MATHS
fifth edition
Higher Student Book
Kevin Evans
Keith Gordon
Trevor Senior
Brian Speed
Michael Kent
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 practical expertise. They place you at the centre of a world of possibility and give you exactly what you need to explore it.
Collins. Freedom to teach
Published by Collins
An imprint of HarperCollinsPublishers
The News Building, 1 London Bridge Street, London SE1 9GF, UK
Macken House, 39/40 Mayor Street Upper, Dublin 1 D01 C9W8, Ireland
Browse the complete Collins catalogue at www.collins.co.uk
© HarperCollinsPublishers Limited 2024
10 9 8 7 6 5 4 3 2 1
ISBN 978-0-00-864730-8
A catalogue record for this book is available from the British Library
This edition updated by Kath Hipkiss, Michael Kent, Chris Pearce, Anne Stothers and Trevor Senior. Based on the previous edition by Kevin Evans, Keith Gordon, Brian Speed and Michael Kent
Commissioned by Katie Sergeant
Copyedited by Jessica Ashdale, Julie Bond, Jane Glendening
Proofread and collated by Mike Harman
Answers checked by Steven Matchett
Typeset by Jouve India Private Limited
Picture research by Hart McLeod Ltd
Illustrations by Ann Paganuzzi and Jouve
Designed by Ken Vail Graphic Design
Cover design by Amparo at Kneath Associates
Production by Alhady Ali
Printed and bound by
All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by
Endorsement Statement
any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission in writing of the Publisher. This book is sold subject to the conditions that it shall not, by way of trade or otherwise, be lent, re-sold, hired out or otherwise circulated without the Publisher’s prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser.
Acknowledgements
With thanks to the following people for their input into the careers profiles, in partnership with WISE: Muneebah Quyyam, Senior Systems Engineer; Jesie Dyos, Water Production Manager; Nintse Dan-The, Graduate Systems Engineer; Ruth Scott-Bolt, IS Portfolio Analyst; Fiona Duffy, Plant Modelling Engineer. The publishers gratefully acknowledge the permissions granted to reproduce copyright material in this book. Every effort has been made to contact the holders of copyright material, but if any have been inadvertently overlooked, the publisher will be pleased to make the necessary arrangements at the first opportunity.
The publishers would like to thank the following for permission to reproduce photographs in these pages:
Cover (bottom) Procy/Shutterstock, cover (top) joingate/Shutterstock, p 8 xtock/Shutterstock, p 18 megainarmy/Shutterstock, p 38 leungchopan/Shutterstock, p 45 Used with kind permission of Casio Electronics Co. Limited, p 58 leungchopan/Shutterstock, p 86 (main) apiguide/Shutterstock, p86 (inset) lrafael/Shutterstock p 116 David Madison/Getty, p 150 hxdbzxy/Shutterstock, p 178 chungking/Shutterstock p182 alkkdsg/Shutterstock, p 216 Lonely/ Shutterstock, p 248 Ant Clausen/Shutterstock, p 274 Yellowj/ Shutterstock, p 304 Lewis Tse Pui Lung/Shutterstock, p 344 ARCHITECTEUR/Shutterstock, p 362 GrandeDuc/Shutterstock, p 386 Maciej Sojka/Shutterstock, p 402 Humannet/Shutterstock, p 436 science photo/Shutterstock, p 470 300dpi/Shutterstock, p 508 Vinap/Shutterstock, p 538 ronfromyork/Shutterstock, p 558 Digital Vision/Getty, p 580 Sethislav/Shutterstock, p 596 Dmitry Kalinovsky/Shutterstock, p 628 solarseven/Shutterstock, p 666 fongfong/Shutterstock, p 686 Celso Diniz/Shutterstock.
In order to ensure that this resource offers high-quality support for the associated Pearson qualification, it has been through a review process by the awarding body. This process confirms that this resource fully covers the teaching and learning content of the specification or part of a specification at which it is aimed. It also confirms that it demonstrates an appropriate balance between the development of subject skills, knowledge and understanding, in addition to preparation for assessment.
Endorsement does not cover any guidance on assessment activities or processes (e.g. practice questions or advice on how to answer assessment questions), included in the resource nor does it prescribe any particular approach to the teaching or delivery of a related course.
While the publishers have made every attempt to ensure that advice on the qualification and its assessment is accurate, the official specification and associated assessment guidance materials are the only authoritative source of information and should always be referred to for definitive guidance.
Pearson examiners have not contributed to any sections in this resource relevant to examination papers for which they have responsibility.
Examiners will not use endorsed resources as a source of material for any assessment set by Pearson.
Endorsement of a resource does not mean that the resource is required to achieve this Pearson qualification, nor does it mean that it is the only suitable material available to support the qualification, and any resource lists produced by the awarding body shall include this and other appropriate resources.
