f1 business maths aug 06

8/14/2019 F1 Business Maths Aug 06

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NOTESAnswer 5 questions.

(Only the first 5 questions answered 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 that candidates are expected to pay particular attention to their 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 the candidates' answers and the extent to which answers are supported with

relevant legislation, case law or examples where appropriate.

BUSINESS MATHEMATICS &QUANTITATIVE METHODS

FORMATION 1 EXAMINATION - AUGUST 2006

The Institute of Certified Public Accountants in Ireland, 9 Ely Place, Dublin 2.


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THE INSTITUTE OF CERTIFIED PUBLIC ACCOUNTANTS IN IRELAND


FORMATION 1 EXAMINATION - AUGUST 2006.

Time Allowed: 3 hours, plus 10 minutes to read the paper Answer 5 questions

Only the first five questions answered will be marked.

All questions carry equal marks.

1. As a financial consultant you are providing advice to an investor on the following proposal. He wishes toinvest 25,000 in a managed fund for 5 years with an expected rate of return of 5%. If he leaves the

investment, or any part of it, in a particular plan for a further 4 years the rate of return will increase to 7%. He

wants, however, the option to withdraw half the original sum of 25,000 after the 5 year period. You are

required to advise the investor on

(i) The total value of the investment after the 9 year period if nothing is withdrawn

(8 Marks)

(ii) The total value of the investment if half the money is withdrawn after the 5 year period

(8 Marks)

(iii) Write a note on the time value of money (4 Marks)

[Total: 20 Marks]

2. Trainees for DIB Manufacturing Co. Ltd are paid a salary during training based on both their experience andstage of training. The number of trainees and the allowances paid are set out in the following table.

Payment Number of Trainees

100 and less 110 1

110 and less 120 4

120 and less 130 7

130 and less 140 13140 and less 150 7

150 and less 160 3

160 or more 1

You are required to

(i) Present the data on a cumulative frequency graph (8 Marks)

(ii) Estimate the median allowance from the graph (4 Marks)

(iii) Compare the median with the mean and modal allowances. (8 Marks)

[Total: 20 Marks]

1


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3. The SECO hotel, a small family owned enterprise, is attempting to estimate the expected daily profit from thebusiness. Although the hotel has a prestigious restaurant the daily profit depends on room occupancy. As the

adviser to the hotel you have estimated the probability of the number of guests arriving each night. The data

is set out in the following table.

No guests 1 2 3 4 5

Probability 0.1 0.2 0.4 0.2 0.1

Profit / room 35 75 125 175 225

As part of your report to the hotel owner you are asked to include details on the following.

(i) Explain how the expected profit is measured and what it represents ? (8 Marks)

(ii) Calculate the expected daily profit from the data provided (8 Marks)

(iii) Write a note on probability and expectation (4 Marks)

[Total: 20 Marks]

4. On the international commodities market the movement in the price of particular foods will have a majorimpact on local prices to consumers in Ireland. The following data from the EU Bureau of Statistics indicates

the trend in prices for three high consumption products over the past 3 years.

Price ( per 1000 kgs)

Commodity 2003 2004 2005

Coffee 20 25 35

Bananas 12 14 18Tea 6 8 12

As an analyst for the Consumers Association you are required to

(i) Develop a simple price index for each product (8 Marks)

(ii) Develop a simple aggregate price index (8 Marks)

(iii) Explain the principles underlying your calculations for both indexes and the results obtained.

(4 Marks)

[Total: 20 Marks]

2


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5. CPA office refurbishment is due to commence in September 2006. The contractor has stated that the projectwill be finished within 10 days. You have been asked to confirm the duration of the work since it will be

necessary to vacate the premises for that time. The following list of activities and durations have been

provided by the contractor.

