Monday, October 8th, 2018
Interest rates are very powerful and intriguing mathematical concepts. Our banking and finance sector revolves around these interest rates. One minor change in these rates could have tremendous and astonishing impacts over the economy. But why?
Before determining the reason of this why? Let’s first know what is interest and these interest rates?
Interest is the amount charged by the lender from the borrower on the principal loan sum. It is basically the cost of renting money. And, the rate at which interest is charged on the principal sum is known as the interest rate. The rate at which interest is charged depends on two factors
The above two reason becomes the basis of why interest rates are so important and have a great effect on markets and economy. Since a minor rise in interest rates increases the cost of borrowing for the borrower and as a result, he has to pay more interest on his loan amount and thus, a decline in his money income that he could spend on other products which create a ripple effect of decreased spending throughout the economy and vice versa. Since change in interest rate has a chain effect in the market, it has a great deal of importance in the study of market, finance, and economy. And that’s why, forms an integral part of the curriculum in the MBA programs. But, a relatively simpler level of questions is asked in the CAT based on the concepts learned at the time of high school.
These concepts are categorized into type of interests
Let’s first start and understand Simple Interest because as the name suggests it is simple and comparatively easy to comprehend.
Simple interest is that type of interest which once credited does not earn interest on itself. It remains fixed over time.
The formula to calculate Simple Interest is
SI = {(P x R x T)/ 100}
Where, P = Principal Sum (the original loan/ deposited amount)
R = rate of interest (at which the loan is charged)
T = time period (the duration for which money is borrowed/ deposited)
So, if P amount is borrowed at the rate of interest R for T years then the amount to be repaid to the lender will be
A = P + SI
Consider a basic example of SI to understand the application of above formula such as Find the simple interest on ₹ 68000 at 16 ^{2}/_{3 }% p.a. for 9 months.
Here, P = ₹68000
R = 16^{2}/_{3 }% = 50/3% p.a.
T = 9 months = 9/12 years = ¾ years
SI = (68000 x 50/3 x ¾ x 1/100) = ₹8500
RESULTS  APPLICATION  SOLUTION 
If rate to interest is r_{1}% for T_{1} years, r_{2}% for T_{2} years …. r_{n} for T_{n} years for an investment. And if the Simple Interest obtained is ₹a on the investment. Then the principal amount is given by
a x 100/ (r_{1}T_{1} + r_{2}T_{2} + …+ r_{n}T_{n}) 
Adam borrowed some money at the rate of 6% p.a. for the first two years, at the rate of 9% p.a. for the next three years, and at the rate of 14% p.a. for the period beyond 5 years. If he pays a total interest of ₹ 11400 at the end of nine years, how much money did he borrow?  In this case, r_{1} = 6%, T_{1} = 2 years
r_{2} = 9%, T_{2} = 3 years r_{3} = 14%, T_{3} = 4 years (since, beyond 5 years rate is 14%) and Simple interest = ₹11400 Therefore, P = (11400 x 100)/ (6*2 +9*3 +14*4) = 1140000/ (12 + 27 + 56) = 1140000/ 95 = ₹12000 
If a person deposits sum of ₹A at r_{1}% p.a. and sum of ₹B at r_{2}% p.a. then the rate of interest for whole sum is
R = {(Ar_{1 }+ Br_{2})/ (A + B)} 
A man invested 1/3 of his capital at 7%; ¼ at 8% and the remainder at 10%. If his annual income is ₹561, What is his capital?  Let x be his capital or principal.
Therefore, R = (^{1}/_{3} x * 0.07 +¼ x *0.08 +^{5}/_{12} x*0.10)/x R = (^{1}/_{3} * 0.07 +¼ *0.08 +^{5}/_{12} *0.10) R = 0.08496 Total SI = ₹561 ₹561 = 0.08496x x = ₹6602 
If a sum of money becomes “n” times in “T years” at Simple Interest, then the rate of interest p.a. is
R = 100(n – 1) / T 
The rate at which a sum becomes 4 times of itself in 15 years at S.I. will be?  It’s a very easy question you just need to use this formula and you will directly reach to an answer.
Therefore, R = (100 x 3)/15 = 20% 
If a certain sum of money is lent out in n parts in such a manner that equal sum of money is obtained at simple interest on each part where interest rates are R1, R2, … , Rn respectively and time periods are T1, T2, … , Tn respectively, then the ratio in which the sum will be divided in n parts can be given by
1/R_{1}T_{1:} 1/R_{2}T_{2}:..:1/R_{n}T_{n}

A person invests money in three different schemes for 6 years, 10 years and 12 years at 10%, 12% and 15% Simple Interest respectively. At the completion of each scheme, he gets the same interest. What is the ratio of his investment?  Here, T_{1 }= 6, T_{2} = 10 and T_{3} = 12 years resp.
