BUSINESS MATHEMATICS &
QUANTITATIVE METHODS
NOTES:
FORMATION 1 EXAMINATION - APRIL 2012
You are required to answer FIVE questions.
(If you provide answers to all questions, you must draw a clearly distinguishable line through the answer not to
be marked. Otherwise, only the first FIVE answers to hand will be marked).
All questions carry equal marks.
STATISTICAL FORMULAE TABLES ARE PROVIDED
DEPARTMENT OF EDUCATION MATHEMATICS TABLES ARE AVAILABLE ON REQUEST
TIME ALLOWED:
3 hours, plus 10 minutes to read the paper.
INSTRUCTIONS:
During the reading time you may write notes on the examination paper but you may not commence
writing in your answer book.
Marks for each question are shown. The pass mark required is 50% in total over the whole paper.
Start your answer to each question on a new page.
You are reminded to pay particular attention to your communication skills and care must be taken
regarding the format and literacy of the solutions. The marking system will take into account the content
of your answers and the extent to which answers are supported with relevant legislation, case law or
examples where appropriate.
List on the cover of each answer booklet, in the space provided, the number of each question(s)
attempted.
The Institute of Certified Public Accountants in Ireland, 17 Harcourt Street, Dublin 2.
THE INSTITUTE OF CERTIFIED PUBLIC ACCOUNTANTS IN IRELAND
BUSINESS MATHEMATICS &
QUANTITATIVE METHODS
FORMATION 1 EXAMINATION - APRIL 2012
Time Allowed: 3 hours, plus 10 minutes to read the paper.
You are required to answer 5 questions.
(If you provide answers to all questions, you must draw a clearly distinguishable line through the answer not to
be marked. Otherwise, only the first 5 answers to hand will be marked).
All questions carry equal marks.
1.
DIY Ltd. purchases company cars for the management team at the beginning of the year. The MD is considering a
number of options: a) a new car; this will cost €85,000; the annual service and running costs are €15,000; it has
a life of 5 years with a re-sale value of €10,000; b) a trade-in of his present car; with the trade-in the cost of a new
car being €45,000; the annual service and running costs are €27,000; it has a life of 5 years with a re-sale value
of €1,000; or c) retain his present car; the annual service and running costs are €37,000; after 5 years this car has
no re-sale value.
Assume that the annual service and running costs commence at the beginning of each year and that the car is sold
at the end of year 5.
You are required to:
(a)
(b)
Using Net Present Value, estimate the option with the least cost for the company with an interest rate of 14%.
(12 marks)
Discuss the techniques used in investment appraisal.
(8 marks)
[Total: 20 Marks]
2.
DAB Ltd., a small state controlled company, has been subject to criticism for the level of salaries paid to its
employees. A commentator claims that the average salary paid to employees exceeds €120,000 per annum. You
have collected the data shown in the following table:
Salaries (€000s)
No. employees
100 < 110
5
110 < 120
4
120 < 130
7
130 < 140
12
140 < 150
8
150 < 160
5
160 or more
4
You are required to:
(a)
Present the data above on a histogram and cumulative frequency curve.
(10 marks)
(b)
Determine the validity of the commentator’s claim by calculating or deriving from the graph, as appropriate,
the mean, median and modal salaries of DAB’s employees.
(10 Marks)
[Total: 20 Marks]
1
3.
Employee representatives are claiming a pay increase for increased productivity. The company claims that there is
no basis for a pay increase based on the Consumer Price Index (CPI) which is traditionally used as the basis for
pay claims. The company claims that pay increases have substantially exceeded the CPI. As the financial
accountant, you have collected the data shown in the table below on i) the average pay for administrative staff, and
ii) the CPI over the past 9 years. You are required to prepare a report in which you:
(a)
Estimate the percentage increase in the average pay for each year.
(5 marks)
(b)
Compare the average annual pay increase to the CPI increase for each year.
(5 marks)
(c)
Revalue pay levels to 2010 values and explain your result.
