On a certain sum, the interest is compounded annually. If the compound interest for the second year is ₹400 and the compound interest for the fourth year is 576, then what is the rate of interest per annum?

Solution

Let's assume the principal and rate are 'P' and 'R'.

On a certain sum, the interest is compounded annually. If the compound interest for the second year is ₹400

$$P\left[\left(1+\frac{R}{100}\right)^2\ -\ 1\right]-P\left[\left(1+\frac{R}{100}\right)\ -\ 1\right]\ =\ 400$$    Eq.(i)

The compound interest for the fourth year is 576.

$$P\left[\left(1+\frac{R}{100}\right)^4\ -\ 1\right]-P\left[\left(1+\frac{R}{100}\right)^3\ -\ 1\right]\ =\ 576$$    Eq.(ii)
Eq.(ii) divided by Eq.(i).
$$\frac{P\left[\left(1+\frac{R}{100}\right)^4\ -\ 1\right]-P\left[\left(1+\frac{R}{100}\right)^3\ -\ 1\right]}{P\left[\left(1+\frac{R}{100}\right)^2\ -\ 1\right]-P\left[\left(1+\frac{R}{100}\right)\ -\ 1\right]} = \frac{576}{400}$$

$$\left(1+\frac{R}{100}\right)^2=\frac{36}{25}$$

$$\left(1+\frac{R}{100}\right)^2=\left(\frac{6}{5}\right)^2$$

$$1+\frac{R}{100}=\frac{6}{5}$$
$$\frac{R}{100}=\frac{6}{5}-1$$
$$\frac{R}{100}=\frac{1}{5}$$
Rate of interest per annum = 20%