C03-Fundamentals of business mathematics

Transcription

1 mple Exam Paper Question 1 A retailer buys a box of a product, which nominally contains Q units. The planned selling price of each unit is P. If both P and Q have been rounded to ± 10%, then the maximum rounding error in total revenue is: A. 10% B. 20% C. 21% D. 0.1Q x 0.1P Question 2 The telephone costs of a company last year were 10,000, including Value Added Tax (VAT) at 17.5%. It has been decided to allocate 60% of these telephone costs, excluding VAT, to Central Administration and to allocate 30% of the remainder, excluding VAT, to Finance. The telephone costs (to the nearest ) to be allocated to Finance will be closest to: Question 3 The following formula is used in the financial analysis of dividends: R= (V/P)+G When the formula is rearranged, with P in terms of the other variables, P is equal to: A. (R/V)-G B. (R-G)/V C. (V/R)-G D. V/R-G Question 4 A company s market value has fallen from 32 billion to 2 billion in four years. The average annual percentage decline in market value is closet to: A. 20% B. 40% C. 50% D. 100% Question 5 If 3x + 2y = 6 and x 2y = 2. The solution in the form (x,y), to the above simultaneous equations is 1

2 Question 6 The estimated total cost of each unit of a product is 12 (± 1), and the estimated selling price of each unit is 20 (± 3). The estimate profit per unit will be: A. 8 (± 4) B. 8 (± 3) C. 8 (± 2) D. 8 (± 1) Question 7 A trader s weekly costs, TC, are less than or equal to $100. Weekly revenue, R, is a minimum of $120. Which one of the following statements is true? A. TC < $100 and R > $120 and R > TC B. TC $100 and R $120 and TC > R C. TC $100 and R > $120 and R < TC D. TC $100 and R $120 and R > TC Question 8 In a group of 100 players, 30 are male, 55 are a pro level, and 6 of the males are at beginner level. A player chosen at random is female. What is the probability that she is not a pro level? A..80 B..56 C..44 D..20 Question 9 Three people are carrying out independent functions during an internal audit. It is known that in each of the three separate areas being investigated there is a serious error. From past experience, it is estimated that the (independent) chances of the individuals finding the serious error in their area are 0.8, 0.7 and 0.6 The probability that at least one of the serious errors will be found is: A. (0.8 x 0.3 x 0.4) + (0.2 x 0.7 x 0.4) + (0.2 x 0.3 x 0.6) B. 1- (0.2 x 0.3 x 0.4) C. 1- (0.8 x 0.7 x 0.6) D. None of the above 2

3 Question 10 Mail order buyers of Brand X, classified by area and age (years) Area/Age Under North South East West Total 1,600 1,400 1,200 1,000 The probability that a randomly-selected Brand X buyer is from the North and under 25 years of age is (to 2 decimal places) A B C D Question 11 Mail order buyers of Brand X, classified by area and age (years) Area/Age Under North South East West Total 1,600 1,400 1,200 1,000 The probability that a randomly-selected Brand X buyer is from the West or under 25 years of age is (to 2 decimal places). A B C D. None of these 3

4 Question 12 In an internal audit of 200 invoices, the following numbers of errors were discovered: Number of Errors Number of Invoices or more 0 The expected value of the number of errors per invoice is: A. 1.8 B. 2 C. 2.1 D. 3 Question 13 The following information shows the daily sales revenue ( 000) of a company producing a particular item of clothing, over a period of two years: les 000 Frequency % 0 to under to under to under to under to under 50 5 The expected daily sales in ( 000) is 4

5 Question 14 A broker has estimated the profits or losses for a particular investment and their respective probabilities as follows: Profit ( 000) Probability The expected profit ( 000) on this investment will be Question 15 A cumulative frequency distribution of weekly spending is as follows: Weekly spending Cumulative frequency Less than $75 50 Less than $ Less than $ Less than $ Less than $ A. How many spent between $150 to $200 B. How many spent less than $300 and more than $200 Question 16 In a particular country, a tax at 40% is payable on any gains on house sales not due to inflation. A house was purchased there for $75,000 and sold for $250,000. Over the same 5

6 period, the country s house price (inflation) index rose from 120 to 240. The tax (to the nearest $) payable on the house sale is? Question 17 The number of rejects from 50 samples of the same size is as follows: Number of rejects in each sample Number of samples (frequency of reject) The arithmetic mean number of rejects per sample is: A. 2.2 B. 2.4 C. 3 D. 20 Question 18 For the following set of ten numbers, the median is 15: X This statement is false of X equals: A. -5 B. -4 C. -3 D. -2 6

