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S T U D Y PAPER F2 T E MANAGEMENT ACCOUNTING X T In this edition approved by ACCA x We ddiscuss the bbest strategies for studying for ACCA exams x We hhighlight the mmost important elements in the syllabus and the kkey skills you will need x We ssignpost how each chapter links to the syllabus and the study guide x We pprovide lots of eexam focus points demonstrating what the examiner will want you to do x We eemphasise key points in regular ffast forward summaries xThiNganHang.comWe ttest your knowledge of what you've studied in qquick quizzes x We eexamine your understanding in our eexam question bank x We rreference all the important topics in our ffull index BPP's i-Learn and i-Pass products also support this paper. FOR EXAMS IN DECEMBER 2009 AND JUNE 2010 ThiNganHang.com
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First edition 2007 Third edition June 2009 ISBN 9780 7517 6362 1 (Previous ISBN 9780 7517 4721 8) British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library All our rights reserved. No part of this publication may be Published by reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, BPP Learning Media Ltd photocopying, recording or otherwise, without the prior BPP House, Aldine Place written permission of BPP Learning Media Ltd. London W12 8AA www.bpp.com/learningmedia We are grateful to the Association of Chartered Certified Accountants for permission to reproduce past examination questions. The suggested solutions in the Printed in the United Kingdom exam answer bank have been prepared by BPP Learning Media Ltd, except where otherwise stated. Your learning materials, published by BPPLearning Media Ltd, are printed on paperThiNganHang.com sourced from sustainable, managed forests. © BPP Learning Media Ltd 2009 ii ThiNganHang.com
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Contents Page Introduction How the BPP ACCA-approved Study Text can help you pass v Studying F2 vii The exam paper and exam formulae sheet viii Before you begin… are you confident with basic maths? 1 Part A The nature and purpose of cost and management accounting 1 Information for management 23 Part B Cost classification, behaviour and purpose 2 Cost classification 41 3 Cost behaviour 55 Part C Business mathematics and computer spreadsheets 4 Correlation and regression; expected values 71 5 Spreadsheets 89 Part D Cost accounting techniques 6 Material costs 113 7 Labour costs 137 8 Overhead and absorption costing 159 9 Marginal and absorption costing 183 10 Process costing 197 11 Process costing, joint products and by-products 225 12 Job, batch and service costing 237 Part E Budgeting and standard costing 13 Budgeting 261 14 Standard costing 281 15 Basic variance analysis 289 16 Further variance analysis 307 Part F Short-term decision-making techniques ThiNganHang.com171819 Cost-volume-profit (CVP) analysisLinear programmingRelevant costing and decision-making 323343357 Exam question bank 379 Exam answer bank 399 Index 417 Review form and free prize draw Contents iii ThiNganHang.com
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A note about copyright Dear Customer What does the little © mean and why does it matter? Your market-leading BPP books, course materials and elearning materials do not write and update themselves. People write them: on their own behalf or as employees of an organisation that invests in this activity. Copyright law protects their livelihoods. It does so by creating rights over the use of the content. Breach of copyright is a form of theft – as well being a criminal offence in some jurisdictions, it is potentially a serious breach of professional ethics. With current technology, things might seem a bit hazy but, basically, without the express permission of BPP Learning Media: x Photocopying our materials is a breach of copyright x Scanning, ripcasting or conversion of our digital materials into different file formats, uploading them to facebook or emailing them to your friends is a breach of copyright You can, of course, sell your books, in the form in which you have bought them – once you have finished with them. (Is this fair to your fellow students? We update for a reason.) But the ilearns are sold on a single user license basis: we do not supply ‘unlock’ codes to people who have bought them second hand. And what about outside the UK? BPP Learning Media strives to make our materials available at prices students can afford by local printing arrangements, pricing policies and partnerships which are clearly listed on our website. A tiny minority ignore this and indulge in criminal activity by illegally photocopying our material or supporting organisations that do. If they act illegally and unethically in one area, can you really trust them? ThiNganHang.com iv ThiNganHang.com
