Out of Rs. 50,000, that a man has, he lends Rs. 8000 at `5""(1)/(2)%` per annum simple interest and Rs. 24,000 at 6 % per annum simple interest. He lends the remaining money at a certain rate of interest so that he gets total annual interest of Rs. 3680. The rate of interest per annum, at which the remaining money is lent, is

Options

A. 0.05
B. 0.07
C. 0.1
D. 0.12

Answer

To solve the problem step by step, we need to calculate the total interest earned from the amounts lent at different rates and then find the rate of interest for the remaining amount.

### Step-by-Step Solution:

1. **Identify the Total Amount and Amounts Lent:**
- Total amount = Rs. 50,000
- Amount lent at 5.5% = Rs. 8,000
- Amount lent at 6% = Rs. 24,000

2. **Calculate the Remaining Amount:**
- Remaining amount = Total amount - (Amount lent at 5.5% + Amount lent at 6%)
- Remaining amount = 50,000 - (8,000 + 24,000) = 50,000 - 32,000 = Rs. 18,000

3. **Calculate the Interest from the First Two Loans:**
- Interest from Rs. 8,000 at 5.5% per annum:
\[ \text{Interest} = \frac{P \times R \times T}{100} = \frac{8000 \times 5.5 \times 1}{100} = 440 \]
- Interest from Rs. 24,000 at 6% per annum:
\[ \text{Interest} = \frac{24000 \times 6 \times 1}{100} = 1440 \]

4. **Calculate the Total Interest Earned from the First Two Loans:**
- Total interest from the first two loans = Interest from Rs. 8,000 + Interest from Rs. 24,000
- Total interest = 440 + 1440 = Rs. 1880

5. **Determine the Total Interest Required:**
- Total interest required = Rs. 3,680

6. **Calculate the Interest from the Remaining Amount:**
- Interest from the remaining Rs. 18,000:
\[ \text{Interest from remaining} = \text{Total interest required} - \text{Total interest from first two loans} \]
- Interest from remaining = 3680 - 1880 = Rs. 1,800

7. **Set Up the Equation for the Remaining Amount:**
- Let the rate of interest for the remaining amount be \( r \% \).
- Using the formula for simple interest:
\[ 1800 = \frac{18000 \times r \times 1}{100} \]

8. **Solve for \( r \):**
- Rearranging gives:
\[ r = \frac{1800 \times 100}{18000} = 10 \]

9. **Conclusion:**
- The rate of interest per annum at which the remaining money is lent is **10%**.