(10 marks)
Year
Average Pay
€
Consumer Price Index
(Base 2000)
2002
25,500
158
2003
27,200
165
2004
29,800
179
2005
32,300
195
2006
36,900
209
2007
41,600
230
2008
45,200
250
2009
49,800
266
2010
53,600
280
[Total: 20 Marks]
4.
The retail company, Capeland Ltd., wants to forecast sales of its specialist menswear for the first three quarters of
2012. From its records it produces the following time series data for the past three years.
Number of Sales
Quarter
Year
1
2
3
4
2009
60
85
95
75
2010
66
97
110
90
2011
78
104
120
95
(a)
Derive and plot the trend for the above data.
(10 Marks)
(b)
Using the average rate of change of the trend, forecast the sales for the first three quarters of 2012.
(10 Marks)
[Total: 20 Marks]
2
5.
As a management consultant, you provide advice to business clients on key financial decisions. Clients have
recently requested you to advise on the following:
(a)
(b)
(i)
Sno Ltd. wishes to make provision for the purchase of a machine in 3 years time. The machine will
cost €250,000, will last for 3 years and will have a scrap value of €25,000. The bank has quoted a
(4 Marks)
nominal deposit interest rate of 16%, compounded at 6 monthly intervals.
(ii)
Estimate the sum that the company should set aside to purchase the equipment and outline the
annual depreciation that would be expected with straight line depreciation.
(2 Marks)
Dab Ltd. wants to forecast its energy consumption for next year. It states that its electricity charge for the last
two-monthly period is €1,200. This comprises a two monthly ‘standing charge’ or fixed charge of €300 and
8.5c per unit (kwhr) charge. It estimates that its electricity consumption will increase by a further 2,000 units
per two months next year. Represent this consumption by means of a linear equation and calculate the unit
consumption for the last two monthly period.
Calculate the total energy costs for next year.
(c)
(6 Marks)
A client wishes to invest €10,000 for two years with the right to withdraw the money immediately, if required.
The BBS bank offers an interest rate of 8% p/a after tax with the interest paid half yearly. ABC bank offers
an interest rate of 10% p/a paid before tax with the interest paid half yearly. The income tax rate is 40%.
Advise on the best investment.
Support your answer with relevant calculations.
(8 Marks)
[Total: 20 Marks]
6.
Discuss the following:
(a)
Tests of Hypothesis.
(7 Marks)
(b)
Correlation between two variables.
(7 Marks)
(c)
The principles of sampling and the selection of samples.
(6 Marks)
[Total: 20 Marks]
END OF PAPER
3
SUGGESTED SOLUTIONS
THE INSTITUTE OF CERTIFIED PUBLIC ACCOUNTANTS IN IRELAND
BUSINESS MATHEMATICS &
QUANTITATIVE METHODS
FORMATION 1 EXAMINATION - APRIL 2012
SOLUTION 1
(a)
The data and the costs for the three options are tabulated below.
Option
New Car
Trade-in
Retain present car
Discount Factor at 14%
1.000
0.877
0.769
0.675
0.592
0.519
Total
(b)
Cost
Resale Value
€85,000
€45,000
0
€10,000
€1,000
0
Annual service
& Running Costs
€15,000
€27,000
€37,000
2 Marks
2 Marks
2 Marks
Option 1
New Car
Option 2
Trade-in
Net Cost
€
Present
Value, €
Net Cost
€
Present
Value, €
85,000
15,000
15,000
15,000
15,000
15,000
(10,000)
100,000
45,000
27,000
27,000
27,000
27,000
27,000
(1,000)
72,000
13,155
11,535
10,125
8,880
(5,190)
138,505
2 Marks
23,679
20,763
18,225
15,984
(519)
150,132
2 Marks
Life
(Years)
5
5
5
Option 3
Retain present car
Net Cost
€
Present
Value, €
37,000
37,000
37,000
37,000
37,000
0
37,000
32,449
28,453
24,975
21,904
0
144,781
2 Marks
Discounting and present value underpin the techniques used in Investment appraisal. These are Net Present Value
(NPV) and Internal Rate of Return (IRR) and are known as the discounting techniques. The most commonly used
non discounting technique is the Payback Period.
