public holiday on the 25th day of December. c The sun will rise tomorrow. d You will win a major lottery in your lifetime. e Your birthday in three years’ time will fall on a weekend day. f You will get homework in at least one subject tonight. g You will be struck by lightning next January. 2 Below is a number line. Copy it and add the following words.

impossible

a

doubtful

b

50-50 chance

very rarely

c

highly likely

almost certain

certain

a little more than even chance

d

3 A bag contains 50 marbles, of which 49 are black and one is white. A marble is randomly chosen from the bag. a How likely is the marble to be black? b Is it certain that the marble is going to be black? c True or false? “There is a 1 in 50 chance it will be white.” 4 A tin contains 10 red and 11 green discs and one disc is randomly selected from the tin. a Is it more likely that the disc is red than it is green? Explain. b What colour is more likely to be selected? c True or false? “There is a 10 in 21 chance that the disc is green.” 5 Describe the following events as either certain, possible or impossible: a When tossing a coin, a tail faces upwards.

a die

b When tossing a coin, it falls on its edge. c When tossing a coin ten times, it falls heads every time. a pair of dice

d When rolling a die, a 3 results. e When rolling a die, a 7 results. ii 3 2

6

1

2

3

i

2

g When twirling these square spinners a 4 results:

4

CHAPTER 13

f When rolling a pair of dice a sum of 1 results.

CHANCE

(CHAPTER 13)

199

Assigning numbers to chance If an event cannot occur, i.e., it has no chance of occurring, we give it the number 0. If an event is certain to occur we give it the number 1. Every other event has a probability between 0 and 1. Events which may or may not happen with equal chance are given the probability number 0:5 or 12 . This is because they happen, on average, once every two times. If you roll a die, the probability of getting a number: ² less than 7 is 1, i.e., it is certain ² less than 4 is 0:5 or

1 2

² greater than 6 is 0, i.e., it is impossible

f3 out of 6 numbers are less than 4g

6 A container has 5 green and 5 blue balls and one ball is randomly selected from it. a What is the probability of selecting a green ball? b If all blue balls are now removed, what is the probability of selecting: i a green ball ii a blue ball? 7 Here is a probability number line:

0

0.5

1

Draw your own probability line and mark on it the approximate probabilities of: a c e g 8

a newborn baby being a girl the Sun rising tomorrow winning Lotto rain falling tomorrow

b d f h

it snowing in January in Darwin a holiday on January 1st being born on a Monday being born on any week day.

a Which of the following events are equally likely to occur? i Getting a head or a tail with a single toss of a coin. ii Winning a 100 m sprint contested by 8 athletes. iii Any team in a 10-team netball competition winning the competition. iv Any of the results 1, 2, 3, 4, 5 or a 6 occuring when a die is rolled. v Two people selecting a card from two separate full decks and each getting the Queen of Hearts. vi Either of two people in a raffle with 100 tickets winning it. b Discuss the difference between events in which outcomes are equally likely and those events in which outcomes are not equally likely. Did you begin your answer to this question with “It depends ........”?

The consequences (risks) of chance Chance is the likelihood or probability of events happening. Chance can also mean risk. With every risk there is a consequence. In groups, discuss the following: What are the risks and possible consequences of taking a chance at: a being out in the sun without proper protection b speeding in a car c playing the pokies d not doing your homework e “putting all your eggs in one basket”?

200

CHANCE (CHAPTER 13)

Defining probability

Unit 94 In general, if outcomes are equally likely to occur, Pr[an event] =

number of outcomes in that event total number of possible outcomes

Pr[...] means the probability of ... occuring.

A die has 4 faces painted blue and 2 faces painted grey. When the die is rolled we can find the chance of getting a blue or a grey face uppermost, in this way. 4 faces are blue, 2 faces are grey, 6 faces in all. So, Pr[a blue] =

total blue faces total outcomes possible

4 6

and Pr[a grey] =

2 6

Exercise 94 1 Blue and white discs are placed in a bag and one disc is randomly selected from it. For the various bags of discs given, answer these questions: i How many of each disc are there in the bag? ii What is the probability of selecting a blue disc? iii What is the probability of selecting a white disc? a

b

c

2 Find the probability that the spinning needle will finish on blue in: a b c

d

3 A 10-cent coin is tossed. Find the probability that it will finish with the uppermost face: a a head (queen) b a tail (other side).

D

5

C

4 The given spinner has sector angles of 120o , 90o , 50o , 60o and 40o . a After a spin, are the outcomes equally likely? b What outcome do you expect to occur: i most often ii least often? c What is the probability of getting dark blue? d If you spin the spinner 40 times, how often would you expect it to finish on dark blue?

A

B

DEMO

40°

120°

60° 50° 90°

A square spinner has A, B, C or D on its equal sides. After one spin, what is the probability of getting: a aB b a B or a C c an E d an A, B, C or D?

When a die is rolled, the probability of a 3 or a 4 is Wy_\, that is, Qe_\.

CHANCE

(CHAPTER 13)

201

A hat contains three blue, three white and one grey ticket. One ticket is selected at random from the hat. Look at the likelihood of these outcomes: There are 3 + 3 + 1 = 7 tickets, all equally likely to be selected. Pr[a blue] =

3 7

fthree blues out of 7 possibleg

Pr[a grey] =

1 7

fone grey out of 7 possibleg

Pr[grey or white] =

4 7

f4 are either grey or whiteg

For a truly random selection, each ticket has the same chance of being selected.

8

67

1

23

6 Consider the illustrated spinner (a regular octagon). If the spinner is spun once, find the probability of getting: a a6 b a 3 or a 4 c a 1, 2 or 3 d a result less than 6 e a result more than 8?

45

7 A hat contains 4 red, 3 white and 2 grey discs and one disc is randomly selected from it. Find the likelihood that it is: a red b white c grey d green e not red f not white g not grey h not red or grey i red, white or grey 8 This illustration shows a full pack of playing cards. For this exercise, the pack is well shuffled and placed face down. For our pack of cards hearts and diamonds are blue. Spades and clubs are black. Jacks, Queens and Kings are called picture cards. Jason picks one card at random from the shuffled pack. Find the chance of getting:

a heart b a blue 4 e a green card h

the 7 of c a black ace f an ace i

A 2 3 4 5 6 7 8 9 10 J Q K A 2 3 4 5 6 7 8 9 10 J Q K A 2 3 4 5 6 7 8 9 10 J Q K

a club | a 5 or a 6 a picture card

K

a d g

A 2 3 4 5 6 7 8 9 10 J Q K

9 A coin is tossed and a die is rolled at the same time. One possible result is ‘a tail with the coin and a 5 with the die’, and this result could be represented by T5. a Using this shorthand notation, list the possible results from this experiment. b How many different results are possible from performing this experiment once?

b Find the probability of getting: i a C and a 4 iii a B

ii iv

B

a Using this shorthand notation, list the possible outcomes from the experiment.

C

10 The two illustrated spinners are twirled together. One possible result is A3, ‘an A with the first spinner and a 3 with the second’. This result is shown.

a tail and an even number T5 or T6

4

ii iv

3

c Find the probability of getting: i H6 iii a head and a prime number

A

B

1

2

a B and a 2 or 3 an A or B and a 1

202

CHANCE (CHAPTER 13)

Unit 95

Tree diagrams and probability generators

Tree diagrams We can use a tree diagram to show the possible outcomes of an experiment. This tree diagram shows the 4 possible outcomes when one coin is tossed twice.

1 st toss 2 nd toss H

All outcomes are equally likely to occur.

branch

From the tree diagram you can see that:

T

X

²

H

HH

T

HT

£ ²

H

TH

£ ²

T

TT

The probability of getting two heads is 1 out of 4, since one outcome has two heads (X) and there are four possible outcomes. The probability of getting a head and a tail is two out of four as two outcomes have a head and a tail in them. (£) The probability of getting at least one head is three out of four. (²)

Exercise 95 1 Draw a tree diagram to show each of these situations, and list all possible outcomes: a A coin is tossed and a die is rolled at the same time. (A start to this tree diagram is shown alongside.)

H

b Three coins are tossed at the same time.

T

c A bag contains some red, blue and green counters. One counter is selected, replaced and another counter is selected. d The given disc is spun two times with the letter recorded each time.

coin

die

A

C

B

2 Draw a tree diagram for these situations and use it to work out the probabilities requested. (You may use your diagram from question 1d to help answer 2a.)

a This spinner is twirled twice. Find the probability that: i you get an A followed by a B ii you get a C followed by a C iii you have at least one C in your outcomes iv you have a B and a C

A C

B

b A family has three children in it. Assuming a boy and a girl are equally likely to happen, find the probability that the family has: ii two girls iii at least 1 girl i three boys iv no boys v no more than 2 girls c A bag contains the same number of red, blue, pink and green counters. One counter is selected. It is then replaced and another counter is selected. Find the probability that: i two green counters are selected ii one red counter is selected iii at least 1 blue counter is selected iv no pink counters are selected v a red and a pink counter are selected

CHANCE

203

(CHAPTER 13)

Probability Generators Suppose you wish to make a device which generates probabilities of so one event has a

2 3

2 3

and 13 ,

chance of occurring and the alternative event has

For example, Pr[blue] =

2 3

1 3

chance of occurring.

and Pr[grey] = 13 .

We could use a spinner or a die.

Grey Blue

o

120 is

1 3

240o

2 3

is

o

of 360 of

120°

120°

4 faces are blue and 2 faces are grey

120°

360o

Blue Grey

3 Design two devices which generate Pr[A] = 4 Design a device which generates Pr[red] =

2 5

1 2

and Pr[B] = 12 . and Pr[blue] = 35 . (Hint:

5 Design two devices which generate Pr[A] = 16 , Pr[B] = 6

Blue

Blue

a b

2 6

1 5

of 360o = 72o .)

and Pr[C] = 36 .

What can this net be used to make? What would you do to the squares to make a device which generates

Pr[A] = 16 , Pr[B] =

and

Pr[C] = 12 ?

1 3

7 You want a way of generating probabilities in the ratio 1 : 2 : 4 and do not wish to make a spinner. What simple way could you do this?

Activity

Tossing one coin When a single coin is tossed our theoretical probabilities for the chances of getting a head or a tail are based on the symmetry of the coin. We have observed that Pr[H] = Pr[T] = 12 .

What to do:

But does this mean that when we toss a coin a large number of times we will get 50% of heads and 50% of tails?

Toss a coin 20 times and record the number of Expect to get Actual result heads (H) and tails (T) resulting. Experiment H T H T 2 Repeat this a second time and a third time. First 20 tosses 3 Did you always get 10 heads and 10 tails? Second 20 tosses 4 How many heads did you expect to get for the Third 20 tosses 60 tosses? How many heads did you actually get for the 60 tosses? TOSSING 5 How accurate were your expectations against the actual result? ONE COIN 6 Rather than toss coins you could use the computer simulation. Click on the icon to load a one coin toss simulation. Use the software to simulate tossing a coin ² 1000 times and repeat 4 times. ² 10 000 times and repeat 4 times. Record all the results in a table. 7 Write a sentence or two about your discoveries. 1

204

CHANCE (CHAPTER 13)

Expectation

Unit 96

When you toss a coin a number of times, say 20, you do not always get equal numbers of heads and tails. However, after tossing the coin a large number of times we expect about 50% of each. The expectation of an event occurring is found by multiplying the probability of the event occurring by the number of times the experiment is conducted. For example:

1 2

£ 100 = 50 heads, as Pr[H] = 12 .

²

If we toss a coin 100 times, we expect

²

If we roll a die 120 times, we expect

²

If in July the probability of rain on any particular day is 0:68, we expect 0:68 £ 31 + 21 rainy days. (31 days in July)

1 6

£ 120 = 20 of them to be 5s.

Exercise 96 1 A coin is tossed 50 times. a How many heads do you expect to get? b If 31 are heads and 19 are tails, which of the following is the most likely explanation?

A the coin is biased B the result is due to chance D the person doing the coin tossing tricked you

C your counting is not accurate

2 A die has 3 green, 2 red and 1 blue faces. If the die is rolled 30 times, what is your expectation for a: a green result b red result c blue result? 3 The chance of hail on any one day is 1:5%. On how many days in a year would you expect hail to fall? 4 The chance of rain falling on a September day in Melbourne is 0:3. On how many September days do you expect rain in Melbourne? 5 If three quarters of broadbeans germinate and Rohan planted 240 beans: a What is Pr[a broadbean germinating]? 6 This square spinner is spun 100 times. a Find: i Pr[A] ii Pr[B or C]

D

A

b How many of each result do you expect?

4 3

2 3

2 1

7

2 1

C

b How many broadbeans should Rohan expect to germinate?

B

The circular spinner is spun 400 times. a Determine: i Pr[1] ii Pr[2] iii Pr[3] iv Pr[4] b How many of each result do you expect?

Click on the icon for the activity ‘Probability of rainfall’.

ACTIVITY

Something to think about ² ² ²

It costs hundreds of millions of dollars to build and run casinos. Where does the money come from? For every winning $1 ticket in a million dollar lottery, there will be more than one million losing tickets. If poker machines allowed every player to win, where would the prize money come from?

CHANCE

Discussion

(CHAPTER 13)

205

To gamble or not to gamble

Discuss some of these ideas to do with gambling: POKER MACHINE

1 If you play enough games you must have a ‘big’ win. 2 If you have a win on a poker machine then that machine will not have another win for a long time, so you should swap to another machine. 3 You only gamble if you have better than average chance of winning. 4 You should have control of the result that you are gambling on. 5 If you have a win you should continue to play to win more. 6 It is all a matter of luck. 7 It is better to gamble on poker machines than horses. 8 In the long term you will lose more money than you win. 9 You are taking a risk with your money.

10 The organisation controlling the gambling must make a profit, so winnings must be less than the amount players put in. 11 Money from gambling goes to the government to support charities, so you should gamble. 12 If you are betting on, for example, coin tossing, always wait for three heads in a row before betting on tails because you have a greater chance of winning ‘by the law of averages’. 13 Buying raffle tickets is not really gambling. 14 If you have a big loss you will make up the money with a big win another day. From the discussion you should understand that gambling is taking a risk with your money and you cannot control the outcome.

Investigation

The game of heads and tails

A coin is tossed. You can play the game for a $1 entry fee. If you bet on a ‘head’ and it comes up, you win and get your $1 back plus an extra 90 cents. If a ‘tail’ results you lose your $1. Jack’s foolproof gambling system Wait until 3 tails come up in a row and then place your bet for a ‘head’ on the next throw. This is because 4 tails in a row is very unlikely.

To examine this system we investigate results of many coin tosses. Consider the following results of tossing a coin: H T T H H H T H T T T H T T H T T T H H T H T T H T H ..... We look for 3 consecutive tails and the result following them. So far the result is

What to do:

TOSSING ONE COIN

Fourth Toss H T 2 0

1 Click on the icon and do a coin tossing simulation with at least 1000 results printed out. Use the results to test whether Jack’s Foolproof Gambling System is valid or not. 2 Also test Jack’s modified system which tells you to bet on a head after 4 tails have come up in a row. 3 Do you think that this is a successful gambling system? If so, why? If not, why not?

206

CHANCE (CHAPTER 13)

Review of chapter 13

Unit 97 Review set 13A

1 Use the words no chance, little chance, some chance, good chance or certain to describe the likelihood of the following events occurring: a A manned space ship from Earth will land on another planet before 2010. b A $5 coin will replace the $5 note. c Your mathematics teacher will not set any homework this week. d A blue moon will occur twice this year. 2

a If the chance of rain falling on an April day in Darwin is 40%, on how many days would you expect rain during April? b In a scientific trial of 500 people with coughs, 75% of the people who took a cough mixture found that their cough cleared. i What was Pr[cough clears]? ii How many people found their cough cleared?

3 The given spinner is spun 40 times. a Determine

5 6

2 8

ii Pr[odd number] iv Pr[even black number]

1 4

i Pr[blue] iii Pr[even blue number] v Pr[odd black number]

7 3

b How many of each result do you expect? 4 In each of the following examples there are two possible events. i ratio of one result to the other For each example write the ii possible result of the 11th outcome. a c

b

(Odds or Evens) - OEEOOOEEOE

Q

C

R

B

b In the spinners above, what is the probability of C occurring in example: i

P

ii

Q

iii

R

iv

6 The cards displayed are shuffled and placed face down. a What is the probability of selecting i a blue card ii a diamond iii an ace iv a blue jack v a king of hearts? b Two cards are to be selected and the first card chosen is the queen of spades. The queen of spades is not replaced and a second card is chosen. What is the probability of selecting with the second choice: ii a queen i a spade iii the king of spades iv the queen of spades?

S?

C A

E

A

E

A

C

P

F

B

D

C B

E A

B

C

C B

D

A

a List these spinners in the order from where A is most likely to occur to where A is least likely to occur.

B A

5

(Heads or Tails) - HTHTHHHHHH (Red or Black) - RRRBRRBBBR

D A

S

CHANCE

(CHAPTER 13)

207

Review set 13B 1 Assign the probabilities 0, 0:5 or 1 to best describe the chance that: a school will be cancelled for the rest of the term b the next person entering your classroom will be a male c you will have a drink today d tomorrow’s date is an odd number. 2 There are 8 blue, 5 red and 3 green discs in a bag. One disc is taken at random from the bag. a How many different events are possible? b What is the probability that a blue disc is taken? c What is the probability that a red disc is taken? d What is the probability that a non-blue disc is taken? 3 Complete the following for the spinners shown: a Spinner A, Pr[dark blue] = ....

Spinner B, Pr[light blue] = ....

Spinner C, Pr[black] = .... . b From 60 spins, with which spinner would you expect to get: i 30 whites ii 20 whites

iii

c From 120 spins, with which spinner are you most likely to get: i 20 whites ii iii 30 dark blues, 30 whites, 30 light blues and 30 greys?

15 whites?

50 dark blues

d Use the words reasonably, likely, highly likely, unlikely or impossible to describe the probability of spinning: i ii iii

dark blue with spinners C, D or E grey with spinners B, C or D black with spinners C, D or E

A

C

E

B

4

D

a A gambler bets $1 and loses then ‘doubles up’ to bet $2 and loses then ‘doubles up’ to bet $4 and loses again etc. Which of the following amounts represent his total loss if he has 6 losses in a row? i

$6

ii

$32

iii

$33

iv

$63

v

$64

b What would be the gambler’s total loss if he doubled up: i 5

one more time and lost

ii

two more times and lost?

a Use a tree diagram to illustrate all the possible outcomes when two six-sided dice are rolled. For each outcome, write the sum of the two numbers rolled. (For example, for a 5 followed by a 6 , the sum is 11). b Use the tree diagram to rank in order from highest probability to the lowest probability the following totals which could occur: 3, 5, 7, 8, 12 .

