The difference between compound interest and simple interest is Rs. 620 for 3 years at 10% per year, then what would be principal amount ?

Question

Accepted Answer

To determine why Option 4 (Rs. 20,000) is the correct principal amount, we need to understand the relationship between compound interest (CI) and simple interest (SI) over a period of time.

### Explanation:

1. **Formula for Simple Interest (SI):**
   $$
   \text{SI} = \frac{P \times R \times T}{100}
   $$
   where $ P $ is the principal, $ R $ is the rate of interest per annum, and $ T $ is the time in years.

2. **Formula for Compound Interest (CI):**
   $$
   \text{CI} = P \left(1 + \frac{R}{100}\right)^T - P
   $$

3. **Difference between CI and SI for 3 years:**
   The difference between compound interest and simple interest for 3 years can be calculated using the formula:
   $$
   \text{Difference} = \frac{P \times R^2}{100^2} \times \left(\frac{3R + 100}{100}\right)
   $$

Given that the difference is Rs. 620, $ R = 10\% $, and $ T = 3 $ years, we substitute these values into the formula:
   $$
   620 = \frac{P \times 10^2}{100^2} \times \left(\frac{3 \times 10 + 100}{100}\right)
   $$

Simplifying:
   $$
   620 = \frac{P \times 100}{10000} \times \left(\frac{30 + 100}{100}\right)
   $$
   $$
   620 = \frac{P}{100} \times \frac{130}{100}
   $$
   $$
   620 = \frac{130P}{10000}
   $$
   $$
   620 \times 10000 = 130P
   $$
   $$
   6200000 = 130P
   $$
   $$
   P = \frac{6200000}{130}
   $$
   $$
   P = 20000
   $$

Therefore, the principal amount $ P $ is Rs. 20,000, which corresponds to Option 4.

### Why Other Options are Incorrect:

- **Option 1 (Rs. 40,000):** If we substitute Rs. 40,000 into the formula, the calculated difference between CI and SI would be much larger than Rs. 620, indicating that this is not the correct principal.

- **Option 2 (Rs. 12,000):** Substituting Rs. 12,000 into the formula would result in a difference that is significantly less than Rs. 620, thus making this option incorrect.

- **Option 3 (Rs. 10,000):** Using Rs. 10,000 as the principal would also yield a difference less than Rs. 620, which does not satisfy the given condition.

In conclusion, the calculations clearly show that the principal amount must be Rs. 20,000 to result in a difference of Rs. 620 between compound interest and simple interest over 3 years at a 10% interest rate. Therefore, Option 4 is the correct answer.