Kamala borrowed from Ratan a certain sum at a certain rate for two years simple interest. She lent this sum at the same rate to Hari for two years compound interest. At the end of two years she received Rs 210 as compound interest, but paid Rs 200 only as simple interest. Find the sum and the rate of interest.
Let the sum be Rs P and the rate of interest be R\%.
We know that Kamla paid Rs 200 as simple interest.
$\therefore 200=\frac{\operatorname{PR}(2)}{100}$
$\mathrm{PR}=10,000 \quad \ldots(1)$
Also, Kamla received Rs 210 as compound interest.
$\therefore 210=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{2}-1$
$210(10,000)=\mathrm{P}\left(\mathrm{R}^{2}+200 \mathrm{R}\right)$
$210 \mathrm{R}=\mathrm{R}^{2}+200 \mathrm{R} \quad[$ from $(1)]$
$\mathrm{R}=10 \%$ p. $\mathrm{a}$
Putting the equation in $(1)$, we get:
$\mathrm{P}=1,000$
Thus, the required sum is Rs 1,000 and the rate of interest is $10 \%$