208

REVIEW OF CHAPTERS 11, 12 AND 13

TEST YOURSELF: Review of chapters 11, 12 and 13 1 Give the next three members of the pattern: a 6, 10, 14, 18, ..... b 3, 6, 12, 24, ..... 2 Write down the coordinates of the point: a A b B c C d D

5 4 3

c

42, 39, 36, 33, .....

A C

2

B

1

D 1

3

2

4

5

6

7

8

9

3 A spinner has 12 equal sectors. 4 are red, 3 are green, 2 are orange, 2 are blue and one is purple. If the spinner is spun once, what is the probability that it will stop on a sector which is: a red b green c not blue d green or orange? 4 The following pattern is built out of matchsticks:

,

,

, .....

a Draw the next 2 members of the pattern. b Copy and complete:

Figure number (n) Matchsticks needed (M)

1

2

3

4

5

c Find the rule connecting M and n. d How many matchsticks are needed to build figure: i 8 ii 30? 5 A scale on a map is 1 : 2000. a If on the map a distance is 13:2 cm, what is the actual distance? b If an actual distance is 400 m, what will be the map distance? 6 P Q

S R

7 Solve the equations: a x ¡ 15 = 20 d 24 ¥ t = 3

The sector angles for this spinner are 140o for P, 40o for Q, 120o for R and 60o for S. If the spinner is spun, what is the probability it will finish on a P b R c T d P or Q e P, Q, R or S? b e

17 ¡ d = 9 3 £ n + 7 = 34

8 On a number plane, plot the points: a A(3, 5) b B(4, 0)

c f

p¥7=6 4 £ y ¡ 7 = 25

c

C(0, 2)

9 The chance of rain on a day in June is 0:3 . a How many days does June have? b On how many days would you expect rain? 10 In an office, the ratio of people who supported cricket to those who supported tennis was 3 : 4. If 28 people gave their opinions, how many supported cricket? 11

a Draw the axes of symmetry of a square. b What is the order of rotational symmetry of a square?

REVIEW OF CHAPTERS 11, 12 AND 13

12 Fred sells refrigerators. He is paid $400 per week plus $80 for every refrigerator he sells. a Copy and complete the table of values showing n 0 1 2 3 4 the amount Fred earns (E dollars) for selling n E refrigerators.

209

5

b Write a formula for the amount that Fred earns. c How much would Fred earn if he sold 8 refrigerators in a week? 13

a

Draw the mirror image of:

b

Draw the axes of symmetry (if they exist) for:

d

Translate the given figure one unit to the left and 4 units down.

m

c

Rotate the given figure through 90o clockwise.

O

14 There are 7 blue, 4 green and 3 red tickets in a bag. One ticket is taken at random from the bag. a What is the probability that a blue ticket is taken? b What is the probability that a red ticket is taken? c What is the probability that a non-blue ticket is taken? 15 A disc jockey charges $200 to set up the equipment and $40 for each 12 hour to provide entertainment for local clubs. a Complete the table of values: Number of 12 hours (n) 0 1 2 3 4

Charge ($C)

200

b What is the disc jockey’s fee formula? c How much would the disc jockey charge for: i 5 12 hours ii 14 hours? 16

a b

How many sectors does this figure have? If the spinner is spun, what is the probability that it will finish on: i white ii black iii black or white iv non-black v black, white or blue?

210

ANSWERS

14 b 54 c 21 d 75 e 901 f 619

Exercise 1

9 a

1 a 8 b 80 c 8 d 800 e 80 f 8000 g 800 h 8000 i 8 j 80 000 k 8000 l 80 000

10 a 6 b 25 c 52 d 48 e 208 f 817 11

2 a 7 units b 7 thousands c 7 hundreds d 7 hundred thousands 3 a b c d

3 thousands, 5 ten thousands, 8 tens 3 thousands, 5 hundreds, 8 tens 3 units, 5 ten thousands, 8 tens 3 hundreds, 5 thousands, 8 hundred thousands

4 a 864 b 974 210 c 997 722 d 345, 354, 435, 453, 534, 543 (6 numbers) 5 a c d e f

8, 16, 19, 54, 57, 75 b 6, 60, 600, 606, 660 1008, 1080, 1800, 1808, 1880 40 561, 45 061, 46 051, 46 501, 46 510 207 653, 227 635, 236 705, 265 703 545 922, 554 922, 594 522, 595 242

6 a b c d

631, 613, 361, 316, 163, 136 9877, 9787, 8977, 8779, 7987, 7897, 7789 498 321, 498 231, 492 813, 428 931, 428 391 675 034, 673 540, 607 543, 576 304, 563 074

7 a 86 b 674 c 9638 d 50 240 e 27 003 f 73 298 g 500 375 h 809 302 8 a c d e f g h

9 £ 100 + 7 £ 10 + 5 £ 1 b 6 £ 100 + 8 £ 10 3 £ 1000 + 8 £ 100 + 7 £ 10 + 4 £ 1 9 £ 1000 + 8 £ 10 + 3 £ 1 5 £ 10 000 + 6 £ 1000 + 7 £ 100 + 4 £ 10 + 2 £ 1 7 £ 10 000 + 5 £ 1000 + 7 £ 1 6 £ 100 000 + 8 £ 100 + 2 £ 10 + 9 £ 1 3 £ 100 000 + 5 £ 10 000 + 4 £ 1000+ 7 £ 100 +1 £ 10 + 8 £ 1

a 22:5 b 15:75 c 16:2 d 12:125 e 71:5 f 68:75 g 87:8 h 82:625 i 477:5 j 258:25 k 240:2 l 587:375 m 3674:5 n 2115:75 o 1599:8

12 a 348 b 16 c 13

386 + 125 511

or

32 386 + 135 521

396 + 125 521

or

Exercise 3 1 22 m 2 $432 3 6 kg 4 $41 5 3923 km 6 1178 cm 7 90 kg 8 $80 9 82 min 10 2450 mm 11

672 km 12 81 min 13 $26 14

$201

15 22 minutes 16 11 oranges 17 54 goats 18 $11 19 $30 20 $1:44 21 a goal 22 $1860 23 $743 24 $550 25 48 km 26 600 g

Exercise 4 1 a 80 b 80 c 300 d 2380 e 3990 f 1650 g 9800 h 61 020 i 49 570 j 30 940 k 999 570 l 128 670 2 a 100 b 500 c 1000 d 3000 e 25 400 f 14 800 g 130 000 h 44 000 3 a 1000 b 6000 c 10 000 d 44 000 e 65 000 f 123 000 g 435 000 h 571 000

9 a 27 b 80 c 608 d 1016 e 8200 f 19 538 g 75 403 h 602 818

4 a $45 000 b 330 kg c $500 d 4800 km e 362 kL or 362 000 L f $490 000 g 37 000 h 600 000 i 36 000 000 j $2 000 000 000

10 a 7 b 13 c 21 d 299 e 4007 f 9997 g 400 004 h 209 026

5 a $200 b $60 c $60 d $30 e $50 f $28 000

11

a c e g

12 a c e g h

375 + 836 + 1200 b 79 £ 8 + 640 978 ¡ 463 = 515 d 7980 ¥ 2 + 400 455 + 544 = 999 f 50 £ 400 = 20 000 2000 ¡ 1010 = 990 h 3000 ¥ 300 = 10 5268 ¡ 3179 < 4169 b 29 £ 30 < 900 672 + 762 < 1444 d 720 ¥ 80 > 8 20 £ 80 > 160 f 700 £ 80 > 54 000 5649 + 7205 > 12 844 6060 ¡ 606 > 5444

6 a 180 b 420 c 540 d 1500 e 2800 f 15 000 g 2400 h 4200 i 9000 j 90 000 k 360 000 l 720 000 7 a 100 b 1000 c 10 000 d 300 e 2000 f 200 g 10 h 100 i 200 8 a

553 b 3560 c 33 831

9 a

61 b 135 c 98 10 400 books

11 12 000 words 12 16 000 bricks 13 80 min 14 10 000 vines

Exercise 5

Exercise 2 1 a 807 b 1330 c 3995 d 1644 e 1597 f 13 059

1 a

2 a 79 b 107 c 748 d 696 e 2155 f 6565 g 814 h 4955 i 4619

2 a 3 000 000, 600 000, 40 000, 8000, 500, 90, 7 b 30 000 000, 4 000 000, 800 000, 60 000, 5000, 200, 70, 1

3 a 82 b 44 c 109 d 453 e 665 f 3656

3 a 37 000 000 b 200 000 000, 17 000 000 c d $111 240 463:10 e 21 240 657

4 a 34 b 48 c 6 d 22 e 182 f 476 g 376 h 3767 i 1980 5 a i 9142 ii 5696 b i

15 732 ii 3844

6 a 500 b 5000 c 50 000 d 6900 e 69 000 f 690 000 g 12 300 h 246 000 i 96 000 j 490 000 k 49 000 l 490 000 7 a 2400 b 240 c 24 d 4500 e 450 f 45 g 7200 h 720 i 72 j 600 k 60 l 6 8 a 120 b 148 c 496 d 1272 e 405 f 2744 g 14 580 h 23 112 i 5754 j 45 026 k 10 413 l 26 864

80 b 50 000 000 c 600 d 400 000 e 70 000 f 2

150 000 000

4 a 20 000 b 4000 c 2500 d 500 e 10 000 f 12 500 g 8000 h 25 000 i 6250 j 125 000 5 Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto 6 a Asia b Africa, Asia, North America c Antarctica, Australia 7 10 000 hours 8 20 000 hours 9 10 a 135 km b 27 hours 11

125 hours (= 5 days 5 hours)

2000 hours

211

ANSWERS

12 a 370 coins b

DI8-200 Tokyo, 296 000 coins; Sears Tower, 165 000 coins; Empire State, 141 000 coins; Sky Tower, 121 000 coins; Centrepoint, 113 000 coins; Eiffel Tower, 111 000 coins

Exercise 6 Statement 1

Directed number

Opposite to statement

Directed number

Review set 1B 1 a

F b F c T d F

2 a

30 b 4000 c 500 000 3 74 310

4 a

4 tens b 4 thousands c 4 hundreds d 4 millions

5 a 8 b 66 c 480 d 219 e g 159 000 h $200

6946 f 586

6 79 562, 96 572, 569 207, 652 097, 795 602 7 a

194 257 b 19 355 c 0

8 a

40 000 b 5 c $2688 10 72 years

a

20 m above sea level

+20

20 m below sea level

¡20

b

45 km south of the city

¡45

45 km north of the city

+45

c

a loss of 2 kg in weight

¡2

a gain of 2 kg in weight

+2

d

a clock is 2 minutes fast

+2

a clock is 2 minutes slow

¡2

1 a 60 b 19 c 392 d 22 e g 100 h 359 i 185

+5

2 1534 m 3 $97 196 4 23 124 5 $3457

e f g

she arrives 5 minutes early a profit of $4000 2 floors above ground level o

¡5 +4000 +2

she arrives 5 minutes late a loss of $4000 2 floors below ground level o

¡4000 ¡2

h

10 C below zero

¡10

10 C above zero

+10

i

an increase of $400

+400

a decrease of $400

¡400

j

winning by 34 points

+34

losing by 34 points

¡34

2 lift +1, car ¡3, parking attendant ¡2, rubbish skip ¡5 3 A ¡2, B ¡6, C +5, D +3, E 0 4 a +11 b

¡6 c ¡8 d +29 e ¡14

5 a ¡30 b

+200 c ¡431 d

¡751 e +809

6 a +7 b ¡15 c ¡115 d +362 e ¡19:6 7 a ¡6 b +4 c ¡16 d 0 e

+2 f +40

8 a 11 b ¡1 c 9 d ¡5 e ¡4 f 8 g 1 h ¡3 i ¡1 j 1 k ¡5 l ¡1 m 6 n 2 o ¡2 p 2 q ¡6 r 4 s ¡4 t 4 9 a A 5, B 3, C 0, D ¡2, E ¡5 b i true ii false iii true iv false v true vi true vii false viii true 10 a b c d

A i i i

35o C, B 5o C, C ¡10o C, D 25o C, E 10o C, F ¡5o C 15o C ii 20o C iii 30o C iv 35o C 45o C ii 20o C iii 5o C iv 15o C 30o C ii 15o C iii 20o C iv 5o C v 10o C vi 30o C

Review set 1A 1 a F b

T c T

d T 2 874 310

3 a 8 hundreds b 8 thousands c 8 hundred thousands d 8 millions 4 654 662, 673 502, 674 551, 750 467, 765 442 5 a 40 b 4000 c 500 000 6 a 179 b 27 c 442 d 173 e 4305 f 821 g 3500 h $150 7 4 126 350 8 560 000 9 11

38 tickets 10 1200 golf balls

$184:80 12 $29 13 ¡2 + 6 = 4

14 a a deposit of $30 b i ¡3 ii ¡7

11

200 g 12 No ($1 short) 13 5 ¡ 8 = ¡3

14 a winning by 2 goals b i 26o C ii 4o C Exercise Exercise8 8

6 5600 people 7 10 358 11

100 f 3840

51 840 cans 8 $16 280 9 $3000

27 840 plants 12 2717 passengers

13 a 8 b 46 c 45 d 15 e 49 f 11 g 11 h 16 i 24 j 3 k 21 l 10 m 29 n 36 o 16 14 a 30 b 32 c 2 d 48 e 18 f 96 g h 88 i 30

0

15 a 9 b 2 c 4 16 a c e g i

(6 + 3) £ 2 = 18 b 8+4¡3£2 = 6 d 5 £ (3 ¡ 1) + 7 = 17 50 ¥ 5 + 5 = 15 h 9 £ 7 + 5 ¡ 2 = 66

21 ¡ 7 £ 3 = 0 50 ¥ (5 + 5) = 5 f (4 + 4) £ 4 ¥ 16 = 2 9 £ (7 + 5) + 2 = 110

17 a b c d

96 ¥ (4 + 8) £ 10 ¡ 9 = 71 96 ¥ 4 + 8 £ (10 ¡ 9) = 32 96 ¥ 4 + 8 £ 10 ¡ 9 = 95 96 ¥ (4 + 8) £ (10 ¡ 9) = 8

18 a true b false c true d false e false f true g false h true i false One example is: Malika ate 2 oranges on the weekend and 3 oranges during the school week. She did this for 4 weeks. ii One example is: Suneetha had 8 dollar coins which she divided equally into 2 separate equal piles. Then she added 5 more dollar coins to one of the piles. b i + + + +

19 a i

ii

$1 $1 $1 $1 $1 $1 $1 $1 + $1 $1 $1 $1 $1

Exercise Exercise9 9 1 a d 2 a c e f h i k l 3 a d

1, 3, 5, 15 b 1, 2, 4, 8, 16 c 16 = 2 £ 8 1 £ 16, 2 £ 8, 4 £ 4 1, 2, 4, 8 b 1, 2, 3, 4, 6, 9, 12, 18, 36 1, 2, 4, 5, 8, 10, 20, 40 d 1, 2, 3, 6, 7, 14, 21, 42 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 1, 3, 7, 9, 21, 63 g 1, 2, 3, 5, 6, 10, 15, 30 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 1, 3, 13, 39 j 1, 5, 7, 35 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 1, 3, 9, 27, 81 33 = 3 £ 11 b 55 = 5 £ 11 c 28 = 4 £ 7 50 = 10 £ 5 e 27 = 9 £ 3 f 42 = 2 £ 21

212

ANSWERS

g 35 = 5 £ 7 h 72 = 8 £ 9 i 99 = 11 £ 9 j 49 = 7 £ 7 k 121 = 11 £ 11 l 48 = 6 £ 8 m 64 = 16 £ 4 n 108 = 12 £ 9 o 88 = 2 £ 44

6 a 210 , 84 , 38 , 65 , 104 , 57 b 127 , 273 , 95 , 58 , 10002 or 1003 7

4 a 6 b 9 c 9 d 24 e 22 f 25 g 45 h 13 5 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

28

6 a

2

80

b

14 2

2 7

40 2

3 20

2 35 5

7

5

3

2 7

24 2

12 2

6 2

7 a d g i

3

2£2£2£2 b 2£2£3£3 c 2£2£7 2£2£2£7 e 3£3£7 f 3£5£5 2£2£2£3£7 h 2£2£3£3£7 5 £ 61 j 2 £ 2 £ 2 £ 7 £ 7

8 a 3 b 7 c 6 d 7 e 20 f 25 g 9 h 11 i 14 j 9 k 12 l 4 m 13 n 5 o 8 p 2 9 a 14, 16, 18 b

35, 37, 39, 41, 43 c 2, 8

10 a even b even c even d odd e odd f odd g odd h odd i even

Exercise 10

10 a b c d e f

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 c 12, 24

g h i j

45, 90, 135 c 60, 120

3 a 6 b 12 c 40 d 60 e 24 f 12 g 60 h 60 4 36 m 5 18 seconds 6 20 tokens 7 30 km 8 60th bar 9 a T b F c F

d T e T f F g F h

T

10 a, b, c, f, g, h, i, j, k, l 11

a 2 = 0, 3, 6, 9 b 2 = 0, 3, 6, 9 c 2 = 1, 4, 7 d 2 = 2, 5, 8 e 2 = 1, 4, 7 f 2 = 2, 5, 8 g 2 = 1, 4, 7 h 2 = 2, 5, 8

12 a and b are divisible by 4

Exercise 11 1 a 64 b 46 c 135 d 32 £ 53 e 24 £ 92 f 33 £ 83 g 22 £ 43 £ 52 h 32 £ 6 £ 113 i 33 £ 52 £ 92 2 a 30 b 12 c 54 d 90 e 396 f 2200 g 480 h 2880 i 648 3 a c d e f

5 £ 5 £ 5 £ 5 = 625 b 7 £ 7 £ 7 = 343 3 £ 3 £ 3 £ 3 £ 3 £ 3 £ 3 = 2187 12 £ 12 £ 12 £ 12 £ 12 = 248 832 100 £ 100 £ 100 = 1 000 000 14 £ 14 £ 14 £ 14 £ 14 = 537 824

4 a 2433 b 30 375 c 46 683 d f 1 953 125

The resulting numbers and 1 words are all palindromes 121 - they are written the same 12 321 forwards as backwards. 1 234 321 123 454 321 12 345 654 321 1 234 567 654 321