Activity Preceding Duration

Activity (Days)

A Remove all furniture - 2B Clear all areas - 1

C Put in new utilities (gas,water, electricity) - 6

D Rewire rooms A,B 5

E Add more power points A 2

F Add new fittings D 3

G Modify wiring layout A,B 3

H Change lighting E 2

I Test lights, fittings C,G,H 1

J Test all systems F,I 1

You are required to:

(i) Draw a network to illustrate the activities (10 Marks)

(ii) Confirm the duration of the project and indicate the critical path. (10 Marks)

[Total: 20 Marks]

6. "A number of methods are available to attempt to achieve a representative sample of the population" Outlinethe principles of four of the following methods of sampling:

(i) simple random sampling

(ii) stratified random sampling

(iii) systematic sampling

(iv) multistage sampling

(v) quota sampling

(vi) cluster sampling

[Total: 20 Marks]

END OF PAPER

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FORMATION 1 EXAMINATION - AUGUST 2006.SOLUTION 1.

(i) The general formula for calculation of compound interest is

Sn = P(1 + r)n

where Sn = sum accrued, P = principal, r = interest rate, n = number of time periods.

P = 25,000, n = 5, r = 5% = .05

S5 = 25,000 (1 + 0.05)5 = 25,000 (1.2763)

= 31,907.5 4 Marks

This is reinvested for 4 years

S9 = 31,907 (1 + 0.07)4 = 31,907 (1.3108)

=

41,823.7 This is the final value after 9 years.

4 Marks

(ii) If half the investment is withdrawn after 5 years, that is,12,500. the balance remaining is 31,907 - 12,500= 19,407 (P).

S9 = 19,407 (1 + 0.07)4 = 19,407 (1.3108)

= 25,438.7 8 Marks

(iii) Time value of money. Capital investment decisions are those decisions which involve current outlays of cashin return for a stream of benefits in future years. Projection of company expenditures are made in theexpectation of realising future benefits. However, because money can be used to earn interest, waiting forthe recoupment of this money has a cost. Since we are waiting for the return of out investment funds so thatthey can be invested elsewhere we favour projects which give us the earliest cash flows. This implies thatmoney has a time value. The concept of a cost of capital relies on this. In order for us to forgo the use ofmoney for a period an amount should be paid in compensation. That is money received at different points intime is of differing significance. The longer we have to wait for it the less valuable a given sum is. The extentof this is governed by the rate of interest.

4 Marks

[Total 20 Marks]

4

Suggested Solutions


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SOLUTION 2

(i) The ogive is constructed from the following data. From the open ended distribution the end point has beenset at 180.

Class No Mid Payment CumulativeBoundaries Trainees Point less than frequency

(f) (x) fx

100 0

100 - 110 1 105 110 1 105

110 - 120 4 115 120 5 460

120 - 130 7 125 130 12 875

130 - 140 13 135 140 25 1755

140 - 150 7 145 150 32 1015

150 - 160 3 155 160 35 465

160 + 1 165 180 36 165

36 4840

2 Marks

Cum Frequency

40 x

x

x

30

x

20

x

10 Median

x

x

10 0 110 120 130 140 150 160 180

Payment / week6 Marks

(ii) The median can be derived from the graph and is approx. 137. 4 Marks

(iii) Mean = x = fx = 4820 = 134 2 Marks

f 36

Mode: found from the payment range with the greatest frequency, 130 - 140.

2 Marks

5


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The appropriate value:

Since the mode is the value that occurs most often, it may be accepted as typical of the data; it is notconsistent in that there may be more than one mode in a set of data

The median is considered to give a true middle of the data set and is preferable if there are extremevalues in the data

The mean uses every data equally in its calculation but is influenced by exceptionally high or low values

in the data.

In the present case there is very little difference between the various values.

4 Marks

[Total 20 Marks]

6


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SOLUTION 3.

