Basic Concepts: Simple Interest & Compound Interest

# Basic Concepts: Simple Interest & Compound Interest | CSAT Preparation - UPSC PDF Download

 Table of contents What are Interest Rates? Types of Interest Rates Table: Some useful results based on Simple Interest Solved Examples on Compound Interest Compound Interest Installments

## What are Interest Rates?

• 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 value of money doesn’t remain same over time. It changes with time. The net worth of ₹ 100 today will not be same tomorrow i.e. If 5 pens could be bought presently with a INR 100 note then in future, maybe only 4 pens can be bought with the same ₹ 100 note. The reason behind this the inflation or price rise. So, the interest rate includes this factor of inflation
• The credibility of the borrower, if there is more risk and chance of default on the borrower’s part then more interest will be charged. And, if there is less chance of payment failure on the part of borrower then the rate of interest would be lower.
• 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.

## Types of Interest Rates

Interests are of two types:

• Simple Interest
• Compound Interest

### Simple Interest

• 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/% p.a. for 9 months.
Here, P = ₹68000
R = 162/% = 50/3% p.a.
T = 9 months = 9/12 years = ¾ years
SI = (68000 x 50/3 x ¾ x 1/100) = ₹8500

Question for Basic Concepts: Simple Interest & Compound Interest
Try yourself:Find the simple interest on Rs. 5200 for 2 years at 6% per annum.

## Table: Some useful results based on Simple Interest

Question for Basic Concepts: Simple Interest & Compound Interest
Try yourself:How much time will it take for an amount of Rs.450 to yield Rs. 81 as interest at 4.5% per annum of simple interest?

Question for Basic Concepts: Simple Interest & Compound Interest
Try yourself:Mr. Thomas invested an amount of Rs. 13,900 divided in two different schemes A and B at the simple interest rate of 14% p.a. and 11% p.a. respectively. If the total amount of simple interest earned in 2 years be Rs. 3508, what was the amount invested in Scheme B?

### Compound Interest

• 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 1-time 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 1-year interest for the 2nd 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)= ₹119.1016.

### Important Formulas

• When the interest is compounded Annually:
Amount = P (1 + R/100) n
• When the interest is compounded Half-yearly:
Amount = P (1 + (R/2)/100)2n
• When the interest is compounded Quarterly:
Amount = P (1 + (R/4)/100)4n
• When the rates are different for different years, say R1%, R2% and R3% for 1 year, 2 years and 3-year resp. Then,
Amount = P (1 + R1/100) (1 + R2/100) (1 + R3/100)
• Present worth of ₹ x due n years hence is given by:
Present worth = x/ (1 + R/100)n
• If a certain sum becomes “x” times in n years, then the rate of compound interest will be R = 100(x1/n – 1)
• If a sum of money P amounts to A1 after T years at CI and the same sum of money amounts to A2 after (T + 1) years at CI, then
R = (A2 – A1)/ A1 x 100

## Solved Examples on Compound Interest

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 the third year.

• Here, P = ₹5000, T = 3 years, r = 5%. Therefore, Interest at the end of 1st 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 1st year = ₹5000 + 200 = ₹5200
• Interest at the end of 2nd year = 5200 (1 + 0.05) – 5200 = ₹260
Interest income after Income tax = 0.8 * ₹260 = ₹208
Total Amount at the end of 2nd year = ₹5200 + 208 = ₹5408
• Interest at the end of 3rd year = ₹5408 (1.05) – 5408 = ₹270.4
Interest income after Income tax = 0.8 * ₹270.4 = ₹216.32
Total Amount at the end of 2rd 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?

• Principal, P = Rs. 12000;
Rate of interest = r%;
Number of years, n = 5;
Amount, A = Rs (2 × 12000) = Rs. 24000
According to the question,
24000 = 12000 × (1 + r/100)5
⇒ (1 + r/100)5 = 2     ...(i)
• For next part,
Principal, P = Rs. 12000;
Rate of interest = r%;
Number of years, n = 20;
Amount = Rs. x
According to the question,
x = 12000 × (1 + r/100)20
⇒ x = 12000 × (1 + r/100)5 × 4
⇒ x = 12000 × [(1 + r/100)5]4
⇒ x = 12000 × 24     ...[from (i)]
⇒ x = 192000
∴ After 20 years it will become Rs. 1,92,000

Shortcut Trick
12000 becomes twice in 5 years = 12000 × 2 = 24000
After another 5 years = 24000 × 2 = 48000
After another 5 years = 48000 × 2 = 96000
After another 5 years = 48000 × = 1,92,000
∴ After 20 years sum will be = 192000

### Compound Interest Installments

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)

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 half-yearly. If the flat is purchased under an installment plan, find the value of each installment.

• 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 Half-yearly 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 document Basic Concepts: Simple Interest & Compound Interest | CSAT Preparation - UPSC is a part of the UPSC Course CSAT Preparation.
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## FAQs on Basic Concepts: Simple Interest & Compound Interest - CSAT Preparation - UPSC

 1. What are interest rates?
Ans. Interest rates refer to the cost of borrowing money or the return on investment earned on an investment. It is expressed as a percentage and represents the price paid for the use of money over a certain period of time.
 2. What are the types of interest rates?
Ans. There are various types of interest rates, including: - Prime Rate: The interest rate set by banks and used as a benchmark for other interest rates. - Fixed Interest Rate: An interest rate that remains the same throughout the entire loan or investment period. - Variable Interest Rate: An interest rate that can change over time, usually based on a benchmark rate. - Nominal Interest Rate: The stated interest rate before taking into account compounding or inflation. - Real Interest Rate: The interest rate adjusted for inflation, providing a more accurate measure of the cost or return on investment.
 3. What is compound interest?
Ans. Compound interest is the interest calculated on both the initial principal amount and the accumulated interest from previous periods. It means that the interest earned or charged in one period is added to the principal, and interest is then calculated on the new total amount. This compounding effect leads to exponential growth or accumulation of interest over time.
 4. How are compound interest installments calculated?
Ans. Compound interest installments are calculated using the formula: Installment = (Principal * (1 + Interest Rate) ^ Number of Periods) / Number of Periods Here, the principal refers to the initial amount, the interest rate is the rate of interest per period, and the number of periods represents the total number of compounding periods.
 5. What are the basic concepts of simple interest and compound interest?
Ans. The basic concepts of simple interest and compound interest are as follows: - Simple Interest: It is calculated only on the initial principal amount and remains the same throughout the investment or loan period. The formula for simple interest is: Interest = (Principal * Interest Rate * Time) - Compound Interest: It takes into account both the initial principal amount and the accumulated interest from previous periods. The interest earned or charged in one period is added to the principal, resulting in exponential growth. The compound interest is calculated using the formula: Compound Interest = Principal * (1 + Interest Rate)^Number of Periods - Principal.

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