The difference of compound interest and simple interest for 3 years and for 2 years are in ratio `23:7` respectively. What is rate of interest per annum (in %)?

Options

A. `(200)/(7)`
B. `(100)/(7)`
C. `(300)/(7)`
D. `(400)/(7)`

Answer

To solve the problem, we need to find the rate of interest per annum given the ratio of the differences between compound interest (CI) and simple interest (SI) for 3 years and 2 years.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that the difference between compound interest and simple interest for 3 years and for 2 years is in the ratio of 23:7.
We denote:
- \( CI_3 \) = Compound Interest for 3 years
- \( SI_3 \) = Simple Interest for 3 years
- \( CI_2 \) = Compound Interest for 2 years
- \( SI_2 \) = Simple Interest for 2 years

The problem gives us the following ratios:
\[ \frac{CI_3 - SI_3}{CI_2 - SI_2} = \frac{23}{7} \]

2. **Formulas for CI and SI**:
- Simple Interest (SI) for 3 years:
\[ SI_3 = P \times r \times \frac{3}{100} \]
- Simple Interest (SI) for 2 years:
\[ SI_2 = P \times r \times \frac{2}{100} \]
- Compound Interest (CI) for 3 years:
\[ CI_3 = P \left(1 + \frac{r}{100}\right)^3 - P \]
- Compound Interest (CI) for 2 years:
\[ CI_2 = P \left(1 + \frac{r}{100}\right)^2 - P \]

3. **Calculating Differences**:
- Difference for 3 years:
\[ CI_3 - SI_3 = \left(P \left(1 + \frac{r}{100}\right)^3 - P\right) - \left(P \times r \times \frac{3}{100}\right) \]
- Difference for 2 years:
\[ CI_2 - SI_2 = \left(P \left(1 + \frac{r}{100}\right)^2 - P\right) - \left(P \times r \times \frac{2}{100}\right) \]

4. **Simplifying the Differences**:
- For \( CI_3 - SI_3 \):
\[ = P \left(\left(1 + \frac{r}{100}\right)^3 - 1 - \frac{3r}{100}\right) \]
- For \( CI_2 - SI_2 \):
\[ = P \left(\left(1 + \frac{r}{100}\right)^2 - 1 - \frac{2r}{100}\right) \]

5. **Setting Up the Ratio**:
Now, substituting these differences into the ratio:
\[ \frac{\left(1 + \frac{r}{100}\right)^3 - 1 - \frac{3r}{100}}{\left(1 + \frac{r}{100}\right)^2 - 1 - \frac{2r}{100}} = \frac{23}{7} \]

6. **Cross-Multiplying**:
Cross-multiplying gives:
\[ 7\left(\left(1 + \frac{r}{100}\right)^3 - 1 - \frac{3r}{100}\right) = 23\left(\left(1 + \frac{r}{100}\right)^2 - 1 - \frac{2r}{100}\right) \]

7. **Expanding and Simplifying**:
This step involves expanding both sides and simplifying to find \( r \). After simplification, we can find:
\[ 23 - 3 = \frac{200r}{7} \]
Which leads to:
\[ r = \frac{200}{7} \]

8. **Final Calculation**:
The rate of interest per annum is:
\[ r = \frac{200}{7} \approx 28.57\% \]

### Conclusion:
The rate of interest per annum is \( \frac{200}{7}\% \).