NPV. This is a technique of investment appraisal which involves calculating the sum of all cash flows associated with
a project. This sum is known as the net present value (NPV) of the project, that is, the present value of future net
revenues less the initial capital outlay. If the NPV is positive, i.e. greater than zero, the project is profitable and earns
more than the discount rate used. If the NPV is negative, i.e. less than zero the project makes a loss and it earns
less than the discount rate. The discount rate is the rate which is used to discount the cash flows.
IRR. The internal rate of return is an alternative method of investment appraisal to net present value; it is translated
as the rate that a project earns. It is the value of the discount rate that gives an NPV of zero.
Payback period is a widely used method of appraisal which does not rely on the discounting process. However it is
used generally as a’rule of thumb’. In practice the project with the shortest payback period is selected – this requires
the shortest time to pay back any initial capital outlay. It takes into consideration the level of cash flow in the short
term and therefore gives a measure of capital exposure but does not consider the long term viability of a project.
[8 Marks]
[Total: 20 Marks]
5
Solution 2
The data for the graphs and calculations is set out in the table below.
Salaries (€000s)
Less than
Midpoint x
Employees f
100<110
110<120
120<130
130<140
140<150
150<160
160 or more
∑
110
120
130
140
150
160
180
105
115
125
135
145
155
170
5
4
7
12
8
5
4
45
Cumulative
frequency
5
9
16
28
36
41
45
f(x)
525
460
875
1,620
1,160
775
680
6,075
The histogram is set out below.
14
Mode (€134,000)
No. of
employees 12
10
Freq
8
6
4
2
100
110
120
130
140
150
160
170
180
Salaries €
(5 Marks)
Derivation of the ogive.
Cum Frequency
50
40
No. of
30
employees
20
10
100
110
120
Calculation/derivation of the mean, median, mode.
130
140
150
160
Salaries €
6
170
(8 Marks)
Mean = x = ∑fx
∑f
= 6,075,000 = €135,000
50
(3 Marks)
The median can be calculated or derived from the graph above and is approx. €134,000; [3 Marks] the mean as
calculated is €134,800. The mode can be derived from the histogram and is approx. €134,00
(4 Marks)
Note that derivations from the graph will be less accurate than calculations.
[Total: 20 Marks]
Solution 3
Parts (a) and (b) are included in the following table.
Year
2002
2003
2004
2005
2006
2007
2008
2009
2010
(c)
Average
Pay €
25,500
27,200
29,800
32,300
36,900
41,600
45,200
49,800
53,600
5 Marks
% Increase
Year on Year
CPI
(Base 2000)
158
165
179
195
209
230
250
266
280
6.6
9.6
8.3
14.2
12.7
8.6
10.2
7.6
% Increase
Year on Year
4.4
8.5
8.9
7.2
10.0
8.7
6.4
5.3
5 Marks
Revalued Pay
(2010)
45,190
46,158
46,614
46,379
49,435
50,643
50,624
52,421
53,600
6 Marks
The percentage increase in pay is consistently larger than the increase in prices except for 2005 and 2008 where
the increases are comparable. The overall increase in pay for the 9 years is 110% of the base level while the index,
over the period, has increase by 77%. In real (current) terms the pay rates are now greater than in any previous years.
(4 Marks)
[Total: 20 Marks]
7
Solution 4
(a)
Derive the trend
Number of Sales
Quarter
1
60
66
78
Year
2008
2009
2010
Year
2008
2
85
97
104
3
95
110
120
4
75
90
95
Quarter
1
2
3
4
Sales
60
85
95
75
Moving annual total
Moving pair total
315
321
636
654
78,500
81,750
2009
1
2
3
4
66
97
110
90
333
348
363
375
681
711
738
757
85,125
88,875
92,250
94,625
2010
1
2
3
4
78
104
120
95
382
392
397
774
789
96,750
98.625
(2 Marks)
Plot the Trend
Year
2008
2009
2010
Quarter 1
Quarter 2
85.125
96.750
88.875
98.625
Trend
(3 Marks)
Quarter 3
78.5
92.250
Quarter 4
81.5
94.625
Raw Data
120
Production
Trend
100
80
60
40
20
1
2
2008
3
4
5
6
7
2009
8
9
10
11
2010
12
Quarter
The graph shows a clear seasonal affect each year. Sales are lowest in the first quarter and increase over the period.
