2/3 part of my sum is lent out at 3%, 1/6 part is lent out at 6% and remaining part is lent out at 12%. All the three parts are lent out at simple interest. If the annual income is 25, what is the sum?

Options

A. 500
B. 650
C. 600
D. 450

Answer

To solve the problem step by step, we will follow the information provided and calculate the sum based on the simple interest formula.

### Step-by-Step Solution:

1. **Define the Total Sum**:
Let the total sum be \( x \).

2. **Determine the Parts of the Sum**:
- The amount lent at 3% is \( \frac{2}{3}x \).
- The amount lent at 6% is \( \frac{1}{6}x \).
- The remaining part lent at 12% can be calculated as follows:
\[ \text{Remaining part} = x - \left(\frac{2}{3}x + \frac{1}{6}x\right) \]
To simplify this, we need a common denominator:
\[ \frac{2}{3} = \frac{4}{6}, \quad \text{so} \quad \frac{2}{3}x + \frac{1}{6}x = \frac{4}{6}x + \frac{1}{6}x = \frac{5}{6}x \]
Thus, the remaining part is:
\[ x - \frac{5}{6}x = \frac{1}{6}x \]

3. **Calculate the Simple Interest for Each Part**:
- For the part lent at 3%:
\[ \text{SI}_1 = \frac{2}{3}x \times \frac{3}{100} \times 1 = \frac{2}{100}x = \frac{1}{50}x \]
- For the part lent at 6%:
\[ \text{SI}_2 = \frac{1}{6}x \times \frac{6}{100} \times 1 = \frac{6}{600}x = \frac{1}{100}x \]
- For the part lent at 12%:
\[ \text{SI}_3 = \frac{1}{6}x \times \frac{12}{100} \times 1 = \frac{12}{600}x = \frac{1}{50}x \]

4. **Total Simple Interest**:
The total simple interest from all parts is given as:
\[ \text{Total SI} = \text{SI}_1 + \text{SI}_2 + \text{SI}_3 = \frac{1}{50}x + \frac{1}{100}x + \frac{1}{50}x \]
To combine these, we need a common denominator, which is 100:
\[ \text{Total SI} = \frac{2}{100}x + \frac{1}{100}x + \frac{2}{100}x = \frac{5}{100}x = \frac{1}{20}x \]

5. **Set Up the Equation**:
We know the total simple interest is 25, so we set up the equation:
\[ \frac{1}{20}x = 25 \]

6. **Solve for \( x \)**:
Multiply both sides by 20:
\[ x = 25 \times 20 = 500 \]

### Final Answer:
The total sum \( x \) is **500**.