The difference between compound interest, compounded annually, and simple interest for 3 years at a rate of 20% per annum is ₹256. What is the principal lent?

The correct option is ₹2,000.

Let us understand the step-by-step derivation to find the principal amount lent.

Let the principal amount be denoted by P, the annual interest rate by r, and the time period by n years.

Here, we are given:
Rate of interest, r = 20% per annum = 0.20
Time period, n = 3 years
Difference between Compound Interest (CI) and Simple Interest (SI) = ₹256

The formula for Simple Interest (SI) for 3 years is given by:
$SI=\frac{P\times r\times n}{100}$

Substituting the values of rate and time:
$SI=\frac{P\times 20\times 3}{100}=\frac{60P}{100}=0.6P$

The formula for Compound Interest (CI) compounded annually for 3 years is:
$CI=P{(1+\frac{r}{100})}^3-P$

Substituting the rate value into the compound interest formula:
$CI=P{(1+\frac{20}{100})}^3-P$
$CI=P{\left(1.2\right)}^3-P$
$CI=1.728P-P=0.728P$

We are given that the difference between compound interest and simple interest is ₹256:
$CI-SI=256$

Substitute the expressions we obtained for CI and SI:
$0.728P-0.6P=256$
$0.128P=256$

Solving for P:
$P=\frac{256}{0.128}$
$P=\frac{256000}{128}$
$P=2000$

Thus, the principal lent is ₹2,000.