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How the BPP ACCA-approved Study Text can help you pass – AND help you with your Practical Experience Requirement! NEW FEATURE – the PER alert! Before you can qualify as an ACCA member, you do not only have to pass all your exams but also fulfil a three year practical experience requirement (PER). To help you to recognise areas of the syllabus that you might be able to apply in the workplace to achieve different performance objectives, we have introduced the ‘PER alert’ feature. You will find this feature throughout the Study Text to remind you that what you are learning to pass your ACCA exams is equally useful to the fulfilment of the PER requirement. Tackling studying Studying can be a daunting prospect, particularly when you have lots of other commitments. The different features of the text, the purposes of which are explained fully on the Chapter features page, will help you whilst studying and improve your chances of exam success. Developing exam awareness Our Texts are completely focused on helping you pass your exam. Our advice on Studying F2 outlines the content of the paper, the necessary skills the examiner expects you to demonstrate and any brought forward knowledge you are expected to have. Exam focus points are included within the chapters to provide information about skills that you will need in the exam and reminders of important points within the specific subject areas. Using the Syllabus and Study Guide You can find the syllabus, Study Guide and other useful resources for F2 on the ACCA web site: www.accaglobal.com/students/study_exams/qualifications/acca_choose/acca/fundamentals/ma The Study Text covers all aspects of the syllabus to ensure you are as fully prepared for the exam as possible. ThiNganHang.comTesting yourself helps you develop the skills you need to pass the exam and also confirms that you canTesting what you can do recall what you have learnt. We include Exam-style Questions – lots of them - both within chapters and in the Exam Question Bank, as well as Quick Quizzes at the end of each chapter to test your knowledge of the chapter content. Introduction v ThiNganHang.com
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Chapter features Each chapter contains a number of helpful features to guide you through each topic. Topic list Tells you what you will be studying in this chapter and the Topic list Syllabusreference relevant section numbers, together with the ACCA syllabus references. Introduction Puts the chapter content in the context of the syllabus as a whole. Study Guide Links the chapter content with ACCA guidance. Exam Guide Highlights how examinable the chapter content is likely to be and the ways in which it could be examined. Summarises the content of main chapter headings, FAST FORWARD allowing you to preview and review each section easily. Examples Demonstrate how to apply key knowledge and techniques. Definitions of important concepts that can often earn you Key terms easy marks in exams. Provide information about skills you will need in the exam Exam focus points and reminders of important points within the specific subject area. Formulae that are not given in the exam but which have to Formula to learn be learnt. This is a new feature that gives you a useful indication of syllabus areas that closely relate to performance objectives in your PER. Question Give you essential practice of techniques covered in the chapter. Case Study Provide real world examples of theories and techniques. ThiNganHang.comChapter Roundup A full list of the Fast Forwards included in the chapterproviding an easy source of review. , Quick Quiz A quick test of your knowledge of the main topics in the chapter. Exam Question Bank Found at the back of the Study Text with more comprehensive chapter questions. vi Introduction ThiNganHang.com
