The simple interest on a certain principal amount for 4 years at 10% per annum is half of the compound interest on ₹ 1000 for 2 years at 20% per annum. Find the principal amount

The correct answer is ₹550.

We solve this in two stages: first finding the Compound Interest (CI), then using it to find the Simple Interest (SI), and finally working backwards to find the unknown principal.

Stage 1: Calculate the Compound Interest on ₹1000 for 2 years at 20% per annum

The formula for Compound Interest is:

$$\mathrm{CI}=P{(1+\frac{R}{100})}^n-P$$

Substituting the values P = 1000, R = 20, n = 2:

$$\mathrm{CI}=1000{(1+\frac{20}{100})}^2-1000$$

$$\mathrm{CI}=1000{\left(\frac{6}{5}\right)}^2-1000$$

$$\mathrm{CI}=1000\times \frac{36}{25}-1000$$

$$\mathrm{CI}=1440-1000=\mathrm{₹440}$$

Stage 2: Find the Simple Interest (SI) on the unknown principal

According to the problem, the Simple Interest is half of the Compound Interest calculated above:

$$\mathrm{SI}=\frac{1}{2}\times 440=\mathrm{₹220}$$

Stage 3: Find the unknown principal using the Simple Interest formula

The formula for Simple Interest is:

$$\mathrm{SI}=\frac{P\times R\times T}{100}$$

We know SI = 220, R = 10% per annum, and T = 4 years. Substituting:

$$220=\frac{P\times 10\times 4}{100}$$

$$220=\frac{40P}{100}$$

$$220=\frac{2P}{5}$$

Solving for P by cross-multiplying:

$$P=\frac{220\times 5}{2}=\frac{1100}{2}=\mathrm{₹550}$$

Conclusion: The required principal amount is ₹550. We verified this by first computing the CI of ₹440 on ₹1000 at 20% for 2 years, halving it to get SI = ₹220, and then back-calculating the principal using the SI formula with R = 10% and T = 4 years.