CIMA c3 ELSEVIER fundamentals of business mathematics

FUNDAMENTALS OF BUSINESS MATHEMATICS 9.13.1 Using Venn diagrams to assist with probability 371 Solutions to Revision Questions 385... How to use your CIMA Learning System This Funda

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Contents

1.15 Using Excel to produce graphs of Linear and Quadratic Equations 21 1.15.1 Producing a single linear equation in Excel 21 1.15.2 Drawing multiple equations on a single graph 22

1.15.4 Two quadratic equations on one graph 24

Solutions to Revision Questions 33

iii

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2.1.1 The difference between information and data 42

3.13 Using spreadsheets to produce histograms, ogives and pie charts 98

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4.15 A practical example of descriptive statistical analysis using Excel 152

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FUNDAMENTALS OF BUSINESS MATHEMATICS

9.13.1 Using Venn diagrams to assist with probability 371

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How to use your CIMA Learning System

This Fundamentals of Business Mathematics Learning System has been devised as a resource

for students attempting to pass their CIMA computer-based assessments, and provides:

● a detailed explanation of all syllabus areas;

● extensive ‘ practical ’ materials;

● generous question practice, together with full solutions;

● a computer-based assessments preparation section, complete with computer-based ments standard questions and solutions

This Learning System has been designed with the needs of home-study and learning candidates in mind Such students require very full coverage of the syllabus topics, and also the facility to undertake extensive question practice However, the Learning System is also ideal for fully taught courses

This main body of the text is divided into a number of chapters, each of which is ised on the following pattern:

organ-● Detailed learning outcomes expected after your studies of the chapter are complete You

should assimilate these before beginning detailed work on the chapter, so that you can appreciate where your studies are leading

● Step-by-step topic coverage This is the heart of each chapter, containing detailed

explana-tory text supported where appropriate by worked examples and exercises You should work carefully through this section, ensuring that you understand the material being explained and can tackle the examples and exercises successfully Remember that in many cases knowledge is cumulative: if you fail to digest earlier material thoroughly, you may struggle to understand later chapters

● Activities Some chapters are illustrated by more practical elements, such as comments

and questions designed to stimulate discussion

● Question practice The test of how well you have learned the material is your ability to

tackle exam-standard questions Make a serious attempt at producing your own answers, but at this stage do not be too concerned about attempting the questions in computer-based assessments conditions In particular, it is more important to absorb the material thoroughly by completing a full solution than to observe the time limits that would apply in the actual computer-based assessments

The CIMA

Learning System

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THE CIMA

● Solutions Avoid the temptation merely to ‘ audit ’ the solutions provided It is an illusion

to think that this provides the same beneﬁ ts as you would gain from a serious attempt

of your own However, if you are struggling to get started on a question you should read the introductory guidance provided at the beginning of the solution, where provided, and then make your own attempt before referring back to the full solution

Having worked through the chapters you are ready to begin your ﬁ nal preparations for the

computer-based assessments The ﬁ nal section of the CIMA Learning System provides you

with the guidance you need It includes the following features:

● A brief guide to revision technique

● A note on the format of the computer-based assessments You should know what

to expect when you tackle the real computer-based assessments, and in particular the number of questions to attempt

● Guidance on how to tackle the computer-based assessments itself

● A table mapping revision questions to the syllabus learning outcomes allowing you to quickly identify questions by subject area

● Revision questions These are of computer-based assessments standard and should be led in computer-based assessments conditions, especially as regards the time allocation

tack-● Solutions to the revision questions

Two mock computer-based assessments You should plan to attempt these just before the date of the real computer-based assessments By this stage your revision should be com-plete and you should be able to attempt the mock computer-based assessments within the time constraints of the real computer-based assessments

If you work conscientiously through the CIMA Learning System according to the

guide-lines above you will be giving yourself an excellent chance of success in your based assessments Good luck with your studies!

