Amit had invested same amount of sums at simple interest as well as compound interest, compounded annually. The time period of both the sums was 2 years and rate of interest too was the same 10% per annum. At the end, he found a difference of ₹45 in both the interests received. What were the sums (in ₹) invested in each case?

The correct option is 4,500.

Let us understand how to solve this step-by-step.

Let the sum of money invested in each case be $P$ rupees.
The rate of interest ($R$) is 10% per annum, and the time period ($T$ or $n$) is 2 years.

For Simple Interest (SI) for 2 years:
$\mathrm{SI}=\frac{P\times R\times T}{100}$
Substituting the given values:
$\mathrm{SI}=\frac{P\times 10\times 2}{100}=\frac{20P}{100}=0.2P$

For Compound Interest (CI) compounded annually for 2 years:
$\mathrm{CI}=P{(1+\frac{R}{100})}^n-P$
Substituting the given values:
$\mathrm{CI}=P{(1+\frac{10}{100})}^2-P$
$\mathrm{CI}=P{\left(1.1\right)}^2-P$
$\mathrm{CI}=1.21P-P=0.21P$

According to the problem, the difference between the compound interest and simple interest is ₹45:
$\mathrm{CI}-\mathrm{SI}=45$
$0.21P-0.2P=45$
$0.01P=45$
$P=\frac{45}{0.01}$
$P=4500$

Thus, the sum invested in each case was ₹4,500.