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Studying F2 This paper introduces you to costing and management accounting techniques, including those techniques that are used to make and support decisions. It provides a basis for Paper F5 – Performance Management. The examiner for this paper is David Forster who was previously the examiner for Paper 1.2 under the previous syllabus. His aims are to test your knowledge of basic costing and management accounting techniques and also to test basic application of knowledge. 1 What F2 is about F2 is one of the three papers that form the Knowledge base for your ACCA studies. Whilst Paper F1 – Accountant in Business gives you a broad overview of the role and function of the accountant, Papers F2 – Management Accounting and F3 – Financial Accounting give you technical knowledge at a fundamental level of the two major areas of accounting. Paper F2 will give you a good grounding in all the basic techniques you need to know in order to progress through the ACCA qualification and will help you with Papers F5 – Performance Management and P5 – Advanced Performance Management in particular. 2 What skills are required? The paper is examined by computer-based exam or a written exam consisting of objective test questions (mainly multiple-choice questions). You are not required, at this level, to demonstrate any written skills. However you will be required to demonstrate the following. x Core knowledge – classification and treatment of costs, accounting for overheads, budgeting and standard costing, decision-making. x Numerical and mathematical skills – regression analysis, linear programming. x Spreadsheet skills – the paper will test your understanding of what can be done with spreadsheets. This section will be particularly useful to you in the workplace. 3 How to improve your chances of passing You must bear the following points in mind. x All questions in the paper are compulsory. This means that you cannot avoid studying any part of the syllabus. The examiner can examine any part of the syllabus and you must be prepared for him to do so. x The best preparation for any exam is to practise lots of questions. Work your way through the Quick Quizzes at the end of each chapter in this Study Text and then attempt the questions in the ThiNganHang.comx In the exam,before – it is probably different in some way that you haven’t spotted.Exam Question Bankread the questions carefully. You should also make full use of the. Beware any question that looks like one you have seenBPP Practice and Revision Kit. x If you really cannot answer something, move on. You can always come back to it. x If at the end of the exam you find you have not answered all of the questions, have a guess. You are not penalised for getting a question wrong and there is a chance you may have guessed correctly. If you fail to choose an answer, you have no chance of getting any marks. Introduction vii ThiNganHang.com
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The exam paper Format of the paper Guidance The exam is a two hour paper that can be taken either as a paper-based or computer-based exam. There are 50 questions in the paper – 40 questions will be worth two marks each whilst the remaining 10 questions are worth one mark each. There are therefore 90 marks available. The two mark questions will have a choice of four possible answers (A/B/C/D) whilst the one mark questions will have a choice of two (A/B) or three possible answers (A/B/C). The one and two mark questions will be interspersed and questions will appear in random order (that is, not in Study Guide order). Questions on the same topic will not necessarily be grouped together. Questions will be a mix of calculation and non-calculation questions in a similar mix to the pilot paper. The pilot paper can be found on the ACCA web site: www.accaglobal.com/students/study_exams/qualifications/acca_choose/acca/fundamentals/ma/past_papers. The examiner has indicated that the pilot paper is an extremely useful guide to the mix of questions that you might expect to find in the ‘real’ exams. You should therefore study the pilot paper carefully to get an idea of the weighting that each syllabus area will be given in the exam. Exam formulae sheet You will be given an exam formulae sheet in your exam. This is reproduced below, together with the chapters of the Study Text in which you can find the formulae. Regression analysis Economic order quantity (Chapter 4 of Study Text) (Chapter 6 of Study Text) 6Y b6x 2C D a = 0 n n C h n6xy 6x6y Economic batch quantity b = n6x2 (6x)2 (Chapter 6 of Study Text) n6xy 6x6y 2C D r = 0 (n6x2 (6x)2 )(n6y2 (6y)2 ) D C (1 ) h R ThiNganHang.com viii Introduction ThiNganHang.com