Guide to the Icons used within this Text

Key term or deﬁ nition

Exam tip or topic likely to appear in the computer-based assessments Exercise

Question Solution Comment or Note Discussion points Equations to learn

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Passing exams is partly a matter of intellectual ability, but however accomplished you are

in that respect you can improve your chances signiﬁ cantly by the use of appropriate study and revision techniques In this section we brieﬂ y outline some tips for effective study dur-ing the earlier stages of your approach to the computer-based assessments Later in the text

we mention some techniques that you will ﬁ nd useful at the revision stage

Planning

To begin with, formal planning is essential to get the best return from the time you spend studying Estimate how much time in total you are going to need for each paper you are studying for the Certiﬁ cate in Business Accounting Remember that you need to allow time for revision as well as for initial study of the material The amount of notional study time for any paper is the minimum estimated time that students will need to achieve the speciﬁ ed learning outcomes set out below This time includes all appropriate learning activities, for example, face-to-face tuition, private study, directed home study, learning in

the workplace, revision time, etc You may ﬁ nd it helpful to read Better Exam Results: a Guide for Business and Accounting Students by S A Malone, Elsevier, ISBN: 075066357X

This book will provide you with proven study techniques Chapter by chapter it covers the building blocks of successful learning and examination techniques

The notional study time for the Certiﬁ cate in Business Accounting paper

Fundamentals of Business Mathematics is 130 hours Note that the standard amount

of notional learning hours attributed to one full-time academic year of approximately 30 weeks is 1,200 hours

By way of example, the notional study time might be made up as follows:

Now split your total time requirement over the weeks between now and the exam This will give you an idea of how much time you need to devote to study each week Remember

to allow for holidays or other periods during which you will not be able to study (e.g because of seasonal workloads)

With your study material before you, decide which chapters you are going to study in each week, and which weeks you will devote to revision and ﬁ nal question practice

Prepare a written schedule summarising the above – and stick to it!

The amount of space allocated to a topic in the Learning System is not a very good guide as to how long it will take you For example, the material relating to Section A ‘Basic Mathematics’ and Section C ‘Summarising and Analysing Data’ both account or 15% of

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THE CIMA

the syllabus, but the latter has more pages because there are more illustrations, which take

up more space The syllabus weighting is the better guide as to how long you should spend

on a syllabus topic It is essential to know your syllabus As your course progresses you will become more familiar with how long it takes to cover topics in sufﬁ cient depth Your time-table may need to be adapted to allocate enough time for the whole syllabus

Tips for effective studying

1 Aim to ﬁ nd a quiet and undisturbed location for your study, and plan as far as possible

to use the same period of time each day Getting into a routine helps to avoid ing time Make sure that you have all the materials you need before you begin so as to minimise interruptions

2 Store all your materials in one place, so that you do not waste time searching for items around your accommodation If you have to pack everything away after each study period, keep them in a box, or even a suitcase, which will not be disturbed until the next time

3 Limit distractions To make the most effective use of your study periods you should

be able to apply total concentration, so turn off all entertainment equipment, set your phones to message mode, and put up your ‘ do not disturb ’ sign

4 Your timetable will tell you which topic to study However, before diving in and ing engrossed in the ﬁ ner points, make sure you have an overall picture of all the areas that need to be covered by the end of that session After an hour, allow yourself a short break and move away from your Learning System With experience, you will learn to assess the pace you need to work at

5 Work carefully through a chapter, making notes as you go When you have covered a suitable amount of material, vary the pattern by attempting a practice question When you have ﬁ nished your attempt, make notes of any mistakes you made, or any areas that you failed to cover or covered only skimpily

6 Make notes as you study, and discover the techniques that work best for you Your notes may be in the form of lists, bullet points, diagrams, summaries, ‘ mind maps ’ , or the written word, but remember that you will need to refer back to them at a later date,

so they must be intelligible If you are on a taught course, make sure you highlight any issues you would like to follow up with your lecturer

7 Organise your notes Make sure that all your notes, calculations etc can be effectively

ﬁ led and easily retrieved later

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The CBA system can ensure that a wide range of the syllabus is assessed, as a pre-determined number

of questions from each syllabus area (dependent upon the syllabus weighting for that particular area) are selected in each assessment

In every chapter of this Learning System we have introduced these types of questions but obviously we have to label answers A, B, C etc rather than using click boxes For convenience, we have retained quite a lot of questions where an initial scenario leads to

a number of sub-questions There will be questions of this type in the CBA but they will rarely have more than three sub-questions In all such cases the answer to one part does not hinge upon a prior answer

