A man invests half his capitial at the rate of 10% per annum. One third at 9% and the rest at 12% per annum. The average rate of interset per annum. Which he gets, is

Options

A. 0.09
B. 0.1
C. 10.5%
D. 0.12

Answer

To find the average rate of interest per annum from the investments made by the man, we will follow these steps:

### Step 1: Define the total capital
Let the total capital be \( C = 100 \) (this simplifies calculations).

### Step 2: Calculate the amounts invested at different rates
- Half of the capital is invested at 10%:
\[ \text{Amount at 10%} = \frac{1}{2} \times 100 = 50 \]

- One third of the capital is invested at 9%:
\[ \text{Amount at 9%} = \frac{1}{3} \times 100 = \frac{100}{3} \approx 33.33 \]

- The rest of the capital is invested at 12%. First, we need to find out how much is left after investing in the first two parts:
\[ \text{Remaining amount} = 100 - 50 - \frac{100}{3} = 100 - 50 - 33.33 = 16.67 \]
This amount is invested at 12%.

### Step 3: Calculate the interest from each investment
- Interest from the amount invested at 10%:
\[ \text{Interest at 10%} = 50 \times \frac{10}{100} = 5 \]

- Interest from the amount invested at 9%:
\[ \text{Interest at 9%} = \frac{100}{3} \times \frac{9}{100} = \frac{9}{3} = 3 \]

- Interest from the amount invested at 12%:
\[ \text{Interest at 12%} = 16.67 \times \frac{12}{100} = 2 \]

### Step 4: Calculate the total interest earned
Now, we sum up all the interests earned:
\[ \text{Total Interest} = 5 + 3 + 2 = 10 \]

### Step 5: Calculate the average rate of interest
The average rate of interest can be calculated using the formula:
\[ \text{Average Rate} = \frac{\text{Total Interest}}{\text{Total Capital}} \times 100 \]
Substituting the values:
\[ \text{Average Rate} = \frac{10}{100} \times 100 = 10\% \]

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

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