Edex_Found Math_00.qxd 16/03/06 17:07 Page i BRIAN SPEED KEITH GORDON KEVIN EVANS This high quality material is endorsed by Edexcel and has been through a rigorous quality assurance programme to ensure it is a suitable companion to the specification for both learners and teachers This does not mean that the contents will be used verbatim when setting examinations nor is it to be read as being the official specification – a copy of which is available at www.edexcel.org.uk This book provides indicators of the equivalent grade level of maths questions throughout The publishers wish to make clear that these grade indicators have been provided by Collins Education, and are not the responsibility of Edexcel Ltd Whilst every effort has been made to assure their accuracy, they should be regarded as indicators, and are not binding or definitive Edex_Found Math_00.qxd 17/03/06 10:25 Page ii William Collins’ dream of knowledge for all began with the publication of his first book in 1819 A self-educated mill worker, he not only enriched millions of lives, but also founded a flourishing publishing house Today, staying true to this spirit, Collins books are packed with inspiration, innovation and a practical expertise They place you at the centre of a world of possibility and give you exactly what you need to explore it Collins Do more Published by Collins An imprint of HarperCollinsPublishers 77–85 Fulham Palace Road Hammersmith London W6 8JB Browse the complete Collins catalogue at www.collinseducation.com Acknowledgements With special thanks to Lynn and Greg Byrd The Publishers gratefully acknowledge the following for permission to reproduce copyright material Whilst every effort has been made to trace the copyright holders, in cases where this has been unsuccessful or if any have inadvertently been overlooked, the Publishers will be pleased to make the necessary arrangements at the first opportunity © HarperCollinsPublishers Limited 2006 Edexcel material reproduced with permission of Edexcel Limited Edexcel Ltd accepts no responsibility whatsoever for the accuracy or method of working in the answers given 10 ISBN-13: 978-0-00-721560-7 ISBN-10: 0-00-721560-6 Grade bar photos © 2006 JupiterImages Corporation and Photodisc Collection / Getty Images The author asserts his moral right to be identified as the author of this work All rights reserved 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 written consent of the Publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP © 2006 JupiterImages Corporation, p1, p22 Main, p23 Middle and BR, p49, p67, p95, p143, p167, p195, p213, p257, p275, p299, p327, p355, p427, p453, p473, p483, p523, p549 © Bernd Klumpp / Istock, p22 TL © Karen Town / Istock, p22 TR © David Wall / Alamy, p22 BL © Neale Haynes/Buzz Pictures, p23 TL © Christian Kretz / Istock, p22 TR British Library Cataloguing in Publication Data A Catalogue record for this publication is available from the British Library © PCL / Alamy, p119 © Images Etc Ltd / Alamy, p225 Commissioned by Marie Taylor, Vicky Butt and Michael Cotter Project managed by Penny Fowler Edited by Joan Miller and Peta Abbott Additional proof reader: Ruth Burns Indexer: Dr Laurence Errington Internal design by JPD Cover design by JPD Cover illustration by Andy Parker, JPD Page make-up by Gray Publishing Page make-up of Really Useful Maths! spreads by EMC Design Illustrations by Gray Publishing, EMC Design, David Russel, Lazlo Veres, Lisa Alderson, Roger Wade Walker, Bob Lea, Peter Cornwell, Martin Sanders and Peters and Zabranksy Production by Natasha Buckland Printed and bound in Italy by Eurografica SpA © Dave Roberts / Istock, p373 © Michal Galazka / Istock, p505 © Agence Images / Alamy, p539 Edex_Found Math_00.qxd 15/03/06 17:18 Page iii CONTENTS Chapter Basic number Chapter Fractions 25 Chapter Negative numbers 49 Chapter More about number 67 Chapter Perimeter and area 95 Chapter Statistical representation 119 Chapter Basic algebra 143 Chapter Further number skills 167 Chapter Ratios, fractions, speed and proportion 195 Chapter 10 Symmetry 213 Chapter 11 Averages 225 Chapter 12 Percentages 257 Chapter 13 Equations and inequalities 275 Chapter 14 Graphs 299 Chapter 15 Angles 327 Chapter 16 Circles 355 Chapter 17 Scale and drawing 373 Chapter 18 Probability 397 Chapter 19 Transformations 427 Chapter 20 