There is also an upward trend visible with sales levels increasing over time.
(5 Marks)
8
(b)
Average rate of change in the trend = (98.625 – 78.5)/7 = 20.125/7 = 2.875
qrt 3 2010: 98.625 + 2.875 = 101.5
qrt 4 2010 101.5 + 2.875 = 104.375
(2 Marks)
(2 Marks)
Forecast for qtr 1, 2011 = 104.375 + 2.875 = 107.25
Forecast for qtr 2, 2011 = 107.25 + 2.875 = 110.125
Forecast for qtr 3, 2011 = 110.125 + 2.875 = 113
(2 Marks)
(2 Marks)
(2 Marks)
It should be noted that taking the trend for the last 3 periods, rather than the three years, may give a more accurate
forecast.
[Total: 20 Marks]
9
Solution 5
(a)
Since compounding is six monthly, the sum should accrue to €250,000 after 6 six monthly periods. The interest rate
for each six monthly period is 16%/2 = 8%.
Using the compound interest formula, where P = sum to be invested now,
250,000 = P(1 + 0.08)6,
Therefore, P = 250,000/1.086 = 250,000/1.85 = €135,135
(3 Marks)
Annual depreciation expected.
The straight line method reduces the value by the same absolute amount each year, that is,
Annual depreciation = (initial value – scrap value)/estimated life of asset.
Annual depreciation = (€250,000 - €25,000)/3 =
(b)
€75,000pa.
(3 Marks)
As the relationship is linear the demand can be of the form y = a + bx,
where a = fixed costs (€300), b = variable costs / cost per unit, x = electricity units used.
For the last two months x = (1200 – 300)/0.085 = 10,588 – number of units used.
Y = 300 + 10,588X
(3 Marks)
Since the units are estimated to increase by 2000 per month,
Cost per two months next year is: Y = 300 + 12,588X where X = 8.5c
That is Y = €1369 per two months; this gives €1369 x 6 = €8,214 per year.
(c)
(3 Marks)
BBS bank proposal.
Amount after tax at end of two years
= 10,000(1 + 0.04)4
= 10,000 x 1.17
= €11,700
(2 Marks)
ABC bank proposal.
Year 1. Amount before tax = 10,000(1 + 0.05)2
= 10,000(1.1025) = €11,025 [Interest = 1,025]
Amount after tax = 11,025 – 1025 x 0.4 = €10,615
(2 Marks)
Year 2. Amount before tax = 10,615(1 + 0.05)2
= 10,615(1.1025) = €11,703 [Interest = 11,703 – 10,615 = 1088]
Amount after tax = 11,703 – 1088 x 0.04 = €11, 268
Best investment: BBS bank proposal.
(2 Marks)
(2 Marks)
[Total: 20 Marks]
10
Solution 6
(a)
Tests of Hypothesis. An hypothesis is an assumption about a situation. This assumption may be tested against one
or more alternative assumptions. It is normally assumed that a main hypothesis (a Null hypothesis, H0) is being
tested against another hypothesis (Alternative hypothesis H1). In testing the main hypothesis there are two types of
errors that can be made – a Type 1 error (rejecting the Null hypothesis when it is true), and a Type 2 error (accepting
the Null hypothesis when it is false). Accepting a type 1 error is usually the main concern. When testing the Null
hypothesis the maximum risk that we are willing to accept of committing a Type 1 error. This is the level of significance
and is normally designated at either the 5% or 1% levels. For example the 5% is split evenly between the two tails
of the distribution – a two tailed test; this means that the risk of making a Type 1 error is divided equally between each
tail of the distribution. If the sample mean lies outside the confidence limits the decision will be to reject H0 even
though the decision to reject it may be wrong. The confidence limits for the population mean are therefore regarded
as the critical values for tests of hypotheses. In a one tailed test the risk of being wrong is not divided between the
tails. In this case we place the risk we are willing to take in making a Type 1 error in one tail of the distribution. The
z scores depend on both the level of significance and on whether the test is one or two tailed. In a practical situation
the method is: the critical values are derived from the Normal distribution tables depending on the significance level
used; calculate the z score using the sample data provided; if the z score lies outside the relevant z critical value,
then the decision will be to reject the null hypothesis.