Contents (Higher tier only material appears bold) How to use this book 6 1 Basic number 8 1.1 Solving real-life problems 9 1.2 Multiplication and division with decimals 12 1.3 Approximation of calculations 15 1.4 Multiples, factors, prime numbers, powers and roots 22 1.5 Prime factors, LCM and HCF 25 1.6 Negative numbers 30 Worked exemplars 34 Ready to progress? 36 Review questions 36 2 Fractions, ratio and proportion 38 2.1 One quantity as a fraction of another 39 2.2 Adding, subtracting and calculating with fractions 40 2.3 Multiplying and dividing fractions 42 2.4 Fractions on a calculator 44 2.5 Increasing and decreasing quantities by a percentage 48 2.6 Expressing one quantity as a percentage of another 51 Worked exemplars 54 Ready to progress? 56 Review questions 56 3 Statistical diagrams and averages 58 3.1 Statistical representation 59 3.2 Statistical measures 65 3.3 Scatter diagrams 74 Worked exemplars 79 Ready to progress? 82 Review questions 82 4 Number and sequences 86 4.1 Patterns in number 87 4.2 Number sequences 89 4.3 Finding the nth term of a linear sequence 92 4.4 Special sequences 95 4.5 General rules from given patterns 100 4.6 The nth term of a quadratic sequence 105 4.7 Finding the nth term for quadratic sequences 108 Worked exemplars 113 Ready to progress? 114 Review questions 114 5 Ratio and proportion 116 5.1 Ratio 117 5.2 Direct proportion problems 125 5.3 Best buys 128 5.4 Compound measures 132 5.5 Compound interest and repeated percentage change 140 5.6 Reverse percentage (working out the original amount) 143 Worked exemplars 146 Ready to progress? 148 Review questions 148 6 Angles 150 6.1 Angle facts 151 6.2 Triangles 154 6.3 Angles in a polygon 157 6.4 Regular polygons 160 6.5 Angles in parallel lines 163 6.6 Special quadrilaterals 166 6.7 Scale drawings and bearings 168 Worked exemplars 174 Ready to progress? 176 Review questions 176 7 Transformations, constructions and loci 178 7.1 Congruent triangles 179 7.2 Rotational symmetry 181 7.3 Transformations 183 7.4 Combinations of transformations 195 7.5 Bisectors 198 7.6 Defining a locus 201 7.7 Loci problems 203 7.8 Plans and elevations 208 Worked exemplars 212 Ready to progress? 214 Review questions 214 8 Algebraic manipulation 216 8.1 Basic algebra 217 8.2 Factorisation 223 8.3 Quadratic expansion 225 8.4 Expanding squares 231 8.5 More than two binomials 232 8.6 Quadratic factorisation 235 8.7 Factorising ax2 + bx + c 239 8.8 Changing the subject of a formula 241 Worked exemplars 244
Ready to progress? 246 Review questions 246 9 Length, area and volume 248 9.1 Circumference and area of a circle 249 9.2 Area of a parallelogram 252 9.3 Area of a trapezium 253 9.4 Sectors 256 9.5 Volume of a prism 259 9.6 Cylinders 262 9.7 Volume of a pyramid 264 9.8 Cones 266 9.9 Spheres 268 Worked exemplars 270 Ready to progress? 271 Review questions 271 10 Linear graphs 274 10.1 Drawing linear graphs from points 275 10.2 Gradient of a line 278 10.3 Drawing graphs by gradientintercept and cover-up methods 282 10.4 Midpoint of a line segment 286 10.5 Finding the equation of a line from its graph 286 10.6 Real-life uses of graphs 290 10.7 Solving simultaneous equations using graphs 295 10.8 Parallel and perpendicular lines 297 Worked exemplars 300 Ready to progress? 302 Review questions 302 11 Right-angled triangles 304 11.1 Pythagoras’ theorem 305 11.2 Finding the length of a shorter side 307 11.3 Applying Pythagoras’ theorem in real-life situations 309 11.4 Pythagoras’ theorem and isosceles triangles 311 11.5 Pythagoras’ theorem in three dimensions 314 11.6 Trigonometric ratios 316 11.7 Calculating angles 319 11.8 Using the sine and cosine functions 321 11.9 Using the tangent function 326 11.10 Which ratio to use 328 11.11 Solving problems using trigonometry 332 11.12 Trigonometry and bearings 336 11.13 Trigonometry and isosceles triangles 338 Worked exemplars 340 Ready to progress? 342 Review questions 342 12 Similarity 344 12.1 Similar triangles 345 12.2 Areas and volumes of similar shapes 351 Worked exemplars 358 Ready to progress? 360 Review questions 360 13 Exploring and applying probability 362 13.1 Experimental probability 363 13.2 Mutually exclusive and exhaustive outcomes 368 13.3 Expectation 372 13.4 Probability and two-way tables 374 13.5 Probability and Venn diagrams 377 Worked exemplars 382 Ready to progress? 384 Review questions 384 14 Powers and standard form 386 14.1 Powers (indices) 387 14.2 Rules for multiplying and dividing powers 389 14.3 Standard form 391 Worked exemplars 399 Ready to progress? 400 Review questions 400 15 Equations and inequalities 402 15.1 Linear equations 403 15.2 Elimination method for simultaneous equations 408 15.3 Substitution method for simultaneous equations 410 15.4 Balancing coefficients to solve simultaneous equations 411 15.5 Using simultaneous equations to solve problems 413 15.6 Linear inequalities 416 15.7 Graphical inequalities 421 15.8 Trial and improvement 426 Worked exemplars 430 Ready to progress? 433 Review questions 433 16 Counting, accuracy, powers and surds 436 16.1 Rational numbers, reciprocals, terminating and recurring decimals 437 16.2 Estimating powers and roots 440 16.3 Negative and fractional powers 442 16.4 Surds 447 16.5 Limits of accuracy 452 16.6 Problems involving limits of accuracy 455