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1 Using this introductory chapter The Paper F2 – Management Accounting syllabus assumes that you have some knowledge of basic mathematics and statistics. The purpose of this introductory chapter is to provide the knowledge required in this area if you haven't studied it before, or to provide a means of reminding you of basic maths and statistics if you are feeling a little rusty in one or two areas! Accordingly, this introductory chapter sets out from first principles a good deal of the knowledge that you are assumed to possess in the main chapters of the Study Text. You may wish to work right through it now. You may prefer to dip into it as and when you need to. You may just like to try a few questions to sharpen up your knowledge. Don't feel obliged to learn everything in the following pages: they are intended as an extra resource to be used in whatever way best suits you. 2 Integers, fractions and decimals 2.1 Integers, fractions and decimals An integer is a whole number and can be either positive or negative. The integers are therefore as follows. .....,-5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5..... . Fractions (such as 1/2, 1/4, 19/35, 101/377, .....) and decimals (0.1, 0.25, 0.3135 .....) are both ways of showing parts of a whole. Fractions can be turned into decimals by dividing the numerator by the denominator (in other words, the top line by the bottom line). To turn decimals into fractions, all you have to do is remember that places after the decimal point stand for tenths, hundredths, thousandths and so on. 2.2 Significant digits Sometimes a decimal number has too many digits in it for practical use. This problem can be overcome by rounding the decimal number to a specific number of significant digits by discarding digits using the following rule. If the first digit to be discarded is greater than or equal to five then add one to the previous digit. Otherwise the previous digit is unchanged. 2.3 Example: Significant digits (a) 187.392 correct to five significant digits is 187.39 Discarding a 2 causes nothing to be added to the 9. (b) 187.392 correct to four significant digits is 187.4 Discarding the 9 causes one to be added to the 3. (c) 187.392 correct to three significant digits is 187 ThiNganHang.comQuestionDiscarding a 3 causes nothing to be added to the 7. Significant digits What is 17.385 correct to four significant digits? Answer 17.39 Basic maths 3 ThiNganHang.com
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3 Mathematical notation 3.1 Brackets Brackets are commonly used to indicate which parts of a mathematical expression should be grouped together, and calculated before other parts. In other words, brackets can indicate a priority, or an order in which calculations should be made. The rule is as follows. (a) Do things in brackets before doing things outside them. (b) Subject to rule (a), do things in this order. (i) Powers and roots (ii) Multiplications and divisions, working from left to right (iii) Additions and subtractions, working from left to right Thus brackets are used for the sake of clarity. Here are some examples. (a) 3 + 6 u 8 = 51. This is the same as writing 3 + (6 u 8) = 51. (b) (3 + 6) u 8 = 72. The brackets indicate that we wish to multiply the sum of 3 and 6 by 8. (c) 12 – 4 y 2 = 10. This is the same as writing 12 – (4 y 2) = 10 or 12 – (4/2) = 10. (d) (12 – 4) y 2 = 4. The brackets tell us to do the subtraction first. A figure outside a bracket may be multiplied by two or more figures inside a bracket, linked by addition or subtraction signs. Here is an example. 5(6 + 8) = 5 u (6 + 8) = 5 u 6 + 5 u 8 = 70 This is the same as 5(14) = 5 u 14 = 70 The multiplication sign after the 5 can be omitted, as shown here (5(6 + 8)), but there is no harm in putting it in (5 u (6 + 8)) if you want to. Similarly: 5(8 – 6) = 5(2) = 10; or 5 u 8 – 5 u 6 = 10 When two sets of figures linked by addition or subtraction signs within brackets are multiplied together, each figure in one bracket is multiplied in turn by every figure in the second bracket. Thus: (8 + 4)(7 + 2) = (12)(9) = 108 or 8 u 7 + 8 u 2 + 4 u 7 + 4 u 2 = 56 + 16 + 28 + 8 = 108 3.2 Negative numbers When a negative number (–p) is added to another number (q), the net effect is to subtract p from q. ThiNganHang.com(a) 10 + (–6) = 10 – 6 = 4 (b) –10 + (–6) = –10 – 6 = –16 When a negative number (-p) is subtracted from another number (q), the net effect is to add p to q. (a) 12 – (–8) = 12 + 8 = 20 (b) –12 – (–8) = –12 + 8 = –4 When a negative number is multiplied or divided by another negative number, the result is a positive number. –8 u (–4) = +32 –18/(–3) = +6 If there is only one negative number in a multiplication or division, the result is negative. 4 Basic maths ThiNganHang.com