7 Question 19 Details of an index number are given below: Group Base Weight Index Food & Drink Travel & Leisure Housing All items ? The All items index number is closest to: A. 130 B. 133 C. 135 D. 146 Question 20 Weekly money wages index 1998 = 100 Index of inflation 1990 = Read the following statements about the period 1998 to 2001: (i) Inflation has increased by more than money wages (ii) Money wages have increased by 5% each year, year on year Which one of the following is true? A. (i) only B. (ii) only C. Both (i) and (ii) D. Neither (i) or (ii) 7

8 Question 21 An index number is made up of two items, food and non-food. Sub-group Weight Index Non-food Food 3? All items The index number for the sub-group Food is closest to: Question 22 On the basis of the scatter diagram above, which of the following equations would best represent the regression line of Y on X? A. Y = -X + 8 B. Y = X + 8 C. Y = X-8 D. Y = -X -8 Question 23 For a certain group of students, the coefficient of rank correlation between their performance in Accounting and their performance is Law is -1. The coefficient of rank correlation between their performances in Law and FBSM is also -1. Therefore, the coefficient of rank correlation between their performance in Accounting and their performance in FBSM is 8

9 A. -2 B. Zero C. +1 D. Impossible to determine from the information given Question 24 The number of daily complaints to a railway company has an average (arithmetic mean) of 12 and a standard deviation of 3 complaints. The coefficient of variation, measured as a percentage, is therefore; A. 0.25% B. 4 % C. 25% D. 400% Question 25 For a set of six pairs of observations for the variables X (number of employees in hundreds) and Y (product sales in thousands of units), the following results were obtained: X = 1 Y = 15 X2 = 15 Y2 = 65 XY = 7 The correlation coefficient is nearest to: A B C D Question 26 The Personnel department of a large manufacturing company wishes to measure the correlation between the performance of its employees on an aptitude test, and their ability to catty out a specific work-related task. The following table shows the rankings of 7 employees at both the test and the task: Employee A B C D E F G Test Rank Test Rank Spearman s rank correlation coefficient for this data is: Question 27 9

10 On one particular checkout in a supermarket, the service times in minutes of five customers were: 3, 2, 1, 5, 4 The standard deviation of these service times, correct to 1 decimal place, is closest to: Question 28 At a second checkout in the same supermarket as in question 27, the service time has an arithmetic mean of 5 minutes and a standard deviation of 1 minute. The coefficient of variation will be: A. 50% B. 20% C. 5% D. 2% Question 29 The sales of a product are recorded monthly for 24 months. The four-point (centred) moving averages are calculated and plotted on a graph. How many moving average points are plotted? A. 20 B. 21 C. 22 D. 24 Question 30 If a company has sales value of $1,800 at a certain point and the seasonal factor is 1.13, using the multiplicative model the adjusted figure to the nearest $00 will be? Question 31 The underlying trend in the demand for a particular product is constant (flat), and is subject to quarterly seasonal variations as follows: Quarter Q1 Q2 Q3 Q4 Seasonality +50% +50% -50% -50% Assume a multiplicative model is appropriate. If the demand for the last quarter, Q2, was 240 units, then the forecasted demand for the next quarter, Q3 is: A. 80 units B. 100 units C. 120 units 10

11 D. 140 units Question 32 A product has a constant trend in its sales and is subject to the following periodical seasonal variations. Period P1 P2 P3 P4 Seasonality +45% +65% -50% -35% Assuming a multiplicative model for the time series, what should be the sale for the Period 3, if the sales in the last period, P2 were 350? Question 33 A multiplicative time series model should be assumed. Quarterly sales (units) of Brand X, 2001 Q1 Q2 Q3 les (units) 1,600 4,400 1,680 Seasonal variation -20% +100% -30% The trend value for Q1 sales (units) is: A. 1,280 B. 1,920 C. 2,000 D. None of these Question 34 A multiplicative time series model should be assumed. Quarterly sales (units) of Brand X, 2001 Q1 Q2 Q3 les (units) 1,600 4,400 1,680 Seasonal variation -20% +100% -30% The seasonal variation for Q4 is: A. -50% B. 0% 11

12 C. +50% D. None of these Question 35 A multiplicative time series model should be assumed. Quarterly sales (units) of Brand X, 2001 Q1 Q2 Q3 les (units) 1,600 4,400 1,680 Seasonal variation -20% +100% -30% The forecast for the fourth quarter s sales (units), Q4, in 2001, assuming the trend pattern continues, is closet to: A. 1,300 B. 2,300 C. 3,800 D. 5,200 Question 36 An annual (year-end) income of 10,000 is required in perpetuity. If there is a fixed interest rate of 8% each year and administrative charges are ignored, the lump sum investment necessary now is closest to: A. 9,260 B. 80,000 C. 100,000 D. 125,000 Question 37 An annual (year-end) income of 15,000 is required in perpetuity. Assuming a fixed rate of interest of 9% each year, and ignoring administrative charges, the sum required now to purchase the annuity is closest to: A. 13,650 B. 135,000 C. 150,000 D. 167,000 Question 38 2,000 is invested in a bank account. The account earns compound interest at 5% per year. The cash value of the account, to the nearest, at the end of five years will be: 12

