If the compound interest on a certain sum for 2 years at `3%` per annum is `Rs. 101.50`, then the simple interest on the same sum at the same rate and for the same time will be

Options

A. A) `Rs. 90.00`
B. B) `Rs.95.50 `
C. C) `Rs.100.00`
D. D) `Rs. 98.25`

Answer

To find the simple interest on a certain sum for 2 years at a rate of 3% per annum, given that the compound interest for the same period and rate is Rs. 101.50, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Compound Interest (CI) for 2 years = Rs. 101.50
- Rate of Interest (R) = 3% per annum
- Time (T) = 2 years

2. **Calculate the Effective Rate of Interest**:
- The formula for effective rate of interest for 2 years is:
\[ R_{effective} = R_1 + R_2 + \frac{R_1 \times R_2}{100} \]
- Here, both R1 and R2 are 3%.
- Therefore,
\[ R_{effective} = 3 + 3 + \frac{3 \times 3}{100} = 6 + 0.09 = 6.09\% \]

3. **Relate Compound Interest to Effective Rate**:
- We know that the compound interest for 2 years at the effective rate is Rs. 101.50.
- This means:
\[ 6.09\% \text{ of the principal} = 101.50 \]

4. **Calculate the Principal Amount**:
- Let the principal amount be P.
- Then,
\[ \frac{6.09}{100} \times P = 101.50 \]
- Rearranging gives:
\[ P = \frac{101.50 \times 100}{6.09} \approx 1667.24 \]

5. **Calculate Simple Interest (SI)**:
- The formula for simple interest for 2 years at 3% per annum is:
\[ SI = \frac{P \times R \times T}{100} \]
- Substituting the values:
\[ SI = \frac{1667.24 \times 3 \times 2}{100} = \frac{100.0334}{100} = 100.00 \]

6. **Conclusion**:
- The simple interest on the same sum at the same rate and for the same time is Rs. 100.

### Final Answer:
The simple interest is **Rs. 100**.

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