Constructions 453 Chapter 21 Units 473 Chapter 22 Pie charts, scatter diagrams and surveys 483 Chapter 23 Pattern 505 Chapter 24 Surface area and volume of 3-D shapes 523 Chapter 25 Quadratic graphs 539 Chapter 26 Pythagoras’ theorem 549 Answers 563 Index 599 iii Edex_Found Math_00.qxd 15/03/06 17:18 Page iv Welcome to Collins GCSE Maths, the easiest way to learn and succeed in Mathematics This textbook uses a stimulating approach that really appeals to students Here are some of the key features of the textbook, to explain why Each chapter of the textbook begins with an Overview The Overview lists the Sections you will encounter in the chapter, the key ideas you will learn, and shows how these ideas relate to, and build upon, each other The Overview also highlights what you should already know, and if you’re not sure, there is a short Quick Check activity to test yourself and recap Maths can be useful to us every day of our lives, so look out for these Really Useful Maths! pages These double page spreads use big, bright illustrations to depict real-life situations, and present a short series of real-world problems for you to practice your latest mathematical skills on Each Section begins first by explaining what mathematical ideas you are aiming to learn, and then lists the key words you will meet and use The ideas are clearly explained, and this is followed by several examples showing how they can be applied to real problems Then it’s your turn to work through the exercises and improve your skills Notice the different coloured panels along the outside of the exercise pages These show the equivalent exam grade of the questions you are working on, so you can always tell how well you are doing iv Edex_Found Math_00.qxd 15/03/06 17:18 Page v Every chapter in this textbook contains lots of Exam Questions These provide ideal preparation for your examinations Each exam question section also concludes with a fully worked example Compare this with your own work, and pay special attention to the examiner’s comments, which will ensure you understand how to score maximum marks Throughout the textbook you will find Puzzles and Activities – highlighted in the green panels – designed to challenge your thinking and improve your understanding Review the Grade Yourself pages at the very end of the chapter This will show what exam grade you are currently working at Doublecheck What you should now know to confirm that you have the knowledge you need to progress Working through these sections in the right way should mean you achieve your very best in GCSE Maths Remember though, if you get stuck, answers to all the questions are at the back of the book (except the exam question answers which your teacher has) We hope you enjoy using Collins GCSE Maths, and wish you every good luck in your studies! Brian Speed, Keith Gordon, Kevin Evans v Edex_Found Math_00.qxd 15/03/06 17:18 Page vi ICONS You may use your calculator for this question You should not use your calculator for this question Indicates a Using and Applying Mathematics question Indicates a Proof question Edex_Found Math_01.qxd 13/03/06 14:37 Adding with grids Times table check Page This chapter will show you … ● how to use basic number skills without a calculator Visual overview In grids Addition Subtraction Order of operations and BODMAS Up to four digits BODMAS Place value Rounding off Multiplication Single-digit numbers Division Place value and ordering numbers Rounding Adding and subtracting numbers with up to four digits Multiplying and dividing by single-digit numbers What you should already know ● Times tables up to 10 × 10 ● Addition and subtraction of numbers less than 20 ● Simple multiplication and division ● How to multiply numbers by 10 and 100 Quick check How quickly can you complete these? 