13 A. 2,680 B. 2,553 C. 2,431 D. 2,335 Question 39 2,000 is invested in a bank account. The account earns compound interest at 5% per year. The investment will have almost doubled in value after: A. 11 years B. 12 years C. 13 years D. 14 years Question 40 A 100,000 mortgage, with interest compounded at 11% each year, is to be repaid by 10 equal year-end payments of X, the first being due one year after the mortgage was contracted. The first payment X is closest to? Question 41 A fixed-interest $200,000 mortgage, with annual interest compounded at 6% each year, is to be repaid by 15 equal year-end repayments of $R. The annual repayment $R will be closest to: A. $14,133 B. $20,593 C. $31,954 D. $83,400 Question 42 You borrow 3,000 and pay 10% each year interest. Ignoring capital, if you pay this interest at the end of each year, what is the present value of the interest payable at the end of the third year? A. (3/10) x 300 x 3 B. (7/10) x 300 C. (10/11)^3 x 300 D. 3,000 x (11/10)^3 Question 43 13

14 A group of 10 people share the cost of a new year s eve for 40 friends. The price of the meal per person is 20 plus 4 for wine and 2 for coffee. Calculate the amount each of the 10 will need to pay. Write down a formula which could be input into one cell in Excel to calculate the amount that the 10 will need to pay. Question 44 Enter the formula required in Excel to perform the following calculation to the specified decimal places in each case: 37/9 x 4.34 (to two decimal places) Question 45 Below is an extract from an Excel spreadsheet For the given data, give the Excel formulae that would be required in Excel to calculate the ROI and NPV in cells B10 and B11 respectively. 14

15 C03-Answers Question Answer Question Answer 1 C 24 C25% 2 1, A 3 D % X=2, Y=0 28 B 6 A 29 B 7 D 30 1,593 8 B 31 A 9 B A 33 C 11 B 34 A 12 A 35 A ( 000) 36 D D 15 A20 B20 38 B 16 40, D 17 A 40 16, A 41 B 19 B 42 C 20 A 43 40* (20+4+2)/10) ROUND(37/9*4.34,2) 22 A 45 ROI= Average(B2:B5)/B1 NPV = NPV(B7,B2:B5)-B1 23 C 15

16 Explanations: 1. 21%. The maximum rounding error in the total revenue is the error that is most deviated from the error free revenue. Error free revenue R=PQ Deviation +10% in P and Q. R=1.1PX1.1Q R= 1.21PQ this is 21% higher than the error free revenue. Deviation -10% in P and Q. R=.9PX.9Q R=.81PQ this is 19% lower than the error free revenue. 2. 1,021. Steps 1 collate the data. Total Costs 10,000. Vat 17.5%. 60% cost to central admin. 30% cost to Finance. Step 2 Exclude the Vat figure. 10,000/1.175 = Step 3 split figure in to 60% & 40% 60%= %= 3404 Steps 4 30% or the 40% 3404X.3 = Re arrangement of the formula % Using the Future value formula calculate the negative growth rate. 5. X=2, Y=0 3X + 2Y = 6 X-2Y = 2 4X = 8 X = 8/4 X=2 (i) Replace the value of x in any equation. 3X+2Y = 6 replace and X= 2 3(2) +2y = 6 6+2y = 6 2y= 6-6 y =0 16

17 X-2y =2 4X = 8 (ii) 6. Estimate SP 20 (± 3). Estimate CP 12 (± 1). Estimated Profit = 8 (± 4) 7. Basic understating of the signs in Mathematics. 8. Eliminate the distracters from the question, only deal with the female number of students. Total no of Female = 70 Total no of Female at beginner level = 39 If a female is not at pro level she has to be from beginner level. Chances are 39/ Sum of probability is 1. The problem is to find at least one serious error is 1- probability of finding no error at all. The independent chances of each of the error not to occur (0.2 x 0.3 x 0.4) 10. The probability of the buyer being under 25 and from north. Multiplication of the probabilities. No of buyers under 25 and from North is 400 Total number of buyers 5200 Chances are as 400/ rounded up to two decimal The probability of the buyer being under 25 is 1600/ The probability of the buyer being from west is 1300/ The probability of the buyer being under 25 or from west is [1600/ /5200] [1600/5200 X 1300/5200] 12. fx/ f 360/ ,000 Take average of class interval and multiply with the relevant frequency percentage. Take sum of all the expected values to calculate the final. les 000 Frequency % X 000 EV 17