9 a (9 £ 1000) + (7 £ 100) + (3 £ 10) + (8 £ 1) b (2 £ 10 000) + (9 £ 1000) + (7 £ 100) + (8 £ 10) +(2 £ 1) c (4 £ 10 000) + (4 £ 100) + (4 £ 1) d (6 £ 100 000) + (5 £ 10 000) + (7 £ 1000) + (9 £ 100) +(3 £ 10) + (1 £ 1) e (8 £ 100 000) + (8 £ 100) + (8 £ 10) + (8 £ 1) f (1 £ 1 000 000) + (2 £ 100 000) + (4 £ 10 000) + (7 £ 1000) + (9 £ 10) + (1 £ 1) g (4 £ 10 000 000) + (9 £ 1 000 000) + (7 £ 100 000) + (5 £ 10 000) + (5 £ 1000) + (4 £ 100) h (6 £ 1 000 000) + (7 £ 100 000) + (7 £ 10 000) + (7 £ 1000) + (7 £ 100) + (7 £ 10) + (7 £ 1)

1 a, b

2 a 30, 60, 90, 120, 150 b d 60, 120

= = = = = = =

8 a 862 953 b 6 987 096 c 3 050 709 d 4 892 260 e 20 369 068 f 1 011 190 g 9836 h 890 637 i 50 875 000

48

f

21 3

5

5

63

e

25

10 2

d

75

c

12 112 1112 11112 11 1112 111 1112 1 111 1112

5184 e 2838

5 a 7 b 17 c 513 d 63 e 228 f 3584 g 924 h 110 i 0

(6 £ 102 ) + (5 £ 101 ) + (8 £ 1) (3 £ 103 ) + (8 £ 102 ) + (7 £ 101 ) + (4 £ 1) (9 £ 104 ) + (5 £ 103 ) + (6 £ 102 ) + (3 £ 101 ) + (6 £ 1) (1 £ 105 ) + (1 £ 102 ) (5 £ 105 ) + (5 £ 103 ) + (7 £ 102 ) + (5 £ 101 ) (1 £ 106 ) + (2 £ 105 ) + (7 £ 104 ) + (4 £ 103 ) + (9 £ 102 ) + (4 £ 101 ) + (7 £ 1) (3 £ 107 ) + (6 £ 106 ) + (6 £ 105 ) (4 £ 106 ) + (2 £ 105 ) + (9 £ 104 ) + (3 £ 103 ) +(3 £ 102 ) + (7 £ 101 ) + (5 £ 1) (4 £ 105 ) + (6 £ 102 ) + (8 £ 101 ) + (7 £ 1) (2 £ 107 ) + (3 £ 106 ) + (6 £ 105 ) + (9 £ 104 ) +(7 £ 103 ) + (5 £ 102 )

a i 107 ii 109 iii 108 b i 1029 ii 1045 iii 1085

11

Exercise 12 1 a 16 b 25 c 49 d 100 e g 21 h 9 i 20 j 36

1024 f 5184

2 a

18 496 b 166 464 c 1 361 889 d 5 313 025

3 a

0, 1, 4, 5, 6, 9 b no (as it ends in 8)

4 a

1 b 4 c 6 d

5 a

7 b 8 c 10 d 0 e 20

9 e 12

6 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 7 a

4 b 120 c 72 d 294

8 a

represents 4 3

b Each block is n units long £ n units wide £ n units high =n£n£n = n3

ANSWERS

KL, LM, MN, NK b LM, MN

Exercise 13

7 a

1 a 25 and 26 or 16, 17, 18 or 6, 7, 8, 9, 10, 11 b 4 spiders and 9 beetles c 36 d 2 + 3 £ 4 ¡ 5

8 a AB, BC, CD, DE, EA b AB, BC c AC, BC, DC, EC

2 a 128 b 12 c 105 d 54, dodecagon 3 a 6 b 9.42 am

Exercise 16

Stephen

c 43 d

4 a 4 months before the end of last month b 12 c 15 d 9.25 am

1 a

Review set 2A

2 a

1 a 8 b 8 c h 7

5 d 100 e 4 f 15 g 36

2

5 6

15 3

5

41 a 1, 2, 4, 7, 8, 14, 28, 56 b 20 c 6

10 a 360 b 3200 c 1 000 000 11

a 50 730 b 4 020 305

12 1000, 10 000, 100 000, 1 000 000 13 23 $1 coins

Review set 2B 1 a 7 b 29 c 13 d 18 2 a (32 ¡ 4) £ 2 = 10 b (4 + 2) £ (8 ¡ 5) = 18

36

3

2

4

18 2

9 3

a b c e

1, 2, 4, 8, 16, 32, 64 11, 13, 17, 19 105 d 2 £ 2 £ 2 £ 3 £ 5 9

3

5 37, 67 6 a 216, 732 b 9316, 22 000 7 36 cm p 81 = 9 d 27 = 33 8 a 49 = 72 b 121 = 112 c p e 400 = 20 9 a 2000 b 63 000 c 1 000 000 d 37 012 e 6 504 308 10 27 cm

1 a (1) a speck of dust (2) corner where two walls and the floor meet b (1) where two walls meet (2) bottom of blackboard B 2 a A vertex is a corner point of vertex a figure. A, B and C are all vertices. C A b A point of intersection is the X point where two lines meet. c Two lines which are always the same distance apart. Ã ! Ã ! Ã ! Ã ! Ã ! Ã ! CD, CE, DE, DC, EC, ED

4 a B b C 5 a B b AB 6 a PQ b Q c

P

E

M

c

Q

d

T

S N

T

P

5 ]ABC, ]BCD, ]ADC, ]BAD, ]BAC, ]BCA, ]CAD, ]ACD 6 a

]ABC b ]BCD c ]BAD, ]ADC

7 a

70o

b

128o

c 151o

8 a ]BAC = 83o , ]ACB = 31o , ]ABC = 66o b ]FDE = 119o , ]DEF = 32o , ]DFE = 29o ]ABC = 94o , ]BCD = 78o , ]CDA = 78o , ]DAB = 110o b ]PQR = 53o , ]QRS = 127o , ]RST = 127o , ]STP = 92o , ]TPQ = 141o

9 a

Exercise 17 1 a a = 85 b b = 25 c c = 46 d d = 53 e e = 37 f f = 46 g g = 22:8 h h = 117 i i = 42, j = 48, k = 63 2 a p + q + r = 180 b a + b = 90 c a + b = 140 3 a

a = 45 b b = 60 c a = 30

4 a a = 110 b b = 90 c c = 40 d x = 80 e x = 90 f x = 90 g p = 70 h q = 100 i r = 90 5 a a + b + c + d = 360 b m + n = 170 c x + y = 180

Exercise 15

à ! à ! 3 a LM or ML b

b

C

D yes

d

acute b obtuse c straight d reflex e reflex

D

7 a 32, 40, 48, 56 b 36, 49 c 31, 37, 41, 43, 47, 53, 59 8 60 000 km 9 a no b yes c yes d

c

3 a ]ABC, acute angle b ]PQR, obtuse angle c ]KLM, reflex angle

6 £ (7 ¡ 5) ¥ 3 = 4

30

4

b

4 a

2 a 14 b 24 c 23 3 a (5 + 2) £ 4 = 28 b

213

QR d QR

Exercise 18 1 a not closed b not all straight sides c some sides cross each other d all curved e some sides cross each other f one side is curved 2 a no, angles not equal b yes, is regular c no, angles not equal d no, angles not equal e yes, is regular f no, sides not equal 3 a d h l 4 a

regular quadrilateral b quadrilateral c triangle quadrilateral e decagon f pentagon g quadrilateral hexagon i hexagon j decagon k decagon quadrilateral b

c

214

ANSWERS

d

e

Exercise 20

f

1

5 a

b

regular hexagon

equilateral triangle

5 cm

4 cm

c

equilateral non-regular pentagon

6 cm C

2

6 a I b I c R d R e R f I g I h I

Exercise 19 1 a scalene b isosceles c scalene d equilateral 2 a

b

5 cm

c

3 a square b rectangle c parallelogram d rhombus e kite f trapezium

A

RQ ? QP b AB k DC HI k KJ, HI ? IJ, IJ ? KJ d KM ? LN PQ k SR, PQ ? SP, SP ? RS WX k ZY, WZ k XY, WX ? XY, XY ? YZ, YZ ? ZW, ZW ? WX

3

B 5 cm

C

6 a

B

5 cm

a equilateral triangle (all sides 5 cm) b ]ABC = ]BCA = ]CAB = 60o c All angles of an equilateral triangle measure 60o .

4 a True b True c False d True e False 5 a c e f

5 cm

A

3 cm

5 cm A

4 cm

R

b

B

4 cm

C a isosceles (2 sides equal) b ]ABC = ]ACB c In an isosceles triangle the angles opposite the equal sides are equal.

S

4 a

3 cm

P c

5 cm A

B

Q 6 cm

D

6 cm

C 6 cm

215

ANSWERS

7 a x = 46 b x = 66 c x = 40, y = 5 d x = 70, y = 4

b

Exercise 21 1 a

4.8 cm

4.8 cm

b

30° 60°

45°

c

d

4.8 cm c

6 cm

22\Qw_\°

15° C

2

4 cm

(50% reduction)

13 cm

5 cm

6 cm

A

d 3 a

4.4 cm

B

A

b OA = OB ) ¢ABO is isosceles

O

4.4 cm

6 cm

B

12 cm

C 4 a diameter b f sector

centre c radius d arc e

5 a

b

radius

arc

5

R

Q

6

6 cm

60° 60°

A

60° 60°

O

B

6

S

P

AQRBSP is a regular hexagon since all sides are 3 cm and all angles are 120o . 7

A

D The angle bisectors of any triangle appear to meet at a point.

B

C

O E

F

ABCDEF is a regular hexagon since all sides are equal in length and all angles are equal in size.

216

ANSWERS

c

Exercise 22 1 a

b

1 2 1

c

top

front

back

left

right

Assume no blocks hidden from view. d

e

d

f

1 2 1 top

2 A sphere has a curved surface; a cone and a cylinder have a combination of flat and curved surfaces.

e

3 a sphere b cone c cylinder d cube e rectangular prism f cylinder 4 a

f

b

6 a c

d

front

back

left

right

1 2 1 1 top

front

back

left

right

2 1 1 1 top

front

back

left

right

The numbers indicate how many blocks are underneath each other within the solid.

b 10 - this is the total number of blocks in the solid. 7 a 5 a A, B, C, D, E, F, G, H b ABCD, BCFE, CDGF, ADGH, ABEH, EFGH c AB, DC, GF, HE, BC, AD, EF, HG, AH, BE, CF, DG 6 a rectangles b triangles 7 a 4 b 4 c

8

b

Any 3 of these:

6

8 a C, (2) b A, (3) c B, (4) d D, (1)

Exercise 23 1

2

Review set 3A

1

3 2 a

1 a

b

c

BC b BC

2 a parallel lines b equal angles c equal sides 3 a

90o

b

180o

5 a

3 a

b

4 a

a = 60 b b = 27

b

sector

c 6 To draw the bisector (NR) of ]MNO. 7

8

4 Any 2 of these: triangular prism 9 a

1 2 1

d

b

c

e

5 a

1 1 2 top b

front

back

left

2 2 2 1 top

front

back

left

right

right

Review set 3B 1 a b c d

Lines continue indefinitely in both directions. AB is the line segment joining A to B. Ã ! CD is the line passing through C and D. Ã ! Ã ! X is the point of intersection of AB and CD.

217

ANSWERS

2 a pentagon b quadrilateral c octagon

Exercise 25

3 a scalene b equilateral c isosceles

1

4 a a = 94 b c = 112 5 a A, B, C, D, E, F b AB, AC, BC, AE, BF, CD, DF, DE, EF c ABC, EFD, ABFE, ACDE, BCDF 6 a

Symbol

Words

Num.

Denom.

a

1 2

one half

1

2

b

3 4

three quarters

3

4

c

2 3

two thirds

2

3

d

2 7

two sevenths

2

7

e

7 9

seven ninths

7

9

f

5 8

five eighths

5

8

g

7 11

seven elevenths

7

11

b

7

8

9

1 2

0 one half

b

One whole divided into four equal parts and three are being considered.

0 1 three quarters

c

One whole divided into three equal parts and two are being considered.

0 two thirds

1

d

One whole divided into seven equal parts and two are being considered.

0 two sevenths

1

e

One whole divided into nine equal parts and seven are being considered.

0 seven ninths

1

f

One whole divided into eight equal parts and five are being considered.

0 five eights

1

g

One whole divided into eleven equal parts and seven are being considered.

a

Review of chapters 1, 2 and 3 1 a 4607 < 4670 b

Meaning One whole divided into two equal parts and one is being considered.

2

50 £ 1000 < 50 021

2 a 7 b 11 c 9 3 a isosceles b equilateral c scalene 4 987 510 5 a

9892 b 108

6 a a = 56 b a = 120 c

a = 30

7 a 60 b 6000 c 6 000 000 8 a 1, 2, 3, 4, 6, 8, 12, 24 b

23, 29 c 24 d 5

9 a 430 b 400 10 24 m 11

12

a 204 b

201

13

a 64 = 82 p b 100 = 10 3

c 8=2 14 a

b 2 a

15 500 grams

5 6

2 5

b

7 8

c

16

36 c 29 or 512 d

50 356

1 4

c

ii

1 16

4 b

d i

1 4

0

1

seven elevenths d

13 16

5 6

e

f

1 8

iii

iv

or

15 18

1 8

5

a Qr_

21 31 368, 31 836, 33 681, 36 183, 38 631 22 a x = 23 b

x = 60 c x = 65

23 a

b

Qw_ eA_w_

c

yA_r_

3 2 1

Qi_

63 64

b

shaded square is one square out of 64

c

64 d

e

8 64

=

1 8

f

4 64

=

1 16

qA_y_

e

Exercise 26 24 a losing by 6 goals b 24 C c 38 C

1 a

5£2 6£2

25 a True b True

2 a

8¥2 10¥2

3 a

2 8

4 a

15 30

b

24 30

c

25 30

d

9 30

5 a

2 16

b

12 16

c

16 16

d

0 16

o

o

c False

26 12 000 27 100 times 28 a 22 b 2 c 10 29 a b c d

5 6

1 4

18 10 278 19 $26 20 $64:80

d

1

3 b Each part is one third of the whole shape. c Each part is one quarter of the whole shape. d circle and triangle 4 a

17 a 36 b

10 12

or

Number line

These are lines and not line segments. The line segment joining P to Q. The line passing through R and S. The point of intersection of the two lines.

= =

b

10 12

b

5£5 7£5

4 5

b

10¥5 15¥5

4 8

6 8

c

d

=

25 35

c

4£10 5£10

=

40 50

2 3

c

18¥2 20¥2

=

9 10

= 8 8

1 2

218

ANSWERS

6 a

50 100

25 100

b

80 100

c

90 100

d

7 a 2 = 1 b 2 = 12 c 2 = 3 d 2 = 3 e 2 = 20 f 2 = 8 g 2 = 40 h 2 = 50 8 a 9 a 10 a 11

4 5

1 4

b

4 5

9 10

b 8 11

3 7

d

3 4

c 9 16

b

3 4

c

5 7

d 3 5

c

3 4

f 1 4

e

5 7

f

7 13

e 1 3

d

4 7

e

1 17

f

4 7

h

3 4

h

3 11

8 27

g

1 14

b 1 16

g

2 27

2 78

11 12

7 8

c

2 5

8 15

c

5 12

b

h

h

7 2 13 2

9 20

2 a

1 2

3 a

1 2

4 a

1 24

b b

1 4

c

1 13

b

1 6

c

3 4

1 6

b

e

3 13

d

4 13

e

5 26

f

1 a

h 7

1 5 1 10

13 40

39 100 27 200

b

1 1440

d

i

1 2

c

2 a

1 4

d

3 a 5 24

e

23 60

f

1 15

g

1 2

14 a

$25

1 10

b

1 2

7 a h

11 25 cents

15 cents

f

o

o

ii 180

90

5 3 48 7

d k

5 d 2 4 5 = 4 5

3 2 23 12

e l 2 1

=

3 5

c

5 6

d

6 7

c

3 4

d

3 5

1 e 1 12

iii 270

1 12

b i

ii

1 6

iii

2 3

7 j 1 18

9 14

k

c 1 13 18

b 6 c 4 d 2 38

1 e 4 12

1 3

b i

1 12 7 15

1 2 1 8

b i

1 1 10

d 3 17

j

11 20 2 15

c j 7 24

b

7 8

1 4

c

3 2 10

k

4 7

1 5 1 6

k

c 1 15

b 1 58

e

d 7 12

c

1 6

2 13

l 3 8

e

d 1 16

e

7 10

g 4 a c

=

7 , 10 4 6

21 , 30

=

5 8

=

4 7

1 2

8 14

=

d

5 , 8

3 4

25 30

f

3 4

=

h

5 6

=

25 , 40

8 10

=

32 40

=

5 , 10

2 5

=

8 , 12

5 a Sanjay b Sanjay

7 , 12

1 2

=

4 10 6 12

1 4

b d

16 = 45 , Robert 20 24 , Robert 25 30 30

=

3 a

6 8

27 , 36

7 9

=

=

25 , 100

7 20

=

3 4

= 68 ,

5 , 8

1 2

3 5

=

12 , 20

25 30

=

5 6

h

28 36

1 2

35 , 100

=

=

9 25

=

36 100

4 8

10 , 20

9 20

c Robert

6 a 4 b 4 c 3 d 5 e 5 f 10 g 3 h 30 i 1 j 8 k 5 l 9

4 a

g

1 2

3 g 2 10

1 3

2

1 23

1 34

3

2 34

7

5

3

2 23

1 13

0

3 12

2 12

1 12

4 12

4

6 12

1

2 3

2 13

2 14

2

3 14

1 10

1 b 1 4 8 15

g 1 13

3 5

c

6 34

7 15

d

e 1 12 l 3 15

e 2 12

f 2 58 m 2 17

6

i b

3 10

h 1 17

c

5 12

d

4 9

i 6 j 6 k

8 21

e 3 4

g 2 23 n

f 10 g 1 f 2 27 l 2

5 a 8 b 6 c 3 d 6 e 6 f 6 g 48 9 k 1 12 l 35 = 8 34 h 12 i 21 j 16 4 6 a 75 cm b 16 hours c 60 years d 50 minutes e 700 mL f 150 g 7 a

5 h 1 12

5 12

p 3 13

7 , 14

1 3

6

o 7 15

=

f

3 12

k 1 56

1 2

4 12

5 8

c

j 2 45

9 12

g

f 1 23

i 4 12

=

1 h 3 12

3 10

e 2 38

h 3 13

3 4

g 3 56

17 24

d

d

8 , 12

g 1 16

1 2

l

c 3 13

=

4 13

g

1 f 1 30

f

b 3 37

2 a 8 b 12 c 36 d 63

f 1 25

4 7

1 45

c

= 62 ,

1 b 3 c 7 12 m

2 a

2 3

3 1

f 4 16

1 a 21 b 15 c 6 d 36 e 72 f 30 g 330 h 36

b

= 63 ,

7 d 2 24

2 = 10 b 2 = 18 c 2 = 21

2 4

9 5

38 45

l

Exercise 28

=

g

e 1 59

Exercise 30

1 2

4 2

=

1 a

1 , 4

15 4

36 = 66 ,

16 18 children 17 2 h 40 min

3 a

1 l 9 12

f

=

b 1 11 20

1 25

10 a

15 a i

k

3 3 6 1

11 12

i

9 a 1 b 1 13

$2:50

d

o

f 5 23

10 14

3 i 2 10

1 1 12

6 a

$5

b

$12:50

e

j

3 b 1 10

7 3 10

8 a c

1 2

5 a

h

5 games 12 49 students 13 37 cars

11

11 4 47 8

c

c

6 10

b

4 a 3 13

9 a 5 b 18 c 4 d 15 e 5 f 11 g 6 h 24 i 65 g j 60 cents k 15 min l 50 10 a

3 4

h 1 11 24

1 3

8

i

h 3 12

1 5

d

1 1 5 a 57 b 14 c 84 d 10 1080 The denominator became larger.

