At what rate of interest per annum will a sum of ₹ 62500 earn a compound interest of ₹ 5100 in one year ? The interest is to be compounded half-yearly ?

Let rate of interest be r% per annum.

When interest is compounded half-yearly.

$A = P\Big(1 + \dfrac{r}{2 \times 100}\Big)^{n \times 2}$

Substituting values we get :

$\Rightarrow C.I. = A - P \\[1em] \Rightarrow C.I. = P\Big(1 + \dfrac{r}{2 \times 100}\Big)^{n \times 2} - P \\[1em] \Rightarrow 5100 = 62500 \times \Big(1 + \dfrac{r}{200}\Big)^{1 \times 2} - 62500 \\[1em] \Rightarrow 5100 + 62500 = 62500 \times \Big(1 + \dfrac{r}{200}\Big)^2 \\[1em] \Rightarrow 67600 = 62500 \times \Big(1 + \dfrac{r}{200}\Big)^2 \\[1em] \Rightarrow \dfrac{67600}{62500} = \Big(1 + \dfrac{r}{200}\Big)^2 \\[1em] \Rightarrow \dfrac{676}{625} = \Big(1 + \dfrac{r}{200}\Big)^2 \\[1em] \Rightarrow \Big(\dfrac{26}{25}\Big)^2 = \Big(1 + \dfrac{r}{200}\Big)^2 \\[1em] \Rightarrow 1 + \dfrac{r}{200} = \dfrac{26}{25} \\[1em] \Rightarrow \dfrac{r}{200} = \dfrac{26}{25}- 1 \\[1em] \Rightarrow \dfrac{r}{200} = \dfrac{1}{25} \\[1em] \Rightarrow r = \dfrac{200}{25} = 8\%.$

Hence, rate of interest = 8%.