Compound Interest Formula: Compound interest has great significance in the study of mathematics which plays a vital role not only in the school syllabus but also in competitive examinations like SSC, Railway, Banking, etc. In this article, we discuss the basics of compound interest, the compound interest formula, derivation of the compound interest formula, interest compounded for different years, some tricks and tips of compound interest, and compound interest solved questions.

## Basics of Compound Interest

Compound interest is defined as the interest incurred on a loan or deposit amount of money. It is a familiar concept applied in day-to-day life. The compound interest for the deposit sum is dependent on both the Principal amount and Interest obtained over a period of time. It is the key point that makes the difference between compound interest and simple interest.

You generally notice that some amount of interest is credited to your bank account on yearly basis. This interest amount changes with time for having the same principal amount. You can notice that the interest amount augments for successive years. So you can get the conclusion that the interest amount charged by the bank is not a simple interest rather it is compound interest (CI). Here you can also learn why the return on compound interest is greater than that on simple interest.

## Compound Interest Formula

Compound interest is a type of interest that is calculated on the principal amount and the interest amount incurred over the previous time interval. In the study of Mathematics as a subject, compound interest is generally represented by its acronym C.I.

In most of the financial transactions in the banking and finance sectors, compound interest plays an important role in this field. Some of the other applications of Compound Interest can be summarized by the following points:

1. Population data varies whether increasing or decreasing.
2. The Bacterial Growth.
3. Inflation and Depreciation in the value of commodities.

## Compound Interest Formula Tricks & Tips

For finding out the compound interest, you require some quantities like the Principal amount, Rate of interest, and Time period. Some simple tricks and tips are explained below.

1. If principal amount P is given on the compound interest at the rate of interest R% per annum then amount A delivered after t years of time is expressed as

2. When the interest amount is compounded on a half-yearly basis then

3. When the interest amount is compounded on a quarterly basis then

4. If the rates of interest are different for different time periods like R₁, R₂, R₃ percent for the first, second, and third year respectively then

5. If the time period is mentioned in the fractional form like 2³/⁴ years then

6. (a) The difference between the simple interest and compound interest on a fixed sum of money for two years at the rate of interest R% per annum is expressed as

(b) The difference between the compound interest and the simple interest on a fixed sum of money for three years at the rate of interest R% per annum is expressed as

7. When a fixed sum of money becomes n times in t time period at the compound interest then the same sum of money becomes nm times in mt time period.

8. When a fixed sum of money becomes n times in t time period then the rate of compound interest is expressed as

9. When a fixed sum at the compound interest amounts to Rs. x in ‘A’ time and to Rs. y in ‘B’ time then the rate of compound interest per annum is expressed as

10. When a loan of Rs. P at the rate of compound interest of R% per annum is to be re-paid in ‘n’ number of same annual instalments then the value of each instalment is expressed as

## Compound Interest Solved Questions

Question 1: A village had 50,000 residents in 2010. Its population decreases at the rate of 10% per year. What will be the total population of that village in the year 2015?

Solution: The population of a given village is declined by the rate of 10% every annum. Therefore it has new population statistics each year. Hence the population of that village for the next year is enumerated on the basis of the present year's population data.

For the decreasing population,  A = P(1 – R/100)n

Hence the population at the end of 5 years from 2010 to 2015 = 50000(1 – 10/100)^5

= 50000 (1 – 0.1)^5 = 50000 x 0.95 = 29524.5

Question 2: The count of a certain species of birds was estimated to rise at the rate of 3% per day. Calculate the bird species after 2 days when the count was originally 300000.

Solution: As per the question, the population of bird species increases at the rate of 3% per day, then A = P(1 + R/100)n

Hence the population of bird species after 2 days = 300000 (1 + 3/100)2

= 300000 (1 + 0.03)² = 300000 (1.03)² = 318270

Question 3: The price of a TV is Rs. 2500 and it’s value is depreciated by 9% per month. Find out the value of that TV after 5 months.

Solution: For the depreciating TV, the value of the TV will be expressed as A = P(1 – R/100)n

Hence the price of the TV after 5 months = 2500 (1 – 9/100)5

= 2500 (1 – 0.09)5 = 2500(0.91)^5 = Rs. 1560 (Approx.)

Question 4: What compound interest amount is to be charged on a loan of Rs. 5000 for 2 (3/2) years at the rate of 20% per year and if compounded half-yearly basis?

Solution: Given that interest is compounded on half-yearly basis,

Principal Amount P = Rs. 5000

Time Period T’ = 2 (3/2) years = 3 years

Rate of Interest R’ = 20% / 2 = 10%

Amount A is expressed as

A= P(1+R′/100)T’

A= 5000 × (1 + 10/100)³

= 5000 × (1.331) = Rs. 6655

CI = A – P =  Rs. 6655 – Rs. 5000 = Rs. 1655

Question 5: What the compound interest will be on Rs. 3000 for 4 years at the rate of 15% per year, if compounded on annual basis?

Solution: Principal Amount P = Rs. 3000 , Time Period T= 4 year, Rate of Interest R = 15 %

Amount A is expressed as,

A= P(1+ R/100)^T

A= 3000 (1+ 15/100)^4

= 3000 (1.15)^4 = Rs. 5247 (Approx.)

Interest Amount (2nd Year) = A – P = 5247 – 3000 = Rs. 2247

Question 6:  A sum of Rs. 20000 is borrowed by Sohit for 2 years at the interest rate of 12% compounded yearly. Calculate the compound interest and amount that Sohit needs to pay after 2 years.

Solution: Given that,

Principal Amount (Sum) = Rs. 20000,  Rate of  Interest = 12%, and Time Period = 2 years

By scrolling up your screen to the table section of this article, you can easily find out the amount and interest for the second year, which is expressed as

Amount (A2) = P (1 + R/100)²

Substituting the given values in the aforesaid formula,

A2 = 20000 (1 + 12/100)²

= 20000 (1.12)² = Rs.25088

Compound Interest (for the second year) will be = A2 – P =  25088 – 20000 = Rs. 5088

Question 7: The difference between Simple Interest and Compound Interest on the sum of Rs. 50000 for 2 years is Rs. 267. What is the rate of Interest here?

Solution: CI - SI= P (R/100)²

As per the question, 267 = 50000 (R/100)²

We get, (267/50000) = (R/100)²

R = 7.30 % (Approx.)

Compound Interest Formula: FAQs

Ans. The Compound Interest (CI) is enumerated by multiplying the original principal amount P by one plus the annual rate of interest R raised to the compound time periods (nt) minus one. CI = P [ (1 + R)nt-1 ]

Ans. The formula for the compounded interest quarterly is A = P (1 + (R/4)/100)4T

Ans. Compound interest means the interest on a loan or deposit money that accrues on both the original principal amount and the earned interest amount from the previous time periods.

Ans. The interest charged by the financial sector and banking sector is the compound interest.

Ans. The investors are generally beneficiaries of the compound interest as the amount earned by them is expressed as Amount = Principal + Interest.