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–8 u 4 = –32 3 u (–2) = –6 12/(–4) = –3 –20/5 = –4 Question Negative numbers Work out the following. (a) (72 – 8) – (–3 +1) 888 (2911) (b) 12 2 (c) 8(2 – 5) – (4 – (–8)) 36 84 81 (d) 93 310 3 Answer (a) 64 – (–2) = 64 + 2 = 66 (b) 8 + (–9) = –1 (c) –24 – (12) = –36 (d) –6 – (–12) – (–27) = –6 + 12 + 27 = 33 3.3 Reciprocals The reciprocal of a number is just 1 divided by that number. For example, the reciprocal of 2 is 1 divided by 2, ie ½. 3.4 Extra symbols You will come across several mathematical signs in this book and there are six which you should learn right away. (a) > means 'greater than'. So 46 > 29 is true, but 40 > 86 is false. (b) t means 'is greater than or equal to'. So 4 t 3 and 4 t 4. (c) < means ' is less than'. So 29 < 46 is true, but 86 < 40 is false. (d) d means ' is less than or equal to'. So 7 d 8 and 7 d 7. (e) z means 'is not equal to'. So we could write 100.004 z 100. ThiNganHang.com(f) 6 means ‘the sum of’. 4 Percentages and ratios 4.1 Percentages and ratios Percentages are used to indicate the relative size or proportion of items, rather than their absolute size. For example, if one office employs ten accountants, six secretaries and four supervisors, the absolute values of staff numbers and the percentage of the total work force in each type would be as follows. Accountants Secretaries Supervisors Total Absolute numbers 10 6 4 20 Percentages 50% 30% 20% 100% Basic maths 5 ThiNganHang.com
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The idea of percentages is that the whole of something can be thought of as 100%. The whole of a cake, for example, is 100%. If you share it out equally with a friend, you will get half each, or 100%/2 = 50% each. To turn a percentage into a fraction or decimal you divide by 100. To turn a fraction or decimal back into a percentage you multiply by 100%. Consider the following. (a) 0.16 = 0.16 u 100% = 16% (b) 4/5 = 4/5 u 100% = 400/5% = 80% (c) 40% = 40/100 = 2/5 = 0.4 There are two main types of situations involving percentages. (a) You may be required to calculate a percentage of a figure, having been given the percentage. Question: What is 40% of $64? Answer: 40% of $64 = 0.4 u $64 = $25.60. (b) You may be required to state what percentage one figure is of another, so that you have to work out the percentage yourself. Question: What is $16 as a percentage of $64? 16 1 Answer: $16 as a percentage of $64 = u100% u100% 25% 64 4 In other words, put the $16 as a fraction of the $64, and then multiply by 100%. 4.2 Proportions A proportion means writing a percentage as a proportion of 1 (that is, as a decimal). 100% can be thought of as the whole, or 1. 50% is half of that, or 0.5. Consider the following. Question: There are 14 women in an audience of 70. What proportion of the audience are men? Answer: Number of men = 70 – 14 = 56 56 8 Proportion of men = 80% 0.8 70 10 (a) 8/10 or 4/5 is the fraction of the audience made up by men. (b) 80% is the percentage of the audience made up by men. (c) 0.8 is the proportion of the audience made up by men. 4.3 Ratios Suppose Tom has $12 and Dick has $8. The ratio of Tom's cash to Dick's cash is 12:8. This can be cancelled down, just like a fraction, to 3:2. Usually an examination question will pose the problem the other way around: Tom and Dick wish to share ThiNganHang.com$20 out in the ratio 3:2. How much will each receive?Because 3 + 2 = 5, we must divide the whole up into five equal parts, then give Tom three parts and Dicktwo parts. (a) $20 y 5 = $4 (so each part is $4) (b) Tom's share = 3 u $4 = $12 (c) Dick's share = 2 u $4 = $8 (d) Check: $12 + $8 = $20 (adding up the two shares in the answer gets us back to the $20 in the question). This method of calculating ratios as amounts works no matter how many ratios are involved. Here is another example. Question: A, B, C and D wish to share $600 in the ratio 6:1:2:3. How much will each receive? 6 Basic maths ThiNganHang.com
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Answer: (a) Number of parts = 6 + 1 + 2 + 3 = 12. (b) Value of each part = $600 y 12 = $50 (c) A: 6 u $50 = $300 B: 1 u $50 = $50 C: 2 u $50 = $100 D 3 u $50 = $150 (d) Check: $300 + $50 + $100 + $150 = $600. Question Ratios (a) Peter and Paul wish to share $60 in the ratio 7 : 5. How much will each receive? (b) Bill and Ben own 300 and 180 flower pots respectively. What is the ratio of Ben's pots: Bill's pots? (c) Tom, Dick and Harry wish to share out $800. Calculate how much each would receive if the ratio used was: (i) 3 : 2 : 5; (ii) 5 : 3 : 2; (iii) 3 : 1 : 1. (d) Lynn and Laura share out a certain sum of money in the ratio 4 : 