If the ratio of difference between SI and CI for 3 years and 2 years is 22:7 then find the rate of interest?

Options

A. `16 (2)/(3)%`
B. `14 (2)/(7) %`
C. `15 (3)/(7)%`
D. `15 (2)/(7)%`

Answer

To solve the problem, we need to find the rate of interest (r) given that the ratio of the difference between Simple Interest (SI) and Compound Interest (CI) for 3 years and 2 years is 22:7.

### Step-by-Step Solution:

1. **Understanding the Formulas**:
- The formula for Simple Interest (SI) for 2 years is:
\[ \text{SI}_{2} = \frac{P \cdot r \cdot 2}{100} \]
- The formula for Compound Interest (CI) for 2 years is:
\[ \text{CI}_{2} = P \left(1 + \frac{r}{100}\right)^{2} - P \]
- The formula for CI for 3 years is:
\[ \text{CI}_{3} = P \left(1 + \frac{r}{100}\right)^{3} - P \]

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

3. **Expressing the Differences**:
- For 2 years:
\[ \text{Difference}_{2} = P \left(1 + \frac{r}{100}\right)^{2} - P - \frac{P \cdot r \cdot 2}{100} \]
Simplifying this gives:
\[ \text{Difference}_{2} = P \left[\left(1 + \frac{r}{100}\right)^{2} - 1 - \frac{2r}{100}\right] \]
- For 3 years:
\[ \text{Difference}_{3} = P \left(1 + \frac{r}{100}\right)^{3} - P - \frac{P \cdot r \cdot 3}{100} \]
Simplifying this gives:
\[ \text{Difference}_{3} = P \left[\left(1 + \frac{r}{100}\right)^{3} - 1 - \frac{3r}{100}\right] \]

4. **Setting Up the Ratio**:
- According to the problem, the ratio of the differences is given as:
\[ \frac{\text{Difference}_{3}}{\text{Difference}_{2}} = \frac{22}{7} \]

5. **Substituting the Differences**:
- Substitute the expressions for the differences into the ratio:
\[ \frac{P \left[\left(1 + \frac{r}{100}\right)^{3} - 1 - \frac{3r}{100}\right]}{P \left[\left(1 + \frac{r}{100}\right)^{2} - 1 - \frac{2r}{100}\right]} = \frac{22}{7} \]
- The P cancels out:
\[ \frac{\left(1 + \frac{r}{100}\right)^{3} - 1 - \frac{3r}{100}}{\left(1 + \frac{r}{100}\right)^{2} - 1 - \frac{2r}{100}} = \frac{22}{7} \]

6. **Cross Multiplying**:
- Cross multiply to solve for r:
\[ 7 \left[\left(1 + \frac{r}{100}\right)^{3} - 1 - \frac{3r}{100}\right] = 22 \left[\left(1 + \frac{r}{100}\right)^{2} - 1 - \frac{2r}{100}\right] \]

7. **Solving the Equation**:
- After simplifying and solving the above equation, we find that:
\[ r = \frac{100}{7} \approx 14.2857 \]
- This can be expressed as \(14 \frac{2}{7}\%\).

### Final Answer:
The rate of interest is \(14 \frac{2}{7}\%\).