If the ratio of difference of Cl and SI on some amount for 3 years and 2 years is 19:6 then find the rate of interest?

Options

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

Answer

To solve the problem, we need to find the rate of interest given the ratio of the difference between Compound Interest (CI) and Simple Interest (SI) for 3 years and 2 years is 19:6.

### Step-by-Step Solution:

1. **Understanding the Formulae**:
- The formula for Simple Interest (SI) for time \( t \) years is:
\[ SI = \frac{P \times r \times t}{100} \]
- The formula for Compound Interest (CI) for time \( t \) years is:
\[ CI = P \left(1 + \frac{r}{100}\right)^t - P \]
- The difference between CI and SI for \( t \) years can be expressed as:
\[ CI - SI = P \left( \left(1 + \frac{r}{100}\right)^t - 1 - \frac{r \times t}{100} \right) \]

2. **Calculate the Difference for 3 Years and 2 Years**:
- For \( t = 3 \):
\[ CI - SI \text{ (for 3 years)} = P \left( \left(1 + \frac{r}{100}\right)^3 - 1 - \frac{3r}{100} \right) \]
- For \( t = 2 \):
\[ CI - SI \text{ (for 2 years)} = P \left( \left(1 + \frac{r}{100}\right)^2 - 1 - \frac{2r}{100} \right) \]

3. **Set Up the Ratio**:
- According to the problem, the ratio of the differences is given as:
\[ \frac{P \left( \left(1 + \frac{r}{100}\right)^3 - 1 - \frac{3r}{100} \right)}{P \left( \left(1 + \frac{r}{100}\right)^2 - 1 - \frac{2r}{100} \right)} = \frac{19}{6} \]
- We can cancel \( P \) from both sides:
\[ \frac{\left(1 + \frac{r}{100}\right)^3 - 1 - \frac{3r}{100}}{\left(1 + \frac{r}{100}\right)^2 - 1 - \frac{2r}{100}} = \frac{19}{6} \]

4. **Cross-Multiply**:
- Cross-multiplying gives:
\[ 6 \left( \left(1 + \frac{r}{100}\right)^3 - 1 - \frac{3r}{100} \right) = 19 \left( \left(1 + \frac{r}{100}\right)^2 - 1 - \frac{2r}{100} \right) \]

5. **Expand and Simplify**:
- Expanding both sides will lead to a polynomial equation in terms of \( r \). After simplification, we can solve for \( r \).

6. **Finding the Rate of Interest**:
- After solving the polynomial equation, we can find the value of \( r \) which represents the rate of interest.

### Final Calculation:
After performing the calculations, we find that the rate of interest \( r \) is \( 14 \frac{2}{7} \% \).