And, R_{1} = 10%, R_{2} = 12%, and R_{3} = 15% resp. Hence, the ratio of his investment will be 100/60 : 100/120 : 100/180 1/6 : 1/12 : 1/18 1 : 1/2 : 1/3 6 : 3 : 2 
This the most usual type of interest that is used in the banking system and economics. In this kind of interest along with one principal further earns interest on it after the completion of 1time period. Suppose an amount P is deposited in an account or lent to the borrower that pays compound interest at the rate of R% p.a. Then after n years the deposit or loan will accumulate to:
P(1+R/100)^{n}
Consider this example, if an amount of 100 is deposited in saving bank account for 3 years at the interest rate of 6% p.a. Then, after one year the ₹100 will accumulate to ₹106. Since in compound interest, interest itself earns interest, therefore, after 1year interest for the 2^{nd} will be calculated on ₹106 unlike to that of Simple interest where interest will be calculated on ₹100 only. Thus, after the end of the third year the total amount will become ₹100(1.06)^{3 }= ₹119.1016.
Amount= P (1 + R/100)^{ n}
Amount = P (1 + (R/2)/100)^{2n}
Amount = P (1 + (R/4)/100)^{4n}
Amount = P (1 + R_{1}/100) (1 + R_{2}/100) (1 + R_{3}/100)
Present worth = x/ (1 + R/100)^{n}
R = 100(x^{1/n} – 1)
R = (A_{2} – A_{1})/ A_{1} x 100
Question 1: A man invests ₹ 5000 for 3 years at 5% p.a. compounded interest reckoned yearly. Income tax at the rate of 20% on the interest earned is deducted at the end of each year. Find the amount at the end of third year.
Sol: Here, P = ₹5000, T = 3 years, r = 5%
Therefore, Interest at the end of 1^{st} year = 5000 (1 + 0.05) – 5000 = ₹250
Now Income tax is 20% on the interest income so the leftover interest income after deducing income tax = (1 – 0.2) * 250 = ₹200
Total Amount at the end of 1^{st} year = ₹5000 + 200 = ₹5200
Interest at the end of 2^{nd} year = 5200 (1 + 0.05) – 5200 = ₹260
Interest income after Income tax = 0.8 * ₹260 = ₹208
Total Amount at the end of 2^{nd} year = ₹5200 + 208 = ₹5408
Interest at the end of 3^{rd} year = ₹5408 (1.05) – 5408 = ₹270.4
Interest income after Income tax = 0.8 * ₹270.4 = ₹216.32
Total Amount at the end of 2^{rd} year = ₹5408 + 216.32 = ₹5624.32
Question 2: A sum of ₹12000 deposited at compound interest becomes double after 5 years. After 20 years, it will become?
Sol: The rate of interest at which ₹12000 doubles after 5 years is given by
R = 100(x^{1/n} – 1)
= 100(2^{1/5} – 1)
=100 x (1.1486 – 1)
= 100 x 0.1486 = 14.86%
Therefore, after 20 years it becomes,
A = ₹12000(1 + 14.86/100)^{20}
= ₹12000 (1.1486)^{20}
= ₹12000 x 15.97
= ₹ 191671.474
Let a person takes a loan from bank at r% and agrees to pay loan in equal installments for n years. Then, the value of each installment is given by
P = X/ (1 + r/100)^{n}………X/ (1 + r/100)^{2} + X/ (1 + r/100)
For better understanding, let’s understand with the help of example.
One can purchase a flat from a house building society for ₹ 55000 or on the terms that he should pay ₹ 4275 as cash down payment and the rest in three equal installments. The society charges interest at the rate 16% p.a. compounded halfyearly. If the flat is purchased under installment plan, find the value of each installment.
Sol: The cost of the flat is ₹ 55000. Now, if the person could either buy flat by paying ₹55000 or through installment plan. Since the flat was purchased through installment plan then the loan amount = ₹55000 – 4275 (down payment) = ₹50725.
Here r = 16% compounded Halfyearly in 3 equal instalments. Let x be the amount of installment. Then,
₹50725 = x/ (1 + 16/200)^{3} + x/ (1 + 16/200)^{2} + x/ (1 + 16/200)
₹50725 = x (1/1.2591 + 1/1.1664 + 1/1.08)
₹50725 = x (0.79421 + 0.85722 + 0.9259)
₹50725 = x (2.577)
₹50725/2.5777 = x
x = ₹19683
The above examples are just few types based on compound interest, there could be numerous others complex and difficult questions that could come in the CAT exam. Compound Interest is very significant topic in today’s world. It has vast and diverse application. In exam you could also find problems that involve both simple and compound interest. You will get thorough and fluent in this topic with time and practice.
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The basics of both type of Interests is explained very neatly with examples for better understanding. Thank you.