**Question:**

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

**Solution:**

Let the principal amount be $P=$ Rs. 65536 .

Annual rate of interest, $R=\frac{25}{2} \%$

Rate of interest for a half year $=\frac{25}{4} \%$

Time, $n=2$ years $=4$ half year

Then the amount with the compound interest is given by $A=P \times\left(1+\frac{R}{100}\right)^{n}$

$=$ Rs. $65536 \times\left(1+\frac{25}{100 \times 4}\right)^{4}$

$=$ Rs. $65536 \times\left(\frac{400+25}{400}\right)^{4}$

$=$ Rs. $65536 \times\left(\frac{425}{400}\right)^{4}$

$=$ Rs. $65536 \times\left(\frac{17}{16}\right)^{4}$

$=$ Rs. $65536 \times\left(\frac{17}{16}\right) \times\left(\frac{17}{16}\right) \times\left(\frac{17}{16}\right) \times\left(\frac{17}{16}\right)$

$=$ Rs. $(17 \times 17 \times 17 \times 17)$

$=$ Rs. 83521

Now. CI $=A-P$$=\operatorname{Rs} \cdot(83521-65536)=$ Rs. 17985

Therefore, interest earned when compounded half yearly $=$ Rs. 17985

Amount when the interest is compounded yearly is given by $A=P \times\left(1+\frac{R}{100}\right)^{n}$

$=$ Rs. $65536 \times\left(1+\frac{25}{100 \times 2}\right)^{2}$

$=$ Rs. $65536 \times\left(\frac{200+25}{200}\right)^{2}$

$=$ Rs. $65536 \times\left(\frac{225}{200}\right)^{2}$

$=$ Rs. $65536 \times\left(\frac{9}{8}\right)^{2}$

$=$ Rs. $65536 \times\left(\frac{9}{8}\right) \times\left(\frac{9}{8}\right)$

$=$ Rs. 82944

Therefore, $\mathrm{CI}=A-P=$ Rs. $(82944-65536)=$ Rs. 17408

$\therefore$ Difference between the interests compounded half yearly and yearly $=$ Rs. $(17985-17408)=$ Rs. 577