6 a

j

e 7 12

Exercise 29

1 48

c

14 3 41 9

b

1 b 56 6 1 = 22 = 1 4 , 51 , 1

10 a

Exercise 27 1 a

3 89

b

9 a

1 4

7 8

d

i

3 16

d

3 17

1 5

b, c, h, j, k

12 a

h

c 4 12

8 a

g g

7 a

40 b 25 c 49 d 175

8 $39 9 84 plants 10 a 1875 kg b 50 boxes

2 29

ANSWERS

a

12

a

Qr_

g

g

Qe_

g

11

Qe_

g 2 4

13 a

£

2 3

£

1 2

=

1 4

2 9

= 1 4

1 4

b

£

2 3

£

1 3

1 6

=

8 b 1 15

1 4

c

9 70

d

1 7

1 4

4

1 6

5

of a tank 6 a

b

3 10

c

10 69 litres 11

195 blocks 12 17 out of 20

1 12

a

b

3 20

7 b 7200 cases 14 2 30 metres 15 50 kg

Review set 4A 3 5

2 a 2 16 6

2 3

=

5 12

b

c 1 89 or

b 1 79

12 , 18

7 9

=

1 8 5 6

3 14 , 18

3 4

9 a $21 b

5 8

b

c

2 5

5+

7 a

15 18

4 10

7 b 2 12 3 7

50 minutes c 625 g 10 a

b

c 1 23

f 1+

7 20

3 10

i 4+

2 100

682 spectators b 355 students c 360 students

8 10

+

8 10

Review set 4B

m 9+

8 1000

1 a For example

o 800 + 8 + or

b For example

4 a

3 b 4 10

4 78

1 5

b

8 9

4 5

c

3 4

5

b

5 8

=

15 , 20

9 a

1 9

5 8 a 1 18

b 600 sheep c

6 16

6 100

1 100

j 2+

2 10

2 1000 9 10

+

6 100

+

9 10

h 7 100

4 1000

+ +

5 100

+

3 1000

+

7 10

+

7 100

+

7 1000

n 100 + 50 + 4 +

4 10

+

5 100

+

1 1000

8 10

+

8 1000

9 a

6 13

c 13 , 20

2 5

2 100 2 10

2 100

b 2 c h

2 1000

i

2 10 000

d 20 e

f 200

2 1000

10 a i 1:6558 or one point six five five eight 6 5 5 ii 1 + 10 + 100 + 1000 + 10 8000 =

b i 1:6459 or one point six four five nine 6 4 5 ii 1 + 10 + 100 + 1000 + 10 9000

8 20

c i 1:7332 or one point seven three three two 7 3 3 + 100 + 1000 + 10 2000 ii 1 + 10

9 b 5 10

d i 1:5884 or one point five eight eight four 5 8 8 ii 1 + 10 + 100 + 1000 + 10 4000

hours (6 h 10 min)

10 a 2 23 pages b 4 pages

11

Exercise 33

a i

d ii c iii b iv d

b c, a, b, d

1 a $205:05 b $12:70 c $120:65

Exercise 35

2 a $4:47 b $15:97 c $7:55 d $36:00 e $150:00 f $32:80 g $85:05 h $30:03

1 a

2:4

2 a

4:24 b 2:73 c 5:63 d 4:38 e 6:52

3 a

0:5

b 0:49 4 a 3:8 b 3:79

5 a

0:2

b 0:18 c 0:184 d 0:1838

6 a

3:9

b 4 c 6:1 d

3 a $0:35 b $0:05 c $4:05 d $30:00 e $4:87 f $2:95 g $38:75 h $6384:75 4 a 1 34 49 g 6 100

b 3 14

c 52 25

7 h 4 25

1 i 1 20

5 a 3:243 b 2:071 c

87 100

d j

7 100

2 1000

l 7+

+

6 1000

3 100

c 2+ 2+

e

g 3+

4 1000

+

9 10

c

6 a 28 b ¤ = 15, ¢ = 36 7 a T b F c F

e 5000 f 5

8 a 0:6 b 0:19 c 0:43 d 0:809 e 0:052 f 0:052 g 0:568 h 0:0023 i 0:094 j 0:101 k 4:387 l 0:0308 g

3 a

5 100

d

3 100

e 3 f

or

or

2 a 1 34

+

8 1000

+

k 20 +

11

5 10

b 10 + 4 +

d 30 + 2 +

3 F c T d T 8 a 6 10

7 a F b

5 b 500 c 5 a 1000 g 50 000 h 10 5000

3 1000

6 a i 346:584 ii 0:3607 iii 4:0082 b 350:9529 c 342:2151

17 9

4 8 5 a =

2:983 c 32:752 d 12:096 e 3:694

3 4 a 300 b 10 c 30 d 3 g 10 000 h 3000

7 km 18 4 16 hours 16 a twelfths b twenty fourths 17 10 24

1 a

i 837 ii 0:837 b i 318 ii 0:318

3 Final written numeral is a 0:8 b 0:003 c 70:8 d 9000:002 e 209:04 f 8000:402 g 5020:3 h 36:42

4 21

6 35

9

1 4

8 21

7 754 hours 8

13 a

8 a

2 a 17:465 b f 4:22

2 a ¤ = 6 b ¤ = 15 c ¤ = 3, ¢ = 10 d ¤ = 24 e ¤ = 70, ¢ = 28 f ¤ = 24 3

b 0:2 c 0:33 d 0:46

1 a zero point six, six tenths b zero point four five, forty five hundredths c zero point nine zero eight, nine hundred and eight thousandths d eight point three, eight and three tenths e fifty six point eight six four, fifty six and eight hundred and sixty four thousandths

1 12

=

Exercise 31 2 1 a 3 15

0:7

Exercise 34

We_ 1 3

7 a

219

13 e 31 100

k 37 11 20

1:056 d 4:009

6 a $32:43, $20:71, $10:56, $40:09 b $103:79

2 f 243 25 1 l 1000 10

b 3:6 c 4:9 d 6:4 e

4:3

0:462 e 2:95 f 0:176

7 a 4:3 b 9:13 c 11:2 d 0:0 e 0:73 f 0:025 g 0:5 h 6:17 i 2:429 8 a

$38 700 b $43 200 c $98 900

220

ANSWERS

f 3:1004 g 18:867 h 7:782 i 4:258 j $5:30 k $5:97 l $4:60

9 a $56:3 K b $32:5 K c $23:2 K 10 a $70:8 K ¡ $73:2 K b $158:7 K ¡ $165:7 K c $327:9 K ¡ $348:4 K 11

a 3:18 m b 91:73 m c 23:46 m d 1:49 m e 30:08 m f 9:48 m

12 a 21 650 000 b 1 930 000 c 16 030 000 d 212 450 000 e 970 000 13 a 3:87 bn b 2:71 bn c 97:06 bn d e 4209:47 bn f 549:00 bn

2:02 bn

7 a

2:62 cm b 7:75 km

8 a

55:1183 b 42:266 c 1:19 d $118:10

9 a

15:867 b

10 0:37 m 11

2:731 c 0:681 d $6:85 17:3o C 12 $17:10 13 13:079 m

14 27:95 kg 15 3:38 kg 16 $8:10 17 No, he has only $59:05 and needs another $3:45.

14 a 3 860 000 000 b 375 090 000 000 c 21 950 000 000 d 4 130 000 000

18 337:4 m 19 69:4 kg

15 $58 500, $82 700 - increase in salary of $24 200

Exercise 38

16 $170 200, $295 800, $672 100 - total is $1 138 100

1 a b c d e

17 a i $67 400 000 ii $69 900 000 b i McLaren Vale ii $2 500 000 18 a i

$1 200 000 000 ii $86 700 000 b $1 113 300 000

Exercise 36 1 a 6:7 b 13:6 c 47:8 c 151:2 e 3:77 f 2:01 g 1:953 h 4:195 i 0:404 j 0:17 k 0:099 l 0:042

a b c d e

2 a 2:5 b 2:48 c 2:54 3 a 1:69, 1:70, 1:73 b 0:76, 0:77, 0:79 c 5:425, 5:427, 5:431 4 a d g j m

0:7 < 0:8 b 0:06 > 0:05 c 0:2 > 0:19 4:01 < 4:1 e 0:81 > 0:803 f 2:5 = 2:50 0:304 < 0:34 h 0:03 < 0:2 i 6:05 < 60:50 0:29 = 0:290 k 5:01 < 5:016 l 1:15 > 1:035 21:021 < 21:210 n 8:09 = 8:090 o 0:904 < 0:94

5 a grapes b pears c bananas d C e D 6 a d f h

0:4, 0:6, 0:8 b 0:1, 0:4, 0:9 c 0:06, 0:09, 0:14 0:46, 0:5, 0:51 e 1:06, 1:59, 1:61 0:206, 2:06, 2:6 g 0:0905, 0:095, 0:905 15:05, 15:5, 15:55

7 a c e g

0:9, 0:8, 0:4, 0:3 b 0:51, 0:5, 0:49, 0:47 0:61, 0:609, 0:6, 0:596 d 0:42, 0:24, 0:04, 0:02 6:277, 6:271, 6:27, 6:027 f 0:311, 0:31, 0:301, 0:031 8:880, 8:088, 8:080, 8:008 h 7:61, 7:061, 7:06, 7:01

8 a

10.015 10.000

10.055

10.55 10.5 10.600

10.05

b 10:015 sec, 10:05 sec, 10:055 sec, 10:5 sec, 10:55 sec c Sam, Matthew, Eli, Jason, Saxby

Number 0:0943 4:0837 0:0008 24:6801 $57:85 £1000 94:3 4083:7 0:8 24 680:1 $57 850

£10 0:943 40:837 0:008 246:801 $578:50 £104 943 40 837 8 246 801 $578 500

£100 9:43 408:37 0:08 2468:01 $5785 £106 94 300 4 083 700 800 24 680 100 $57 850 000

2 a 430 b 8000 c 5 000 000 d 6 e 46 f 58 g 309 h 250 i 80 j 324 k 900 l 845 m 240 n 208:5 o 8940 p 53 q 0:094 r 71 800 3 a 100 b 1000 c 10 000 d 100 e 10 f 10 g 10 h 1 000 000 i 10 4 $135 5 a 475 cents b 12 560 L c 38:6 mm d 13 860 kg e 9847 mm f 2080 g 6

a b Num. 647:352 93 082:6 ¥10 64:7352 9308:26 ¥100 6:473 52 930:826 ¥1000 0:647 352 93:0826 ¥105 0:006 473 52 0:930 826

7 a g l q

c d 42 870 10:94 4287 1:094 428:7 0:1094 42:87 0:010 94 0:4287 0:000 109 4

0:23 b 0:036 c 0:426 d 0:3 e 5:8 f 0:58 39:4 h 0:07 i 0:458 j 0:8007 k 0:024 05 0:0632 m 5:79 n 0:579 o 0:0579 p 0:003 0:0003 r 0:000 046

8 a 10 b 100 c 100 d 1000 e 10 f 10 000 g 100 h 1000 9 $133:52

9 a 0:4, 0:5, 0:6 b 0:6, 0:5, 0:4 c 0:8, 1:0, 1:2 d 0:11, 0:13, 0:15 e 0:55, 0:5, 0:45 f 2:05, 2:01, 1:97 g 4:8, 4:0, 3:2 h 3:789, 3:9, 4:011

10 a 4:975 km b 5:685 kg c 309:5 cm d $754 e 47:85 kL f 2:348 t g 0:26 m h 5:655 m i 0:5 km

Exercise 37

11

1 a 0:9 b 3:3 c 1:13 d 1:13 e 27:82 f 18:43 g 5:2 h 0:444 i 10:92 j 32:955 k 0:7006 l 4:748

9655 cents 12 15:5 cm 13 1:2 kg (or 1200 g)

Exercise 39 1 a

1 b 3 c 2 d

3 a i 39:012 ii 2:134 iii 3:076 iv 8 b i 1:101 ii 0:099 iii 11:754 iv 22:694

2 a g l q

0:56 b 0:36 c 0:24 d 7:2 e 26:0 f 13:5 0:006 h 0:000 16 i 0:035 j 0:144 k 1:32 0:4545 m 0:09 n 0:36 o 6:4 p 840 0:0081 r 0:024

4 a 64:892 b 27:493 c 12:214 d e 408:488 f 209:7442

3 a 95:2 b 9:52 c 0:952 d 0:952 e 0:952 f 0:0952 g 0:0952 h 0:000 952 i 0:952

2 a 0:8 b 1:5 c 0:4 d 1:4 e 2:3 f 2:26 g 2:67 h 0:01 i 9:02 j 5:593 k 0:001 l 0:001

21:2919

5 a 0:001 b 2:374 c 2:721 d 2:222 e 8:149 f 21:91 6 a 5:981 b 1:011 c

4:481 d 167:348 e 58:626

5 e 4 f 6

4 a 1339:5 b 133:95 c 13:395 d 1:3395 e 13:395 f 1:3395 g 0:133 95 h 0:001 339 5 i 133:95

221

ANSWERS

b $35:7 K c 8:1 m

5 a 2:4 b 0:88 c 2:5 d 0:27 e 2:7 f 15:2 g 0:72 h 0:0063 i 0:0016 j 0:08 k 0:04 l 0:0009 m 0:072 n 1:21 o 0:01

3 a

0:5

4 a

272:6 b 2:726 c 27:26

6 a 12:27 m b 123 cm c 30:225 km d 219 mm e 17:92 m f 30 cm g 11:25 mm h 36:8 m i 93:61 cm

5 a

6200 b 215:8 c 0:56 d

6 a

35 276 cents b

7 44:8 kg 8

7 a

6:16 b 0:015 c 14:8 m d $20:60

$15:30

8940 mL

9 6 £ 3:9 m = 23:4 m, so Fred needs to find another 1:6 m.

8 0:026, 0:062, 0:206, 0:216, 0:621

10 a $39:51 b $36:96 c 90 L

9 a

a 912 kg b 576 kg c 1488 kg d 3 vans e $2762:10

11

12 a $56:80 b $45:00 c $30:00 d $94:80 e $56:00 f $69:30

11

0:042

1937:88 t b 645:96 t 10 28:2 km a i

3 5

0:5 ii 0:375 iii 0:6 b i

17 20

ii

2 9

iii

Review set 5B 6 1000

1 a

1:493 b 2:058 c 3:551 2 a 8:709 b

Exercise 40

3 a

1 a 0:8 b 1:5 c 0:42 d 0:51 e 3:02 f 0:41 g 0:08 h 20:4 i 0:556 j 10:5 k 1:334 l 0:007 m 0:059 n 0:43 o 0:015 p 0:85

9:44 b $29:76 K c 3:47 bn

4 a

13:78 b 0:1378 5 a 0:189 b 1114 c 0:0217

6 a

74:08 m b 97 820 000 mg

7 a

5:04 b 28:754 c 147:05 d $109:90

2 a $0:21 b $0:19 c $0:67 d $0:78 e $0:65 f $10:13 g $27:52 h $0:58 i $12:75 j $112:50 k $130:00 l $31:66 3 a $1:55 b $8:50 c 0:7 m d 12 bags e $16:08 f $23:65 g $7:70 4 a 12:22 m b 5:12 km c 17:34 cm 5 a ii b

d 6:84 mm

6 a 45, 43:86 b 100, 129:36 c 100, 145:18 d 6, 6:0564 e 160, 124:797 51 f 80, 81:2032 g 90, 90:5728 h 2000, 2389:296 i 50 000, 71 143:2 $3649:75 9 $18:55 10 3:165 m

150 lengths 12 1:425 m 13 $281

11

14 a 17 full lengths b 0:28 m

b 0:5 c h 0:75 m 0:65 s 1:28

0:4 d 0:3 e 0:8 f 0:25 i 0:125 j 0:625 k 0:35 n 0:088 o 4:25 p 2:2 q 5:6 t 0:716

2 a 0:6 b 1:8 c 0:375 d 1:125 e 2:75 f 5:8 g 4:875 h 5:375

5 a 0:718 75 b 0:6875 c 0:2125 d 0:44 e 1:1875 f + 0:214 285 7 g 0:13 h 0:81 i 2:23 j 1:94 k 0:461 538 l 0:306 25 m 3:416 n 0:252 03 o 0:51

i

9 50

7 a g

4 5

7 10

b j

13 20

8 a 2 45 g 28 21 50

h

1 20

k

22 25

b

8 125

1 2

c

1 5

l

7 100

111 125

c 12 125

b 4 12

d

i

c 3 35

1 h 5 500

d

17 200

9 10

e

j

m 551 1000 351 500

d 7 15

i 3 13 50

f 3 4

3 5

n

e k

19 100

g 1 40

49 100 1 3

o f l

8 e 22 25

7 j 4 500

4 100

+

19 f 46 100

1 k 13 40

9 1000

p

3 8

3 50 2 3

1 l 12 1000

Review set 5A 1 a 1:8 b 0:54 2 a 2 +

1 4

h 1 25

b

2 1000

3 8

ii

b i 0:75 ii 0:16 iii 0:¹ 7

1 20

iii

b

2 a M=

11 100

46 100

b C=

17 100

c L=

10 100

d X=

35 100

e V=

27 100

3 a i 14% ii 86% b i 38% ii 62% c i 67% ii 33% d i 95% ii 5% 50% b

85% c 25% d + 40%

6 a 70% b 10% c 90% d 50% e 25% f 75% g 40% h 80% i 35% j 55% k 28% l 76% m 46% n 94% o 100% 7 a Fourteen percent means fourteen out of every hundred. b If 53% of the students in a school are girls, 53% means 53 the fraction 100 . c 39 out of one hundred is 39%. 52% b

48% 9 60%

10 a i 24% ii 52% iii 24% iv 32% b i 28% ii 36% iii 36% c i 50% ii + 23% iii 25% iv 50%