5, and Laura ends up with $6. (i) How much was shared out in the first place? (ii) How much would have been shared out if Laura had got $6 and the ratio had been 5 : 4 instead of 4 : 5? Answer (a) There are 7 + 5 = 12 parts Each part is worth $60 y 12 = $5 Peter receives 7 u $5 = $35 Paul receives 5 u $5 = $25 (b) Ben's pots: Bill's pots = 180 : 300 = 3 : 5 (c) (i) Total parts = 10 Each part is worth $800 y 10 = $80 Tom gets 3 u $80 = $240 Dick gets 2 u $80 = $160 Harry gets 5 u $80 = $400 (ii) Same parts as (i) but in a different order. Tom gets $400 Dick gets $240 ThiNganHang.comHarry gets $160 (iii) Total parts = 5 Each part is worth $800 y 5 = $160 Therefore Tom gets $480 Dick and Harry each get $160 (d) (i) Laura's share = $6 = 5 parts Therefore one part is worth $6 y 5 = $1.20 Total of 9 parts shared out originally Therefore total was 9 u $1.20 = = $10.80 (ii) Laura's share = $6 = 4 parts Therefore one part is worth $6 y 4 = $1.50 Therefore original total was 9 u $1.50 = $13.50 Basic maths 7 ThiNganHang.com
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5 Roots and powers 5.1 Square roots The square root of a number is a value which, when multiplied by itself, equals the original number. 9 = 3, since 3 u 3 = 9 Similarly, the cube root of a number is the value which, when multiplied by itself twice, equals the original number. 3 64 = 4, since 4 u 4 u 4 = 64 The nth root of a number is a value which, when multiplied by itself (n – 1) times, equals the original number. 5.2 Powers Powers work the other way round. Thus the 6th power of 2 = 26 = 2 u 2 u 2 u 2 u 2 u 2 = 64. Similarly, 34 = 3 u 3 u 3 u 3 = 81. Since 9 = 3, it also follows that 32 = 9, and since 3 64 = 4, 43 = 64. When a number with an index (a 'to the power of' value) is multiplied by the same number with the same or a different index, the result is that number to the power of the sum of the indices. (a) 52 u 5 = 52 u 51 = 5(2+1) = 53 = 125 (b) 43 u 43 = 4(3+3) = 46 = 4,096 Similarly, when a number with an index is divided by the same number with the same or a different index, the result is that number to the power of the first index minus the second index. (a) 64 y 63 = 6(4-3) = 61 = 6 (b) 78 y 76 = 7(8-6) = 72 = 49 Any figure to the power of zero equals one. 10 = 1, 20 = 1, 30 = 1, 40 = 1 and so on. Similarly, 82 y 82 = 8(2-2) = 80 = 1 An index can be a fraction, as in 161 . What 161 means is the square root of 16( 16 or 4). If we multiply 2 2 1 1 (11) 162 by 162 we get 16 2 2 which equals 161 and thus 16. 1 1 1 1 1 1 1 Similarly, 2163 is the cube root of 216 (which is 6) because 2163 u 2163 u 2163 = 216 ( ) 3 3 3 = 2161 = 216. An index can be a negative value. The negative sign represents a reciprocal. Thus 2-1 is the reciprocal of, ThiNganHang.com=or one over, 2211 = 21 1 5.3 Example: Roots and powers -2 1 1 -3 1 1 (a) 2 = = and 2 = = 22 4 23 8 -6 1 1 (b) 5 = = 56 15,625 8 Basic maths ThiNganHang.com
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5 -2 5 1 5-2 3 (c) 4 u 4 = 4 u = 4 = 4 = 64 42 When we multiply or divide by a number with a negative index, the rules previously stated still apply. 2 -2 (2+(-2)) 0 2 1 (a) 9 u 9 = 9 = 9 = 1 (That is, 9 u = 1) 92 (b) 45 y 4-2 = 4(5-(-2)) = 47 = 16,384 (c) 38 u 3-5 = 3(8-5) = 33 = 27 -5 -2 -5-(-2) -3 1 1 1 1 1 2 1 (d) 3 y 3 = 3 = 3 = = . (This could be re-expressed as y u3 .) 33 27 35 32 35 33 Question Calculations Work out the following, using your calculator as necessary. (a) (18.6)2.6 (b) (18.6)–2.6 (c) 2.6 18.6 (d) (14.2)4 u (14.2) 1 4 (e) (14.2)4 + (14.2) 1 4 Answer (a) (18.6)2.6 = 1,998.64 –2.6 1 2.6 (b) (18.6) = ( ) = 0.0005 18.6 (c) 2.6 18.6 = 3.078 (d) (14.2)4 u (14.2) 1 = (14.2)4.25 = 78,926.98 4 (e) (14.2)4 + (14.2) 1 = 40,658.69 + 1.9412 = 40,660.6312 4 6 Equations ThiNganHang.com6.1 Introduction So far all our problems have been formulated entirely in terms of specific numbers. However, think back to when you were calculating powers with your calculator earlier in this chapter. You probably used the xy key on your calculator. x and y stood for whichever numbers we happened to have in our problem, for example, 3 and 4 if we wanted to work out 34. When we use letters like this to stand for any numbers we call them variables. Today when we work out 34, x stands for 3. Tomorrow, when we work out 72 , x will stand for 7: its value can vary. The use of variables enables us to state general truths about mathematics. For example: x = x x2 = x u x If y = 0.5 u x, then x = 2 u y Basic maths 9 ThiNganHang.com