4×6 3×7 5×8 9×2 6×7 13 + 14 15 + 15 18 – 12 19 – 10 11 – 11 50 ÷ 12 48 ÷ 13 35 ÷ 14 42 ÷ 15 36 ÷ 16 × 10 17 × 100 18 × 10 19 14 × 100 20 17 × 10 © HarperCollinsPublishers Limited 2007 Edex_Found Math_01.qxd 13/03/06 1.1 14:37 Page Adding with grids In this section you will learn how to: ● add and subtract single-digit numbers in a grid ● use row and column totals to find missing numbers in a grid Key words add column grid row Adding with grids You need a set of cards marked to 9 Shuffle the cards and lay them out in a by grid You will have one card left over Copy your grid onto a piece of paper Then add up each row and each column and write down their totals Finally, find the grand total and write it in the box at the bottom right 17 19 18 13 13 44 Look out for things that help For example: • in the first column, + make 10 and 10 + = 18 • in the last column, + = + + = 10 + = 13 Reshuffle the cards, lay them out again and copy the new grid Copy the new grid again on a fresh sheet of paper, leaving out some of the numbers 8 17 17 21 42 13 8 17 21 42 Pass this last grid to a friend to work out the missing numbers You can make it quite hard because you are using only the numbers from to Remember: once a number has been used, it cannot be used again in that grid Example Find the numbers missing from this grid 19 © HarperCollinsPublishers Limited 2007 17 17 11 Edex_Found Math_01.qxd 13/03/06 14:38 Page CHAPTER 1: BASIC NUMBER Clues The two numbers missing from the second column must add up to 1, so they must be and The two numbers missing from the first column add to 11, so they could be and or and Now, or won’t work with or to give 17 across the top row That means it has to be: 19 17 17 giving 17 11 11 11 17 39 19 as the answer You can use your cards to try out your ideas EXERCISE 1A Find the row and column totals for each of these grids 8 8 9 8 9 8 7 g d a 9 b e h c f i Find the numbers missing from each of these grids Remember: the numbers missing from each grid must be chosen from to without any repeats a 16 11 14 16 36 16 d b 15 17 13 13 19 13 10 g f 42 11 17 h 17 17 13 42 18 11 19 38 16 12 12 15 i 38 13 14 18 35 e 24 14 15 16 c 10 20 12 © HarperCollinsPublishers Limited 2007 15 15 36 Edex_Found Math_01.qxd 1.2 13/03/06 14:38 Page Times table check In this section you will: ● recall and use your knowledge of times tables Special table facts You need a sheet of squared paper Start by writing in the easy tables These are the ×, ×, ×, 10 × and × tables Now draw up a 10 by 10 tables square before you go any further (Time yourself doing this and see if you can get faster.) Once you have filled it in, shade in all the easy tables You should be left with something like the square on the right × 10 12 18 21 24 12 16 24 28 32 18 24 36 42 48 21 28 42 49 56 24 32 48 56 64 10 Now cross out one of each pair that have the same answer, such as × and × This leaves you with: × 12 16 18 24 36 21 28 42 49 24 32 48 56 64 10 Now there are just 15 table facts Do learn them The rest are easy tables, so you should know all of them But keep practising! © HarperCollinsPublishers Limited 2007 10 Found Maths_ANS.qxd 15/03/06 16:29 Page 589 Quick check Exercise 18E a – b – c – 4 d – a – 2 b – c – –– d 18 e – f – e – GR, GPu, GPi, YR, YPu, YPi Exercise 18A a unlikely b certain c likely d very unlikely e impossible f very likely g evens d c e a b 1 a g a b a b a a a b c a d g Exercise 18B 1 1 1 1a – b – c – d –– e – f – g – 6 13 1 h –– i –– j 26 13 1 1 – – c – 2a b d –– e –– f –– 52 13 52 3a b 1 2 a –– b – c – d – e – 10 5 1 5a – b – c – 3 6 –– –– a 11 b 11 c –– 11 1 7a – b – c – d –– 2 10 8 a –– b –– c –– d e –– 15 15 15 15 –– 50 10 a AB, AC, AD, AE, BC, BD, BE, CD, CE, DE 3 –– –– b c 10 d e – f 10 5 11 a b c i – ii – 9 1 12 a –– b –– c – d –– e –– f –– 13 13 13 13 26 –– 13 a i 12 ii –– iii –– 25 25 25 b They add up to c All possible outcomes used 14 35% 15 0.5 B b B c C d A e B f A B h B 0.2, 0.08, 0.1, 0.105, 0.148, 0.163, 0.1645 c d – e 1000 0.095, 0.135, 0.16, 0.265, 0.345 40 c No 0.2, 0.25, 0.38, 0.42, 0.385, 0.3974 b Caryl, threw greatest number of times 0.39, 0.31, 0.17, 0.14 Yes; all answers should be close to 0.25 not likely b impossible c not likely certain e impossible f 50–50 chance 50–50 chance h certain i quite likely Exercise 18F a c d a a a a a b and 12 1 1 5 1 1 ––, ––, ––, –, ––, –, ––, –, ––, ––, –– 36 18 12 36 36 12 18 36 1 –– i –– ii – iii – iv –– v 12 12 36 11 –– b –– c – d – 12 36 –– b 11 –– c –– 36 36 18 1 –– b – c – d e – 18 1 – b – c – d – 4 4 –– b i –– ii 13 iii – iv – 25 25 5 –– vi 18 Exercise 18G 1 a – b 25 a – b 1000 1 1 a i – ii –– iii – iv –– 13 52 b i 260 ii 40 iii 130 iv 10 a –– b 37 a 150 b 100 c 250 d a 167 b 833 1050 a 10, 10, 10, 10, 10, 10 b 3,5 21 c Find the average of the scores ( –– ) Exercise 18C –– a 19 b 55% 20 –– a i –– ii 10 13 13 3 a i – ii – 4 c 0.2 b i – b i –– 11 ii – ii –– 11 c i –– 13 –– ii 