Fundamentals of Business Mathematics and

Computer-Based Assessments

The computer-based assessments for Fundamentals Business Mathematics is a 2-hour computer-based assessments comprising 45 compulsory questions, with one or more parts Single part questions are generally worth 1–2 marks each, but two and three part questions may be worth 4 or 6 marks There will be no choice and all questions should be attempted

if time permits CIMA are continuously developing the question styles within the CBA system and you are advised to try the on-line website demo at www.cimaglobal.com, to both gain familiarity with assessment software and examine the latest style of questions being used

Fundamentals of Business Mathematics

Syllabus outline

The Syllabus comprises:

Topic and study weighting

D Inter-relationships between variables 15%

Learning Aims

This syllabus aims to test the candidate’s ability to:

● demonstrate the use of basic mathematics, including formulae and ratios;

● identify reasonableness in the calculation of answers;

● demonstrate the use of probability where risk and uncertainty exist;

● apply techniques for summarising and analysing data;

● calculate correlation coefﬁ cients for bivariate data and apply the technique of simple regression analysis;

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THE CIMA

● demonstrate techniques used for forecasting;

● apply ﬁ nancial mathematical techniques;

● apply spreadsheets to facilitate the presentation of data, analysis of univariate and ate data and use of formulae

Assessment strategy

There will be a computer-based assessments of 2 hours duration, comprising 45 sory questions, each with one or more parts

A variety of objective test question styles and types will be used within the assessment

Learning outcomes and indicative syllabus

content

A Basic Mathematics – 15%

Learning Outcomes

On completion of their studies students should be able to:

(i) demonstrate the order of operations in formulae, including brackets, powers and roots;

(ii) calculate percentages and proportions;

(iii) calculate answers to appropriate number of decimal places or signiﬁ cant ﬁ gures; (iv) solve simple equations, including two variable simultaneous equations and quadratic equations;

(v) prepare graphs of linear and quadratic equations

Indicative syllabus content

● Use of formulae, including negative powers as in the formula for the learning curve

● Percentages and ratios

(i) calculate a simple probability;

(ii) demonstrate the addition and multiplication rules of probability;

(iii) calculate a simple conditional probability;

(iv) calculate an expected value;

(v) demonstrate the use of expected value tables in decision making;

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(vi) explain the limitations of expected values;

(vii) explain the concepts of risk and uncertainty

● The relationship between probability, proportion and percent

● Addition and multiplication rules in probability theory

● Venn diagrams

● Expected values and expected value tables

● Risk and uncertainty

C Summarising and Analysing Data – 15%

(i) explain the difference between data and information;

(ii) identify the characteristics of good information;

(iii) tabulate data and prepare histograms;

(iv) calculate for both ungrouped and grouped data: arithmetic mean, median, mode, range, variance, standard deviation and coefﬁ cient of variation;

(v) explain the concept of a frequency distribution;

(vi) prepare graphs/diagrams of normal distribution, explain its properties and use tables

of normal distribution;

(vii) apply the Pareto distribution and the ‘ 80:20 rule ’ ;

(viii) explain how and why indices are used;

(ix) calculate indices using either base or current weights;

(x) apply indices to deﬂ ate a series

● Data and information

● Tabulation of data

● Graphs and diagrams: scatter diagrams, histograms, bar charts and ogives

● Summary measures of central tendency and dispersion for both grouped and ungrouped data

(i) prepare a scatter diagram;

(ii) calculate the correlation coefﬁ cient and the coefﬁ cient of determination between two variables;

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THE CIMA

(iii) calculate the regression equation between two variables;

(iv) apply the regression equation to predict the dependent variable, given a value of the independent variable

● Scatter diagrams and the correlation coefﬁ cient

● Simple linear regression

E Forecasting – 15%

Learning outcomes

(i) prepare a time series graph;

(ii) identify trends and patterns using an appropriate moving average;

(iii) identify the components of a time series model;

(iv) prepare a trend equation using either graphical means or regression analysis;

(v) calculate seasonal factors for both additive and multiplicative models and explain when each is appropriate;

(vi) calculate predicted values, given a time series model;

(vii) identify the limitations of forecasting models

● Time series analysis – graphical analysis

● Trends in time series – graphs, moving averages and linear regression

● Seasonal variations using both additive and multiplicative models

● Forecasting and its limitations

F Financial Mathematics – 15%

(i) calculate future values of an investment using both simple and compound interest; (ii) calculate an annual percentage rate of interest given a monthly or quarterly rate; (iii) calculate the present value of a future cash sum using formula and CIMA Tables; (iv) calculate the present value of an annuity and a perpetuity using formula and CIMA Tables;

