At a certain rate of interest the compound interest of 3 years and simple interest of 5 years for certain sum of money is respectively Rs. 1513.2 and Rs. 2400. Find the common rate of interest per annum ?

Explanation

To solve the problem, we need to find the common rate of interest per annum given the compound interest (CI) for 3 years and simple interest (SI) for 5 years.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Compound Interest (CI) for 3 years = Rs. 1513.2
- Simple Interest (SI) for 5 years = Rs. 2400

2. **Use the Formula for Simple Interest:**
The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

For our case, substituting the values:
\[ 2400 = \frac{P \times R \times 5}{100} \]
Rearranging gives:
\[ P \times R = \frac{2400 \times 100}{5} = 48000 \quad \text{(1)} \]

3. **Use the Formula for Compound Interest:**
The formula for Compound Interest (CI) is:
\[ CI = P \left(1 + \frac{R}{100}\right)^n - P \]
For our case, substituting the values for 3 years:
\[ 1513.2 = P \left( \left(1 + \frac{R}{100}\right)^3 - 1 \right) \]
Rearranging gives:
\[ 1513.2 = P \left( \left(1 + \frac{R}{100}\right)^3 - 1 \right) \quad \text{(2)} \]

4. **Divide Equation (2) by Equation (1):**
We can express \( P \) from equation (1):
\[ P = \frac{48000}{R} \quad \text{(3)} \]
Substitute \( P \) from (3) into (2):
\[ 1513.2 = \frac{48000}{R} \left( \left(1 + \frac{R}{100}\right)^3 - 1 \right) \]
Simplifying gives:
\[ 1513.2R = 48000 \left( \left(1 + \frac{R}{100}\right)^3 - 1 \right) \]

5. **Solve the Equation:**
This equation can be solved for \( R \). However, to simplify, we can assume \( R \) is a small percentage and use the approximation:
\[ \left(1 + \frac{R}{100}\right)^3 \approx 1 + \frac{3R}{100} \]
Thus:
\[ 1513.2R = 48000 \left( \frac{3R}{100} \right) \]
Simplifying gives:
\[ 1513.2R = 1440R \]
Therefore:
\[ R \approx \frac{1513.2}{1440} \times 100 \approx 5\% \]

### Final Answer:
The common rate of interest per annum is **5%**.