11 13 – – 4 –– 11 – a Everton 2a b Man Utd, Everton, Liverpool Shape Exercise 18D – – –– 11 – Exercise 18H – – –– 11 11 –– 15 a b c a b c a b c 11 a b c d –– e – 15 a 0.6 b 120 a 0.8 b 0.2 17 a –– b – c – 20 Because these are three separate events Also probability cannot exceed 1 b – a 40 4a b 16 Circle Triangle c 40% A Letter on card B C b – c – © HarperCollinsPublishers Limited 2007 Shaded d 10% No 4 c Leeds Unshaded e 16 on disc 5 6 589 Found Maths_ANS.qxd 15/03/06 16:29 Page 590 ANSWERS: CHAPTER 18 b 20% c –– 25 b c 14% a 23 a 10 7a Spinner B c i – b 16 b a 16 9a d 480 d 15% Spinner A 8 9 10 10 11 –– ii 16 iii – 51 c 73 d –– 73 ( –1 ) –3 v ( –2 ) –4 c i ( ) –2 v ( –5 ) d i (3) v (5) b x=1 e y=x f y = –x c y=2 d y = –1 Exercise 19A a yes b yes c no d yes e no f yes a triangle ii b triangle iii c sector i a 1, 3, b 2, c 1, d 1, 2, 3, Q P PQR to QRS to RSP to SPQ; SXP to PXQ to QXR to RXS X R E C X D ABC to CDA; BDC to DBA; BXA to DXC; BXC to DXA –1 O AXB to AXC C Exercise 19B All quadrilaterals tessellate The interior angle for each shape divides exactly into 360° Q 590 (1) ii (4) iii iv ( –1 ) iii ( –4 ) iv ( –2 ) –7 C A B ( –3 ) –1 (1) ( –1 ) iv (5) v ( –1 ) vi R x ( –5 ) –2 f h ( –4 ) –7 10 × 10 = 100 (including ( )) ( –4 ) g ( –4 ) b c Exercise 19D a Exercise 19C a i ( –2 ) –3 –4 a X iii S A B ( –2 ) B iv P G A ( –4 ) H iii ii y EGF to FHE to GEH to HFG; EFX to HGX; EXH to FXG F X ( –1 ) vi ( ) –3 ii ( ) vi ( ) ii ( –4 ) vi ( ) –5 b i a x = –2 on dice 6 10 12 b (1 and 4) c – 10 a larger mean diameter b smaller range, so more consistent Quick check S H T Coin 10 11 12 Number 3 1 (4) © HarperCollinsPublishers Limited 2007 b d (4) e ( –1 ) Found Maths_ANS.qxd 15/03/06 16:29 Page 591 ANSWERS: CHAPTER 19 c d Exercise 19E a b c X d X X a a b c X b d X e c g f X d X X a b c 3 R y B S C A R –4 -3 -2 -1 -1 D -2 x Q -3 -4 C –4 –3 –2 –1 –1 y A a X B C 6x –2 –10 –1 –2 –3 –4 –5 x P –3 x P –2 Q B f reflection in y-axis c congruent –2 –3 –4 –5 y A –5 –4 R d 90° turn clockwise about O y –4 a A(1, 2), B(3, 1), C(4, 3) b (2, –1), (1, –3), (3, –4) c (–1, –2), (–3, –1), (–4, –3) d (–2, 1), (–1, 3), (–3, 4) y e corresponding vertices have same pairs of numbers C U switching round and changing A signs B –4 –2 x T S –3 –4 Y b a a b c ii i ii i a b c ii ii a–i U T y –5 –4–3–2–1 S –2 R –3 –4 –5 i j reflection in y = x W A i C B x P b rotation 90° anticlockwise rotation 270° clockwise Q 10 c always a reflection in y = x © HarperCollinsPublishers Limited 2007 591 Found Maths_ANS.qxd 15/03/06 16:29 Page 592 ANSWERS: CHAPTER 19 c y Q –4 P X –1 –2 Y Z x –3 –4 –5 B′ C′ A′ –2 –3 –4 –5 8x a c a rotation of 90° clockwise about the point (0, –1) y d rotation 180° about O C e yes f yes y A –4–3–2 –2 –3 –4 –5 B b y y x x –3–2 x –2 –3 –4 d they are all congruent Exercise 19F b a a b y 1 8x a d c y y 10 A b 3:1 c 3:1 d 9:1 B x –2 10 1 iii Ratios of perimeters = – , –, – 1 Ratios of areas = –, –, — 16 iv Ratios of perimeter go up according to scale factors, ratios of area go up according to square of scale factors 8x Quick check a cm a 30° b 7.5 cm c 11 cm b 135° Exercise 20A a d b a a BC = 2.9 cm, ∠B = 53°, ∠C = 92° b EF = 7.4 cm, ED = 6.8 cm c ∠G = 105°, ∠H = 29°, ∠I = 46° ∠J = 48°, ∠L = 32°, JK = 4.3 cm e ∠N = 55°, ON = OM = cm f ∠P = 51°, ∠R = 39°, QP = 5.7 cm ∠ABC = 44°, ∠BCA = 79°, ∠CAB = 57° 5.9 cm b 18.8 cm2 BC = 2.6 cm, 7.8 cm 4.5 b 11.25 cm2 a 4.3 cm b 34.5 cm2 Exercise 20B [No answers needed for this exercise] Exercise 20C Circle with radius 592 a cm b cm c cm © HarperCollinsPublishers Limited 2007 Found Maths_ANS.qxd 15/03/06 16:29 Page 593 ANSWERS: CHAPTER 20 a b P c cm P A cm cm A B cm A cm a Circle with radius m B cm cm B cm b 2m a b A B D C B D Diagram c A c d B D C A C A D e B A C D f B C A B D C Exercise 20D Fence Stake Fence cm cm Stake cm cm Stake Fence cm Fence Fence Stake cm Shed A 1.5 cm Stake Pen B 10 a Glasgow Newcastle upon Tyne Tree Irish Sea Power line Leeds York Manchester b No Sheffield c Yes Norwich Birmingham Tree Bristol London Exeter © HarperCollinsPublishers Limited 2007 593 Found Maths_ANS.qxd 15/03/06 16:29 Page 594 ANSWERS: CHAPTER 20 11 a No 12 Glasgow Glasgow Newcastle upon Tyne Newcastle upon Tyne Irish Sea Irish Sea Leeds York Manchester Drawing is shown at half size Leeds York Manchester Sheffield Birmingham Sheffield Norwich London Bristol Norwich Birmingham Bristol