If the ratio of difference of CI and SI on some amount for 3 years and 2 years is 29:9 then find the rate of interest?

Options

A. a. `14 (2)/(7) %`
B. b. `16 (2)/(3) %`
C. c. `11(1)/(9) %`
D. d. `22 (2)/(9) %`

Answer

To solve the problem, we need to find the rate of interest (R) given the ratio of the difference between Compound Interest (CI) and Simple Interest (SI) for 3 years and 2 years is 29:9.

### Step-by-Step Solution:

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

2. **Calculate the difference between CI and SI for 2 years and 3 years**:
- For 2 years:
\[ CI_2 = P \left(1 + \frac{R}{100}\right)^2 - P = P \left[\left(1 + \frac{R}{100}\right)^2 - 1\right] \]
\[ SI_2 = \frac{P \times R \times 2}{100} \]
\[ CI_2 - SI_2 = P \left[\left(1 + \frac{R}{100}\right)^2 - 1\right] - \frac{P \times R \times 2}{100} \]

- For 3 years:
\[ CI_3 = P \left(1 + \frac{R}{100}\right)^3 - P = P \left[\left(1 + \frac{R}{100}\right)^3 - 1\right] \]
\[ SI_3 = \frac{P \times R \times 3}{100} \]
\[ CI_3 - SI_3 = P \left[\left(1 + \frac{R}{100}\right)^3 - 1\right] - \frac{P \times R \times 3}{100} \]

3. **Set up the ratio**:
- According to the problem, the ratio of the differences is given as:
\[ \frac{(CI_3 - SI_3) - (CI_2 - SI_2)}{(CI_2 - SI_2)} = \frac{29}{9} \]

4. **Simplify the expression**:
- Substitute the expressions for \( CI \) and \( SI \) into the ratio and simplify:
\[ \frac{P \left[\left(1 + \frac{R}{100}\right)^3 - 1 - \left(1 + \frac{R}{100}\right)^2 + 1\right] - \frac{P \times R \times 3}{100} + \frac{P \times R \times 2}{100}}{P \left[\left(1 + \frac{R}{100}\right)^2 - 1 - \frac{P \times R \times 2}{100}\right]} = \frac{29}{9} \]

5. **Cross-multiply and solve for R**:
- This will lead to a polynomial equation in terms of \( R \). Solve this equation to find the value of \( R \).

6. **Final Calculation**:
- After solving the equation, you will find the rate of interest \( R \).