Quantitative Methods for Business
1) George & Jim were brothers who were each left a will inheritance of ,000. George invested his money in an account for 5 years at an annual interest rate of 6.0% compounded quarterly. Jim invested his money for 5 years at an annual simple interest rate of 6.75%. At the end of that time they agreed to poor their accumulated principals into one account for a further 5 years with an annual interest rate of 5.0% compounded monthly.
a) How much accumulated principal did George and Jim each have after 5 years? Hence, determine how much money they had to pool.
Answers:
To determine the accumulated principal did George and Jim each have after 5 years, we will use the simple interest formula and compound interest formula.
i.e. for George we have,
Where F is the final amount or the sum due at the end of the transaction; P is the original principal; r is the interest rate in one period of time; and t is the time in which the number of period during which the principal is used expressed in years.
Then we have
On the other hand, we have to use the compound interest formula for Jim’s investment for 5 years at an annual simple interest rate of 6.75% to determine his final amount. Then we have,
Where F is the final amount or the sum due at the end of the transaction; P, is the original interest rate per period; and n, the number of conversion periods within the entire length of the transaction. The compound interest rate j is usually expressed as the rate per year and the interest is understood to be converted once a year if the frequency of conversion is not specified. The frequency of conversion m is the number of times interest compounded in year.
The interest rate per period is expressed as:
From this we have,
Thus the amount of money they have to pool is ,000 + , 949.97 = ,949.97.
b) What was the accumulated principal of their money at the end of the 10-year period?
Answer:
Using the formula we have,
c) In exercise (b), when George and Jim received their inheritance, the executor of the will suggested that they pool their resources immediately for 10 years and invest in an account that paid an annual interest rate of 6.75% compounded monthly. In view of your answer (b), should they have taken this advice? Explain.
Answer:
With respect to the answer in (b), let’s try to compare the final amount to be accumulated in years using 6.75% of interest compounded monthly of the principal ,000. Then we have,
From these values George and Jim should take the advice of the executor since the amount of money to be accumulated at the end of the 10th year is much bigger than their preferred investment. Actually, the suggested investment preference of the executor is ,175.7 bigger compared to preference of George and Jim at the end of 10th year.
2) A cricket team decide to purchase a house where they can hold their functions near their oval. They take out a mortgage of 0,000 from the Ruptcy Bank, which charges on annual interest rate of 5%, compounded semi-annually over 10 years.
a) What payments will the team have to make every 6 months on this loan?
Answers:
Let I = ? P= 0,000 t=10 j = 5% m = 2
i = 0.025 n = 20
Then we have,
6,633.97/20 = ,831.6985
The team have to pay ,831.6985 every 6 months on this loan.
b) What is the total amount of interest they will pay?
Answers:
3) In Exercise 2, the rival Fleece Finance offers the cricket team a loan for which it will charge on annual interest rate of 4.8% compounded monthly over 10 years.
a) What payments will the team have to make every month on this loan?
Answers:
Let I = ? P= 0,000 t=10 j = 4.8% m = 12
i = 0.004 n = 120
Then we have,
3,743.34/120 = ,614.53
The team have to pay ,614.53 every month on this loan.
b) What is the total amount of interest they will pay?
Answers:
4) In view of your answers to Exercise 2 and 3, from which lending institution should the cricketers take their loan?
From the previous answers, Fleece Finance should be considered by the cricket team for this load since the interest is much lower compared to Ruptcy Bank.
5) Scott, Anne and Jane form a partnership in which Scott invests ,000. Anne invests ,000 and Jane invests ,000.
a) Scott and Jane are each to receive a salary of ,000 out of the profit, while the remaining profit is to be divided between the three in the ratio of their investments. Find out how much each partner will receive in total if the profit is 0,000.
Answers:
From the given amount of investment (i.e. 0,000), Scott actually invested 35% of their total investment, 18% for Anne and 47% for Jane.