Exeter London Exeter b Shaded region a The line 13 14 b The region c This part of line 15 Newcastle upon Tyne Newcastle upon Tyne Glasgow Irish Sea Newcastle upon Tyne Irish Sea Leeds York Leeds York Irish Sea Manchester Leeds York Manchester Manchester Sheffield Sheffield Birmingham Bristol Sheffield Birmingham Norwich London Bristol Norwich London 16 Exeter West Pier East Pier Really Useful Maths!: The street Number of end-terraced bungalows Number of mid-terraced bungalows 15 40 3.5 3.5 60 Number of chimney pots needed 200 Number of doors needed 200 Number of windows needed 440 Number of gates needed 140 Length of fencing needed in metres 4820 Number of 1s for the doors 2 Bungalow Drawing is shown at a reduced size 20 Number of 0s for the doors 21 Number of 8s for the doors 594 11 © HarperCollinsPublishers Limited 2007 Found Maths_ANS.qxd 15/03/06 16:29 Page 595 Quick check 100 1000 1000 1000 35 0.000 94 l 36 2160 cl 38 14 000 l 39 0.19 ml 37 15 200 g Exercise 21C Exercise 21A 11 15 metres kilograms or tonnes kilograms millimetres kilometres millimetres grams litres kilograms millilitres centilitres 10 metres 12 litres 13 grams 14 centilitres Exercise 21B 13 17 20 23 27 31 1.25 m 8.2 cm 0.55 m 2.1 km 2.08 cm 1.24 m 4.2 kg 5.75 t 8.5 cl 10 2.58 l 11 3.4 l 12 0.6 t 0.755 kg 14 0.8 l 15 l 16 63 cl 8.4 m3 18 35 cm3 19 1.035 m3 0.53 m3 21 34 000 m 22 3400 mm 135 mm 24 67 cm 25 7030 m 26 7.2 mm 25 cm 28 640 m 29 2400 ml 30 590 cl 84 ml 32 5200 l 33 580 g 34 3750 kg b 62 A(1, 4), B(2, 1), C(5, 2) it b Rabb Cat Peugeot Ford Vauxhall Dog Toyota Gu ard ian Nissan c Sun Exercise 21D 13 17 21 22 20 cm 13.2 lb 48 km 67.5 l 2850 ml 10 gal 12 in 50 miles kg 10 pints 11 160 km 12 123.2 lb 180 l 14 90.9 kg 15 1100 yd 16 30 cm 6.3 kg 18 90 cm 19 ton 20 metre a i 1000 g ii kg b i 4500 g ii 4.5 kg a 135 miles b 50 mph c h 42 Exercise 22B Exercise 22A a 24 in 12 ft 3520 yd 80 oz 56 lb 6720 lb 40 pt 48 in 36 in 10 30 ft 11 64 oz 12 ft 70 lb 14 12 yd 15 224 oz 16 miles 120 pt 18 5280 ft 19 ft 20 st gal 22 lb 23 yd 24 tons 63 360 in 26 lb 27 gal 28 10 st miles 30 35 840 oz a 124°, 132°, 76°, 28° b Split of total data seen at a glance a 55° b 22 c 33–% Quick check a 27 13 17 21 25 29 Mirror Times a positive correlation b negative correlation c no correlation d positive correlation a a person’s reaction time increases as more alcohol is consumed b as people get older, they consume less alcohol c no relationship between temperature and speed of cars on M1 d as people get older, they have more money in the bank c about 20 cm/s d about 35 cm b yes, usually (good correlation) c Greta d about 70 e about 72 b No, becauses there is no correlation Exercise 22D a 36°, 90°, 126°, 81°, 27° b 90°, 108°, 60°, 78°, 24° c 168°, 52°, 100°, 40° 60°, 165°, 45°, 15°, 75° a 36 b 50°, 50°, 80°, 60°, 60°, 40°, 20° d Bar chart, because easier to make comparisons a leading question, not enough responses b simple ‘yes’ and ‘no’ response, with a follow-up question, responses cover all options and have a reasonable number of choices a overlapping responses b □ £0–£2 □ over £2 up to £5 □ over £5 up to £10 □ over £10 © HarperCollinsPublishers Limited 2007 595 Found Maths_ANS.qxd 15/03/06 16:29 Page 596 ANSWERS: CHAPTER 22 Exercise 22E Price: 78p, 80.3p, 84.2p, 85p, 87.4p, 93.6p a 9.7 million b 4.5 years c 12 million d 10 million a £1 = $1.88 b Greatest drop was from June to July c There is no trend in the data so you cannot tell if it will go up or down £74.73 a holiday month b i 138–144 thousand ii 200–210 thousand Really Useful Maths!: Riding stables Horse Weight (kg) Summer 875 6.1 Sally 350 3.5 Skip 550 5.4 Simon 500 a b 2x b Barney 350 650 Exercise 23A 596 9, 11, 13: add b 10, 12, 14: add 80, 160, 320: double 81, 243, 729: multiply by 28, 34, 40: add f 23, 28, 33: add 78° 30° MN g 20 000, 200 000, 000 000: multiply by 10 h 19, 22, 25: add i 46, 55, 64: add j 405, 1215, 3645: multiply by k 18, 22, 26: add l 625, 3125, 15 625: multiply by a 16, 22 b 26, 37 c 31, 43 d 46, 64 e 121, 169 f 782, 3907 g 22 223, 222 223 h 11, 13 i 33, 65 j 78, 108 a 48, 96, 192 b 33, 39, 45 c 4, 2, d 38, 35, 32 e 37, 50, 65 f 26, 33, 41 g 14, 16, 17 h 19, 22, 25 i 28, 36, 45 j 5, 6, k 0.16, 0.032, 0.0064 l 0.0625, 0.031 25, 0.015 625 a 21, 34: add previous terms b 49, 64: next square number c 47, 76: add previous terms d 216, 343: cube numbers 15, 