CIMA - C3 Fundamentals Of Business Mathematics

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320

Financial mathematics

Compound Interest (Values and Sums)

Future Value of S

of a sum X, invested for n periods, compounded at r% interest:

S= X

[]

r1+

Annuity

Present value of an annuity of $1 per annum receivable or payable, for n years, commencing in one year,

discounted at r% per annum:

PV =

⎥

⎦

⎤

⎢

⎣

⎡

−

r]1[

Perpetuity

Present value of $1 per annum, payable or receivable in perpetuity, commencing in one year discounted at r% per

annum

PV =

321

Formulae to learn

Fraction =

Numerator

Denominator

Percentage change =

'Change'

Original value

× 100%

The

equation of a straight line:

y = a + bx

where a = the intercept of the line on the y axis

and b = the slope (gradient) of the line

b = gradient =

xin Change

y in Change

= (y2 – y1 )/(x2 – x1) where (x1 , y1) and (x2, y2) are two points on the straight line.

Histogram with unequal intervals:

Adjustment factor =

widthclass Current

widthclass Standard

Coefficient of variation =

mean

deviationStandard

Fixed base index =

year base in Value

yeargivenany in Value

× 100

Chain base index =

value syear' Last

value syear' This

× 100

Deflated/inflated cash flow = Actual cash flow in given year ×

year given for number Index

year base fornumberIndex

Probability of achieving the desired result =

outcomes possible of number Total

result desiredachievingofwaysofNumber

Expected value (EV) = ∑np

Where

∑ = sum of

n = outcome

p = probability of outcome occurring

Simple interest formula:

S = X + nrX

Where X = the original sum invested

r = the interest rate (expressed as a proportion, so 10% = 0.1)

n = the number of periods (normally years)

S = the sum invested after n periods, consisting of the original capital (X) plus interest earned.

322

Compound interest formula when there are changes in the rate of interest:

S = X(1 + r

)

(1 + r

)

n-y

Where r

= the initial rate of interest

y = the number of years in which the interest rate r

applies

= the next rate of interest

n – y = the (balancing) number of years in which the interest rate r

applies.

Effective annual rate of interest:

(1 + R) = (1 + r)

Where R is the effective annual rate

r is the period rate

n is the number of periods in a year

The terminal value of an investment to which equal annual amounts will be added (the sum of a geometric

progression):

S =

)1R(A

−

Where S is the terminal value

A is the first term

R is the common ratio

n is the number of terms

Discounting formula:

X = S

)r1(

Where S is the sum to be received after n time periods

X is the present value (PV) of that sum

r is the rate of return, expressed as a proportion

n is the number of time periods (usually years)

Internal rate of return (IRR) =

NPV

a(ba)%

NPV NPV

⎡

⎤

⎛⎞

+−

⎢

⎥

⎜⎟

−

⎝⎠

⎣

⎦

Where a is one interest rate

b is the other interest rate

NPV

is the NPV at rate a

NPV

is the NPV at rate b

323

Question bank

324

Question bank 325

1 The expression

)(x

equals

A 0

B 1

C x

D x

2 The term

can also be written as

A –x

B x

–1

C x

D x –1

3 Lynn and Laura share out a certain sum of money in the ratio 4 : 5, and Laura ends up with $6.

(a) How much was shared out in the first place?

(b) How much would have been shared out if Laura had got $6 and the ratio had been 5 : 4 instead of

4 : 5?

4 What is 23% of $5,000?

5 What is $18 as a percentage of $45?

6 Deirdre Fowler now earns $25,000 per annum after an annual increase of 2.5%. What was her annual salary

to the nearest $ before the increase?

7 If purchases at Watkins Ltd are $200,000 in 20X0 and there was a percentage decrease of 5% in 20X1,

what are purchases in 20X1?

8 An accountant charges $X per hour to which sales tax of 17.5% is added. If her final hourly charge is $235,

what is the value of X?

326 Question bank

Questions 9 to 11 refer to the spreadsheet shown below.

9 The cell F5 (column F row 5) shows the opening position for month 3. The value in this cell is a formula.

Which of the following would not be a correct entry for this cell?

A =E7-E30+D32

B =E5+E7-E30

C =E32

D =2860

10 The formula in D17 (column D row 17) adds a percentage national insurance charge to the sub total of staff

costs. Which of the formulae shown below would be the best formula for cell D17?

A =SUM(D11:D16)*0.1

B =SUM(D11:D16)*$C17

C =SUM(D11:D16)*10%

D =SUM(D11:D16)*C17

11 The cell D30 (column D row 30) shows the total costs. Which of the following is the correct formula for this

cell?

