A sum compounded annually becomes $\dfrac{25}{16}$ times of itself in two years. Determine the rate of interest per annum.

Let rate of interest = r.

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

Let Principal = P.

Given, sum becomes $\dfrac{25}{16}$ times of itself in two years,

∴ A = $\dfrac{25}{16}$P.

Putting values in formula we get,

$\Rightarrow \dfrac{25}{16}P = P\Big(1 + \dfrac{r}{100}\Big)^2 \\[1em] \Rightarrow \dfrac{25}{16} = \Big(1 + \dfrac{r}{100}\Big)^2\\[1em] \Rightarrow \Big(\dfrac{5}{4}\Big)^2 = \Big(1 + \dfrac{r}{100}\Big)^2\\[1em]$

Taking square root on both sides,

$\Rightarrow \dfrac{5}{4} = 1 + \dfrac{r}{100} \\[1em] \Rightarrow \dfrac{5}{4} - 1 = \dfrac{r}{100} \\[1em] \Rightarrow \dfrac{5 - 4}{4} = \dfrac{r}{100} \\[1em] \Rightarrow \dfrac{1}{4} = \dfrac{r}{100} \\[1em] \Rightarrow r = \dfrac{100}{4} \\[1em] \Rightarrow r = 25\%.$

Hence, rate of interest = 25%.