The difference of compound in terest 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 step by step, we will first establish the formulas for the difference between Compound Interest (CI) and Simple Interest (SI) for 2 years and 3 years.

### Step 1: Understanding the Formulas
The difference between Compound Interest and Simple Interest for:
- 2 years is given by:
\[ \text{Difference}_{2 \text{ years}} = \frac{P \cdot r^2}{100^2} \]
- 3 years is given by:
\[ \text{Difference}_{3 \text{ years}} = \frac{P \cdot r^2}{100^2} + \frac{P \cdot r^2 \cdot r}{100^3} = \frac{P \cdot r^2}{100^2} + \frac{P \cdot r^3}{100^3} \]

### Step 2: Setting Up the Ratios
According to the problem, the ratio of the differences for 3 years and 2 years is given as:
\[ \frac{\text{Difference}_{3 \text{ years}}}{\text{Difference}_{2 \text{ years}}} = \frac{23}{7} \]

### Step 3: Substituting the Formulas into the Ratio
Substituting the formulas we derived:
\[ \frac{\frac{P \cdot r^2}{100^2} + \frac{P \cdot r^3}{100^3}}{\frac{P \cdot r^2}{100^2}} = \frac{23}{7} \]

### Step 4: Simplifying the Ratio
Cancelling \( P \) and \( \frac{r^2}{100^2} \) from both the numerator and denominator:
\[ \frac{1 + \frac{r}{100}}{1} = \frac{23}{7} \]

### Step 5: Cross Multiplying
Cross multiplying gives:
\[ 7 \left(1 + \frac{r}{100}\right) = 23 \]

### Step 6: Expanding and Rearranging
Expanding the left side:
\[ 7 + \frac{7r}{100} = 23 \]
Now, subtract 7 from both sides:
\[ \frac{7r}{100} = 16 \]

### Step 7: Solving for r
Multiplying both sides by 100:
\[ 7r = 1600 \]
Now, divide by 7:
\[ r = \frac{1600}{7} \approx 228.57 \]

### Step 8: Final Result
Thus, the rate of interest per annum is approximately:
\[ r \approx 228.57\% \]