21, 28, 36 61, 91, 127 Exercise 23C a 3, 5, 7, 9, 11 b 1, 4, 7, 10, 13 c 7, 12, 17, 22, 27 d 1, 4, 9, 16, 25 e 4, 7, 12, 19, 28 a 4, 5, 6, 7, b 2, 5, 8, 11, 14 c 3, 8, 13, 18, 23 d 0, 3, 8, 15, 24 e 9, 13, 17, 21, 25 –, –, –, –, – a 6, 10, 15, 21, 28 b Triangular numbers a 2, 6, 24, 720 b Exercise 23D a c d e 122° FE 6.2 c Exercise 23B C 2.8 Teddy c 2x + 11111 × 11111 = 123454321, 111111 × 111111 = 12345654321 99999 × 99999 = 9999800001, 999999 × 999999 = 999998000001 × = 72 + 7, × = 82 + 50 × 51 = 2550, 60 × 61 = 3660 + + + + + + + + = 25 = 52, + + + + + + + + + + = 36 = 62 21 + 23 + 25 + 27 + 29 = 125 = 53, 31 + 33 + 35 + 37 + 39 + 41 = 216 = 63 + + 15 + 20 + 15 + + = 64, + + 21 + 35 + 35 + 21 + + = 128 12 345 679 × 45 = 555 555 555, 12 345 679 × 54 = 666 666 666 13 + 23 + 33 + 43 = (1 + + + 4)2 = 100, 13 + 23 + 33 + 43 + 53 = (1 + + + + 5)2 = 225 10 362 + 372 + 382 + 392 + 402 = 412 + 422 + 432 + 442, 552 + 562 + 572 + 582 + 592 + 602 = 612 + 622 + 632 + 642 + 652 11 12345678987654321 12 999999998000000001 13 122 + 12 14 8190 15 81 = 92 16 512 = 83 17 512 18 999 999 999 19 (1 + + + + + + + + 9)2 = 2025 40° 4.0 Quick check a x+3 FN Feed (kg) 1a c e g i 13, 15, 2n + 33, 38, 5n + 20, 23, 3n + 21, 25, 4n – 17, 20, 3n – © HarperCollinsPublishers Limited 2007 b d f h j 25, 29, 4n + 32, 38, 6n – 37, 44, 7n – 23, 27, 4n – 42, 52, 10n – ME Found Maths_ANS.qxd 15/03/06 16:29 Page 597 ANSWERS: CHAPTER 23 24, 28, 4n + l 29, 34, 5n – 3n + 1, 151 b 2n + 5, 105 c 5n – 2, 248 4n – 3, 197 e 8n – 6, 394 f n + 4, 54 5n + 1, 251 h 8n – 5, 395 i 3n – 2, 148 3n + 18, 168 k 7n + 5, 355 l 8n – 7, 393 i 4n + ii 401 i 2n + ii 201 c i 3n + ii 301 i 2n + ii 206 e i 4n + ii 405 i 5n + ii 501 g i 3n – ii 297 i 6n – ii 596 i i 8n – ii 799 i 2n + 23 ii 223 64, 128, 256, 512, 1024 i 2n – ii 2n + iii × 2n – They are the same b i 10n – ii × 10n Odd + even = odd, even + odd = odd, even + even = even b Odd × even = even, even × odd = even, even × even = even k 2a d g j 3a b d f h j 4a b 5a c 6a Quick check cube cuboid b b a b a b a a b cylinder 4n – c 97 d 50th diagram 2n + c 121 d 49th set 18 b 4n + c 12 2n + c 101 d 149th diagram i 20 cm ii (3n + 2) cm iii 152 cm 332 i 20 ii 162 b 79.8 km 2n i The quantity doubles ii 1600 ml 0.75 g/cm3 156.8 g 2.72 g/cm3 1.79 g/cm3 120 cm3 3200 cm3 36 800 kg (36.8 t) 1.6 g/cm3 Exercise 24D Exercise 24A 12 cm 23 cm3 Exercise 23E Exercise 24C triangular prism a 36, 49, 64, 81, 100 b i n2 + ii 2n2 iii n2 – + + + = 16 = , + + + + = 25 = 52 a 28, 36, 45, 55, 66 b i 210 ii 5050 c They produce square numbers 10 a even b odd c odd d odd e odd f odd g even h odd i odd 11 a either b either c either d odd e either f even a i a 20 cm 32 cm3 b c Exercise 24B a i 198 cm3 ii 234 cm2 b i 90 cm ii 146 cm2 c i 1440 cm ii 792 cm2 d i 525 cm ii 470 cm2 24 litres a 160 cm3 b 480 cm3 c 150 cm3 a i 64 cm3 ii 96 cm2 b i 343 cm3 ii 294 cm2 c i 1000 mm ii 600 mm2 d i 125 m ii 150 m2 e i 1728 m ii 864 m2 86 a 180 cm3 b cm c cm d 10 cm e 81 cm3 1.6 m 48 m2 a cm b 5m c mm d 1.2 m 10 a 148 cm3 b 468 cm3 d e f ii a 21 cm2 b 48 cm2 c 36 m2 2 d 108 m e 25 m f 111 m2 3 iii a 63 cm b 432 cm c 324 m3 3 d 432 m e 225 m f 1332 m3 3 a 21 cm , 210 cm b 54 cm , 270 cm2 525 000 litres 1.024 t solid b heavier (2880 g) than solid a (2851.2 g) Exercise 24E a 251 cm3 d 25 m3 a 226 cm3 d 1060 cm3 £80 1.71 g/cm3 a 360π cm3 © HarperCollinsPublishers Limited 2007 b 445 cm3 b 15 cm3 c 2150 cm3 c 346 cm3 1229 kg 7.78 g/cm3 b 300π cm3 332 l 597 Found Maths_ANS.qxd 15/03/06 16:29 Page 598 Quick check a a 16 b a 20 b a 24 –1 –1 –1 –1 –1 –1 –1 –1 y –1 –2 –1 14 b 0.3 c –2.7, 0.7 b a 14 x –3 –2 –1 x2 +2x –6 –4 –2 b Exercise 25A x y x y a b a b a b –3 27 –5 27 –2 12 –4 18 –5 25 15 40 –1 3 –3 –2 –1 11 x –4 –3 –2 x2 16 –3x 12 y 28 18 10 1.8 c –1.2, 4.2 x –5 –4 –3 –2 x2 25 16 –2x 10 –8 –8 –8 –8 –8 y 27 16 –8.8 c –1.5, 3.5 x –2 –1 y 18 10 6.8 c 0.2, 4.8 12 27 11 18 27 –1 1 16 25 –3 –6 –9 –12 –15 –2 –2 10 –1 1 16 25 –2 –4 –6 –8 –10 –8 –8 –8 –8 –8 –8 –8 –5 –8 –9 –8 –5 –2 –2 246.5 16.1 0.7 0.6 2.8 Exercise 26A 11 12 –3 x = ±2 –5 –8 x = ±3 –3 –4 x = –4 or 16 –5 x = or 10 –2 x = –3 or 10 –2 –5 x = 0.5 or 5.5 5.9 cm 18.6 cm 2.4 m 13 cm 12 8.5 cm 17.5 cm 500 m 10 cm 11 13 598 15 cm 5.4 cm 20.8 m 22.9 m 5m b –3 –9 –8 –5 –3 –8 –9 –2 –6 –5 12 12 –8 –5 10 –2 18 10 6.6 m 2.1 m 10.8 m 11.3 m 9.2 m 19.2 km 147 km a 127 m b 99.6 m c 27.4 m 2.4 km 10 