If the compound interest on a certain sum for two years at 3% p.a. be ₹ 101.50, then what would be the simple interest on the same sum for same period ?

Question

Accepted Answer

To determine why Option 2 (₹100) is the correct answer for the simple interest on the same sum for the same period, let's go through the calculations and reasoning step by step.

### Explanation:

1. **Understanding Compound Interest:**
   - The compound interest (CI) for two years at a rate of 3% per annum is given as ₹101.50.
   - The formula for compound interest for two years is:
     $$
     CI = P \left(1 + \frac{r}{100}\right)^n - P
     $$
     where $ P $ is the principal, $ r $ is the rate of interest, and $ n $ is the number of years.

2. **Calculating the Principal:**
   - We know:
     $$
     CI = P \left(1 + \frac{3}{100}\right)^2 - P = ₹101.50
     $$
   - Simplifying:
     $$
     P \left(\left(1 + \frac{3}{100}\right)^2 - 1\right) = ₹101.50
     $$
   - Calculating $\left(1 + \frac{3}{100}\right)^2$:
     $$
     \left(1.03\right)^2 = 1.0609
     $$
   - Therefore:
     $$
     P \times (1.0609 - 1) = ₹101.50
     $$
     $$
     P \times 0.0609 = ₹101.50
     $$
   - Solving for $ P $:
     $$
     P = \frac{₹101.50}{0.0609} = ₹1666.67
     $$

3. **Calculating Simple Interest:**
   - The formula for simple interest (SI) is:
     $$
     SI = \frac{P \times r \times n}{100}
     $$
   - Substituting the values:
     $$
     SI = \frac{₹1666.67 \times 3 \times 2}{100}
     $$
     $$
     SI = \frac{₹10000}{100} = ₹100
     $$

### Conclusion:

- **Option 2 (₹100) is correct** because the calculated simple interest for the same principal and period is ₹100.
- **Why other options are incorrect:**
  - **Option 1 (₹50):** This is incorrect because the calculated simple interest is not ₹50.
  - **Option 3 (₹125):** This is incorrect because the calculated simple interest is not ₹125.
  - **Option 4 (₹147):** This is incorrect because the calculated simple interest is not ₹147.

Thus, based on the calculations and logical reasoning, Option 2 (₹100) is indeed the correct answer for the simple interest on the same sum for the same period.