16.7 Choices and outcomes 459 Worked exemplars 467 Ready to progress? 468 Review questions 468 17 Quadratic equations 470 17.1 Plotting quadratic graphs 471 17.2 Solving quadratic equations by factorisation 474 17.3 Solving a quadratic equation by using the quadratic formula 480 17.4 Solving quadratic equations by completing the square 483 17.5 The significant points of a quadratic curve 487 17.6 Solving one linear and one non-linear equation using graphs 492 17.7 Solving quadratic equations by the method of intersection 494 17.8 Solving linear and non-linear simultaneous equations algebraically 498 17.9 Quadratic inequalities 501 Worked exemplars 504 Ready to progress? 506 Review questions 506 18 Sampling and more complex diagrams 508 18.1 Sampling data 509 18.2 Frequency polygons 513 18.3 Cumulative frequency graphs 517 18.4 Box plots 523 18.5 Histograms 526 Worked exemplars 533 Ready to progress? 535 Review questions 535 19 Combined events 538 19.1 Addition rules for outcomes of events 539 19.2 Combined events 541 19.3 Tree diagrams 544 19.4 Independent events 549 19.5 Conditional probability 552 Worked exemplars 555 Ready to progress? 556 Review questions 556 20 Properties of circles 558 20.1 Circle theorems 559 20.2 Cyclic quadrilaterals 566 20.3 Tangents and chords 569 20.4 Alternate segment theorem 573 Worked exemplars 576 Ready to progress? 578 Review questions 578 21 Variation 580 21.1 Direct proportion 581 21.2 Inverse proportion 587 Worked exemplars 592 Ready to progress? 594 Review questions 594 22 Triangles 596 22.1 Further 2D problems 597 22.2 Further 3D problems 600 22.3 Trigonometric ratios of angles between 0° and 360° 604 22.4 Solving any triangle 611 22.5 Using sine to calculate the area of a triangle 621 Worked exemplars 624 Ready to progress? 626 Review questions 626 23 Graphs 628 23.1 Distance–time graphs 629 23.2 Velocity–time graphs 635 23.3 Estimating the area under a curve 640 23.4 Rates of change 643 23.5 Equation of a circle 646 23.6 Other graphs 649 23.7 Transformations of the graph y = f (x) 655 Worked exemplars 661 Ready to progress? 663 Review questions 663 24 Algebraic fractions and functions 666 24.1 Algebraic fractions 667 24.2 Changing the subject of a formula 672 24.3 Functions 674 24.4 Composite functions 677 24.5 Iteration 678 Worked exemplars 682 Ready to progress? 684 Review questions 684 25 Vector geometry 686 25.1 Properties of vectors 687 25.2 Vectors in geometry 692 Worked exemplars 698 Ready to progress? 699 Review questions 699 Glossary 702 Index 710 Answers 719
How to use this book
Welcome to Collins Edexcel GCSE Maths 5th Edition Higher Student Book. You will find a number of features in the book that will help you with your course of study.
Chapter overview
See what maths you will be doing, what skills you will learn and how you can build on what you already know.
About this chapter
Maths has numerous everyday uses. This section puts the chapter’s mathematical skills and knowledge into context, historically and for the modern world.
This section will show you …
Detailed learning objectives show you the skills you will learn in that section.
Key terms and glossary
Learn the important words you need to know. The explanations for the words in bold in the text can be found in the glossary at the back of the book.
Examples
Understand the topic before you start the exercise by reading the examples in blue boxes. These take you through questions step by step.
Exercises
Once you have worked through the examples you will be ready to tackle the exercises. There are plenty of questions, carefully designed to provide you with enough practice to become fluent.
Hints and tips
These are provided where extra guidance can save you time or help you out.
17.3 Solving a quadratic equation by using the quadratic formula
This chapter is going to show you: how to calculate with integers and decimals • how to round numbers to a given number of signi cant gures • how to work out and recognise multiples, factors, prime numbers and squares, cubes and their roots how to nd the prime factors of a number • how to work out lowest common multiples (LCM) how to work out highest common factors (HCF) how to calculate with negative numbers. You should already know: • how to add, subtract, multiply and divide with integers • what multiples, factors, square numbers and prime numbers are the BIDMAS/BODMAS rule and how to substitute values into simple algebraic expressions. Context and careers Number skills play a signi cant role in different jobs and help us to solve real world problems and make decisions. For any job, you should be competent in the basic number skills of addition, subtraction, multiplication and division, using whole numbers, fractions and decimals. Careers that use this maths Doctors use number skills to work out how much medicine to prescribe to patients with different ages and weights. • Air traf c controllers use multiplication and division to determine the time it will take an aircraft to travel the distance from one airport to another. Accountants and auditors use number skills to calculate a company’s pro t, loss and taxes. • Engineers use numbers to design strong structures and work out the quantity and cost of the materials required. Interior designers apply number skills to work out the area and volume of a room. • Cashiers use numbers to work out what coins to give in change.