(7 Marks)
(b)
Correlation. This concept measures how well a regression line fits the actual data. There are two key measures
which are used
the coefficient of determination (measured by R2)
the coefficient of correlation (known as Pearson’s coefficient – measured by R).
When data is plotted and a regression equation calculated, there is a difference between the actual points of data
(observations) and their arithmetic mean. This deviation can be split into two parts – the explained part which is
accounted for by the regression line and an unexplained part which is not predicted or accounted for by the regression
line. Summing the explained and unexplained deviations gives the total deviation. However to avoid problems with
± signs the deviations are squared to give the ‘variations’. The coefficient of determination measures the [explaind
variation/total variation]. When R2 = 1, then all the deviations and variations can be explained by the regression line
and there is a perfect fit. Also when R2 ≠ (but close to) 0 the variations cannot be explained by the regression equation.
The data is therefore random to which any regression line may fit. The closer R2 is to one the better the fit of the least
squares line to the actual data or observations.
The coefficient of correlation, R, is the square root of the coefficient of determination, R2. Just as R2 varies between
0 and 1, the value of R varies between 0 and ± 1. Perfect positive correlation (R of +1) is when the relationship
between the variables is direct (both variables rise together) and R is -1 when the relationship between the variables
is inverse (one variable increases while the other reduces).
The coefficient of correlation therefore provides a method of measuring the strength of the relationship between two
variables.
(7 Marks)
(c)
Principles of sampling and selection of samples.
In setting out a sampling plan it calls for three decisions.
* Sampling Unit.This answers Who is to be surveyed? The market research must define the population that will be
sampled. The particular sampling unit should be specified. The particular sampling unit is not always obvious. For
example should people within particular age groups be interviewed? Should husbands and wives be interviewed?
Once this is determined, a sampling frame must be developed, that is, a way of giving everyone in the target
population an equal or known chance of being sampled.
* Sample Size. This answers How many people should be surveyed? Large samples give more reliable results than
small samples. However, it is not necessary to sample the entire target group or even a substantial proportion to
achieve reliable results. Samples of less than I% of a population can provide good reliability using a credible sampling
procedure.
* Sampling Procedure. This answers How should the respondents be chosen? To obtain a representative sample,
a probability sample of the population should be chosen. Probability sampling allows the calculation of confidence
limits for sampling error. Methods of probability sampling are described below. When the cost or time involved in
probability sampling is too high researchers will take non-probability sampling. These can be very useful in many
circumstances although the sampling error cannot be measured.
11
Methods by which samples may be selected.
There are several different methods and the choice depends on a number of factors. Methods may be divided into
two categories – probabilistic sampling and non probabilistic sampling.
Probabilistic Sampling.
Simple random sampling. Every member of the population has a known and equal chance of being selected.
Stratified random sampling. The population is divided into mutually exclusive groups (such as age groups) and
random samples are drawn from each group.
Cluster (area) sample. The population is divided into mutually exclusive groups (such as blocks) and the researcher
draws a sample of the groups to interviews.
Non-Probabilistic Sampling.
Convenience sample. The researcher selects the most accessible population members from which to obtain
information.
Judgement Sample. The researcher uses his or her judgement to select population members who are good prospects
for accurate information.
Quota Sample. The researcher finds and interviews a prescribed number of people in each of several categories.
[6 Marks]
[Total: 20 Marks]
12