If the ratio of difference of S.I and C.I. for two years and 3 years is `(4)/(13)`. Find the rate of interest.

Answer

To solve the problem, we need to find the rate of interest when the ratio of the difference between Simple Interest (SI) and Compound Interest (CI) for 2 years and 3 years is given as \( \frac{4}{13} \).

### Step-by-Step Solution:

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

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

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

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

5. **Express the Differences**:
- For 2 years:
\[ D_{2} = P \left(1 + \frac{R}{100}\right)^2 - P - \frac{P \times R \times 2}{100} \]
- For 3 years:
\[ D_{3} = P \left(1 + \frac{R}{100}\right)^3 - P - \frac{P \times R \times 3}{100} \]

6. **Set Up the Ratio**:
- According to the problem, the ratio of the differences is given as:
\[ \frac{D_{2}}{D_{3}} = \frac{4}{13} \]

7. **Substituting and Simplifying**:
- After substituting \( D_{2} \) and \( D_{3} \) into the ratio and simplifying, we can find a relationship involving \( R \).

8. **Solving for R**:
- After simplifying the equation, we will isolate \( R \) to find its value.

9. **Final Calculation**:
- The final value of \( R \) will give us the rate of interest.

### Final Answer:
After performing the calculations, we find that the rate of interest \( R \) is **25%**.

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