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These will be true whatever values x and y have. For example, let y = 0.5 u x If y = 3, x = 2 u y = 6 If y = 7, x = 2 u y = 14 If y = 1, x = 2 u y = 2, and so on for any other choice of a value for y. We can use variables to build up useful formulae. We can then put in values for the variables, and get out a value for something we are interested in. Let us consider an example. For a business, profit = revenue – costs. Since revenue = selling price u units sold, we can say that profit = selling price u units sold – costs. 'Selling price u units sold – costs' is a formula for profit. We can then use single letters to make the formula quicker to write. Let x = profit p = selling price u = units sold c = cost Then x = p u u – c. If we are then told that in a particular month, p = $5, u = 30 and c = $118, we can find out the month's profit. Profit = x = p u u – c = $5 u 30 – $118 = $150 – $118 = $32. It is usual when writing formulae to leave out multiplication signs between letters. Thus p u u – c can be written as pu – c. We will also write (for example) 2x instead of 2 u x. 6.2 Equations In the above example, pu – c was a formula for profit. If we write x = pu – c, we have written an equation. It says that one thing (profit, x) is equal to another (pu – c). Sometimes, we are given an equation with numbers filled in for all but one of the variables. The problem is then to find the number which should be filled in for the last variable. This is called solving the equation. (a) Returning to x = pu – c, we could be told that for a particular month p = $4, u = 60 and c = $208. We would then have the equation x = $4 u 60 – $208. We can solve this easily by working out $4 u 60 – $208 = $240 – $208 = $32. Thus x = $32. (b) On the other hand, we might have been told that in a month when profits were $172, 50 units were sold and the selling price was $7. The thing we have not been told is the month's costs, c. We can work out c by writing out the equation. $172 = $7 u 50 – c ThiNganHang.com$172 = $350 – c We need c to be such that when it is taken away from $350 we have $172 left. With a bit of trial and error, we can get to c = $178. Trial and error takes far too long in more complicated cases, however, and we will now go on to look at a rule for solving equations, which will take us directly to the answers we want. 6.3 The rule for solving equations To solve an equation, we need to get it into the form: Unknown variable = something with just numbers in it, which we can work out. We therefore want to get the unknown variable on one side of the = sign, and everything else on the other side. 10 Basic maths ThiNganHang.com
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The rule is that you can do what you like to one side of an equation, so long as you do the same thing to the other side straightaway. The two sides are equal, and they will stay equal so long as you treat them in the same way. For example, you can do any of the following. Add 37 to both sides. Subtract 3x from both sides. Multiply both sides by –4.329. Divide both sides by (x + 2). Take the reciprocal of both sides. Square both sides. Take the cube root of both sides. We can do any of these things to an equation either before or after filling in numbers for the variables for which we have values. (a) In Paragraph 6.2, we had $172 = $350 – c. We can then get $172 + c = $350 (add c to each side) c = $350 – $172 (subtract $172 from each side) c = $178 (work out the right hand side). (b) 450 = 3x + 72 (initial equation: x unknown) 450 – 72 = 3x (subtract 72 from each side) 450 72 = x (divide each side by 3) 3 126 = x (work out the left hand side). (c) 3y + 2 = 5y – 7 (initial equation: y unknown) 3y + 9 = 5y (add 7 to each side) 9 = 2y (subtract 3y from each side) 4.5 = y (divide each side by 2). 3x2 x (d) = 7 (initial equation: x unknown) 2 x 3x2 x = 49 (square each side) 4x (3x 1) 4 = 49 (cancel x in the numerator and the denominator of the left hand side: this does not affect the value of the left hand side, so we do not need to change the right hand side) ThiNganHang.com3x + 1 = 196 (multiply each side by 4) 3x = 195 (subtract 1 from each side) x = 65 (divide each side by 3). (e) Our example in Paragraph 6.1 was x = pu – c. We could change this, so as to give a formula for p. x = pu – c x + c = pu (add c to each side) x c = p (divide each side by u) u x c p = (swap the sides for ease of reading). u Given values for x, c and u we can now find p. We have re-arranged the equation to give p in terms of x, c and u. Basic maths 11 ThiNganHang.com
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