A sum of money becomes `6.400 in 2 years and `8,100 in 4 years on compound interest. Find the rate of compound interest.

​Given:

Amount after 2 years = ₹6,400

Amount after 4 years = ₹8,100

Time difference between the two amounts = 4 years - 2 years = 2 years

Compound Interest formula:

​$$A = P \times \left(1 + \frac{r}{100}\right)^t$$​​

A is the amount, P is the principal, r is the rate of interest, and t is the time in years.

Let the amount after 2 years (₹6,400) be the principal for the next 2 years to obtain ₹8,100.

Thus, for the 2-year period from year 2 to year 4:

​$$\begin{aligned}&8100 = 6400 \times \left(1 + \frac{r}{100}\right)^2 \\&\frac{8100}{6400} = \left(1 + \frac{r}{100}\right)^2 \\&\frac{81}{64} = \left(1 + \frac{r}{100}\right)^2 \\&\sqrt{\frac{81}{64}} = 1 + \frac{r}{100}\\& \frac{9}{8} = 1 + \frac{r}{100}​ \\&\frac{9}{8} - 1 = \frac{r}{100} \\&\frac{1}{8} = \frac{r}{100} \\&r = \frac{1}{8} \times 100 = 12.5\% \\\end{aligned}$$​​

Thus, the rate of compound interest is 12.5% per annum.