Basic number
1.1 Solving real-life problems This section will show you how to: solve problems set in a real-life context. During your maths course, you will meet many problems set in real-life contexts. You will have to read them carefully think about and then plan a strategy This may involve arithmetical skills such as long multiplication and long division You may need to work out the answers with or without a calculator. There are several ways to do long multiplication or long division without using a calculator, so make sure you are familiar with them and con dent in using at least one method. The rst example shows both the grid method (or box method and the standard column method (or traditional method for long multiplication. The second example shows one method for long division. In this type of problem it is important to show your working, as you will get marks for correct methods. Key terms column method (or traditional method) grid method (or box method) long division long multiplication strategy A supermarket receives a delivery of 235 cases of tins of beans. Each case contains 24 tins. a How many tins of beans does the supermarket receive altogether? b 5% of the tins were damaged. These were thrown away. The supermarket knows that it sells, on average, 250 tins of beans a day. How many days will the delivery of beans last before a new delivery is needed? The number of tins is worked out by the multiplication 235 × 24. Using the grid method 200 30 5 20 4000 600 100 4 800 120 20 Using the column method 235 ×2 4 940 4700 5640 So the answer is 5640 tins. b 10% of 5640 is 564, so 5% is 564 ÷ 2 = 282. This leaves 5640 – 282 = 5358 tins to be sold. There are 21 lots of 250 in 5358 (you should know that 4 × 250 = 1000), so the beans will last for 21 days before another delivery is needed. 4000 600 100 800 120 +2 0 5640 Example 1.1 Solving real-life problems 8 9
Number:
1
Exercise 17D 1 Use the quadratic formula to solve these equations, giving your answers to 2 decimal places. a + 5 + 2 = 0 b – 10 = 0 c – 9 + 17 = 0 d = 2 + 4 e + 3 – 6 = 0 2 – 8 = 0 g 7 + 12 + 2 = 0 h 6 + 22 + 19 = 0 4 + 5 = 3 4 – 9 + 4 = 0 k 7 + 3 = 2 5 + 1 = 10 2 A rectangular lawn is 2 m longer than it is wide. The area of the lawn is 21 m The gardener wants to edge the lawn with edging strips, which are sold in lengths of 1 1 m. How many will she need to buy? 3 Daisy is solving a quadratic equation, using the formula. He correctly substitutes values for a b and c to get: 337 2 ± What is the equation Daisy is trying to solve? 4 Yoshio uses the quadratic formula to solve 4 – 4 + 1 = 0. Amaya uses factorisation to solve 4x – 4x + 1 = 0. They both find something unusual in their solutions. Explain what this is, and why. 5 Solve the equation + 3 = 7. Give your answers correct to 2 decimal places. 6 The sum of a number and its reciprocal is 2.05. What are the two numbers? Hints and tips Use brackets when substituting and do not try to work two things out at the same time. MR PS
This section will show you how to: solve a quadratic equation by using the quadratic formula recognise why some quadratic equations cannot be solved. Many quadratic equations cannot be solved by factorisation because they do not have simple factors. For example, try to factorise – 4 – 3 = 0 or 3 – 6 + 2 = 0. One way to solve this type of equation is to use the quadratic formula You can use this formula to solve any quadratic equation that is soluble (Some are not, which the quadratic formula would immediately show. You will learn about this later in this section.) The solution of the equation ax2 + bx + c = 0 is given by: −±bb 4 2 where and b are the coef cients of and respectively and is the constant term. This is the quadratic formula. The symbol ± states that the square root has a positive and a negative value, and you must use both of them in solving for Solve 5 – 11 – 4 = 0, giving solutions correct to 2 decimal places. Substitute a = 5, b = – 11 and c = – 4 into the formula: x −±bb ac4 2 So ( ±− ) ( ) ) 11 11 45 4 25 Note Using brackets can help you to avoid arithmetic errors. A common error is to write – 11 is – 121. x = 11 121 80 10 11 201 10 ±+ ± = 2.52 or – 0.32 Note: The calculation has been done in stages. You can also work out the answer with a calculator, but make sure you can use it properly. If not, break the calculation down. Remember the rule ‘if you try to do two things at once, you will probably get one of them wrong’. Example 8 A rectangle has sides of m and + 4) m. Its area is 100 m Find the perimeter of the rectangle, correct to 1 decimal place. So + 4) = 100 + 4 – 100 = 0 Put = 1, b = 4 and = – 100 into the quadratic formula, which gives x () () () () () 44 ± 41 100 21 2 x = −± −± 416 400 2 4 416 2 = – 12.198 or 8.198 Since is the length of the side of a rectangle, it cannot be negative, so the only valid answer is 8.198. The other side of the rectangle is 8.198 + 4 = 12.198. The perimeter of the rectangle is 2(8.198 + 12.198) = 40.8 cm (1 decimal place). Example 9 Key terms quadratic formula soluble discriminant Hints and tips The reciprocal of the fraction b is 17 Algebra: Quadratic equations 17.3 Solving a quadratic equation by using the quadratic formula 480 481
Colour-coded questions
The questions in the exercises and the review questions are colour-coded, to show you how difficult they are. Most exercises start with more accessible questions and progress through intermediate to more challenging questions.