4 0:1, 0:2, 0:3, 0:4, 0:5, 0:6, 0:7, 0:8, 0:9 = 1

1 10

60 100

1 a

8 a

3 a 0:3 b 0:6 c 0:16 d 0:142 857 e 0:285 714 f 0:083 g 0:2 h 0:83 i 0:27 j 0:583

6 a

23 50

5 a 19% b 3% c 37% d 54% e 79% f 50% g 100% h 85% i 6:6% j 34:5% k 7:5% l 35:6%

Exercise 41 0:7 0:16 0:24 2:45

57:05 seconds b 57:21 seconds

10 a i

4 a

15 a 93 litres b $83:70 c 1860 km d 180 km

1 a g l r

9 a

Exercise 43

iii c i d iii

7 29 hamburgers 8

8 4:44, 4:404, 4:044, 4:04, 0:444

Exercise 44 1 a h o 2 a h

43 37 b 100 c 12 100 1 i 34 j 19 k 4 20 4 1 p 3 q 20 r 5 1 3 1 b 40 c 200 8 1 5000

7 10

d

1 l 11 25

d

e

9 10

3 100

m

s 2 t

173 1000

e

39 40

f

1 5

1 4

n

f

1 500

2 5

g 3 5

7 2

g

1 2000

3 a 0:5 b 0:3 c 0:25 d 0:6 e 0:85 f 0:05 g 0:45 h 0:42 i 0:15 j 1 k 0:67 l 1:25 4 a 0:075 b 0:183 c 0:172 d 1:067 e 0:0015 f 0:0863 g 0:375 h 0:065 i 0:005 j 0:015 k 0:0075 l 0:0425 5 a 10% b 80% c 40% d 60% e 40% f 50% g 15% h 25% i 95% j 6% k 78% l 68% m 37 12 % n 100% o 11% p 62 12 % q 33 13 % r 66 23 %

222

ANSWERS 1 4 2 5 2 3 1 4

6 a b c d

is 25%, is 40%,

1 8 3 5

is 66 23 %, is 25%,

is 12 12 %, is 60%, 3 3

1 2

1 16 4 5

is 6 14 %

is 80%,

2 a 5 5

is 100%

is 100%

b 3 4

is 50%,

is 75%,

4 4

is 100%

7 a 37% b 89% c 15% d 49% e 73% f 5% g 102% h 117% 8 a 20% b 70% c 90% d 40% e 7:4% f 73:9% g 0:67% h 0:18% 9

%

Fraction

Decimal

a

20%

1 5

0:2

b

40%

2 5

0:4

50%

1 2

0:5

75%

3 4

c d

3 a b c d

0:75

e

85%

17 20

0:85

f

f

8%

2 25

0:08

g

g

35%

7 20

0:35

h

12:5%

1 8

0:125

i

62:5%

5 8

0:625

j

100%

1

1:00

k

15%

3 20

0:15

37:5%

3 8

0:375

e

l

68 100

= 0:68 = 68%

58 100

= 0:58 = 58%,

89 100

= 0:89 = 89%

22 100

= 0:22 = 22%,

55 100

= 0:55 = 55%

45 100

= 0:45 = 45%,

74 100

= 0:74 = 74%,

37 100

= 0:37 = 37%,

Number

Fraction

Denom. of 100

%

4

4 20

20 100

20%

9

9 20

45 100

45%

8

8 20

40 100

40%

2

2 20

10 100

10%

0

0 20

0 100

0%

9

9 20

45 100

45%

20

20 20

100 100

100%

70% b i

c i 40% ii e i 65% ii

3 5 13 20

20% ii

1 5

d i 70% ii 9 students f 80%

Exercise 46 1 a

21% b

65% c 14%

2 a

60% b

i 10% ii 36%

3 WA - 32:88%, NT - 17:52%, QLD - 22:48%, SA - 12:80%, NSW - 10:43%, VIC - 2:97%, TAS - 0:089%, ACT - 0:026% 4

1 a

= 0:28 = 28%,

i 30% ii

4 a

Exercise 45 0.65

Et_

70%

0%

Figure

Frac.

% shaded

% unshad.

a

1 2

50%

50%

b

3 4

75%

25%

c

1 4

25%

75%

d

1 6

16 32 %

83 31 %

e

3 8

37:5%

62:5%

f

7 10

70%

30%

g

2 3

66 32 %

33 31 %

h

9 30

30%

70%

100%

b

wL_p_

55%

0.83

0%

100% 79% Qw_Up_

c

0%

0.93

100% Er_

d

0%

0.85 92%

100% Wt_Up_ 0.59 67%

e

f

c

28 100

0%

100% 47%

Qe_Ip_

0.74

0%

100% 42%

g

0% h

0% i

0.65

Er_

100%

0.39 wJ_p_ Wt_

58%

0.47

Ti_ Qw_Ep_

73%

100% 5 a

0%

100%

35 squares b

13 20

ANSWERS

4 Store A - $410:40, B - $420 ) costs less from Store A

6 a 18% b 14% c 28% d 32% 1 4

7 a

1 2

b

1 10

c

7 20 2 5

d

8 a i 20% ii 20% b

3 5

e

223

f 1

Exercise 47 1 a hot drink b fast food c airline d diving costume e public transport f communication g musical instrument h make of car 2 a 20% b 25% c 75% d 33 13 % e 30% f 25% g 20% h 12:5% i 5% j 4:8% k 18% l 6:25% m 50% n 40% o 60% p 2% q 0:25% r 12:5%

5 a

$2 b $80 c $5:60

6 a

$2420 b $10:45 c $148:50

7 a

$10 b $1 c $1:60 d $32

8 a

$110 b $52:80 c $2200 d $704 $16 b $176 10 a $25 b $275

9 a

a $4 b

11

$480

Exercise 50 1

3 a 52% b 90% c 85:4% d 32:5% e 68:5% f 76%

Principal

4 a 70% b 85% c 40% d 5% e 17:5% f 22% g 42% h 66 23 % i 174%

Interest rate

Time (years)

Interest for one year

Total interest

$1000

15% p.a.

1

5 a 72 ha b 1050 m2 c 45 cm d 160 t e 18 min f 640 cm g 600 kg h 480 mm i 108 min j 187 500 mL

0:15 £ $1000 = $150

1 £ $150 = $150

$5000

8% p.a.

4

0:08 £ $5000 = $400

4 £ $400 = $1600

$20 000

12% p.a.

1 21

6 388 students 7 360 kg 8 1215 tonnes 9 2 h 51 min 10 1:584 m 11

8925 L 12 22 games

1 4

of a metre c 8% of $100 13 a 40% of a litre b d 5% of a kilolitre e 13 of 1000 f 315 g

2 a

$300 b $700 c $1600 d $3600 e $14 000

3 a

$2800 b $8450 c $12 320 d $20 000 4 $4400

5 $1440 6 a $14 000 b 5:24% loss

Exercise 48 1 a This fraction represents 15 cents out of every 100 cents 15 = 15%. = 100 b This fraction represents 12:5 dollars for every 100 dollars = 12:5%. = 12:5 100 2 a $4:70 b $54 c $14:70 d $2:20 e $259 f $40:50 g $597:60 h $1:62 i $3:88 j $354 k $4365 l $1350 m $700 n $2:72 o $5450

7 a

$120 b 40% profit 8 a $5100 b 80% c $20400

9 a

$600 b $584 10 $6000

Review set 6A 1 a

28 100

4 a Jessie (76%), Jill (75%), Jan b Jeff (70%), Joel (68%), Jake

(72%), Jenny (62 12 %) (66 23 %), Juan (62 12 %)

5 a Kim 64% b 2% 6 a Tom (60%), Toni (55%), Tao (52%), Tina (50%) b $33:80 c $57:20 7 a 25% b

11 19 %

8 a 70% b

30%

9 Sarah $360, Jack $540, Peter $300 10

2

7 10

1

5

2 5

3

5 6

0

11

7

8

1

15 18

7

5

3

21 25

0 12

5

14

4

8

4

0

0 8

6 0

19

1

22 26

2

4

1 1

6 9

5

7 0

16

3

23

6

2

20

8

27

8

= 12:5%, 52%, 0:8 = 80%

52%

4 a

24

5 5

a $272 b $345 c $127:40 d $2288

3 a $5:60 b $27 c $128:80 d $49:28 e $16:15 f $455 g $91:84 h $340 i $32:90 j $376 k $47:10 l $37:19

4%

6 70 households 7 $30 8 a $94 b $1034 9 $655:50 10 $576

Review set 6B 1 a

2 5

b 45% c 28% d $1020 e 64 cm

2 64% 3 4

= 75%, 0:78 = 78%, 72%

72%

9 6

0.8

12 21 % b

$12 b 47% 5 a

0

Exercise 49 1 $9 2

1 8

10% 20% 30% 40% 50% 60% 70% 80%

13 17

3

Qi_

3

7

b 33 31 % c 45% d 10% e $180 f 2800 m

2 66 32 %

3 a 25% b 10% c 15% d 25% e 20% f 5% g 5% h 25% i 33 13 % j 66 23 % k 1% l 10%

1

0:12 £ $20 000 1:5 £ $2400 = $2400 = $3600

60%

4 a

Er_

0.78

70%

52% b

29% c 10% 5 76% 6 32:5%

7 $3 8 a $126 b 10 a $840 b

$714 9 a $27 b $297

$4340 11

$3360 12 $159 500

Review of chapters 4, 5 and 6 1 a

3 6

3 a

A is

4

or

1 2 2 81 ,

80%

b

1 16

B is 2 85

2 40% b C is 1 53 , D is 2 52

5 $30:75

224

ANSWERS

1 56 ,

15 b 100 2 1 13, 12,

11

a 75% b 60 g

6 a $260:45 b 4:5 km 7 a 8 6+

0 10

+

9 100

10 a 15 23

b

5 + 1000 5 c 100

12

9 2, 16 25

Wt_ =40% 55% 0%

13

24% c 0:4 1 13 ,

1 16 ,

1,

5 6

0.63=63%

50%

100%

9 40

14 a 36 b

7 12

=

16 56 17 a 1 17 18

21 5 , 36 9

=

20 36

b 1 11 12

c True 15 16% c 1 78

18 $8

Exercise 54 1 a

11:5 cm b 10:0 cm c 11:9 cm

2 a

59 cm b 41 m c 12:6 km

3 a

48 cm b 28:6 cm c 26:6 km

4 a 60 cm b 72 cm c 38 km d 96 cm e 42 cm f 60 cm g 72 cm h 56:2 cm i 40 cm 5 a

9:1 cm b 9:0 cm c 10:3 cm d 14:0 cm

6 a

760 m b 7:05 km c $4350

7 a 106:5 mm b 27 cm c 19 m d 117 cm e 112 cm 14 cm b 18 m

19 a 6:333 b 3:793 c 5:67 d 7

8 a

20 a $61:50 b $676:50 21 4:24, 4:15, 4:06

Exercise 55

22 $21 23 $210 24 ¤ = 10 and ¢ = 40

1 a actual object b drawing c actual object d actual object e actual object

25 a 0:9 b $87:5K c 8:7 mill

26 $105:60

27 8:75 L 28 21 apples 29 $34:55 30 800 m

2 a

1 : 100 b 1 : 125 c 200 : 1

31 $640 32 1:45 kg 33 2220 shoppers 34 $33:75

3 a

2:5 m b 100 m c

35 2000 books 36 18 students 37 66 23 % 38 a $9 b

12% 39 $76:50

Exercise 52 1 a 24 cm b 13 cm c 10:2 cm d 16:8 cm e 25:6 cm f 18:5 cm 2 a 35o C b 37:4o C c 38:3o C d 35:7o C 3 a

3 4

full b

1 4

full c

9 16

full

4 a 120 km/h b 95 km/h c 65 km/h 5 a 71:6 kg b

45:05 kg c 63:63 kg

500 m d 250 km

4 a i 200 m ii 290 m iii 120 m iv 630 m b i 10 cm ii 35 mm iii 4 mm iv 2:16 cm 5 a i 6 m ii 9 m iii 16:4 m iv 1:6 m b i 1 m ii 9 cm iii 2:8 cm iv 6:1 cm 6 a

4:5 m b 1:5 m c 4:7 m

7 a

3210 km b 2550 km c 1860 km d 930 km

8 a 12 m b 5 m c 2 m by 4 m d 2:4 m by 2:4 m and 3:6 m by 2:4 m 9 a

5:3 m b 2:3 m

10 a 25 mm, 1 : 2 b

i 6 mm ii 7:6 cm iii 1:1 cm

a 5 km b 40 cm c i 21 km ii 9:5 km iii 10:5 km

6 a 50 381 kWh b 16 443 kWh

11

7 a 700 g b 250 g c 850 g

Exercise 56

8 a 700 mL b 350 mL c 650 mL

1 a kg b tonnes c mg d g e g f kg g mg h tonnes i g j kg k kg l g m g n tonnes o tonnes p g q g r kg s mg t kg

9 a 49 mm b 63 mm c 133 mm d 81:5 mm e 151 mm f 235 mm g 116 mm h 205 mm

Exercise 53

2 a C b None of these devices is suitable. c A d B e B f none g none h D i B j C k none l B m none n D o D p none q B r C s none t D

1 a kilograms b kilometres c metres d milligrams e metres f kilograms g centimetres h tonnes

3 a 2000 mg b 34 000 mg c 350 000 mg d 4500 mg e 300 mg

2 a 9 m b 6:43 m c 47:53 m d 0:35 m e 9 m f 13:5 m g 0:62 m h 0:058 m i 2000 m j 6800 m k 500 m l 826 m

5 a

6000 g b 34 000 g c 2500 g d 256 000 g e 600 g

6 a

3 g b 2:5 g

7 a

4 t b 95 t c

3 a 700 cm b 1380 cm c 34 cm d 2 cm e 8:5 cm f 132:8 cm g 40:2 cm h 0:04 cm i 100 000 cm j 50 000 cm k 2000 cm l 300 cm 4 a 7000 mm b 34 mm c 780 mm d 460 mm e 2:6 mm

4 a 4000 kg b 25 000 kg c 3600 kg d 294 000 kg e 400 kg c 45 g

d 0:0675 g e 0:0095 g

4:534 t d 0:0456 t e 0:0008 t

8 a 8000 g b 3200 g c 14 200 g d 0:38 g e 4:25 g f 75:42 g g 6 800 000 g h 560 000 g 9 a

13:87 kg b 3400 kg c 0:786 kg d 0:003 496 kg

5 a 4:5 km b 17:458 km c 0:2 km d 0:164 km e 6:53 km

10 a 18:62 kg b 2:363 kg c 0:75 kg d 1237 kg e 1500 kg f 7:5 kg

6 7:5 km 7 a

11

10 200 m b 10:2 km

8 a 3110:32 m b 72 043:486 m c 155:218 m d 15 348:727 m 9 a c e g

40 mm, 37 mm b 800 cm, 780 cm, 750 cm 1:3 km, 1:25 km d 5:2 m, 5 m, 4:85 m 3:6 m, 3:5 m, 3:47 m f 134 m, 130 m, 128 m 5:12 m, 4:9 m, 4:82 m h 72 cm, 71:8 m, 71:5 m

10 a 577 cm b 6:5 cm c 2950 m d 8:51 m

a 24 kg b 200 nails c 33 tonnes d 1:2 kg e 10:55 kg

12 a 2000 mg, 2400 mg b 6700 g, 7000 g c 1400 kg, 1420 kg d 3:6 kg, 3:8 kg, 4 kg e 1:5 g, 1:9 g, 2 g 13 a large box costs $2:20/kg, small box costs $2:25/kg ) large box is the better buy b large box costs $16:75/kg, small box costs $18:20/kg ) large box is the better buy

225

ANSWERS

c 64 km by 1 km, 130 km; 32 km by 2 km, 68 km; 16 km by 4 km, 40 km; 8 km by 8 km, 32 km

Exercise 57 1 a 3900 m b

$9360 2 a 180 m b

$1665

d 48 mm by 1 mm, 98 mm; 24 mm by 2 mm, 52 mm; 16 mm by 3 mm, 38 mm; 12 mm by 4 mm, 32 mm; 8 mm by 6 mm, 28 mm

3 7:65 m 4 a 88 m b i 44 sleepers ii 1760 kg 5 9:9 kg 6 a 3600 bricks b 9 t 7 a 87:5 m, $393:75 b $210 c $630:75 8 4 hours 9 a 7:92 m b

$35:64

10 a 16 posts, 32 m b 60 m c 150 pickets, 180 m d $618

Review set 7A 1 a

1 8

full b 75 km/h c 3:2 kg d 3:5 L

e 64 u by 1 u, 130 u; 32 u by 2 u, 68 u; 16 u by 4 u, 40 u; 8 u by 8 u, 32 u f 144 mm £ 1 mm, 48 mm £ 3 mm, 24 mm £ 6 mm, 16 mm £ 9 mm,

72 mm £ 2 mm, 148 mm; 36 mm £ 4 mm, 80 mm; 18 mm £ 8 mm, 52 mm; 12 mm £ 12 mm, 48 mm

5 a

2 a 3560 mm b 3:2 kg c 0:45 km d 83 t e 7630 mm f 6:3 m

A¡=¡5¡mX

3 a 46 cm b 19 m

1m

5m

4 a i 19 km ii 32 km iii 61 km b i 10 cm ii 4:4 cm iii 26 cm 5 90 m £ 50 m, D 6 7:9 km 7 a 270 kg b 30 truckloads 8 a 1000 m b

290 mm; 102 mm; 60 mm; 50 mm;

A¡=¡8¡mX

2m

3000 m c $5250

A¡=¡9¡mX

4m

Review set 7B 1 28 105 kWh

3m

2 a 3:48 kg b 8:623 m c 4600 mg d 540 cm e 13 200 kg f 13 300 m

2

2

2

2

c 17 km2 , 32 km2 , 45 km2 , 56 km2 , 65 km2 , 72 km2 , 77 km2 , 80 km2 , 81 km2

4 a i 120 km ii 17:5 km b i 4:8 cm ii 3:92 cm 5 a 8 cm b 8 cm c 25 cm 6 2:45 m 7 14 rulers

6 65 m2

8 225 kg 9 2000 bricks 10 124 m

8 a

Exercise 59

7 a 27 ha

b $4860

280 tiles b $980

Exercise 61 b 20 units2

c 20 units2

d 28 units2

2 a yes b i 16 units ii 20 units iii 18 units iv 26 units v 34 units vi 24 units vii 26 units viii 22 units c For different shapes of the same area, the perimeter varies. 3 a i 132 ii 302 b

18 m2

c $988:20

4 a m2 b cm2 c ha d cm2 e mm2 f km2 g mm2 or cm2 h m2 i cm2 j ha k km2 m mm2 n mm2 5 a e i n r