(v) calculate loan/mortgage repayments and the value of the loan/mortgage outstanding; (vi) calculate the future value of regular savings and/or the regular investment needed

to generate a required future sum using the formula for the sum of a geometric progression;

(vii) calculate the net present value (NPV) and internal rate of return (IRR) of a project and explain whether and why it should be accepted

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● Simple and compound interest

● Annuities and perpetuities

● Loans and mortgages

● Sinking funds and savings funds

● Discounting to ﬁ nd NPV and IRR and interpretation of NPV and IRR

G Spreadsheets – 10%

● explain the features and functions of spreadsheet software;

● explain the use and limitations of spreadsheet software in business;

● apply spreadsheet software to the normal work of a Chartered Management Accountant

● Features and functions of commonly used spreadsheet software: workbook, worksheet, rows, columns, cells, data, text, formulae, formatting, printing, graphics and macros Note: Knowledge of Microsoft Excel type spreadsheet vocabulary/formulae syntax is required Formulae tested will be that which is constructed by users rather than prepro-

grammed formulae

● Advantages and disadvantages of spreadsheet software, when compared to manual

analy-sis and other types of software application packages

● Use of spreadsheet software in the day-to-day work of the Chartered Management Accountant: budgeting, forecasting, reporting performance, variance analysis, what-if analysis and discounted cashﬂ ow calculations

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Mathematical Tables

LINEAR REGRESSION AND CORRELATION

The linear regression equation of y on x is given by:

∑

( )

FINANCIAL MATHEMATICS

Compound Interest (Values and Sums)

Future Value of S , of a sum X , invested for n periods, compounded at r % interest

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Basic Mathematics

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Business Mathematics – and, indeed, the rest of your CIMA studies After completing

this chapter you should be able to:

The basic mathematical operations are addition, subtraction, multiplication and division; and there is a very important convention about how we write down exactly what operations are to be carried out and in what order Brackets are used to clarify the order of operations and are essential when the normal priority of operations is to be broken The order is:

● work out the values inside brackets ﬁ rst;

● powers and roots (see Section 1.5);

● multiplication and division are next in priority;

● ﬁ nally, addition and subtraction

Basic Mathematics

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Returning to the main problem, if you want to add 2 to 3 and then multiply by 4, you must use brackets to give priority to the addition – to ensure that it takes place ﬁ rst You should write (2 3) 4 The contents of the bracket total 5, and this is then multiplied

(a) 6 8 takes priority, then add 5, so 5 6 8 48 5 53

(b) Work out the bracket ﬁ rst, then multiply by 2; 3 1 4, so (3 1) 2 8

(c) 7 ÷ 2 takes priority, and is then subtracted from 9; 9 3.5 5.5

(d) Work out the bracket ﬁ rst, then divide by 10; 4 5 9, and 9/10 0.9

(e) The multiplication of 7 8 takes priority, giving 56; 5 56 2 59

(f) Work out the brackets ﬁ rst – the order is unimportant but it is usual to work from left to right; 9 1 8, and 6 4 10, so 8 10 80

1.3 Different types of numbers

A whole number such as 5, 0 or 5 is called an integer, whereas numbers that contain parts of a whole number are either fractions – such as 3

4 – or decimals – such as 0.75 Any type of number can be positive or negative If you add a positive number to some-

thing, the effect is to increase it whereas, adding a negative number has the effect of ing the value If you add B to any number A, the effect is to subtract B from A The rules

reduc-for arithmetic with negative numbers are as follows:

● adding a negative is the same as subtracting, that is A ( B ) A B;

● subtracting a negative is the same as adding, that is A ( B ) A B;

● if you multiply or divide a positive and a negative, the result is negative, that is ( ) ( ) and ( ) ( ) and ( ) ( ) and ( ) ( ) are all negative;

● if you multiply or divide two negatives, the result is positive, that is ( ) ( ) and ( ) ( ) are both positive

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(a) Multiplication takes priority; 7 ( 2) 14, so 9 7 ( 2) 9 ( 14) 9 14 23