Mathematical skills
As you progress you will be expected to absorb new ways of thinking and working mathematically. Some questions are designed to help you develop a specific skill. Look for the icons:
Mathematical reasoning – you need to apply your skills and draw conclusions from mathematical information.
Communicate mathematically – you need to show how you have arrived at your answer by using mathematical arguments.
Problem solving and making connections –you need to devise a strategy to answer the question, based on the information you are given.
Evaluate and interpret – your answer needs to show that you have considered the information you are given and commented upon it.
Worked exemplars
Ready to progress?
Review questions
Worked exemplars
Develop your mathematical skills with detailed commentaries walking you through how to approach a range of questions.
Ready to progress?
Review what you have learnt from the chapter with this colour-coded summary to check you are on track throughout the course.
Review questions
Practise what you have learnt in all of the previous chapters and put your mathematical skills to the test. Questions range from accessible through to more challenging.
5 Ben and Mia share a pizza in the ratio of 2 3. They eat it all. What fraction of the pizza did Ben eat? b What fraction of the pizza did Mia eat? 6 7 10 of a campsite is allocated to caravans. The rest is allocated to tents. Write the ratio of space allocated in the form caravans tents. 7 Amy gets of a packet of sweets. Her sister Susan gets the rest. Work out the ratio of sweets that each sister gets. Write it in the form Amy Susan. 8 To make a drink you mix squash and water in the ratio 1 4. How much water do you need to mix with 40 ml of squash? b How much squash do you need to mix with 500 ml of water? c How much squash do you need to make 1 litre of drink? 9 In a safari park at feeding time, the elephants, lions and chimpanzees are given food in the ratio 10 to 7 to 3. What fraction of the total food is given to: a the elephants b the lions c the chimpanzees? 10 A recipe uses 100 g of sugar and 250g of flour. a Find the ratio of sugar to flour. The recipe also uses butter. The ratio of sugar to butter is 2 3. b Work out the amount of butter in the recipe. 11 Andy plays 16 bowls matches. He wins 3 4 of them. He plays another matches and wins them all. The ratio of wins losses is now 4 1. Work out the value of 12 Three brothers share some money. The ratio of Noah’s share to Oscar’s share is 1 2. The ratio of Oscar’s share to Dylan’s share is 1 2. What is the ratio of Noah’s share to Dylan’s share? 13 Alan, Bella and Connor share a block of chocolate in the ratio of their ages. Alan gets and Bella gets 10 a Find the ratio of their shares in the form Alan Bella Connor. b Connor is 8 years old. How old is Bella? 14 Three cows, Gertrude, Gladys and Henrietta, produced milk in the ratio 2 3 4. Henrietta produced 1 1 litres more than Gladys. How much milk did the three cows produce altogether? 15 In a garden, the area is divided into lawn, vegetables and flowers in the ratio 3 2 1. If one-third of the lawn is dug up and replaced by flowers, what is the ratio of lawn vegetables flowers now? Give your answer as a ratio in its simplest form. MR MR EV CM Ratios as fractions You can express ratios as fractions by using the total number of parts in the ratio as the denominator (bottom number) of each fraction. Then use the numbers in the ratio as the numerators. If the ratio is in its simplest form, the fractions will not cancel. Always cancel the ratio to its simplest form before converting it to fractions. Express 25 minutes 1 hour as a ratio in its simplest form. The units must be the same, so change 1 hour into 60 minutes. 25 minutes 1 hour = 25 minutes 60 minutes Cancel the units (minutes). = 25 60 Divide both sides by 5. = 5 12 So, 25 minutes 1 hour simpli es to 5 12. Example 1 A garden is divided into lawn and shrubs in the ratio 3 2. What fraction of the garden is covered by: a lawn b shrubs? The denominator (bottom number) of the fraction is the total number of parts in the ratio each time (that is, 2 + 3 = 5). a The 3 in the ratio becomes the numerator. The lawn covers of the garden. b The 2 in the ratio becomes the numerator. The shrubs cover of the garden. Example 2 Exercise 5A 1 Express each ratio in its simplest form. 6 18 b 15 20 16 24 d 24 36 20 to 50 12 to 30 g 25 to 40 h 125 to 30 2 Write each ratio of quantities in its simplest form. a £5 to £15 b £24 to £16 c 125 g to 300 g d 40 minutes 5 minutes e 34 kg to 30 kg f £2.50 to 70p g 3 kg to 750 g h 50 minutes to 1 hour 1 hour to 1 day Hints and tips Remember to express both parts in a common unit before you simplify. 3 A length of wood is cut into two pieces in the ratio 3 7. What fraction of the original length is the longer piece? 4 Jack is 10 years old. Tom is 15. They share some marbles in the ratio of their ages. What fraction of the marbles does Jack get? 5 Ratio, proportion and rates of change: Ratio and proportion 5.1 Ratio 118 119
1 Find the slant length of a cone with height 10 cm and radius 8 cm. Give your answer correct to 1 decimal place. [3 marks] 8 cm 10 cm The slant length, is the hypotenuse of the right-angled triangle with the radius and the height being the adjacent sides to the right-angle. b 82 + 10 [1] = 64 + 100 = 164 [1] = 164 = 12.806 The rst step is to identify the rightangled triangle. Then apply Pythagoras’ theorem. The slant length of the cone is 12.8 cm (to 1 dp). [1] This is useful as we need to know the slant length to nd the curved surface area of a cone. 11 Geometry and measures: Right-angled triangles 340
can nd the output of a function given an input. can rearrange more complicated formulae where the subject may appear twice or as a power. can nd an inverse function by rearranging. can nd a composite function by combining two functions together. can combine and simplify algebraic fractions. can use iteration to nd a solution to an equation to an appropriate degree of accuracy.