2

b 9 m , 16 m , 21 m , 24 m , 25 m

3 a 43 km b 56 cm

1 a 10 units2

3m

1 a 42 m2 b 20 cm2 c 38:5 cm2 e 6 m2 f 12:48 m2 2 a

78 cm2

4 a

48 ha b $7200 5 a 24 m2

6 a

b 89 m2 2

2625 cm

c 11 m2

b 240 cm

d

37:1 m2

3 178 cm2 b $333:60

2

Exercise 62 2

l m

4:52 cm2 b 75 000 cm2 c 58 000 m2 d 0:3579 m2 630 ha f 36 500 000 mm2 g 0:55 m2 h 520 mm2 0:68 ha j 44 cm2 k 6000 m2 l 2 km2 m 70 mm2 4:8 km2 o 2500 mm2 p 8000 cm2 q 88 cm2 0:66 m2 s 50 ha t 5:5 km2 u 0:001 m2

1 a cm3 b mm3 c cm3 d m3 f cm3 or mm3 g cm3 h m3

e m3

2 a 0:008 cm3 b 60 000 cm3 c 11800 mm3 d 640 mm3 e 3 000 000 000 mm3 f 7 500 000 mm3 3 a 0:5 cm3 b 7 cm3 c 5 m3 e 0:002 m3 f 5:4 m3

d 0:45 m3

6 a i 540 ii 280 b 16:4 m2 c $506:76 d i 164 m ii 82 m e d i is double d ii

4 a mL b mL or L c kL d L e mL f mL g kL h mL i kL j mL k ML l ML m ML n kL o ML p ML q mL r mL

Exercise 60

5 a 5600 L b 3:54 L c 0:76 ML d 7:2 L e 6:3 m3 f 12 400 000 mL g 62:5 mL h 400 mL i 3500 kL

1 a 720 cm2 2 a 225 m2

b

504 mm2

b 70:56 cm2

3 a 200 cm2 b 64 cm2 e 198 m2 f 400 cm2 4 a 12 m by 1 m, 26 m; 4 m by 3 m, 14 m

c 48 km2

6 1650 mL 7

c 4 ha c 28 m2

d

92 m2

6 m by 2 m, 16 m;

b 36 cm by 1 cm, 74 cm; 18 cm by 2 cm, 40 cm; 12 cm by 3 cm, 30 cm; 9 cm by 4 cm, 26 cm; 6 cm by 6 cm, 24 cm

650 mL 8 6:75 L 9 1600 bottles

Exercise 63 1 a

63 units3

2 a

60 cm3

b 64 units3 b 420 m3

c 96 units3

c 110 cm3

3 a 50 m3 b 48:44 cm3 c 315:4 m3 e 73:44 m3 f 108:63 m3 4 a

36 cm3

b 140 m3

d 40 units3

c 80:64 cm3

d 1332 cm3

226

ANSWERS

5 a 48 £ 1 £ 1 b 162 £ 1 £ 1 81 £ 2 £ 1 24 £ 2 £ 1 54 £ 3 £ 1 16 £ 3 £ 1 27 £ 6 £ 1 12 £ 4 £ 1 18 £ 9 £ 1 8£6£1 27 £ 3 £ 2 12 £ 2 £ 2 9£9£2 8£3£2 18 £ 3 £ 3 6£4£2 9£6£3 4£4£3

5 a

a i 24 £ 1 £ 1 ii 40 £ 1 £ 1 iii 64 £ 1 £ 1 32 £ 2 £ 1 20 £ 2 £ 1 12 £ 2 £ 1 16 £ 4 £ 1 10 £ 4 £ 1 8£3£1 8£8£1 8£5£1 6£4£1 16 £ 2 £ 2 10 £ 2 £ 2 6£2£2 8£4£2 5£4£2 4£3£2 4£4£4 b 1 cm 24 cm

6 cm

3 cm 3 cm 2 cm

2 cm

4 cm

2 cm

6 cm

7 a 210 mL b 600 mL c 216 mL 8 72 L 9

) 66 times 10 210 kL

1 a 66 m2 2

3 7:875 m

b $825 2 a 4400 m2 2

4 a 11 m

88 truckloads

b

b 300 rolls

5 a 100 m b 10 m c 40 m d 5 m e 12:5 m f 8 m g 20 m h 800 m 6 a 13 slats b 1:716 m2 11

7L needed

9 posters b 5400 cm2 (or 0:54 m2 )

8 a

100 stamps b $25 9 67:5 kL

b 320 mm2

1 a

0:34 ha

2 a

10:5 cm2

4 a 5 a

45 kL b 3

36 units

b

32 cm2 3

8:9 cm

c

7 200 000 mm2

3 a 92 m2

b 329 m2

c 4600 L 3

b 0:36 m

c 100 m3

6 1000 containers 7 4 prisms 8 1731 kL 9 $42:75 10 37 m2

Exercise 66 1 cm

1 a-f

1 cm

All of these samples are unrepresentative as they are not randomly selected. The sample groups named may be likely to share a particular view on the subject under consideration. For example, farmers may be more likely than the population as a whole to believe that the government should give drought assistance to farmers.

Pick a page at random from the Electoral Roll and then pick a name at random. Repeat until 400 names have been obtained. b Remove a bottle every 2 or 3 minutes from the assembly line. c Use the school’s enrolment list numbers. Place the numbers in a hat and randomly select 30 of them. d

Use a dictionary. Randomly select a page and then randomly select a word on it. Repeat this process.

Mix the tickets in a hat and take out one before identifying it. b Place A on one side of a coin and B on the other. Flip the coin. c Use a die. Roll it.

3 a

Exercise 64

8 18 m2

b 6:27 L )

47 m

7 a

2 a

1 cm

8 cm

66 23

c 7:528 L

6 a

1 cm 4 cm

12 cm

2

Review set 8B

6 Your prisms should have dimensions

2 cm

0:38 L b 5:4 m3

9 180 000 m3

7 1350 cm3 or 1:35 L

13

3 3 3 3

6

6

3

3

3 3

3

i.e., 36 of them.

6

3 3

6

2 a 56 cm2

b

Make a sample batch of 8 or 10 of each colour for your distributors to survey their customers to see which colours are most popular.

c

Conduct some research into the new area to see what shops are currently there and the ages and types of people visiting the area during opening hours.

b 54 m3

5 a

10 000 b 300 c 12% d 1200

6 a

750 b 50 c 34% d 255

7 a

85 b 5231 c 71:8% d 1477

1 a

b 0:357 m2

b 75 cm2

4 a 120 units3

Conduct a survey of people at the local shopping centre at various time intervals on different days.

Exercise 67

Review set 8A 1 a 35 600 m2

4 a

3

3 3 3

3

Shuffle a pack of cards, select one at random before identifying it.

10 30 000 bottles

a 630 L b 78 34 buckets 12 510 fuel tanks

6

d

c 7200 mm3

3 a 24:8 m2 c 199:52 cm3

b 57 m2

Eye Colour Brown Blue Green Grey

Tally © j © jjjj jj j jjjj Total

© © jjjj © © jjjj © © jjjj

Frequency 11 7 6 4 28

b brown

227

ANSWERS

Grade A B C D E

Tally jjj © © jjjj © © © © © j © jjjj jjjj jjjj jjj j Total 5 28

b i 16 ii 3 a

Frequency 3 5 16 3 1 28

6 a

c C

Grade excellent good satisfactory unsatisfactory

Tally jjjj © © © j © jjjj jjjj © jj © jjjj jjj Total

Frequency 4 11 7 3 25

b

b 5 a

b

No. of children 0 1 2 3 4 5 6

i

9 ii

Tally jjjj jjjj © jjjj © jjjj jjjj © j © jjjj jj j Total

20 c 9 d

Tally j jjjj © j © jjjj jjj jjjj j j Total

Frequency 1 4 6 3 4 1 0 1 20

Hockey goals frequency

2 1 0

0 1 2 3 4 c 12 times d 60% 7 a

No. of goals 0 1 2 3 4 5

b i

5

Tally jjj jjjj © j © jjjj © j © jjjj jj j Total

6 games ii

6

7

8

9 10

goals

Frequency 3 4 6 6 2 1 22

9 games

Exercise 68 1 a

Eye colour 15 10 5

Ages of children at party

8

0

6

brown

blue

eye colour green

grey

b brown c 25%

2 0

Frequency 1 2 4 3 3 2 5 1 2 1 1

4

45%

4

Tally j jj jjjj jjj jjj jj © © jjjj j jj j j

3

2 15

Ages of children 10 11 12 13 14 15 16 17

e

Frequency 4 4 9 4 6 2 1 30

6 5

b good c To see if improvement in service is needed. 4 a

Goals 0 1 2 3 4 5 6 7 8 9 10

frequency

2 a

10 11 12 13 14 15 16 17

2 a Mitsubishi 20, Ford 28, Holden 35, Toyota 25, Other 12 b Holden c 23% 3 a 1993, 1994, 1995, 1998 b The business broke even. c $7 million profit 4 a b c d

shop A, June; shop B, January; shop A, $6000; shop B, $5000 Increase in length of bars from February to June for A. Shop A, $28 000; shop B, $27 000

5 a

garden b cleaning c i 156 kL ii 20 kL

6 a

size 14 b 70 women

228

ANSWERS

b

Votes awarded

frequency

7 a The ways (and percentages) in which ‘City Council’ spends its money. b It saves the reader having to do the calculations. c i 8% ii 4% d i $26mill ii $69mill e Water and Sewerage f $1095mill

0

1 4 2 4 3 34:75 4 a X, 6:5; Y , + 7:64 b false c Y

c 30

25:4

7 $230:33 8 25:35 kg 9 a 4 b 7 12 10 mean + 3:1, median

1 12 .

c

4

The mean is better.

a 9 12 runs b 21:3 runs c median

12 a size 8 b 13 a i

size 8 12

2 goals ii 2 goals b i 11 years ii 12 years

14 a 7P , 20 and 11; 7Q, 20 and 10 b 7P , mean 16:4 marks, median 16 marks 7Q, mean + 14:4 marks, median 14 marks c 7P

rainfall (mm)

1 a 120 beats/min b 80 beats/min c from 0 to 3 min d 40 beats/min decrease e 140 beats/min f yes

5 10 d 5 e 32 f 75% g i

votes

15 20 week 1:6 votes ii 2 votes

3 a i 10 days ii 7 days b i day 17 ii day 28 iii day 17 iv day 2 c 5 days 4 a lung cancer b lung cancer and chronic bronchitis/emphysema c i 4800 ii 5400 iii 1600

1 a

... sectors ... b ... frequency c ... height of columns ...

2 a Month 1 2 3 4 5 6 7 8 9 10 11 12

300 200 100

J FM AM J J A S O N D

month

3 a A break in the scale at that place. Scale does not start at 0. b February c June to September d Temperature is getting colder from end of summer to end of winter. e + 21:1o C 4 a Income $9000, costs $5000 b $4000 c from week 5 to week 9 d i weeks 4 and 5 ii week 13 Sales figures appear to rise one quarter and fall the next, for each successive quarter shown. This may be due to customers being more likely to buy airconditioning systems at the beginning of the hot season or at the beginning of the cold season, rather than partway through the season. ii Sep 02 b Sales are increasing each quarter overall.

5 a i

Review set 9A Tally

© © jjjj jjjj © © jjjj © j © jjjj Total

Frequency 5 4 5 6 20

Tally G B j j jj jjj jj jj jjj j j

j j jj jj

jjj jj j j j Total

G 1 0 2 3 2 2 3 1 1 0 0 1 16

Frequency B Total 1 2 0 0 1 3 1 4 2 4 0 2 2 5 0 1 3 4 2 2 1 1 1 2 14 30

2 b i 30 ii 16 iii September iv 15 v February vi 10% vii February, October, November

3 a

177 12 cm b 179 cm

4 a

Teachers ages

12

frequency

8 4 0

b i

1 a True b False c True No. of votes 0 1 2 3

3

total votes

0

b May to August c November d + 50%

2 a

2

10

400

0

1

Review set 9B

Exercise 70

2 a

0

20

6 a Sally, 26:4; Jan, 23:5; Jane, 25:3; Peta, 31:3; Lee, 25:9; Polly, 28:4; Sam, 32:8 b Sam

11

4 2

Exercise 69

5 a 37:4 b

6

20s

30s

40s

50s

60s

ages

45 12 years ii 44:7 years

5 a i airconditioners ii hot summer, or cold winter b $85 000 c $215 000 d $50 Exercise Exercise7272 1 a 2 hours b 3 days c 4 minutes d 2 fortnights e 11 weeks f 12 years g 4 years h 9 decades i 8 centuries j 3 millennia k 3 years l 10 800 sec 2 a 150 sec = 2 min 30 sec b 90 min = 1 hour 30 min c 200 min = 3 hours 20 min

ANSWERS

d e f g h i j

53 days = 7 weeks 4 days 56 months = 4 years 8 months 73 years = 7 decades 3 years 60 hours = 2 days 12 hours 500 sec = 8 min 20 sec 7300 years = 7 millennia 3 centuries 160 months = 13 years 4 months

3 a 1440 min b 10 080 min c 525 600 min 4 a 86 400 sec b 1 209 600 sec c 31 536 000 sec 5 a 1461 days b 35 064 h

c 2 103 840 min

6 a 8 h 30 min b 13 h 17 min c 19 h 23 min d 4 h 17 min e 4 h 49 min f 8 h 22 min 7 a 444 min b 4663 min c 18 216 min d 24 977 min 8 a 2438 sec b 12 927 sec c 51 163 sec d 82 331 sec 9 a 5 days 4 h b 23 days c 36 days 9 h d 90 days 7 h

2 a e i m

3:00 am b 6:15 am f 2:50 pm j 11:45 am

6:30 am c 6:00 pm d 3:45 pm g 8:17 pm h 12:30 am k 7:20 am l n 2:35 pm o 2:00 am

229

12:00 noon 11:48 pm 11:30 pm p 12:15 pm

3 a 1030 h b 1825 h c 1220 h d 1045 h e 2140 h f 0012 h 0930 h b

4 a

1240 h c 1915 h

5 a More than 60 minutes is not possible. b 0713 h is correct. c Greater than 24 hours in a day is not possible. 6 a

Before Christ b

125AD c 1800 years

7 a

Elizabeth II b 14 years c 18 years

8 a

Zhou Warlords b 425 years c 450 years

9 a

1991 b 9 years c 2007

Exercise 75

10 a 47 days b 22 days 37 min

10:35 am, 11:45 am, 12:50 pm, 1:50 pm, 2:25 pm, 2:45 pm, 3:15 pm, 4:10 pm b 2:25 pm c 4:10 pm d i 11:10 pm ii 8:50 pm

1 a

Exercise 73 1 a b c d

Wei joined the club on the 17th December 1999. Jon arrived on the 13th March 2000. Piri is departing for Malaysia on the 30th July 2004. Sam will turn 21 on the 28th May 2009.

2 a 19th August b November 12th c 8th February d 21st September e 15th March f 17th June g 14th May h 24th July i 1st July j 28th Sept. 3 a 27 days b 43 days c 117 days d 111 days e 68 days f 179 days g 119 days h 99 days 4 a 45 days b $6:20 5 a 195 days b $3510 c Yes, $1939 6 19th October 1996 7

Sunday 8 Thursday

9 a 7:00 am b 1:00 am c 6:49 am d 8:06 pm e 10:32 pm f 4:09 pm g 11:05 am h 6:42 am i 11:43 am j 10:44 pm Sunday 10 a 8 h 19 min b 3 h 19 min c 7 h 17 min d 12 h 52 min e 20 h 9 min f 9 h 37 min g 26 h 48 min h 87 h 54 min 11

3 one way trips 12 25 13 1200 sec = 20 min

14 a b c d e f g h i

Bus departs 9:15 am 6:25 pm 7:15 am 12:25 pm 1:50 pm 8:25 pm 9:00 pm 11:18 pm 10:13 am

Bus arrives 12:30 pm 9:50 pm 9:05 am 2:50 pm 3:20 pm 10:40 pm 12:15 am 1:50 am 12:38 pm

Time taken 3 h 15 min 3 h 25 min 1 h 50 min 2 h 25 min 1 h 30 min 2 h 15 min 3 h 15 min 2 h 32 min 2 h 25 min

2 a 6 b 8:45 am c 5:00 pm d i 1 h 55 min ii 40 min e 9 h 30 min f bus C g bus B 3 a c e f

i arrival time ii departure time b 4:50 pm 5:27 pm d 6:20 pm i 4:11 pm ii 4:36 pm iii 6 min i 45 min ii 51 min iii 44 min There would be more trains on the track and more passengers for the 5:23 pm train (peak hour).

4 a 7:21 am b 9:08 pm c 0:9 m, 3:20 am d 1:2 m, 1:46 pm

Exercise 76 1 a

3 pm b 8 pm c 10 pm d

8 am

2 a 2:00 am Tuesday b 5:30 am Tuesday c 9:00 am Tuesday d 12 midnight at the end of Monday 3 a 5:00 pm Tuesday b 2:00 pm Tuesday c 8:00 am Wednesday d 12 midnight at the end of Tuesday 4 a 12:15 pm Sunday b 8:15 am Sunday c 10:45 pm Saturday d 8:45 pm Saturday 5 a 9:30 pm Friday b 8:00 pm Friday c 12:00 noon Friday d 4:00 am Friday 6 a 3:30 pm

b 4:00 pm c 4:00 pm d 4:00 pm

7 a 3:50 am

b 3:50 am c 2:20 am d 4:20 am

8 e.g.,

10:30 am 11:00 am

9:00 am

15 171 sec 16 7:45 am, 2:05 pm, 8:25 pm (Mon), 2:45 am, 9:05 am, 3:25 pm, 9:45 pm (Tue), 4:05 am (Wed)

11:30 am

17 1440 times

a 12:30 pm b 2:00 pm c 2:30 pm d 3:30 pm e 3:30 pm

Exercise Exercise7474 1 a e i m

0313 h 1741 h 0215 h 1830 h

12:00 noon

b 1117 h c 0000 h d 1247 h f 1200 h g 2019 h h 2359 h j 2125 h k 2252 h l 0015 h n 1255 h o 0806 h p 0200 h

9 a

8:10 am b 6:10 am c 7:40 am

10 a 8:30 am b 11

d 8:10 am

5:50 pm c 5:30 pm d 8:00 pm

a 2 pm Sunday b 12:30 am Monday c 4 am Monday d 11 am Sunday

230

ANSWERS

Exercise 77

16 a 120 m3

1 a 90 km/h b 70 km/h c 83 km/h d 94 km/h

17 a five minutes to six in the afternoon b 5:55 pm c 1755 h

2 a 630 km b 450 km c 900 km d 315 km e 1026 km 880 km c

441 km d 171 km

4 a 3 h b 6 h c 5 12 h d 8 h 20 min e

3 h 15 min

7 Charlotte travelled at the greater speed (19:5 km/h) which was + 1:2 km/h faster than Zoe. 122o F b 176o F c 68o F

8 990 km 9 a

20 a tennis b 21

5 9 km/h 6 8 hours

4 3 2 1 0

13 a 127:4o F b 56:7o C

0

1

ii 0015 h

6 360 bricks

7 a 176 days b $2640 c $360 8 31st May 9 a 80 km/h b 0:36 h = 21 min 36 sec c 2:36 h + 2 h 22 min d 196 km e 83:1 km/h 8:30 pm Saturday

Review set 10B

Exercise 79 1 a

,

1 a 107 days b 61 h c 81 min d 300 sec

Unit no. Matches

3 9

c

Unit no. Matches

8 24

2 a

i ii, iii

4 a 21=11=1963 b 17=10=2010 b

h

9 a 8 days 3 h 19 min b Vostok 6 c 97 h 35 min d 8 yr 3 months 4 days

c

b 3550 g c 0:56 m

2

2 a 2:5 cm

b 50 000 m2

3 a 192nd

b 20th c 2nd

4 a height b 7 a 67:2 cm 8 a 43:2 m

2

c 3:8 m3

ii, iii

6 63 years

b 14 km c 30 m 2

b 17:5 m

11

3 8

full b 45 km/h c

5 14 kg d

1 2

L

a true b false, you would conduct a survey

12 a 72 days b 17 hours c 13 88 units3 15 a i

265 min d 3720 sec

14 a 16:5 mm b + 14:3 mm

100 m ii 25 m b i 1 cm ii 2:5 cm

1 5

Unit no. Matches

2 8

3 11

11 33

iii

iii 600 matches

4 14

5 17

1 4

Unit no. Matches

2 9

3 14

4 19

,

5 24

6 29

7 34

6 25

7 29

, .....