(i) Many quantitative techniques treat situations with certainty. However, in many business situations it is notpossible to treat outcomes with certainty. This can be done by developing a probability model and aprobability distribution can be used. This gives a distribution of profits with their associated probabilities. Inthis case there is no single deterministic value for the profit. The expected value is an average of the possiblerange of outcomes weighted according to their probability of occurance, that is, expected value

E(X) = X1P(X1) + X2P(X2) + ------------------ XnP(Xn)

The probability is expressed as a value between 0 and 1. The total probability must equal 1. Since the totaldistribution of values is used then the expected value is equal to the mean of the x values. The expectedvalues themselves represent a long-term average and, as individual values, they are unlikely to occur.However, in business situations they give a single value for comparing alternatives and represent a bestsingle estimate of an uncertain situation.

8 Marks

(ii)

No guests 1 2 3 4 5

Probability 0.1 0.2 0.4 0.2 0.1

Profit / room 35 75 125 175 225

The expected profit is

E(X) = X1P(X1) + X2P(X2) + X3P(X3) + X4P(X4) + X5P(X5)

where X is the profit per occupied room.

E (X) = 0.1 x 35 + 0.2 x 75 + 0.4 x 125 + 0.2 x 175 + 0.1 x 225

= 3.5 + 15 + 50 + 35 + 22.5

= 126

The average expected daily profit is 126 this is the average amount that will be generated per day overa long period.

8 Marks

(iii) Probability and expectation. This process is part of decision analysis and involves using a range oftechniques to assist the manager in choosing the most appropriate decisions in given circumstances. Thereare a number of practical decision making techniques using probability. Such methods are necessary since

there are many circumstances where relevant information is not known with any degree of certainty. Theremay be probabilities associated with the likelihood of an event occurring and this will allow an expected valueto be determined. In many examples the expected value is obtained by multiplying a probability by the totalnumber of values. For example, if the probability of an employee being satisfied with their job is 0.75 andthere are 200 employees in the company, the expected number of satisfied employees would be 0.75 x 200= 150. This process can be extended to relate to more complex problems. In general the expected value ofa variable is obtained by multiplying each probability by the corresponding value and obtaining the sum ofthese products. The expected value can be regarded as being an estimate of the average value for thevariable.

4 Marks

[Total 20 Marks]

7


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SOLUTION 4.

(i) Commodity 2003 2004 2005

Coffee 20 25 35

Bananas 12 14 18

Tea 6 8 12

To construct a simple price index calculate the ratio of the new price to the base year price for each

commodity and then multiply by 100. the ratio of new price to vase year price is the price relative.

Simple price index = Pn/Po x 100 where Pn = new price (year n), Po = base year price (year 0).

Simple price index for each of the commodities.

Year Price Pn/Po Index

2003 20 1.00 100

2004 25 1.25 125

2005 35 1.75 175

2 Marks


2003 12 1.00 100

2004 14 1.16 116

2005 18 1.50 1502 Marks


2003 6 1.00 100

2004 8 1.33 133

2005 12 2.00 200

2 Marks

From the above it can be seen that the price of coffee has risen by 75%, the price of bananas by 50% andthe price of tea by 100% over the whole time period.

2 Marks

(ii) To derive an index for the overall change in price of all three types of commodity, a simple aggregate priceindex is required.

To include all three types of commodity their respective prices in each of the three years is summed, that is,Simple Aggregate Price Index = Pn x 100

Po 2 Marks

Where Pn = new prices (year n) and Po = base year prices (year 0).

Coffee Bananas Tea Pn / Po

P2003 20 12 6 38/38 = 1.00

P2004 25 14 8 47/38 = 1.23

P2005 35 18 12 65/38 = 1.71

6 Marks

This price index shows that the aggregate prices of the three drinks has risen by 71%. However, this indexhas ignored the relative quantities consumed in each of the three years. If consumption is taken into account

the fact that tea experienced the greatest % increase in price over the period should be taken into accountby calculating a weighted aggregate price index. 4 Marks

[Total 20 Marks]

8

Coffee

Bananas

Tea


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SOLUTION 5.

(i) The network for the activities is outlined below. It illustrates the sequence of activities and the steps involved.

10 Marks

The following additional information is provided to give students an appreciation of the process in deriving thenetwork but is not required in the examination.