1 f(x = 20 – 3x Work out the value of f (–2). [1 mark] 2 f( = + 2 and g( = 3 – 1 a Work out the value of fg(3). [2 marks] b Give an expression for g–1 x). [1 mark] 3 A sequence of numbers is formed by this iterative process: 1 2 + 4 = 16 Show that x is the first non-integer term in the sequence. [3 marks] 4 f(x = x + 3x and g(x) = 2x + 1 Work out an expression for fg( ). Give your answer in the form bx [3 marks] 5 a Make the subject of the formula 6 K Cx [3 marks] b Hence find the value of x when a = 5, K = –12 and C = – 8. [1 mark] 6 f(x = x 5 2 + Find an expression for ( ). [1 mark] b Solve 2f( = f ( ). [2 marks] 7 a Write f(x) = xx ) 3 9 3 as a single fraction in its simplest form. [3 marks] b Hence find the inverse function f –1 x). [3 marks] 8 Show that x xx xx xx 8 16 2 15 11 28 10 21 ++ ++ simplifies to ax b cx d where b and d are integers. [4 marks] 9 Solve 8 2 5 3 1 + [4 marks] 10 The iterative formula x + = 613 x + can be used to solve the equation x = 6x + 13. Use the starting value = 2.5 to find the values of 2 and [2 marks] Give your answers to 4 significant figures. b Continue the iteration to find the root to 5 decimal places. [1 mark] 24 Algebra: Algebraic fractions and functions 684
MR CM PS EV
and proportion 2
Number: Fractions, ratio
This chapter is going to show you:
• how to work out one quantity as a fraction of another
• how to add, subtract, multiply and divide fractions with and without a calculator
• how to use a percentage multiplier
You should already know:
• how to work out percentage increase and decrease
• how to work out one quantity as a percentage of another.
• how to cancel fractions to their simplest form
• how to nd equivalent fractions, decimals and percentages
• how to add and subtract fractions with the same denominator
Context and careers
• how to work out simple percentages, such as 10%, of quantities
• how to convert a mixed number to an improper fraction and vice versa.
Fractions and percentages are all around you in your everyday life. For example, salespeople often get a basic salary plus a percentage of the sales they make and the government uses fractions or percentages to set targets or to make claims, such as: ‘Our aim is to cut carbon emissions by one-third by 2030’ or ‘Unemployment has fallen by 1%’.
Careers that use this maths
• Teachers use percentages when reporting student progress.
• Water quality technicians use fractions when describing how much they have opened or closed a valve to allow water through the water main, for example, 1 2 turn or 1 4 turn.
• Engineers use fractions and ratios to calculate the optimum air : fuel ratio to burn inside the engine of a vehicle.
• Architects and set designers model buildings and movie sets to different scales, for example, 1 : 200.
• Fractions, ratios and proportions are used when cooking, for example, when adjusting recipes to cater for more or fewer people.
38
2.1 One quantity as a fraction of another
This section will show you how to:
• nd one quantity as a fraction of another.
Key terms
There are many situations when you may need to describe one amount or quantity as a fraction of another. For example, in one day you might nd that about half the students in your school are boys, you spend about a third of your time in bed or your bus fares cost about a fth of your money each week.
Sometimes you need to be more accurate and work with exact rather than approximate amounts. The next examples will show you how to do this.
Write £5 as a fraction of £20.
£5 as a fraction of £20 is written as 5 20 .
Note that 5 20 15 45 = × × so you can cancel the fraction to 1 4 . So £5 is one-quarter of £20.
Example
A book has 320 pages. 200 of the pages have illustrations. 3 4 of these pages have colour illustrations.
How many of the pages of the whole book have colour illustrations? Express the answer as a fraction of the whole book.
200 pages have illustrations.
3 4 of the pages with illustrations are in colour. 3 4 × 200 = 150 150 of the 320 pages have colour illustrations. This is 150 320 of the book. It cancels to 15 32
Exercise 2A
1 Write the first quantity as a fraction of the second. a 2 cm, 6 cm b 4 kg, 20 kg c £8, £20 d 5 hours, 24 hours e 12 days, 30 days f 50p, £3 g 4 days, 2 weeks h 40 minutes, 2 hours
2 During April, it rained on 12 days. For what fraction of the month did it rain?
3 In a class of 30 students, 3 5 are boys. Of these boys 1 3 are left-handed. What fraction of the whole class is made up of left-handed boys?