12 ft a 3.9 m b 1.7 m 12 3.2 m a (7, –) b 13 cm 14 a (27, 28) b 50 cm 15 a 4.7 m b 4.5 m 16 16.5 cm2 17 12.07 m 18 yes, 25 = 242 + 72 Exercise 26B a e a e a –4 Exercise 26C 5.3 10.3 cm 20.6 cm 32.2 cm cm a b a b a b a b a b a b Quick check 10 Exercise 25B b 14.7 cm c 6.3 cm d 18.3 cm f 218 m g 0.4 cm h 8m b 15.5 cm c 15.5 m d 12.4 cm f 19.8 m g 7.1 m h 0.64 m 6m c 3m d 50 cm © HarperCollinsPublishers Limited 2007 Found Maths_ANS.qxd 15/03/06 16:29 Page 599 Index acute angles 328 addition decimals 174 up to four digit numbers 14–16 fractions 27–8, 36–7, 181–2 in grids/columns/rows 2–3 negative numbers in 56–61 in order of operations addition rule (probability) 405–7 algebra 143–66, 275–98 equations see equations expressions see expressions allied angles 345 alternate angles 345 angles 327–54 bisectors 458 measuring and drawing 328–30 in parallel lines 344–7 polygons (other than triangles/quadrilaterals) 337–44 quadrilaterals 338, 347–9 of rotation 440 of sector in pie chart 484 60°, construction 459 triangles 333–7, 455–6 anticlockwise rotation 440 approximation 187–90, 377–9 calculations 188–90 see also estimation; rounding arc 356 drawn with compasses 459, 460 areas circle 363–6, 367 compound shape 102–3 dimensions of 111–12 irregular shape 98–9 parallelogram 107–8 quadrilateral 107 rectangle 100–1 surface see surface area trapezium 108–9, 159 triangles 103–7, 159 average(s) 225–56 which to use 238–41 see also mean; median; mode average speed 202–5, 304 axis (axes) bar charts 127 graphs see x-axis; y-axis balancing (‘doing same to both sides of’) equations 277, 280–2 bar charts 127–30 base in area calculation parallelogram 107–8 triangles 103–5, 159 bearings 349–51 best buy 207–8 best fit, line of 489 bias 413 bisectors, line/angle 458 BODMAS 6–9, 143 box method, long multiplication 168, 169 brackets, in algebraic expressions 285–6 expanding see expansion brackets, order of operations with 6–9 C angles 345 calculators algebraic equations 160–1 negative numbers 58 powers 76, 80, 83 standard form on 83 statistical keys 234, 242 cancelling see simplifying capacity see volume carry mark (long multiplication) 169 census, national 496 centilitre 476, 528 centimetres 476, 479 cubic 524 centre of circle 356 of enlargement 443, 444, 445, 446 of rotation 439 certainty of outcome 398 chance see probability chord 356 chunking method (repeated subtraction) 169, 170 circles 355–71 area 363–6, 367 circumference see circumference drawing 356–8 see also pie charts circumference 356, 359–63 calculations 359–63, 367 class intervals (classes) 121, 127 modal 228 clockwise rotation 440 coefficient of n 510, 511 co-interior angles 345 column(s) addition in 2–3 long multiplication 168 column vector 432, 433 comment column with equations 289 common factors in algebraic expressions, taking out (=factorisation) 155–8 highest 89 common units 196 compasses circle drawing 357 perpendicular construction 459, 460 60° angle construction 459 triangle construction 454, 455, 456 compound shape 97 area of 102–3 © HarperCollinsPublishers Limited 2007 computers, survey data 492 congruent shapes 428–30 consecutive terms 508 difference between 508, 510, 512 consistency between data sets 236 constructions 453–72 continuous data 248 conversion between fractions and decimals 43, 179–80 between units 300–3, 479–80 graphs of 300–3 coordinates 309 in image enlargement 445–6 correlation, graphs showing 487, 488 corresponding angle 345 cost/price per unit weight 207 retail, index of 496 unit see unit cost cross-section of prism 531 cube (shape) pentomino/hexomino 390 volume 112 cube (third power) 79 cubic units 524 cuboid 386, 526, 527–30 volume 112, 527–30 curves, quadratic 539–48 cylinder, volume 112, 534–5 data continuous 248 discrete 134, 248 experimental 409 extreme vs representative values 238 frequency distribution of see frequency grouped 121, 245–7 historical 409 ordered and unordered 134 raw 134–6 spread 236–7 trends 130–4 data collection sheet 120, 491, 492 decagon 338, 341, 342 decimal numbers (decimals) 172–9 addition/subtraction 174 division 175–7 fractions and, converting between 43, 179–80 multiplication 175–9 percentages expressed as 258–62 recurring 42, 43 rounding 173–4, 187, 188, 190 terminating 42, 43 decimal place 172–4 in equations 288 decimal point 172 hidden 174–5 599 Found Maths_ANS.qxd 15/03/06 