Question

Neeraj lend $\text{Rs.}65536$ for $2$ years at $12\frac{1}{2}\%$ per annum compound annually. How much more could he earn if interest were compounded half-yearly?

Answer 1

To calculate the amount, if interest were compounded annually,

$\begin{array}{cc}A& =P{\left(1+\frac{R}{100}\right)}^n\\ & =65536{\left(1+\frac{25}{100\times 2}\right)}^2\\ & =65536{\left(\frac{9}{8}\right)}^2\\ & =1024\times 81\\ & =\text{Rs.}82944\end{array}$

To calculate amount if interest were compounded half-yearly, rate of interest will become $\frac{25}{4}\%$ and value of $n$ will be $4,$

$\begin{array}{cc}A& =P{\left(1+\frac{R}{100}\right)}^n\\ & =65536{\left(1+\frac{25}{100\times 4}\right)}^2\\ & =65536{\left(\frac{17}{16}\right)}^4\\ & =17\times 17\times 17\times 17\\ & =\text{Rs.}83521\end{array}$

Clearly, we can observe that amount compounded half-yearly is more than compounded annually. So, the difference between the two amounts will be the extra amount that Neeraj can earn,

$\begin{array}{cc}\text{Extra Interest}& =83521-82944\\ & =\text{Rs.}577\end{array}$