Derivation of Compound Interest Formula

# Derivation of Compound Interest Formula

## Compound Interest

When we deposit some money in a bank, every year some interest is added to it. The interest is not the same for every year and it goes on increasing year by year. This is because at the end of the first year, simple interest is calculated and added to the principal to get the amount. This amount becomes the principal for the next year.

At the end of the second year again, the amount is calculated by adding principal and interest. This amount becomes the principal for the third year and so on. Every year, the principal changes and the interest is calculated on the amount of the previous year. When interest is calculated in this manner, we call it compound interest.

## Derivation of Compound Interest Formula

When the time period is longer, calculating simple interest every time and adding it to the principal amount to get the principal for next year is very complex calculation and time consuming.

Using step by step procedure, we will arrive at a formula to find the amount and the compound interest.

Let us take the principal as ₹ P, rate = R% p.a. and time = T years

## What is P, R and T in Compound Interest Formula?

We know that the formula to calculate compound interest is as follows:

In this formula, P is the initial principal for which the compound interest is to be calculated.

R is the rate of interest. It is given in percentage. Suppose the rate is 5% per annum. It means that on a deposit of ₹ 100 or on a loan of ₹ 100, you will receive ₹ 5 or pay ₹ 5 as an interest in one year.

## What is Conversion Period?

Sometimes, interest is calculated and added to the principal after every 6 months or 3 months. This is called interest is compounded half-yearly or quarterly.

The time period T after which the interest is added to the principal each time to form a new principal is called the conversion period.

This period may be 1 year, 6 months or 3 months. According to that, calculations are done by reducing the rate of interest half-yearly or quarterly.

## Time Period and Rate When the Interest is Compounded Half-yearly and Quarterly

If the interest is compounded half-yearly, there are two conversion periods in a year and the rate of interest is half the annual rate.

Let us suppose, a sum of ₹ 10,000 is borrowed for 1 year 6 months at 8% p.a. compounded half-yearly.

In this case, P = ₹ 10,000

In 1 year and 6 months, there are 3 half years, hence we have 3 conversion periods; or T = 3

Rate (R) = half of the annual rate = ½ × 8% = 4%

Therefore, A = P(1 + R/100)T

= 10,000(1 + 4/100)3

= 10,000 (1 + 0.04)3

= 10,000 (1.04)3

= 10,000 × 1.04 × 1.04 × 1.04

= ₹ 11,248.64

Thus, CI = ₹ 11,248.64 – ₹ 10,000

= ₹ 1,248.64

If the interest is compounded quarterly, there are 4 conversion periods in a year. The interest rate is one-fourth of the annual rate.

For example, if a sum of ₹ 10,000 is borrowed for 1 year at 8% p.a. compounded quarterly, the number of conversion periods is 4 quarters and the rate = ¼ × 8% = 2% quarterly

Therefore, A = P (1 + R/100)T

= 10,000 (1 + 2/100)4

= 10,000 (1 + 0.02)4

= 10,000 (1.02)4

= 10,000 × 1.02 × 1.02 × 1.02 × 1.02

= ₹ 10,824.32

Thus, CI = ₹ 10,824.32 – ₹ 10,000

= ₹ 824.32

Example 1: Find the amount and compound interest on ₹ 2400 for 2 years at 10% p.a. compounded half-yearly.

Solution: Here, P = ₹ 2400, R = 10% p.a. = 10/2 = 5% half-yearly, T = 2 years = 4 half-years

Amount (A) = P (1 + R/100)T

= 2400 (1 + 5/100)4

= 2400 (1 + 0.05)4

= 2400 (1.05)4

= ₹ 2917.21

Compound interest = A – P = ₹ 2917.21 – ₹ 2400 = ₹ 517.21

Hence, amount = ₹ 2917.21 and compound interest = ₹ 517.21

Example 2: Find the amount and compound interest on ₹ 12,000 for 9 months at 16% p.a., interest being compounded quarterly.

Solution: Here, P = ₹ 12,000, R = 16% p.a. = 16/4 = 4% quarterly, T = 9 months = 3 quarters

Amount (A) = P (1 + R/100)T

= 12,000 (1 + 4/100)3

= 12,000 (1 + 0.04)3

= 12,000 (1.04)3

= 13,498.37

Compound interest = A – P = ₹ 13,498.37 – ₹ 12,000 = ₹ 1498.37

Hence, amount = ₹ 13,498.37 and compound interest = ₹ 1498.37

Related Post

Difference between Simple Interest and Compound Interest

Simple Interest Questions for Class 6

Simple Interest Worksheet for Class 7

Compound Interest Class 8 Worksheets