4 Reka wins £1200 in a competition and puts £400 in a bank account. She gives 1 4 of what is left to her sister and then spends the rest. What fraction of her winnings did she spend?
5 Jon earns £900 and saves £300 of it. Andray earns £1000 and saves £350 of it. Who is saving the greater proportion of his earnings?
6 In two tests Harry gets 13 out of 20 and 16 out of 25. Which is the better mark? Explain your answer.
2.1 One quantity as a fraction of another
fraction quantity
Example 1
This is 200 320 1045 4108 == ×× ×× 5 8 == of the book.
2
MR CM CM
39
7 In a street of 72 dwellings, 5 12 are bungalows. The rest are two-storey houses. Half of the bungalows are detached and 2 7 of the houses are detached. What fraction of the 72 dwellings are detached?
8 I have 24 T-shirts. 1 6 have logos on them; the rest are plain. 2 5 of the plain T-shirts are long-sleeved. 3 4 of the T-shirts with logos are long-sleeved. What fraction of all my T-shirts are long-sleeved?
9 Three quantities are x, y and z. x y = 3 4 and y z = 4 7 .
Work out the value of x z .
10 Three quantities are a, b and c. a b = 3 4 and b c = 6 11 .
Work out the value of a c
2.2 Adding, subtracting and calculating with fractions
This section will show you how to:
• add and subtract fractions with different denominators.
You can only add or subtract fractions that have the same denominator. If necessary, change one or both to equivalent fractions with the same denominator. Then add or subtract the numerators.
Always look for the lowest common denominator of the fractions you are changing. This is the lowest common multiple (LCM) of both denominators.
The LCM of 4 and 6 is 12, so the problem becomes:
A fraction in which the numerator is bigger than the denominator is called an improper fraction. You know how to change improper fractions to mixed numbers, and mixed numbers to improper fractions. You also know that a mixed number is made up of a whole number and a proper fraction, for example:
Separate the whole numbers from the fractions.
MR PS PS PS
6
Work this out. 5
–3 4
5 6 –3 4 = 5 6 × 2 2 –3 4 × 3 3 = 10 12 –9 12 = 1 12 Example 3
a 2 1 3 + 3 5 7 b 3 1 4 – 1 3 5
Work these out.
a 2 1 3 + 3 5 7 = 2 + 3 + 1 3 + 5 7 = 5 + 7 21 + 15 21 = 5 + 22 21 = 5 + 1 1 21 = 6 1 21 b 3 1 4 – 1 3 5 = 3 – 1 + 1 4 –3 5 = 2 + 5 20 –12 20 = 2 – 7 20 = 1 13 20 Example 4
14 5 = 2 4 5 and 3 2 7 = 23 7 2 Number: Fractions, ratio and proportion 40
Shop A sells a bicycle for £540 including VAT but has an offer of 1 4 off the selling price. Shop B sells the same model of bicycle for £350 (excluding VAT). VAT will add 1 5 to the price. In which shop is the bike cheaper? Show your working.
Shop A: 540 ÷ 4 × 3 = 405
Shop B: 350 × 1 5 = 70 350 + 70 = 420
So the bike is cheaper in shop A.
Exercise 2B
1 Work these out.
+ 1 5
1
–1 10
2 Which is the biggest: half of 96, one-third of 141, two-fifths of 120 or three-quarters of 68?
3 Work these out.
1 3 + 1 9 20
4 a In a class election, half of the students voted for Aminah, one-third voted for Jenet and the rest voted for Pieter. What fraction of the class voted for Pieter?
b One of the numbers in the box is the number of students in the class in part a
How many students are there in the class?
5 A one-litre bottle of milk is used to fill four glasses. Three glasses have a capacity of one-eighth of a litre. The fourth glass has a capacity of half a litre. Priya likes milky coffee so she always has at least 10 cl of milk in her cup. Is there enough milk left in the bottle for Priya to have two cups of coffee?
6 Mick has worked out this sum.
1 1 3 + 2 1 4 = 3 2 7
His answer is incorrect. What mistake has he made? Work out the correct answer.
7 Find the difference between
8 There are 900 students in a school. 11 20 of the students are boys. Of the boys, 2 11 are left-handed. Of the girls, 2 9 are left-handed. What fraction of all the students are left-handed? Show your working.
9 There are 600 counters in a bag. Each counter is red, blue or yellow. 3 8 of the counters are red. 1 5 of the counters are blue.
a What fraction of the counters are yellow?
b How many yellow counters are there in the bag?
Example 5
a
b 1 3 + 1 4 c 2 3 + 1 4 d
e 7 8 –3 4 f 5 6 –3 4
1 3
5
a
b 1 1 8
5 9 c 7 10 + 3 8 + 5 6 d 1 1 3 + 7 10 –4 15
3
–
25 28 30 32
1 4 3 5 + and 3 4 1 5 +
MR EV PS CM CM PS 2.2 Adding, subtracting and calculating with fractions 41
10
A small gym has 200 members. 27 40 of the members are at least 40 years of age. 2 5 of the members are women.
a Show that some of the women are at least 40 years of age.
b 5 8 of the women are at least 40 years old. How many of the men are aged less than 40?
