A man invests half his capital at the rate of 8% per annum, one-third at 9% and the rest at 18% per annum. The average rate of interest per annum, which he gets, is

Options

A. `12%`
B. `18%`
C. `10%`
D. `15.5%`

Answer

To find the average rate of interest per annum that a man receives from his investments, we can follow these steps:

### Step 1: Define the total capital
Let's assume the total capital is \( C \). For simplicity, we can take \( C = 600 \) (the actual value does not affect the average rate).

### Step 2: Calculate the amounts invested at different rates
- Half of the capital is invested at 8%:
\[ \text{Amount at 8%} = \frac{1}{2} \times 600 = 300 \]

- One-third of the capital is invested at 9%:
\[ \text{Amount at 9%} = \frac{1}{3} \times 600 = 200 \]

- The remaining amount is invested at 18%:
\[ \text{Remaining amount} = 600 - (300 + 200) = 100 \]

### Step 3: Calculate the interest earned from each investment
- Interest from the amount invested at 8%:
\[ \text{Interest at 8%} = \frac{300 \times 8 \times 1}{100} = 24 \]

- Interest from the amount invested at 9%:
\[ \text{Interest at 9%} = \frac{200 \times 9 \times 1}{100} = 18 \]

- Interest from the amount invested at 18%:
\[ \text{Interest at 18%} = \frac{100 \times 18 \times 1}{100} = 18 \]

### Step 4: Calculate the total interest earned
Now, we add up all the interest amounts:
\[ \text{Total Interest} = 24 + 18 + 18 = 60 \]

### Step 5: Calculate the average rate of interest
To find the average rate of interest, we use the formula:
\[ \text{Average Rate} = \frac{\text{Total Interest}}{\text{Total Principal}} \times 100 \]
Substituting the values:
\[ \text{Average Rate} = \frac{60}{600} \times 100 = 10\% \]

### Final Answer
The average rate of interest per annum that he receives is **10%**.

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