A certain sum is lent at 5% p.a. for 2 years, 9% p.a. for the next 3 years and 13% p.a. after 5 years. If the simple interest on the sum for a period of 10 years is 8,160, then the sum (in) is:

Options

A. 8000
B. 9000
C. 8400
D. 9600

Answer

To solve the problem, we need to calculate the principal amount (sum) based on the given interest rates and the total simple interest earned over 10 years.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let the principal amount be \( P \).
- The interest rates and their respective time periods are:
- 5% per annum for 2 years
- 9% per annum for 3 years
- 13% per annum for 5 years
- Total simple interest earned over 10 years is \( 8160 \).

2. **Calculate the Simple Interest for Each Period**:
- For the first 2 years at 5%:
\[ \text{SI}_1 = \frac{P \times 5 \times 2}{100} = \frac{10P}{100} = 0.1P \]
- For the next 3 years at 9%:
\[ \text{SI}_2 = \frac{P \times 9 \times 3}{100} = \frac{27P}{100} = 0.27P \]
- For the last 5 years at 13%:
\[ \text{SI}_3 = \frac{P \times 13 \times 5}{100} = \frac{65P}{100} = 0.65P \]

3. **Sum Up the Simple Interests**:
- Total simple interest over 10 years:
\[ \text{Total SI} = \text{SI}_1 + \text{SI}_2 + \text{SI}_3 = 0.1P + 0.27P + 0.65P = (0.1 + 0.27 + 0.65)P = 1.02P \]

4. **Set Up the Equation**:
- We know that the total simple interest is \( 8160 \):
\[ 1.02P = 8160 \]

5. **Solve for \( P \)**:
- Divide both sides by \( 1.02 \):
\[ P = \frac{8160}{1.02} = 8000 \]

6. **Conclusion**:
- The principal amount (sum) is \( 8000 \).

### Final Answer:
The sum is \( 8000 \).