11 This is how Jo works out the fraction at the mid-value of two other fractions.
• Write the two fractions with a common denominator.
• The numerator of the midpoint fraction is the sum of the numerators of the two fractions written with a common denominator.
• The denominator of the midpoint fraction is the sum of the denominators of the two fractions written with a common denominator.
For example, to find the midpoint fraction of 1 5 and 3 4 :
1 5 = 4 20 3 4 = 15 20
So the midpoint fraction is 415 20 20 19 40 + + = .
a Show that the calculation above does give the midpoint fraction of 1 5 and 3 4
b Does the method always work? Explain your answer.
2.3 Multiplying and dividing fractions
This section will show you how to:
• multiply proper fractions
• multiply mixed numbers
• divide by fractions.
Multiplying fractions
To multiply fractions, follow these four steps.
Key term
Step 1: Convert any mixed numbers into improper fractions and rewrite the multiplication if necessary.
Step 2: Simplify the multiplication by cancelling by any common factors in the numerators and the denominators.
Step 3: Multiply the numerators to obtain the numerator of the answer and multiply the denominators to obtain the denominator of the answer.
Step 4: If the answer is an improper fraction, convert this into a mixed number.
Identify any common factors in the numerators and denominators: 2 is a factor of 4 and 10; 3 is a factor of 3 and 9.
Simplify the fractions, cancelling by 2 and 3, before multiplying.
Convert the mixed numbers into improper fractions.
Simplify the fractions, cancelling by 4 and 5.
CM MR CM EV
reciprocal Work these out. a 4 9 × 3 10 b 2 2 5 × 1 7 8 a 2 3 4 9 × 3 10 1 5 = 2 15
b 2 2 5 × 1 7 8 = 12 5 × 15 8 3 1 12 5 × 15 8 3 2 = 9 2 = 4 1 2
Example 6 2 Number: Fractions, ratio and proportion 42
Worked exemplars
1 A car journey takes 4 1 2 hours. It includes a 45-minute break.
What fraction of the time is spent driving? [2 marks]
Be careful to use the same units when you are working with fractions.
4 1 2 hours = 270 minutes
The break is 45 270 1 6 = of the time.
The driving is 5 6 of the time. [2] Simplify the fraction as much as possible.
You could work in 15-minute intervals and say the fraction is 3 18 1 6 =
2 The price of a basket of shopping has increased by 13.8% since last year.
Last year it was £128.44. Find its price now. [3 marks]
The multiplier method is the most ef cient way to calculate percentage changes.
The multiplier for a 13.8% increase is 1.138. [1]
The price now is £128.44 × 1.138 = £146.16. [2]
13.8% is 0.138.
Round the calculator answer to the nearest penny.
3 Over a period of 20 years the value of a painting increased from £2500 to £8500. Calculate the percentage increase. [2 marks]
There are several ways you could answer this question. You could nd the increase in value, or you could nd the multiplier.
The multiplier is 8500 2500 34 = . [1]
The percentage increase is 240%. [1]
The multiplier is new value ÷ old value.
The multiplier is more than 1 so the increase is more than 100%.
3.4 – 1 = 2.4 = 240% increase
2 Number: Fractions, ratio and proportion 54
Ready to progress?
I can write one quantity as a fraction of another. I can add, subtract, multiply and divide fractions. I can calculate percentage increases and decreases.
I can compare proportions using percentages. I can use percentage multipliers to carry out percentage calculations.
I can calculate with mixed numbers. I can solve complex problems involving percentage increases and percentage decreases.
Review questions
1 Mrs Patel earns £520 per week. She is awarded a pay rise of 10%. How much does she earn each week after the pay rise? [2 marks]
2 Here are the times for Ali and Beth in a 200-metre race.
a Ali runs 200 m again and reduces her time by 5%.
Calculate her new time. [2 marks]
b Beth runs 200 m again and reduces her time to 29.0 seconds. Calculate the percentage reduction in her time. [2 marks]
3 The cost of repairing a car is £235. 20% tax is added to this to get the final cost. Work out the final cost. [2 marks]
4 This table shows the height and mass of a baby girl at 6 months and 1 year.
kg
kg
Show that the percentage increase in mass is greater than the percentage increase in height. [3 marks]
5 a In a month of 30 days it rains on 9 days.
Work out the percentage of days when there is no rain. [2 marks]
b During 14 1 2 hours of daylight it is sunny for 8 3 4 hours.
Work out the percentage of the hours of daylight that are sunny. [2 marks]
7 A washing machine normally costs £350. It is reduced by 8% in a sale. How much is the sale price of the washing machine? [2 marks]
Age Height Mass 6 months 73 cm 7.8
1
80 cm 10.1
year
6 a Change 7 8 to a decimal. [1 mark] b Work out 3 5 –2 7 . [1 mark] c Work out 3 1 4 × 1 3 5 . [1 mark]
Ali Beth Time (seconds) 28.0 32.5 MR 2 Number: Fractions, ratio and proportion 56
Runner