C.I. for 2 years $@ 10 \%$ p.a. interest

$\begin{array}{l}=P\left[\left(1+\frac{10}{100}\right)^{2}-1\right] \\\text { S.I. }=\frac{P \times 2 \times 10}{100}=\frac{P}{5}\end{array}$

$(1)-(2) \Rightarrow P\left[\left(\frac{11}{10}\right)^{2}-1-\frac{1}{5}\right]=20$

Or $P\left(\frac{(11)^{2}-(10)^{2}-20}{100}\right)=20$

or $P=\frac{20 \times 100}{1}=₹ 2000$

$\therefore \quad$ If interest is compounded half yearly,

C.I. $-$ S.I. $=P\left[\left(1+\frac{5}{100}\right)^{4}-1\right]-\frac{P}{5}$

$=P\left[\left(1+\frac{5}{100}\right)^{4}-1-\frac{1}{5}\right]$

$=2000\left[\left(\frac{21}{20}\right)^{4}-1-\frac{1}{5}\right]=2000\left[\frac{(21)^{4}-(20)^{4}}{(20)^{4}}-\frac{1}{5}\right]$

$=2000\left[\frac{\left(21^{2}+20^{2}\right)\left(21^{2}-20^{2}\right)}{20^{4}}-\frac{1}{5}\right]=2000\left[\frac{841 \times 41 \times 1}{20^{4}}-\frac{1}{5}\right]$

$=2000\left[\frac{34481}{1,60,000}-\frac{1}{5}\right]=\frac{2000}{1,60,000}[34481-32000]$

$=\frac{1}{80} \times 2481=31.0125$

$\therefore \quad$ Difference of interest $=₹ 31.01$