(b) Work out the bracket ﬁ rst; (5 8) ( 6) 3 ( 6) 18

(c) Division takes priority; 12 8 ÷ ( 4) 12 ( 2) 12 2 14

(d) Work out the bracket ﬁ rst; (4 16)/( 2) ( 12)/( 2) 6

For example, 78.187 78 to the nearest whole number The only other nearby whole number is 79 and 78.187 is nearer to 78 than to 79 Any number from 78.0 to 78 49 will round down to 78 and any number from 78.5 to 78 99 will round up to 79

The basic rules of rounding are that:

1 digits are discarded (i.e turned into zero) from right to left;

2 reading from left to right, if the ﬁ rst digit to be discarded is in the range 0–4, then the previous retained digit is unchanged; if the ﬁ rst digit is in the range 5–9 then the previous digit goes up by one

Depending on their size, numbers can be rounded to the nearest whole number, or 10

or 100 or 1,000,000, and so on For example, 5,738 5,740 to the nearest 10; 5,700 to the nearest 100; and 6,000 to the nearest 1,000

1.4.2 Significant figures

For example, 86,531 has ﬁ ve digits but we might want a number with only three The ‘ 31 ’ will be discarded Reading from the left the ﬁ rst of these is 3, which is in the 0–4 range, so

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as signifi cant fi gures However, zeros sandwiched between non-zeros are signifi cant Hence,

87,000 has two s.f., while 80,700 has three

1.4.4 Rounding up or rounding down

A number to be rounded up will be changed into the next higher whole number so, for example, 16.12 rounds up to 17

A number to be rounded down will simply have its decimal element discarded (or truncated)

Numbers can also be rounded up or down to, say, the next 100 Rounding up, 7,645 becomes 7,700 since 645 is increased to the next hundred which is 700 Rounding down, 7,645 becomes 7,600

(b) (5.9 8.2) ÷ (3.6 7.1) ( 2.3) ÷ ( 3.5) 0.65714 0.7 to one d.p

(c) 8,539 349.1 ÷ (32.548 1) 8,539 349.1 ÷ 31.548 8,539 11.066 8527.934 8,530 to three s.f

(d) 56/5 28 11.2 28 16.8 17 to the nearest whole number

Correct rounding is essential in computer-based assessments Don’t move on to the next topic until you are quite sure about this

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1 The n th power of a number, a, is the number multiplied by itself n times in total, and

is denoted by a n or a^n For example,

4 The n th root of a number, a , is denoted by a 1/n and it is the number that, when

multi-plied by itself n times in total, results in a For example,

81 3 382

Check: 2 2 2 8

The square root, a1/2 , is generally written as a without the number 2

5 a n/m can be interpreted either as the m th root of a n or as the m th root of a multiplied by itself n times For example,

95 2 ( 9)5 35 243

6 The rules for arithmetic with powers are as follows:

(i) Multiplication: a m a n a m n For example,

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(b) m 3 /m 2 m (3 ( 2)) m 5

1.6 Mathematical operations in Excel

When performing calculations in Excel the same mathematical rules that have been cussed in this chapter apply The following examples use the data from Examples 1.2.1 and 1.2.2 and show how formulae in Excel would be created to arrive at the same results

dis-Figure 1.1 Creating basic formulae in Excel

Notice from Figure 1.1 that most of the values have been addressed by the cell reference – but it is also possible to incorporate numbers into the formulae

1.6.1 Rounding numbers in Excel

To accurately round numbers in Excel a built-in function called round () is used This can

be used to set any degree of accuracy required and once the function is incorporated into

a formula any future references to the cell containing the round function will use a value

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Figure 1.2 Rounding numbers in Excel

Figure 1.3 Using the Excel INT function

Figure 1.4 The results of using ROUND and INT

Notice that the third example requested the result be rounded to three signiﬁ cant ﬁ gures, the formula is a little more complex and has been done here in two steps

In the fi rst step in cell f6 the arithmetic has been performed and the result rounded to three decimal places Then in g6 the len and the int functions have been applied to fur-ther round the result to three signifi cant fi gures

It is sometimes preferable to take the integer value of a number as opposed to rounding

it to the nearest whole number The difference is that the integer value is a number out any decimal places Therefore the integer value of 9.99 is 9 and not 10 as it would be

with-if the number had been rounded to the nearest whole number

Figure 1.3 shows the table used in the rounding exercise but with the Excel int function

in place of the round function

Figure 1.4 shows the results of the rounding and the integer formulae used in ﬁ gures 1.2 and 1.3

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BASIC MA

TICS Looking at Figure 1.4 , the different result produced through the use of the int function

as opposed to the round function can be seen In each case the result has been rounded down to the integer value

1.7 Variables and functions

A variable is something which can take different values Variables are often denoted by letters Thus, the set of positive whole numbers can be considered as a variable If we

denote it by x, then this variable can have many values

x x x

123

or

or , and so on.