16:29 Page 600 INDEX denominators (in fractions) 25, 180 in addition and subtraction 27 in cancelling 31 density 530–1 diameter 356, 363, 364 circumference calculation 359–63 difference between consecutive terms 508, 510, 512 digits (in number) addition/subtraction up to digits 14–16 multiplication/division by single-digits 16–18, 175–7 place value 9–11 dimensional analysis 110–13 direct proportion 205–6 direction and bearing 349–51 discrete data 134, 248 distance/time/speed relationships 202–5, 304–8 division (of number/amount) decimals 175–7 fractions 184–5 long 169–70 negative numbers 185–6 in order of operations powered numbers 81–3 by ratios of amounts 198–200 by single-digit numbers 16–18 division(s) (on scale) 374 ‘doing same to both sides’ 277, 280–1 drawing angles 328–30 circles 356–8 graphs see plotting line with gradient 318–20 pie charts 484–7 to scale 380–3 scatter diagrams 487–91 see also constructions dual bar chart 128 edges of 3-D shapes 526 elevation (3-D shape) 387 enlargements 443–8 equally likely outcome 400, 409 equations (algebraic) 159–61, 275–98 quadratic 539–48 setting up 286–7 equations (of line on graphs) 309 equilateral triangles 335 isometric grid 386–9 equivalent fractions 28–33 estimation 377–9 irregularly shaped area 98–9 of length/weight/volume 474 see also approximation even-numbered sequences 513 event 400 outcomes see outcomes 600 expansion of brackets (multiplying out) 152–4 quadratic expression 156–8 reversal of process 155–8 expected outcome 416–17 experiment 120 experimental probability 407–13 expressions, algebraic 144, 159, 290 brackets in see brackets quadratic 156–8 simplifying 148–51, 153–4 substitution in 159–61 exterior angles of polygons 342 triangles 337 F angles 345 faces of 3-D shapes 526 area see surface area factors 71–2, 76, 77 common see common factors conversion 479–80 prime 85–90 scale 380, 443, 444, 445, 446 flow diagrams/graphs 308–14 inverse 277, 278–80 foot 478, 479 formulae algebraic 145, 159, 290–1 in dimensional analysis 112 fractions 25–48, 179–85 addition 27–8, 36–7, 181–2 cancelling 31–3, 182, 258 decimals and, converting between 43, 179–80 division 184–5 equivalent 28–33 multiplication 40–1, 182–4 percentage expressed as 258–62 probability 400 proper 33, 36 of quantity see quantity ratios as 196–8 reciprocals of 43 of shapes 26 subtraction 27–8, 37, 181–2 top-heavy (improper) 33–5, 36 frequency averages see average diagrams/tables/charts etc of 120–30, 241–4, 248–50, 484, 485 relative 409 function 309 gallon 478, 479 gradient 317–18 drawing a line with a 318–20 gradient-intercept method 320–1 gram 476, 479 graphs 299–326 line see line graphs © HarperCollinsPublishers Limited 2007 plotting see plotting quadratic 539–48 see also scatter diagrams ‘greater than’ (>) sign 291 ‘greater than or equal to’ (≥) sign 291 grids 70 in addition 2–3 isometric 386–9 in long multiplication 170 grouped data 121, 245–7 grouped (grouped frequency) table 121, 245 guess column with equations 289 height in area calculation parallelogram 107 trapezium 108 triangles 103–5, 159 height in volume/surface area calculation cuboid 527, 528 cylinder 534 heptagon 338, 341 hexagon 338, 341, 342 hexomino cube 390 highest common factor 89 historical data 409 horizontal axis see x-axis horizontal lines, equations 309 hypotenuse 550, 551 image (in transformation) 432 enlarged 443, 444, 445 reflected see reflections rotated 439, 440 imperial system 474, 478–9 conversion between metric and 479–80 impossibility of event 398 improper (top-heavy) fractions 33–5, 36 inches 478, 479 indices see powers inequality linear 291–4 signs for 54 input values 310 integers and linear inequalities 292 interior angles of polygons 338, 341, 342 triangles 335 inverse flow diagrams 277, 278–80 inverse operations 277–8 isometric grids 386–9 isosceles triangle 335 key in pictogram 124 in stem-and-leaf diagrams 134, 135, 136 kilogram 476, 479 kilometre 476, 479 kite 348 Found Maths_ANS.qxd 15/03/06 16:29 Page 601 INDEX leading question 493, 494 least common multiple 89 least terms (lowest terms) 31–3, 36 length dimensions of 110–11 imperial units of 474, 478 metric-imperial conversion 479 metric units of 474, 476 rectangle, in area calculation 100–1 right-angled triangle’s sides, finding 551–4 in volume calculations 112, 524, 527, 528, 531, 534 see also height ‘less than’ (