18 0 to under to under to under to under to under Sum of expected values = Multiply the profit with estimated probability. Take sum of all the expected values to calculate the final. Profit ( 000) Probability EV Sum of expected value A20:B20 Identify the right group of people by keenly looking at the range values in each row ,000 Step 1. Adjust the price according to the rise in index. Step2. Calculate the taxable profit and apply tax rate to it. Price according to new index is $75000 x 240/120 $150,000 Profit on index adjusted cost is $250,000- $150,000 $100,000 Tax rate 40%, Hence the tax figure is $40,000 18

19 17. Application of the formula 110/ 50 Number of rejects in each sample Number of samples (frequency of reject) fx f = 50 fx = Replace X with the given values as option A, B, C, D. Arrange a new range of values. Apply the definition of median. An example X (-3) (14+16)/2 15 repeat this for other values of X. Note: when the sample is in even number to find the value of the median middle number of values 19. Weighted average = Wx/ W 13,300/ Group Base Weight Index (X) WX Food & Drink Travel & Leisure Housing All items ? 13,300 19

20 20. Wages inflation has been 15%. 15% inflation on the inflation index would have given 207. The actual index is 210, confirming that inflation is higher than wages. Wage inflation is 5 index points year on year, not 5% Step1 Use the weighted average formula Step 2 represent the missing value with X Step 3 Calculate the value by solving the equation for X 127 = (7x X )/ = x 360=3x X= 120 Sub-group Weight Index Non-food Food 3? All items The graph represents the inverse relationship between two variables. As the value of X increases the Value of Y decreases. 23. Both Accounting and Maths are perfectly negatively correlated with Law which suggests that their behaviour is 100% predictable. Therefore, Accounting and Maths must behave predictably and in the same way as each other meaning another perfect correlation, this time positive % C v = Standard Deviation / Mean 3/12 =.25 25% 25. Formula for correlation coefficient is provided with the mathematics table Use the formula for Rank correlation. 20

21 Calculate the mean value. Use the standard deviation formula, =(10/5-1)½ C v = Standard Deviation / Mean 1/5.2 20% 29. Tabulate 24 month sales, deduce the four point moving average, count the number of averages deduced Calculate the value eliminating the seasonal factor in Q2. Apply the seasonal factor of Q3. Demand in Q Q2 seasonality was +50% Elimination of the seasonality 240/1.5 =160 Q3 seasonality -50% Demand in Q3 = 160(.5) = 80 Units Eliminate the seasonal factor from P2. Adjust the P3 figure according to the new period factor. les in P2 350 Seasonality +65% Eliminate the seasonality 350/1.65 = 212. P3 Seasonality -50% 212(.5) 106 Units 33. Trend = 1,600 / ( )= 2, Seasonalities should be balanced to zero. Hence the Q4 should be -50%. 35. Trend calculation Q1Trend = 1,600 / ( )= 2,000 Q2Trend = 4,400 / ( )= 2,200 Q3Trend = 1,680 / ( )= 2,400 If same trend continues then Q4 trend is 2,600 Q4 Forecast = 2,600* 0.50= 1, Use perpetuity formula. Required investment = 10,000/ ,000 21

22 37. Use the perpetuity formula. 15,000/.09 The absolute value we get by calculations 166, this is closest to 167, Use 2000 as present value and apply the future value formula for the calculation. FV= PV (1+r)^n FV = 2000 (1.05) ^5 2, Use 2000 as present value and 4000 as future value. Use compound rate as provided in the data and calculate the value of T by solving the whole equation. FV= PV (1+r)^t 4000= 2000(1+.05) ^t 2=(1.05) ^t Apply logarithm t= 14 years ,981 Use the annuity function/formula to calculate the equal instalments. C=Pr(1+r) ^n/(1+r) ^n-1 C= [(100,000)(.11)(1+.11) ^10]/ (1+.11) ^10-1 C= 16, Use the annuity function/formula to calculate the equal instalments. C=Pr(1+r) ^n/(1+r) ^n-1 C= [(200,000)(.06)(1+.06) ^10]/ (1+.06) ^15-1 C= $20, FV= PV(1+r) ^n PV= FV/ (1+r) ^n PV = 300 X( 1/1+r) ^3 PV= 300 X (1/1.11)^3 PV= 300X (10/11) ^ * (20+4+2)/ ROUND(37/9*4.34,2) 45. ROI= Average(B2:B5)/B1 NPV = NPV(B7,B2:B5)-B1 22

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