The difference between C.I. and S.I. on Rs 7,500 for two years is Rs 12 at the same rate of interest per annum. Find the rate of interest.

Answer

To find the rate of interest given that the difference between Compound Interest (C.I.) and Simple Interest (S.I.) on Rs 7,500 for two years is Rs 12, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values:**
- Principal (P) = Rs 7,500
- Time (T) = 2 years
- Difference between C.I. and S.I. = Rs 12

2. **Recall the formulas:**
- Simple Interest (S.I.) = \( \frac{P \times R \times T}{100} \)
- Compound Interest (C.I.) = \( A - P \), where \( A = P(1 + \frac{R}{100})^T \)

3. **Express C.I. and S.I. in terms of R:**
- S.I. = \( \frac{7500 \times R \times 2}{100} = \frac{15000R}{100} = 150R \)
- C.I. = \( 7500 \left(1 + \frac{R}{100}\right)^2 - 7500 \)

4. **Calculate C.I. using the formula:**
- \( C.I. = 7500 \left(1 + \frac{R}{100}\right)^2 - 7500 \)
- Expanding \( \left(1 + \frac{R}{100}\right)^2 \):
\[ C.I. = 7500 \left(1 + \frac{2R}{100} + \frac{R^2}{10000}\right) - 7500 \]
- Simplifying:
\[ C.I. = 7500 \left(\frac{2R}{100} + \frac{R^2}{10000}\right) = 7500 \cdot \frac{2R + \frac{R^2}{100}}{100} \]

5. **Set up the equation for the difference between C.I. and S.I.:**
- \( C.I. - S.I. = Rs 12 \)
- Thus, we can write:
\[ 7500 \cdot \frac{2R + \frac{R^2}{100}}{100} - 150R = 12 \]

6. **Multiply through by 100 to eliminate the fraction:**
- \( 7500(2R + \frac{R^2}{100}) - 15000R = 1200 \)
- Simplifying:
\[ 7500 \cdot 2R + 75R^2 - 15000R = 1200 \]
- This simplifies to:
\[ 15000R + 75R^2 - 15000R = 1200 \]
- Thus:
\[ 75R^2 = 1200 \]

7. **Solve for R:**
- Divide both sides by 75:
\[ R^2 = \frac{1200}{75} = 16 \]
- Taking the square root:
\[ R = 4 \]

8. **Conclusion:**
- The rate of interest is **4% per annum**.