The ratio of difference of S.I and C.I for 2 years and 3 years is `(8)/(25)`. Find the rate of interest.

Answer

To solve the problem of finding the rate of interest given the ratio of the difference between Simple Interest (S.I) and Compound Interest (C.I) for 2 years and 3 years is \( \frac{8}{25} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Formulas**:
- The formula for Simple Interest (S.I) for \( n \) years is:
\[ \text{S.I} = \frac{P \times R \times n}{100} \]
- The formula for Compound Interest (C.I) for \( n \) years is:
\[ \text{C.I} = P \left(1 + \frac{R}{100}\right)^n - P \]

2. **Calculate S.I and C.I for 2 years**:
- For 2 years:
\[ \text{S.I}_{2} = \frac{P \times R \times 2}{100} \]
\[ \text{C.I}_{2} = P \left(1 + \frac{R}{100}\right)^2 - P \]

3. **Calculate S.I and C.I for 3 years**:
- For 3 years:
\[ \text{S.I}_{3} = \frac{P \times R \times 3}{100} \]
\[ \text{C.I}_{3} = P \left(1 + \frac{R}{100}\right)^3 - P \]

4. **Find the Differences**:
- The difference between C.I and S.I for 2 years:
\[ D_{2} = \text{C.I}_{2} - \text{S.I}_{2} \]
- The difference between C.I and S.I for 3 years:
\[ D_{3} = \text{C.I}_{3} - \text{S.I}_{3} \]

5. **Set Up the Ratio**:
- According to the problem, we have:
\[ \frac{D_{2}}{D_{3}} = \frac{8}{25} \]

6. **Substituting the Differences**:
- Substitute the expressions for \( D_{2} \) and \( D_{3} \):
\[ \frac{\left(P \left(1 + \frac{R}{100}\right)^2 - P - \frac{P \times R \times 2}{100}\right)}{\left(P \left(1 + \frac{R}{100}\right)^3 - P - \frac{P \times R \times 3}{100}\right)} = \frac{8}{25} \]

7. **Simplifying the Equation**:
- Cancel \( P \) from both sides and simplify:
\[ \frac{\left(\left(1 + \frac{R}{100}\right)^2 - 1 - \frac{R \times 2}{100}\right)}{\left(\left(1 + \frac{R}{100}\right)^3 - 1 - \frac{R \times 3}{100}\right)} = \frac{8}{25} \]

8. **Cross-Multiplying**:
- Cross-multiply to eliminate the fraction:
\[ 25 \left(\left(1 + \frac{R}{100}\right)^2 - 1 - \frac{R \times 2}{100}\right) = 8 \left(\left(1 + \frac{R}{100}\right)^3 - 1 - \frac{R \times 3}{100}\right) \]

9. **Solving for R**:
- Expand both sides and solve for \( R \). After simplification, you will find:
\[ R = 12.5\% \]

### Final Answer:
The rate of interest is \( 12.5\% \) per annum.