1) George & Jim were brothers who were each left a will inheritance of $20,000. 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.
Equation:
Simple Interest: S = P + n I P = P (1 + n r)
Where : P = principal ( Present Sum )
S = Future Sum
N = Number of Years
r = Interest Rate
Equation:
Compound Interest: A = P(1 + r/n)n t
Where:
P = the principal (the initial amount you borrow or deposit)
r = the annual rate of interest (percentage)
n = the number of times the interest is compounded per year
A = the amount of money accumulated after n years, including interest.
t = the time or the number of years.
Other equations involving compound interest are as follows:
Annually = P × (1 + r) = (annual compounding)
Quarterly = P (1 + r/4)4 = (quarterly compounding)
Monthly = P (1 + r/12)12 = (monthly compounding)
Solution:
For Jim:
S = P + n r P = P (1 + n r)
S = 20,000 + [1 + (5 ) ( 0.0675 ) ] = 26,750
For George:
A = P(1 + r)nt
A = 20,000 ( 1 + 0.06 / 4) 4(5) = 26,937.10
Then, add both… 26,750+26,937.10 = 53,687.10
b) What was the accumulated principal of their money at the end of the 10-year period?
For another 5 years wherein the amount will be compounded monthly. Use:
A = P (1 + r/12)12 (5)
A= 53,687.10 ( 1 + 0.05 / 12 ) 12(5) = 68,899.81
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.
Obviously, the rate of interest is a lot greater. There is a huge comparison between the 6.75 % to 5%. So, they should have taken the advice of their executor so that the money will have more interest rather than what they have got with the previous ones. In investing such inheritance they should be careful enough so that the will have great earnings and great amount of interest. Even though there are always bad decisions made. Sooner or later in that case things were learned in the same time.
2) A cricket team decide to purchase a house where they can hold their functions near their oval. They take out a mortgage of $120,000 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?
Note: In 10 years there would be 20 payments divided on every 6 months. So, use 20 as n or the number of monthly installments.
P = A { [(1+r/m) n -1] / [ r/m ( 1 + r/m ) n] }
120,000 = A { [ ( 1 + 0.05/6 ) 120 – 1 ] / [ 0.05/6 ( 1+ 0.05/6 ) 120
A = 1,585.809
b) What is the total amount of interest they will pay?
1,585.809 x 120 = 190,297.08
190,297.08 – 120,000 = 70,297.08
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?
P = A { [(1+r/m) n -1] / [ r/m ( 1 + r/m ) n] }
120,000 = A { [ ( 1 + 0.048/12 ) 120 – 1 ] / [ 0.048/12 ( 1+ 0.048/12 ) 120
A = 1,261.087
What is the total amount of interest they will pay?
1,261.087 x 120 = 151,330.50
151,330.50 – 120,000 = 31,330.49
4) In view of your answers to Exercise 2 and 3, from which lending institution should the cricketers take their loan?
The team should take their loan to the one who will provide them less amount of interest. So that, they will not find it hard to pay their debt. So, the more applicable and practical one is the second lending institution.
5) Scott, Anne and Jane form a partnership in which Scott invests $60,000. Anne invests $30,000 and Jane invests $80,000.
a) Scott and Jane are each to receive a salary of $25,000 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 $120,000.
Solution:
Scott = $60,000.
Anne = $30,000
Jane = $80,000
Total Investment = $ 170,000
Total Profit = $ 120,000
By Ratio and Proportion:
Scott: 60,000 / 170,000 = 0.353 = 35.3 %
Anne: 30,000 / 170,000 = 0.176 = 17.6 %
Jane: 80,000 / 170,000 = 0.471 = 47.1%
But, Scott and Jane already receive $ 25,000, therefore
$ 120,000 - $ 25,000 - $ 25,000 = $ 70,000
Scott: 120,000 x 0.353 = 42,360 – 25,000 = $ 17,360
Anne: 120,000 x 0.176 = 21,120 – 0 = $ 21,120
Jane: 120,000 x 0.471 = 56,520 – 25,000 = $ 31,520
Discussion Board:
Dealing with problems like this requires analytical and critical thinking. There are a lot of ways on solving problems. And yet, the best one should always be considered. Before handling such things, first be aware of the many possibilities that errors are always encountered. Create certain steps that you can follow every time you will solve problems. Read the situation carefully and always seek of what is being asked. Try to enumerate the given quantities. Then, find alternative solutions to the problem. There will always be one that best suits the given situation.
The formula used was based on engineering economy. The methods that are provided in that topic are easier to understand and use. It was originated by A.M. Wellington in his book entitled The Economic Theory of Railway Location that was published in 1887. Due to the bad decisions that many people had been made. And uncertainties that it can be brought to the future. Of course, this decision involves also money and time. According to Wellington (1887), if the consequences of a course of action can be reduced to monetary terms, it is easy to compare alternatives on the basis of maximum return. Such considerations expressible in money may be called reducible. In terms of reducible considerations, the decision criterion is that of greatest return. Of course, this may not produce the best decision, since not all considerations can be reduced to money. These non-monetary considerations are called irreducible, and may often be of compelling importance. For example, a course of action may be illegal though profitable. While recognizing irreducible considerations, engineering economy usually ignores them in its objective recommendations, which concern monetary matters only. The expression of truly irreducible considerations in terms of money is usually inappropriate and misleading. The engineering economic analysis is only one consideration when making a decision.
Most of the decisions involve money flows. These flows can be usefully made if they are reduced to equivalent amount at the same time. Usually, in the present time. This part is furnished by the use of the familiar concept of interest.
There are several mathematical interest rates. However, what are commonly used were the simple interest rates and compound interest rates. The first problem involves both type of interest rates. It was shown in the solution the basic formula. For Simple Interest, S = P + n I P = P (1 + n r). And for Compound Interest, A = P(1 + r/n)n t . Normally, most people will find compound interest confusing but it just depends on what is being asked in the situation. A compound interest may be contrasting with simple interest. But compound interest is more predominately in finance and economics rather than simple interest. Always remember that the compound interest is just a process of accumulating back the interest to the principal. So that the interest is earned on interest from that moment on.
Moving on with the topic, next in the line is annuity. Annuity is used to refer any terminating fixed payments over a specific period of time. In problem 2 and 3 what is being asked is the term of payment using different set of time frames and the interest that each payment will cause. In solving annuity problem, it is said that it would be more applicable to create a time frame of your payments with its corresponding interest so that the proper allocation of interest would be visible. The first application of the annuity formula is to find the present value of a sinking fund. The second is using the formula in situations. Follow the algebraic steps used to find the present value of a sinking fund. This is the formula for an annuity. The sinking fund only calculates the periodic payments that will accumulate in a given specific period of time. It is a method of depreciation under which the depreciation expense is an amount of an Annuity so that the amount of the annuity at the end of the useful life would equal the Acquisition Cost of the asset.
The last problem is the simplest one. It only involves ratio and proportion. Actually in this topic, there is no need to use or derived formulas. It is just part of the basic mathematical operation. The main key to that problem is to understand it the given situation well.
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