Another example is the set of the major points of a compass If this variable is denoted by c,

then it can have more than one value, but only a limited number

These examples show that variables can take on non-numerical ‘ values ’ as well as

numeri-cal ones In this text we shall concentrate on numerinumeri-cal variables, that is, those whose values

are numbers, like the ﬁ rst case above

A mathematical function is a rule or method of determining the value of one

numeri-cal variable from the values of other numerinumeri-cal variables We shall concentrate on the case

where one variable is determined by or depends on just one other variable The ﬁ rst able is called the dependent variable, and is usually denoted by y, while the second is called the independent variable, denoted by x The relationship between them is a function of one variable, often referred to as a function, for brevity Note that whilst functions are similar

vari-to formulae (see Section 1.8) there are speciﬁ c conditions relating vari-to the deﬁ nition of a function, but these are outside the scope of this book

A very useful way of stating a function is in terms of an equation, which is an expression containing an ‘ equals ’ sign The equation of a function will thus take the typical form:

y a mathematical expression containing x

If we know the value of the independent variable x, then the expression will completely determine the corresponding value of the dependent variable, y.

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To ﬁ nd the value of y , we write the known value of x (2 in this case) in place of x in the mathematical

expres-sion and perform the necessary arithmetical calculations This is known as the substitution of the x-value into the equation.

(a) Substituting x 2 gives:

y 3 2 2 3 4 7

so the dependent variable has the value 7 in this case

(b) Clearly, this dependent variable has the value 2, the same as x

A formula is a statement that is given in terms of mathematical symbols: it is a

mathemati-cal expression that enables you to mathemati-calculate the value of one variable from the value(s) of one or more others Many formulae arise in ﬁ nancial and business calculations, and we shall encounter several during the course of this text In this chapter, we shall concentrate

on some of the more complicated calculations that arise from the application of formulae

(a) Calculate the value of V when P 10,000, r 0.06 and n 4.

(b) Calculate the value of P when V 1,000, r 0.04 and n 3

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The example above required us to change the subject of the formula from V to P, that is, to end up

with P some expression There are various rules and techniques which help this process

1 If something is added or subtracted at one side of an equation, then it changes its sign when you take it to

the other side For example: P 5 9, so P 9 5 4

2 If something multiplies one side of an equation, then it divides when taken to the other side Similarly, divisions turn into multiplications For example: 5 R 210, so R 210 ÷ 5 42; T ÷ 20 7, so

Gathering the X terms together:

2 1

Y X XY

X Y

( )

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is 4 ∧ 2 The carat can also used to ﬁ nd the square root In this case the formula would

be 4 ∧ (1/2), or to ﬁ nd the cube root the formula would be 4∧ (1/3) The method

is used to ﬁ nd the 4th root, 5th root and so on Some examples are demonstrated in Figures 1.5 and 1.6

Figure 1.5 Examples of the use of the carat ( ∧) symbol

Figure 1.6 Formulae used to produce the results

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BASIC MA

1.10.1 Linear equations with only one variable

An equation is linear if it has no term with powers greater than 1, that is, no squared or cubed terms, etc The method is to use the same techniques as in changing the subject of a formula, so that the equation ends up in the form variable something

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A quadratic equation has the form aX2 bX c 0 where a, b and c are constants The equation can be solved using a formula but if either the bX or c terms or both are missing

the formula is not necessary Examples will be used to illustrate the methods

(d) The only solution is that Y 5 0, so Y 5

You may have noticed that most quadratic equations have two roots, that is, two values for which the two sides of the equation are equal, but occasionally, as in (d) above, they appear to have only one It is, in fact, a

repeated (or double) root For example, Y 2 9 has no real roots We shall consider this again when we look

at quadratic graphs in the next chapter

For quadratic equations all of whose coefﬁ cients are non-zero, the easiest method of solution is the formula If

the equation is aX 2 bX c 0, then the roots are given by:

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