Thus, their shares in the profit of 0,000 are:
Scott: 120,000 x 0.35 = 42,000
Anne: 120,000 x 0.18 = 21,600
Jane: 120,000 x 0.47 = 56,400
Since Scott and Jane also receive a salary of ,000 each out of the profit, then they will have:
Scott: 120,000 x 0.35 = 42,000 + 25,000 = ,000
Anne: 120,000 x 0.18 = ,600
Jane: 120,000 x 0.47 = 56,400 + 25,000 = ,400
6) Next week you have to travel around parts of the country on business and have decided to hire a car from a car-hire company. You have contacted two companies which offer different services. The first company will rent you a car for 5 per day. The second company will charge per day, but with an additional charge of 50 cents per kilometre travelled. You know you will require the car for four days but are unsure of the number of kilometres that you will cover.
(a) Determine mathematically what distance you need to cover to make the second company’s charge cheaper.
Answer:
Let us consider that the car will be used in 4 days. Consider the following breakdown per company.
Company A (5/day) = $ 500 in 4 days
Company B (/day+.50/per km.) = 0 in 4 days.
With these values, there is 0 discrepancy between Company A and B regardless of the per km charges of company B. Meaning to say, I only need to travel (i.e. 180/0.5) 360km to make the total charges of Company B become 0. Thus, I only need to cover 359km. to make the second company’s charge cheaper.
(b) Confirm this using Excel graphs.
Answer:
(c) Both companies now realise that they will need to add 10% GST to the charge for the daily car hire, but not to the charge for the kilometres covered. How will this affect your decision? Justify your answer with both the calculations and Excel graphs.
Answer:
Let us consider that the car will be used in 4 days. Consider the following breakdown per company in accordance to 10% additional GST.
Company A (5/day +10%GST) = $ 550 in 4 days
Company B (/day+10%GST +.50/per km.) = 2 in 4 days.
With these values, there are now 8 discrepancy between Company A and B regardless of the per km charges of company B. Meaning to say, I only need to travel (i.e. 198/0.5) 396 km to make the total charges of Company B become 0. Thus, I only need to cover 395 km. to make the second company’s charge cheaper.
7) You are the manager of a toy company. Your company manufactures a variety of soft toys and you are trying to improve the efficiency of your operations. You wish to adhere to an annual operating budget and to develop operating plans using quantitative methods wherever possible. The overall budget for the year for production of two different soft toys is 0,000. The soft toys are koalas and kangaroos. The budget is intended to pay for labour and materials. Processing requirements for the two toys, on a per unit basis, are shown in the table.
Product
Grey cloth
White cloth
Stuffing
Labour
Koala
1.5 metres
0.5 metres
350gms
1.75 hours
Kangaroo
1.0 metres
1.0 metres
400gms
1.50 hours
The production costs for the toys are:
Material
Cost
Labour/hour
.50
Grey material/metre
.50
White material/metre
.50
Stuffing/100gms
.35
The company has a contractual agreement to produce a minimum of 3,000 toys per year. There must be at least 200 of each toy produced. The company can only get a maximum of 5,000 metres of each material. You wish to minimize the number of labour hours required to meet your contractual obligation.
Use Excel Solver to obtain a solution to this problem. Your submission should include:
· your mathematical set-up of the objective function and constraints
· your Excel spreadsheet
· the sensitivity report
· the answer report
Use your Excel output to answer the following questions:
What is the amount of labour hours required to meet the contractual agreement?
What are the optimal quantities of each toy?
If you only wish to employ staff for a maximum of 5 hours per day. How many staff should you employ if there are 48 working weeks in the year? How many hours per week would each staff member work?
As the manager you realize that labour per unit is actually the same for koalas as it is for kangaroos (1.50 hours). Does this change the optimum number of toys of each kind to be produced? Does the optimum number of labour hours change?
Credit:ivythesis.typepad.com
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