If the difference between SI and CI for 2 years is 35 and three years is 77 then find rate of interest?

Options

A. `15%`
B. `22%`
C. `20%`
D. `25%`

Answer

To solve the problem, we need to find the rate of interest (R) given the differences between Simple Interest (SI) and Compound Interest (CI) over 2 years and 3 years.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- The difference between CI and SI for 2 years is Rs. 35.
- The difference between CI and SI for 3 years is Rs. 77.

2. **Use the Formula for the Difference Between CI and SI:**
- The difference between CI and SI for 2 years can be expressed as:
\[ \text{Difference}_{2 \text{ years}} = \frac{P \cdot R^2}{100^2} \]
- The difference between CI and SI for 3 years can be expressed as:
\[ \text{Difference}_{3 \text{ years}} = \frac{P \cdot R^3}{100^3} \]

3. **Set Up the Equations:**
- From the information given:
\[ \frac{P \cdot R^2}{100^2} = 35 \quad \text{(1)} \]
\[ \frac{P \cdot R^3}{100^3} = 77 \quad \text{(2)} \]

4. **Divide Equation (2) by Equation (1):**
- This will help eliminate P:
\[ \frac{\frac{P \cdot R^3}{100^3}}{\frac{P \cdot R^2}{100^2}} = \frac{77}{35} \]
- Simplifying gives:
\[ \frac{R}{100} = \frac{77}{35} \]

5. **Cross Multiply to Solve for R:**
- Rearranging gives:
\[ R = \frac{77 \cdot 100}{35} \]

6. **Calculate R:**
- Simplifying:
\[ R = \frac{7700}{35} = 220 \]

7. **Find the Rate of Interest:**
- Since we need the percentage, we divide by 100:
\[ R = 22\% \]

### Conclusion:
The rate of interest is **22%**.

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