With a given rate of simple interest, the ratio of principal and amount for a certain period of time is 4 : 5. After 3 years, with the same rate of interest, the ratio of the principal and amount becomes 5: 7. The rate of interest per annum is

Options

A. `4%`
B. `5%`
C. `6%`
D. `7%`

Answer

To solve the problem, we need to find the rate of interest per annum based on the given ratios of principal and amount.

### Step-by-Step Solution:

1. **Understanding the Ratios**:
- The first ratio of Principal (P) to Amount (A) is given as 4:5. This means:
\[ \frac{P}{A} = \frac{4}{5} \]
- From this, we can express the amount A in terms of P:
\[ A = \frac{5}{4}P \]

2. **Using the Simple Interest Formula**:
- The amount A can also be expressed using the formula for simple interest:
\[ A = P + \text{SI} \]
- The simple interest (SI) for a time period of t years at a rate R% is given by:
\[ \text{SI} = \frac{P \cdot R \cdot t}{100} \]
- For the first case, we do not have the time, so we denote it as \( t_1 \).

3. **Setting Up the Equation for the Second Ratio**:
- After 3 years, the new ratio of Principal (P) to Amount (A) is given as 5:7:
\[ \frac{P}{S} = \frac{5}{7} \]
- Here, S is the amount after 3 years, which can be expressed as:
\[ S = P + \frac{P \cdot R \cdot 3}{100} \]
- Substituting S in the ratio gives us:
\[ \frac{P}{P + \frac{P \cdot R \cdot 3}{100}} = \frac{5}{7} \]

4. **Cross-Multiplying**:
- Cross-multiplying the equation:
\[ 7P = 5\left(P + \frac{P \cdot R \cdot 3}{100}\right) \]
- Expanding the right side:
\[ 7P = 5P + \frac{15PR}{100} \]

5. **Rearranging the Equation**:
- Rearranging gives:
\[ 7P - 5P = \frac{15PR}{100} \]
\[ 2P = \frac{15PR}{100} \]

6. **Dividing by P** (assuming P ≠ 0):
- Dividing both sides by P:
\[ 2 = \frac{15R}{100} \]

7. **Solving for R**:
- Rearranging gives:
\[ 15R = 200 \]
\[ R = \frac{200}{15} = \frac{40}{3} \approx 13.33 \]

8. **Finding the Rate of Interest**:
- The rate of interest per annum is:
\[ R \approx 13.33\% \]

### Final Answer:
The rate of interest per annum is approximately **13.33%**.