1 5

Unit no. Matches

2 9

3 13

4 17

5 21

, 1 3

Unit no. Matches

2 7

, ..... 3 11

4 15

5 19

, .....

2, 5, 8, 11 ii 299

,

,

, .....

,

iii

iii

7 23

, .....

c i 6, 10, 14, 18 ii 402

d i

6 20

,

i 4, 7, 10, 13 ii 301 iii , ,

b i

9 a 2600 kg b 500 m c 0:846 m2 10 a

3 a

2

c 20:5 cm

10 30

7 21

, .....

i

tally or frequency

b 8:2 cm3

5 a 0:64 L

d

42 000 cm2

c

9 27

6 18

,

i ii, iii

Review of chapters 7, 8, 9 and 10 1 a 3:62 km

5 15

i ii, iii

6 a 13 h 30 min b 6 days 1 h 3 min c 1 h 18 min 7 94 km/h 8

4 12

,

d i 90 matches ii 150 matches iv 3 000 000 matches

3 a 67 years b 151 years c 39 years 5 a fifty eight minutes after 2 o’clock in the afternoon b 2:58 pm c 1458 h

,

b

2 a 179th decade b 18th century c 2nd millennium

5 12

no. of calls

4

24 a 71 km/h b 5 12 hours

4 a 20 h 41 min b 3 h 38 min

10 a 2:00 pm Saturday b

3

23 a 6 hours 20 minutes b 1 hour 43 minutes

2 a 49 days b 720 sec c 555 min d 1000 years

5 a 8 h 20 min b 12:13 pm

2

22 2000 steps

1 a hours b minutes c seconds d years e centuries f seconds 3 a i 6:45 am ii 0645 h b i 12:15 am c i 9:30 pm ii 2130 h

12 students c 60 students Emergency calls

10 a 38o C b 10o C c 27o F

Review set 10A

c 780 cm3

18 4=12=2003 19 120 m2

frequency

3 a 255 km b

b 346:5 cm3

,

, .....

1, 5, 9, 13 ii 397

,

,

, .....

6 23

7 27

231

ANSWERS

4 a 16, 19, 22; The next number is equal to the previous number plus 3. b 31, 35, 39; The next number is equal to the previous number plus 4. c 37, 44, 51; The next number is equal to the previous number plus 7. d 30, 36, 42; The next number is equal to the previous number plus 6. e 49, 58, 67; The next number is equal to the previous number plus 9. f 59, 72, 85; The next number is equal to the previous number plus 13.

3 a

b i

e 218, 210, 202; The next number is equal to the previous number minus 8.

d i

4 a

c i

d i

The next number is equal to the previous l 3, 1, number divided by 3.

5 12 ,

8 12

7, d 45, 34, 23 6 a 13, 19, 25 b 12, 21, 30 c e 125, 100, 75 f 3:3, 2:8, 2:3 g 11, 25, 53 h 26, 256, 2556 i 49, 25, 13 j 4, 16, 256 7 a 2 = 15 b 2 = 24 c 2 = 45 d 2 = 12 e 2 = 15 f 2 = 81 g 2 = 8 h 2 = 21 i 2 = 120 j 2 = 25 k 2 = 23 l 2 = 24

, n M

1 2

, 2 4

3 6

, .....

4 8

n M

1 4

2 5

3 6

4 7

n M

1 1

2 4

3 7

4 10

n M

1 1

2 5

3 9

4 13

ii The number of matchsticks is four times the unit number minus three. iii , , , , .....

1 ; 3

o 21, 34, 55; After the first 2 members, each number is the sum of the two previous numbers.

M = 4 £ n + 3, where M is the number of matchsticks and n is the unit number.

ii The number of matchsticks is three times the unit number minus two. iii , , , , .....

The next number is equal to the previous k 5, number divided by 2.

n 23, 30, 38; Each number is increased by one more than the previous number is increased.

, .....

ii The number of matchsticks is three more than the unit number. iii , , , , .....

1 14 ;

m 0:025, 0:0025, 0:000 25; The next number is equal to the previous number divided by 10.

i

b i

i 512, 2048, 8192; The next number is equal to the previous number multiplied by 4.

2 12 ,

,

ii The number of matchsticks is two times the unit number. iii , , , , .....

h 162, 486, 1458; The next number is equal to the previous number multiplied by 3:

j 2, 1, 12 ; The next number is equal to the previous number divided by 2.

,

ii

f 45, 41, 37; The next number is equal to the previous number minus 4. g 32, 64, 128; The next number is equal to the previous number multiplied by 2.

,

c i M = 3 £ n ¡ 1, where M is the number of matchsticks and n is the unit number. ii , , , , .....

c 33, 27, 21; The next number is equal to the previous number minus 6. d 88, 85, 82; The next number is equal to the previous number minus 3.

M = 3 £ n + 2, where M is the number of matchsticks and n is the unit number.

ii

5 a 28, 26, 24; The next number is equal to the previous number minus 2. b 17, 14, 11; The next number is equal to the previous number minus 3.

i M = 2 £ n + 1, where M is the number of matchsticks and n is the unit number. ii , , , , .....

5 a b

, n M

1 4

2 6

, 3 8

4 10

, .....

5 12

6 14

c M = 2 £ n + 2 d 48 6 a b

, n M

,

1 3

2 5

3 7

, ..... 4 9

5 11

6 13

c M = 2 £ n + 1 d 87

,

7 a

, .....

Exercise 80 1 a M = 3 £ n, where M is the number of matchsticks and n is the unit number. b

,

,

,

, .....

2 a M = 5 £ n, where M is the number of matchsticks and n is the unit number. b

,

,

,

, .....

b

n M

1 4

2 10

3 16

4 22

5 28

c M = 6 £ n ¡ 2 d 340 8 a M = 4 £ n + 1; when n = 218, M = 873 b M = 7 £ n + 1; when n = 218, M = 1527

232

ANSWERS

Review set 11A

Exercise 81 1 a $60 b 2 a

$100 c $140 0 8 6 26

Hrs of sunlight Height (cm) Hrs of sunlight Height (cm)

1 11 7 29

2 14 8 32

3 17 9 35

4 20 10 38

5 23

1 a 29, 38, 47 b 160, 320, 640 c 7 12 , 9 12 , 11 12 d 5, 2 12 , 1 14 2 a b

b H = 11 + (n ¡ 1) £ 3 where n = hours of sunlight, n > 1 H = height of bean 3 C = 75 + 42 £ n where

n = number of months C = cost

Time (h hours) Flow (F cumecs)

0 8

1 10

2 12

3 14

4 16

Time (h hours) Flow (F cumecs)

5 18

6 20

7 22

8 24

9 26

b F = 8+2£h

where

c 26 cumecs

3

100

2 7

3 10

4 13

5 16

2 50

3 65

4 80

5 95

6 110

b $425 4 a

15

charge ($C)

10 5 0

0

1

2

3

4 5 6 no. of videos (n) b C = 2 £ n + 1 c i $19 ii $35 5 6 £ n ¡ 1 = 47, n = 8 6 a

60

f = 35 b n = 42 c b = 7

Review set 11B

40

1 a

20

2 a

0

1 35

n C

h = time in hours F = flow in cumecs

profit ($P)

80

1 4

Unit no. Matches

a C = 15 £ n + 20 dollars where C is the total cost and n is the number of rooms.

5 a i $17 ii $31 iii $37 b The graph does not show values of n greater than 20. 6

, .....

c M = 3 £ n + 1 d i 22 ii 304

a For n = 7, C = $369 b For n = 12, C = $579 c For n = 18, C = $831 d For n = 60, C = $2595 4 a

,

0

a i $37 ii

10

20 30 no. of spanner sets (n)

3 a

Exercise 82 1 a a = 3 b b = 5 c x = 21 d a = 15 e b = 9 f x = 21 g a = 5 h b = 9 i x = 20 j a = 25 k b = 24 l x = 7 2 a a = 12 b a = 5 c a = 2 d x = 10 e x = 4 f x = 5 g y = 3 h y = 12 i y = 0 3 a x = 7 b x = 12 c x = 4 d x = 2 e x = 7 f x=8 g x=9 h x=0 i x=3 4 a x = 6 b x = 11 c x = 10 d x = 7 e x = 7 f x = 10 g x = 12 h x = 9 i x = 13 5 a d g j

n ¡ 5 = 16 b n ¡ 8 = 30 c n + 3 = 31 n + 8 = 11 e 7 £ n = 35 f 2 £ n = 26 n ¥ 5 = 7 h n ¥ 3 = 9 i 7 £ n ¡ 6 = 64 8 £ n ¡ 5 = 27

6 a c e g h

n ¡ 10 = 11, n = 21 b n + 3 = 17, n = 14 9 £ n = 63, n = 7 d n ¥ 9 = 6, n = 54 8 £ n ¡ 6 = 50, n = 7 f 4 £ n + 3 = 23, n = 5 3 £ n + 1 = 64, n = 21 5 £ n ¡ 11 = 54, n = 13

7 a 21 = 3 £ d + 6 b 5 days 8 a 465 = 45 £ x + 60 b 9 hours 9 a 500 = 800 ¡ 15 £ w

b 20 weeks

1 6

n M

2 11

3 16

4 21

5 26

6 31

b M =5£n+1 c The number of matchsticks is five times the number of units plus one. d 401

$61 iii $73 b iii $112

7 a i 8 cm ii 14 cm iii 32 cm b i 8 weeks ii 12 weeks iii 17 weeks

11, 19, 35 b 59, 115, 227

0 150

n C

1 190

2 230

3 270

4 310

b C = 40 £ n + 150 c i $190 ii $270 iii $350 4 a

i $28 ii $52 b C = 8 £ n + 4 c $116

5 a

637 = 21 £ x + 532 b 5 hours overtime

6 a

k = 35 b m = 3 c d = 15

Exercise 84 1

6

G

5

E

4 3 2 1

B

A I 1

J D F

C 2

3

H 4

5

6

7

8

a rectangle b square c yes

9

2 a B b W c Q d P e Y f T g H h A i G j O k R l U m S n L o I p K q E r D s M t N u X v C w J x V 3 a

i iv b i iv

(7, 5) ii (8:6, 1:6) iii (6:5, 6:5) (2:3, 1:2) and (6:1, 6:2) Treasure Trove ii Lion’s Den iii Mt Ogre Oasis

ANSWERS

4 a i (1:5, 4) ii (10:7, 0:4) iii (11, 9:7) iv (8:4, 3:3) b i Kalgoorlie ii Alice Springs iii Mt Isa iv Brisbane

c

6 a b c e

Vintage Machinery Investigator Science & Technology Centre Ridley Centre d Dairy Cattle Police/Lost children f Horse Warm-up Area

d

270°

O

5 a A8 b B2 and B3 c G5 d G6 e B4 f E8 g E6 h B6 and C6 i F7 j H7

O

e 90°

7 a Main Entrance b Goodwood Rd Entrance

360°

f

O

O

Exercise 85 1 a c e f

translation b reflection along vertical axis translation d rotation 180o rotation or reflection in the hypotenuse enlargement

2 a

b

270°

g

h

c O

MAHS SHAM

3 a i

233

ii

O

180°

3 a

90°

b

O

O

iii

c

4 a

d

b

O

O m e

m c

O

f

O

d

4 a

m

2 b 4 c 2 d

2 e 4 f 4 g 2 h 6

Exercise 87 e

1 a 7 units left, 1 unit up b 7 units right, 1 unit down c 3 units right, 4 units down d 3 units left, 4 units up e 4 units left, 3 units down f 4 units right, 3 units up

f

2 a

b

c

d

Exercise 86 1 a B b D c A d C 2 a

b

90°

O

180° O

234

ANSWERS

e

f

2 a

3 b 1 12

4 a

1 3

1 2

c 1 2

b 3 c

3 a 3 b

1 3

1 4

c 2 d

4

d

Exercise 89 1 a 9 : 7 b 5 : 3 c 14 : 5 d 8 : 7 e 2 : 11 f 3 : 4 g 95 : 100 h 7 : 24 i 500 : 78 j 500 : 2000 k 300 : 1000 l 100 : 20

3 a 4 units right b 3 units down c 3 units left, 2 units up d 4 units right, 2 units down e 4 units right, 2 units up f 2 units right, 2 units down 4 B and C 5 a

2 a b c d

i 2 : 3 ii 6 : 5 i 7 : 5 ii 8 : 7 i 2 : 3 ii 4 : 1 i 18 ii 18 iii

iii 6 : 9 : 5 iii 5 : 7 : 8 iii 1 : 3 : 1 18

3 a c d f g

Tom : Tamara = 8 : 13 b boys : girls = 3 : 1 blue eyes : hazel eyes = 3 : 5 cars : motorbikes = 15 : 1 e books : software = 3 : 1 Power supporters : Crows supporters = 7 : 8 William’s weight : Thomas’ weight = 4 : 5

4 a 12 : 12 b 8 : 16 c 18 : 6 d 16 : 8 e 9 : 15 f 6 : 18 g 6 : 18 h 4 : 8 : 12 i 3 : 6 : 15 j 3 : 9 : 12 5 a 210 pkts b 540 pairs c 125 girls passed d 175 bring lunch e i 175 m ii 500 m 6 a

120 mL b 4 L

7 a

1 4

b

b i $21 000 ii $5250 c $32

Exercise 90 1 a N70o E b f N75o E 2 a

S30o W c N50o W d S60o W e

N

N

b

B

B

40° 6 a 1 unit right, 1 unit down b 3 units right c 3 units down d 1 unit right, 1 unit down

1 a

c

b

scale factor Qw_

45°

f

3 a

N

Q 84°

M

O

X

80°

S

65°

W

C

N

S

N P

Y

e

d

d

D

e

scale factor 3

50°

A

N

S scale factor 2

E

A c

Exercise 88

S65o E

i F5 ii F3 iii D1 b i N ii SW

4 a 070o T b 210o T c 310o T d 240o T e 115o T f 075o T

scale factor Qe_

5 a

b

N 70°

scale factor 4

c

N 160°

N 213°

f d

N

e

312° 6 a

scale factor 2

N 96°

West b True North

7 a 000o T b 045o T c 090o T d 135o T e 180o T f 225o T g 270o T h 315o T

235

ANSWERS

8 a i H9 ii G5 iii K2 iv D2 b i 068o T ii 127o T iii 307o T iv 349o T

5 a

b

Exercise 91 1 a N b SE c i 041o T

O

ii 333o T iii 096o T

2 a i 064o T ii 325o T b i 53 km ii 99 km 3 a i 1560 km ii 3200 km iii 3060 km b i 022o T ii 299o T iii 133o T

6 1 : 4 7 35 pasties

Review set 12A 1

8 a

C

6 5 B 4 3 2 A 1

14 km, South b 34 km, N 62o W c 20 km, N38o E

9 1 cm ´ 50 m

Exercise 93

a parallelogram

1 a highly unlikely b almost certain c certain d highly unlikely e unlikely f probable g highly unlikely

D

2

b

d

c

1 2 3 4 5 6 2 a

b

m

impossible

highly likely

50-50 chance

a

certain

3 a

very likely b no c true

4 a

No, there is one more green disc. b green c

false

5 a possible b possible c possible d possible e impossible f impossible g i possible ii impossible 3 a 6 b

6 a

0:5

g a

b

7

O

b i 1 ii 0

0

4 a

b

1

0.5

e

8 a

c

f

d

h

i Yes ii No, as some athletes are better than others. iii No, as some teams are better than others. iv Yes v Yes vi No, as they can buy more than one ticket.

Exercise 94 3 5 4 7 3 4

1 a i 3 blue, 2 white ii b i 4 blue, 3 white ii c i 6 blue, 2 white ii 5 88 pieces of meat

6 1:3 7

o

6

2 a

o

8 a i 244 T ii 299 T b 119 km

9 5:3 cm

Review set 12B 1

a b c d e

(6, (0, (5, (2, (3,

3) 2) 0) 6) 2)

2

7 6 5 4 3 2 1

D

C D(4, 5)

A

b

m

m

2 3

c

5 8

d

1 4

b

1 2

c 0 d

6 a

1 8

b

1 4

c

7 a

4 9

9 a 3 a

1 2

5 a

h

1 2 3 4 5 6 7

b

3

a

1 2

1 2

b

4 a No, as there are different sized sector angles. b i white ii black c 14 d 10 times

8 a

B

1 2

2 5 3 7 1 4

iii iii iii

1 4 1 13

b b i

1 3

3 8

1 52 3 13

c

5 8

d

2 9

c

1

d 0 e 1 4

e

0

5 9

f

1 26

d

e

2 3

g

7 9

1 26

f

2 13

iv

1 6

H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6. 1 12

b 12 c i

ii

1 4

iii

1 4

10 a A1, A2, A3, A4, B1, B2, B3, B4, B1, B2, B3, B4, C1, C2, C3, C4 b i 4 a

b

2

1 16

ii

1 4

iii

1 2

iv

3 16

h

1 3

g 0

i 1

236

ANSWERS

c

Exercise 95 coin H

T

b

1 2 3 4 5 6

H1 H2 H3 H4 H5 H6

1 2 3 4 5 6

T1 T2 T3 T4 T5 T6 rd

1 st coin H

T

2 nd coin 3 coin H H T H T T H H T H T T

c

1 st pick R

B

G

st

1 spin A

B

C

2 a i b

1 9

ii

1 9

st

1 child B

iii

5 9

1 8

ii

3 8

7 8

iv

G

outcome HHH HHT HTH HTT THH THT TTH TTT

1 16

6 16

ii

7 16

iii

iv

9 16

v

2 16

3

For example, coin with A on one side and B on the other or die with three faces A and three faces B.