Each activity is represented by an arrow on the diagram. Circles are numbered and drawn to indicate the startand of finish activity. The diagram shows the relationships between the activities as set down in the table. Inorder to produce a network diagram a list of activities is required and the interdependence between activities,that is, the activities that precede other activities. In the above diagram a dummy activity is introducedbecause activities D and G both follow A and B and E only follows A. The purpose of this dummy is to maintainthe logic of the sequence.

(ii) The total project duration is an important factor when managing projects. The overall duration can becalculated from the network providing the duration of each activity is known. In order to calculate the overallduration of the project it is necessary to estimate the earliest and latest event times. The earliest time isdetermined by the longest route through the network and the largest value is used. This gives the earliesttime in which the project can be completed. The latest event times are then calculated. In the network thelatest event time equals the earliest event time. Preceding latest event times are calculated by subtracting anactivitys duration from the subsequent latest event time. If two or more activities start from an event, the latesttime for each route is calculated and the lowest value is used. Both values are shown in the diagram. It showsthat the project can be completed in 11 days.

The critical path identifies the route that defines the overall duration. The activities on the critical path haveno flexibility if the project is to finish on time. It is 1-2-4-6-7-8 and takes 11 days.

10 Marks

[Total 20 Marks]

9


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SOLUTION 6.

Sampling Methods. There are a wide range of sampling methods depending on the type of sample required andthe technique being used. A description of four of the following methods is required.

Simple random sampling. In this case every member of the population must have an equal chance of being includedin the sample. For large populations each member is normally given a unique identification number and thenrandom numbers are generated or random tables are used. The sample then comprises the population memberswhose numbers match those generated from the random numbers. This method has minimum bias. In some cases

it may be difficult to contact all members of the chosen sample and occasionally an unrepresentative sample mayoccur. This method is used by large marketing companies to obtain a wide geographical spread in the data.

5 Marks

Stratified random sampling. This method is similar to simple random sampling but is used where the populationcontains distinct groups, that is, groups with different views about the issues under study or of particular interest.The sample can be stratified according to the particular groups so that it has the same proportions, approximately,as the population. For each stratum the sample members are selected randomly as for simple random sampling. Itis a unbiased method and gives a representative sample. The process of stratification can be expensive and incurscosts additional to the survey process.

5 Marks

Systematic Sampling. This method is similar to simple random sampling. In this case the data are assumed to berandom and every nth member is selected where n is determined by (population size) / (sample size). The startpoint of the sample may be chosen randomly. It is an inexpensive and easy method to use and is useful if the exactpopulation is unknown. Often particular steps must be taken to ensure that the sample is not biased.

5 Marks

Multistage sampling. This is a process used for producing a representative sample form a widespread population.The process selects individual sampling units by splitting the sampling process into stages and using the mostrelevant sampling technique. A typical process is a three-stage survey where 1) stage 1 takes a number of primarysampling units the population may be divided into a number of regions and then randomly select a number ofthese 2) Stage 2 where each of the regions is taken and random samples of sub-regions taken (secondary samplingunits) 3) stage 3 where individuals could be selected by using systematic sampling (tertiary sampling units). Thistechnique is used for widely spread data.

5 Marks

Quota sampling. Used where interviewing is the main method of data collection. It is necessary to ensure that thecomposition of the sample matches that in the population. The interviewer is given a predetermined sample profilewhere the number of interviewees in each category is chosen to match the population proportions. The interviewerselects from the population to match the required numbers in each category proportions. This method tends tohave a good response rate. However, it is a non-random technique and is reliant on the interviewer.

5 Marks

Cluster Sampling. This method is used when the population items of interest are widely spread and it is desirableto ensure that the sample elements are grouped together in some way. A number of groups/areas would be selectedand populations within those groups could be interviewed. This is a useful method for widely spread geographicaldata where the population is not defined exactly.

5 Marks

[Total for four: 20 Marks]

10

f1 business maths aug 06

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