Amit had invested same amount of sums at simple as well as compound interest, compounded annually. The time period of both the sums was 2 years and rate of interest too was same 10% per annum. At the end, he found a difference of ₹97 in both the interests received. What were the sums (in ₹) invested?

The correct option is 9,700.

Let the sum invested in each case be represented as P (Principal).
The rate of interest is R = 10% per annum.
The time period is T = 2 years.

The formula for Simple Interest (SI) for 2 years is:
$$SI=\frac{P\times R\times T}{100}$$

Substituting the given values:
$$SI=\frac{P\times 10\times 2}{100}=0.20P$$

The formula for Compound Interest (CI) compounded annually for 2 years is:
$$CI=P{(1+\frac{R}{100})}^2-P$$

Substituting the given values:
$$CI=P{(1+\frac{10}{100})}^2-P$$
$$CI=P{\left(1.1\right)}^2-P=1.21P-P=0.21P$$

The difference between Compound Interest and Simple Interest is given as ₹97:
$$CI-SI=97$$
$$0.21P-0.20P=97$$
$$0.01P=97$$
$$P=\frac{97}{0.01}=9700$$

Alternatively, we can use the direct formula for the difference between CI and SI for 2 years:
$$\mathrm{Difference}=P{\left(\frac{R}{100}\right)}^2$$
$$97=P{\left(\frac{10}{100}\right)}^2$$
$$97=P{\left(0.1\right)}^2$$
$$97=0.01P$$
$$P=9700$$

Thus, the sum invested was ₹9,700.