A =D28+D18

B =SUM(D11:D28)

C =SUM(D7:D28)

D D18+D28

Question bank 327

12 The data that can be entered onto a spreadsheet comprises which of the following?

A Text and numbers

B Formulae, text and numbers

C Numbers and formulae

D Text and formulae

Questions 13, 14, 15 and 16 refer to the Employment targets and bonuses spreadsheet shown below. This

spreadsheet shows the target sales for each employee (column C) and the amount that they actually sold (column

D). The bonus value is calculated by subtracting the actual sales from the target sales. If the employee has failed to

achieve their target level, then there is no bonus. The target figure for next month (column F) is calculated by

multiplying the target figure by a defined growth rate (cell B15).

A B C D E F G

2 Employment targets and bonuses

4 Employee No Target Actual Bonus Next month

5 345 Rianne 2,100.00 2,000.00 0.00 2,121.00

6 567 Sian 4,300.00 4,400.00 100.00 4,343.00

7 543 Claire 2,345.00 2,300.00 0.00 2,368.45

8 231 Mark 5,680.00 5,600.00 0.00 5,736.80

9 890 Danny 345.00 500.00 155.00 348.45

11 14,770.00 14,800.00 14,917.70

12 Sales staff 5

13 Income/staff 2,960

15 Growth rate 1.00%

13 In the spreadsheet what is the formula in C11 (column C row 11)?

A =SUM(C5:C9)

B =TOTAL(C5:C9)

C =SUM(C5:C11)

D =TOTAL (C5:C9)

14 In the spreadsheet the figure in E5 (column E row 5) is calculated by subtracting the value in C5 (column C

row 5) from the value in D5 (column D row 5). However, if the calculated value is negative, then the value

returned in E5 is zero.

In the spreadsheet what is the formula in E5?

A =IF(D5-C5>0,D5-C5,0)

B =IF(C5-D5>0,C5-D5,0)

C =IF(C5-D5>0,0,C5-D5)

D =IF(D5-C5>0,0,D5-C5)

328 Question bank

15 The value in cell B12 (column B row 12) is a formula. It shows how many sales people are in the

spreadsheet.

In the spreadsheet what is the formula in B12?

A =SUM(A5:A9)

B =TOTAL(A5:A9)

C =COUNT(A5:A9)

D =CALCULATE(A5:A9)

16 The value in F5 (column F row 5) is calculated by C5 (column C row 5) by the growth value in B15 (column

B row 15). the formula in F5 is then copied down to rows 6, 7, 8 and 9

What should the formula be in F5 to allow this to be done accurately and quickly?

A =$C5+($C5*$B$15)

B =$C$5+($C5*$B$15)

C =$C5+($C$5*$B$15)

D =$C5+($C5*$B15)

17 In the formula Q =

CoD2

, if C

= $40, D = 48,000 and Q = 200, then C

is closest in value to

A $3.43

B $11.77

C $96.00

D $138.56

18 If demand D = 30 – 3P, how is price (P) expressed in terms of D?

A P = 10 – D

B P = D/27

C P = 10 + D/3

D P = 10 – D/3

19 If x

= 4.913, then x =

20 If 34x – 7.6 = (17x – 3.8)

× (x + 12.5), then x =

21 If y = aX

and a = 9 minutes, X = 7 and b = –0.24. What is the value of Y to two decimal places?

22 Solve the inequality 3 (X + 4) < 2X + 1

23 Solve the following simultaneous equations.

x – y = –9

9x – 13y = –97

x =

y =

Question bank 329

24 The sum of the squares of two positive numbers (x and y) is 890, and the difference between the two

numbers is 4. Find a quadratic equation in terms of either x or y.

25 A rectangle has a perimeter of 44 metres and an area of 112 square metres. Find a quadratic equation in

terms of either length 'a' or breadth 'b'.

26 Solve the following quadratic equation.

–3x

+ 6x + 24 = 0

x =

27 A survey is being carried out to find out the number of goals scored by the ten most popular football clubs

in Britain in their past ten matches to see if there is much variation between the teams. What sort of data is

being collected in such a survey?

A Quantitative Continuous

B Quantitative Discrete

C Qualitative Continuous

D Qualitative Discrete

28 A company which makes rechargeable batteries selects some of the batteries for examination. The

procedure used chooses two random numbers, say n and m. Starting at the n

battery, every battery at an

interval of m is then chosen for examination.

This type of sampling is known as:

A Stratified

B Systematic

C Random

D Multistage

29 Which of the following statements are true?

I In quota sampling, investors are told to interview all the people they meet up to a certain quota.

II In cluster sampling, there is very little potential for bias.

III A sampling frame is a numbered list of all items in a sample.

IV If a sample is selected using random sampling, it will be free from bias.

A I only

B I and IV

C I, II and IV

D I, II, III and IV