Question

Romesh borrowed a sum of $Rs.245760$ at $12.5\%$ per annum compounded annually. On the same day, he lent out his money to Ramu at the same rate of interest but compounded semi-annually. Find his gain after $2$ years.

Answer 1

In first case,
Principal $\left(P\right)=Rs.245760$
Rate $\left(R\right)=12.5\%=\frac{25}{2}\%$ p.a
Period $\left(n\right)=2$ years
We know that,
$\begin{array}{cc}\text{Amount}& =P{(1+\frac{R}{100})}^n\\ & =Rs.245760{(1+\frac{25}{2\times 100})}^2\\ & =Rs.245760{(1+\frac{1}{8})}^2\\ & =Rs.245760{\left(\frac{9}{8}\right)}^2\\ & =Rs.245760\times \frac{9}{8}\times \frac{9}{8}\\ & =Rs.311040\end{array}$
$\begin{array}{cc}\therefore C.I& =A-P\\ & =Rs.311040-Rs.245760\\ & =Rs.65280\end{array}$
In second case,
Rate $\left(R\right)=\frac{25}{4}\%$ half-yearly
Period $\left(t\right)=2$ years
Number of times interest applied $\left(n\right)=2$

We know that,
$\begin{array}{cc}\therefore \text{Amount}& =P{(1+\frac{R}{100n})}^{nt}\\ & =Rs.245760{(1+\frac{25}{2\times 4\times 100})}^{2\times 2}\\ & =Rs.245760{(1+\frac{1}{16})}^4\\ & =Rs.245760{\left(\frac{17}{16}\right)}^4\\ & =Rs.245760\times \frac{17}{16}\times \frac{17}{16}\times \frac{17}{16}\times \frac{17}{16}\\ & =Rs.313203.75\end{array}$
$\therefore C.I=A-P$
$=Rs.313203.75-Rs.245760=Rs.67443.75$
$\therefore \text{Gain}=67443.75-Rs.65280$
$=Rs.2163.75$
Hence, Romesh gain $Rs.2163.75$ after $2$ years.