The ratio of investments of two partners A and B is 7:5 and the ratio of their profits is 7:10. If A invested the money for 5 months, find for how much time did B invest the money ?

Options

A. A) 11 months
B. B) 9 months
C. C) 7 months
D. D) 10 months

Answer

To solve the problem, we need to find out how long partner B invested their money based on the given ratios of their investments and profits.

### Step-by-Step Solution:

1. **Understanding the Ratios**:
- Let the investment of partner A be \( 7x \) and the investment of partner B be \( 5x \), where \( x \) is a common multiplier.
- The ratio of their profits is given as \( 7:10 \).

2. **Setting Up the Profit Equation**:
- The profit is directly proportional to the investment and the time for which the investment is made.
- If A invested for 5 months, we can express the profit for A as:
\[ \text{Profit of A} = \text{Investment of A} \times \text{Time of A} = 7x \times 5 = 35x \]
- Let the time for which B invested be \( t \) months. Then the profit for B can be expressed as:
\[ \text{Profit of B} = \text{Investment of B} \times \text{Time of B} = 5x \times t = 5xt \]

3. **Using the Profit Ratio**:
- According to the profit ratio:
\[ \frac{\text{Profit of A}}{\text{Profit of B}} = \frac{7}{10} \]
- Substituting the expressions for profit:
\[ \frac{35x}{5xt} = \frac{7}{10} \]

4. **Simplifying the Equation**:
- Cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[ \frac{35}{5t} = \frac{7}{10} \]
- Simplifying \( \frac{35}{5} \) gives:
\[ \frac{7}{t} = \frac{7}{10} \]

5. **Cross-Multiplying**:
- Cross-multiplying gives:
\[ 7 \times 10 = 7 \times t \]
- This simplifies to:
\[ 70 = 7t \]

6. **Solving for \( t \)**:
- Dividing both sides by 7:
\[ t = \frac{70}{7} = 10 \]

### Conclusion:
Partner B invested the money for **10 months**.