4

A regular pentagonal spinner with two red and three blue parts or five identically shaped objects in a hat with two red and three blue.

5

A die with one face marked A, two marked B and three marked C or six identically shaped objects in a hat with one marked A, two marked B and three marked C.

6 a

a die b A on one face, B on two faces, C on three faces

7 a

G

RG

Seven identically shaped objects in a hat with one labelled X, two labelled Y, four labelled Z.

R

BR

Exercise 96

B

BB

1 a

G

BG

3 5 days 4 9 days 5 a

R

GR

6 a

B

GB

G

GG outcome AA

25 b B 2 a 15 times b 10 times c 5 times 0:75 b

7 a i 14 ii 38 iii 14 iv 18 b 100 1s, 150 2s, 100 3s, 50 4s

Review set 13A

B

AB

C

AC

2 a

A

BA

B

BB

C

BC

A

CA

B

CB

C

CC

outcome BBB BBG BGB BGG GBB GBG GGB GGG

v

7 8

180

i 0:25 ii 0:5 b 25 of each of A, B, C and D

1 a little chance b some chance c little chance d no chance

2 9

1 8

i

RB

3 rd child B G B G G B B G B G G

iii

P

RR

2 nd child B

G

i

iv

B

RR RB RP RG BR BB BP BG PR PB PP PG GR GB GP GG

B

2 nd spin A

d

R

outcome

2 nd pick R

R B P G R B P G R B P G R B P G

1 pick

die outcome

1 a

2 nd pick

st

12 days b

3 a i b i

1 2

i 0:75 ii

1 2

375 people

1 8

ii iii iv 38 v 18 20 ii 20 iii 5 iv 15 v 5

4 a i H : T = 8 : 2 = 4 : 1 ii H or T b i O : E = 5 : 5 = 1 : 1 ii odd or even c i R : B = 6 : 4 = 3 : 2 ii R or B 5 a

S, P, R, Q b i

6 a i b i

1 2 1 5

ii ii

1 4 1 5

iii iii

3 8 1 4 1 15

ii

1 4 1 8

iii

iv v iv 0

1 6 1 16

iv

1 8

Review set 13B 1 a

0 b 0:5 c 1 d 0:5

2 a

3 b

3 a

Pr[dark blue] = 14 , Pr[light blue] = 13 , Pr[black] =

1 2

c

5 16

d

1 2

b i D ii B iii A and E c i C ii C iii A and E d i C, reasonably likely; D, unlikely; E, unlikely ii B, impossible; C, unlikely; D, unlikely iii C, unlikely; D, impossible; E, impossible 4 a

iv ($63) b i $127 ii $255

1 12

ANSWERS

5 a

die 1

die 2

sum

1

1 2 3 4 5 6

2 3 4 5 6 7

2

1 2 3 4 5 6

3 4 5 6 7 8

3

1 2 3 4 5 6

4 5 6 7 8 9

4

1 2 3 4 5 6

5 6 7 8 9 10

5

1 2 3 4 5 6

6 7 8 9 10 11

6

1 2 3 4 5 6

7 8 9 10 11 12

11

b

1 4

c

4 a b

14 a

,

c M =2£n+1 d i 17 matchsticks ii

1 3

2 5

3 7

4 9

5 11

61 matchsticks

5 a 264 m b 20 cm 6 a

7 18

b

1 3

c 0 d

1 2

e 1

7 a x = 35 b d = 8 c p = 42 d t = 8 f y=8 8

5 4 3 C 2 1

A

B 1 2 3 4 5

9 a 30 days b 9 days 10

12 people

4 720

5 800

b

d

1 2

b

3 14

c

1 2

No. of 12 hours (n) Charge ($C)

0 200

1 240

b C = 40 £ n + 200 dollars c i

5 12

Figure number (n) Matchsticks needed (M)

3 640

O

48, 96, 192 c 30, 27, 24

d

2 560

c

15 a

5 6

1 480

m

13 a

2 a A is at (3, 5) b B is at (8, 2) c C is at (0, 4) d D is at (6, 0) 1 3

0 400

n E

b E = 80 £ n + 400 dollars c $1040

Review of chapters 11, 12 and 13

3 a

b 4

a

12 a

b 7, 8, 5, 3, 12

1 a 22, 26, 30 b

237

e n=9

16 a 12 sectors b i

1 4

ii

1 3

iii

2 280

3 320

4 360

$640 ii $1320 7 12

iv

2 3

v 1

238

ANSWERS

CORRELATION CHART: R-7 SACSA MATHEMATICS TEACHING RESOURCE

Correlation Chart: R-7 SACSA Mathematics Teaching Resource Knowledge, skills and understandings Core Skills STRAND: Exploring, analysing and modelling data Data collection and representation •understand the purpose of taking a sample population •explain the difference between a random sample and a biased sample •plan a range of ways to collect data (e.g., surveys, interviews) •record data using spreadsheets, and use simple formulae to create graphs using graphing software •construct and interpret pie graphs using graphing software •find the mean, median and mode from given data •interpret information from data, graphs and tables

Mathematics 7 Chapter Unit

239

Mathematics for Year 7 (second edition) Chapter Unit

9

66

9

A

9 9

66 66

9 9

A A

9 9 9 9

67 68 69 70

9 9 9 9

B, E C, D B, F D, G

13

94

13

A

13 13

97 97

13 13

F, G F, G

13

94

13

B

13

96

13

E

13 13

95 96, 97

13 13

C, D E, F, G

7 10

53 77

7 10

C G

8

61

8

C

8

59

8

A

8

60

8

A, B

8

60, 61

8

B, C

8 7

54, 57 55

8 7

G E

8 8

62 62

8 8

F E, F

8

64

8

G

8 8

63 62

8 8

E D, E, F

STRAND: Exploring, analysing and modelling data Chance and probability •identify risks and consequences of taking chances •demonstrate an understanding of what constitutes gambling (e.g., lotto, raffles, poker machines) •identify some of the social consequences of gambling •assign numbers and percentages to chance (i.e., if it has no chance of occurring it is assigned 0 or 0%; if it is certain to occur it is assigned 1 or 100%) •make your own probability generator (e.g., a spinner or a die to show P(red) =\ Wt_\) •assign probabilities for given situations (e.g., ‘Five discs are placed in a bag, two are blue and three are black. What is the probability of drawing a blue disc?’) •test predictions (e.g., coin tossing)

STRAND: Measurement Length, perimeter and area

•convert between millimetres, centimetres, metres and kilometres (e.g., 25 mm = 0.025 m) •use formula Distance = Speed × Time to solve problems •develop and use the formula for the area of a triangle (e.g., A = ½ B × H = B × H ÷ 2) •use the appropriate units of measurement (e.g., km², cm², m², mm², ha) •use appropriate strategies and devices to estimate and accurately measure the area of a shape (e.g., using an overlay grid) •calculate the area of irregular shapes by separating them into simple parts •demonstrate understanding of the relationship between perimeter and area through practical problem -solving activities •use scale in ratio form to calculate either original size or drawing size

STRAND: Measurement Volume and capacity •convert mL to L and L to kL and vice versa •use the symbols cm³, m³, mL, L and kL •demonstrate understanding of volume through practical problem solving activities •develop and use formula for volume of rectangular prisms: V = L × W × H or V = L × B × H •demonstrate awareness that capacity is related to volume

240

CORRELATION CHART: R-7 SACSA MATHEMATICS TEACHING RESOURCE

STRAND: Measurement Mass

Core Skills Mathematics 7 Chapter Unit

•choose the appropriate units and tools to measure mass of a variety of objects •identify the relationships between milligrams, grams, kilograms and tonnes (e.g., 1 kg = 1000 g, 1 t = 1000 kg, 1 g = 1000 mg) •apply the knowledge of mass to practical problem-solving situations

Mathematics for Year 7 (second edition) Chapter Unit

7

56

7

F

7 7

56 57

7 7

F H

10 10 10

76 75 73

10 10 10

F E A

10 10

72 77

10 10

B G

10

77

10

H

10

H

STRAND: Measurement Time

•make comparisons between time zones in Australia and calculate changes incorporating daylight saving •read and use a variety of timetables •construct and interpret timelines using an appropriate scale •explain ways in which time is measured in other cultures (e.g., calendars which are calculated by moon cycles) •use Speed = Distance/Time to answer problems

STRAND: Measurement Temperature •interpret the terminology Fahrenheit, °F •use online resources to compare current temperatures in different parts of the world

STRAND: Number Whole numbers

•develop an understanding of number systems across time and place (e.g., Mayan, Chinese) •recognise, use and write in words and numbers beyond 1 000 000 •identify place value of numbers over 1 000 000 •compare numbers and use symbols (e.g., +, (), >, 6) •write numbers up to 1 000 000 in expanded form (e.g. using powers of 10) •use powers or index (exponents) notation •write numbers over 100 000 in ascending and descending order •identify large numbers in everyday use •identify factors, common factors, prime factors, highest common factor and lowest common multiple •use arrays and divisibility rules •identify triangular and cubic numbers •apply square root to square numbers and use symbol ~` •solve a given 2-step number or word problem (e.g., ‘A school has a total of 854 students - 102 boys and 84 girls leave. How many students are left at the school?’) •multiply a 3 digit number by a 2 digit number using the extended form (long multiplication) •divide a number with 3 or more digits by a single digit or multiples of 10 (with a remainder expressed as a decimal) •understand the order of operations using BEDMAS (Brackets, Exponents, Division, Multiplication, Addition, Subtraction) •use and explain appropriate strategies in problem solving (e.g., trial and error, working backwards, looking for patterns) •use calculators to solve problems where the numbers are outside mental and written limits •identify the operations required to solve more complex problems within your experiences (e.g., deposits and withdrawals in banking, and other everyday use) •recognise the existence of negative numbers (e.g. , profit and loss)

1 1 1 1 2 2 1 1

1 5 5 1 11 11 1 4, 5

1 1 1 1 2 2 1 1

A H H B F F, G B G, H

2 2 2 2

9, 10 10 12 12

2 2 2 2

C, D E H H

1

3

1

I

1

2

1

E

1

2

1

E

2

8

2

B

2

13

2

I

5

40

5

M

2 1

13 6

2 1

I J

CORRELATION CHART: R-7 SACSA MATHEMATICS TEACHING RESOURCE

STRAND: Number Fractions, decimals, percentages, ratios and rates •round off decimals to 3 places •divide decimals by a whole number · •use notation for recurring decimals such as 0.3 or 0.235 •multiply decimal numbers by decimal numbers to 2 places (e.g., 0.2 × 0.3 = 0.06) •divide decimals using calculators (e.g., calculating averages) •convert decimals to fractions •use decimals in problem solving •compare the size of fractions (e.g., ‘Which is larger: Wt_ or Qe_\ ?’) •compare and order fractions in ascending or descending order (e.g., Qe_ ' Wt_ ' Ui_\) •add and subtract fractions with different denominators, including improper fractions and whole numbers •multiply fractions including whole numbers and mixed numbers •convert fractions to frequently used decimals and percentages (e.g., Wt_ ' Ti_ ' We_\) •convert percentages to fractions and decimals •convert fractions and decimals to percentages •express fractions of quantities as percentages (e.g., 20 out of 25 is Rt_ is 80%) •find simple percentages of quantities (e.g., 20% of $80) using both pen and paper, and calculator) •find discount as a percentage, especially money •solve practical problems involving percentage (e.g., simple interest, banking problems) •compare quantities using ratio in problem solving

Core Skills Mathematics 7 Chapter Unit

Mathematics for Year 7 (second edition) Chapter Unit

5 5 5

35 40 41

5 5 5

C J K

5 5 5 5 4

39 40 41 37, 39, 40 28

5 5 5 5 4

I M L G, H, I, M G

4

28

4

G

4 4

29 30

4 4

H, I, J K

6 6 6

43 44 44

6 6 6

B, C, D, E C, E B, D

6

43

6

B

6 6

47 49

6 6

I M

6 12

50 90

6 12

N, O C

11

79

11

A, B

11

81

11

E

11 11

81 81

11 11

F, H F

11

81

11

E

11

82

11

G

3 3 3

15 20 20

3 3 3

A H G

12

91

12

D

3 3

19 21

3 3

F, G N

STRAND: Pattern and algebraic reasoning Algebra

•extend and describe the rule for numerical and geometric patterns (e.g., 7, 36, 181, 906) •investigate pattern rules in solving problems (e.g., rates charged by tradespeople 1 hr - $25, 2 hrs - $60, 3 hrs - $95, n ´ 35¡-10 for various hours worked) •investigate and analyse graphs showing the relationship between variables (e.g., analysing winter rainfall patterns and making comparisons and predicting future trends) •predict future trends from linear graphs •construct a number sentence to match a problem that is presented in words and that requires finding an unknown •use inverse operations to solve a number sentence (e.g., 2x = 8, x = 8 ÷ 2)

STRAND: Spatial sense and geometric reasoning Lines and angles •use the terms lines, points, rays, segments, intersections, parallel and perpendicular when constructing diagrams •bisect angles using a compass •construct triangles when only the length of sides is given •use understanding of angles to determine compass bearings and true bearings •draw a 2-D shape given a description of its side and angle properties, using geometric software or a ruler, protractor and set square •identify the terminology of a circle - radius, diameter, circumference

241

242

CORRELATION CHART: R-7 SACSA MATHEMATICS TEACHING RESOURCE

Core Skills Mathematics 7

STRAND: Spatial sense and geometric reasoning 2-D and 3-D shapes

•identify 2-D shapes within patterns across cultures and in nature (e.g., an investigation of Islamic design) •classify solids in terms of their geometric properties (i.e., faces, edges, vertices and cross-sections) •draw 3-D solids •identify and name properties of polyhedra (e.g., tetrahedron, pentagonal prism, hexagonal prism) •construct complex solids from nets (e.g., hexagonal-based pyramid) •draw oblique and isometric projections of cubes using paper or drawing software •recognise the properties of quadrilaterals •construct, name and classify scalene, isosceles and equilateral triangles •determine unknown angles in quadrilaterals and triangles

Chapter

Unit

Mathematics for Year 7 (second edition)

Chapter

Unit

3

H

3 3

22 23

3 3

I M

3 3

22 22

3 3

I J

3 3

22 19

3 3

K F

3 3

19 20

3 3

F C, D

12 12

86 85

12 12

B B

12 12

87 88

12 12

B B

12

87

12

B

3

22, 23

3

K, M

3

22, 23

3

M

3

22, 23

3

K, M

12 12 7

84, 91 91 55

12 12 7

A D E

12 12

84 84

12 12

A A

12

91

12

D

12

91

12

E

12 12

84 91

12 12

D E

STRAND: Spatial sense and geometric reasoning Transformation •rotate a shape about a point (e.g., rotate 90° clockwise) •reflect a complex shape or design on a line •translate shapes over a given distance (e.g., translate the shape 5 squares horizontally to the left on grid paper) •enlarge and reduce shapes using a scale •create a tessellation using rotation, translation and reflection (e.g., using software)

STRAND: Spatial sense and geometric reasoning Location and position

•draw environmental and geometric objects from different perspectives •describe and draw what is seen and not seen from different views of 3-D shapes (e.g., pyramids and prisms) •draw 3-D shapes using solid lines for visible edges and dotted lines for invisible edges •recognise that a location can be represented on maps or plans using different scales •use a scale to calculate the distance between two points on a map •read, write and use scales in words, ratios and diagrams •produce scaled plans (e.g., classroom, bedroom) •evaluate maps and plans in terms of appropriateness of scale, use of symbols, appropriateness for task, clarity of purpose, accuracy, etc. •use coordinate grids to make more complex 2 -D shapes •explain a pathway to a location on models, maps or plans using distance, direction, angle multiples of 45°, compass points and coordinates •find alternative routes using a scale (e.g., to find the shortest route between two points) •follow simple directions to move from point to point on a given path, using maps, a magnetic compass, and written and oral instructions •develop a simple orienteering course

INDEX

INDEX

actual length acute angle angle sum of triangle area area of triangle bar graph base ten BEDMAS bias capacity cardinal points centre of rotation century chance circumference clockwise column graph commission compass bearing composite number cube number cubic units data Daylight Saving Time decade decimal denominator deposit diameter difference digit discount divisibility test enlargement equilateral triangle equivalent fractions Euler's rule expanded form expectation of event exponent factor fraction frequency geometric pattern gram

120 38 40 128 132 146 74 23 142 135 192 184 154 198 49 184, 193 146 109 192 24 30 134 142 163 154 74, 96 58, 66 109 49 10, 22 9 106 27 182 44 60 43 8, 29 204 23, 28 24 91, 96 144 170 116

Greenwich Mean Time GST hectare highest common factor image improper fraction intersecting lines inverse operation isometric projection isosceles triangle line line of symmetry line segment linear graph lowest common denominator lowest common multiple lowest terms MA blocks mass mean median metre millennium mixed number natural number negative number net of solid number line number system numerator oblique projection obtuse angle one figure approximation order of rotational symmetry parallel lines percentage perimeter perpendicular pie graph place value point population positive number prime number principal prism probability generator product pyramid

162 107 128 24 183 65 36, 45 176 52 44 36 182 36 175 64 26, 64 61 75 122 148 149 116 154 65 9 18 51 58, 80, 198 9 58 52 38 15 185 36, 45 94 118 45 147 8, 76 36 142 18 24 108 50 203 11, 22 50

243

244

INDEX

quadrilateral quotient radius ratio ray rectangular prism recurring decimal reflection reflex angle regular polygon revolution right angle rotation rounding off sample scale diagram scale factor scale length scalene triangle simple interest speed formula square number square root Standard Time Zone straight angle sum tally/frequency table terminating decimal tessellation three-point notation timetable transformation translation tree diagram true bearing true north variable vertex volume of solid

41, 44 11, 22 49 190 36 136 90 182 38 42 38 38 182 14, 78 142 120 188 120 44 108 164 30 30 162 38 10, 22 144 90 187 38 160 182 182 202 193 193 172 36 134

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