A and B started a business with initial investments in the ratio of 5:7. If after one year their profits were in the ratio of 1:2 and the period for A’s investment was 7 months, then B invested the money for

Options

A. 6 months
B. 2 months
C. 10 months
D. 4 months

Answer

To solve the problem, we need to determine how long B invested his money based on the given ratios of investment and profit.

### Step-by-Step Solution:

1. **Identify the Ratios**:
- The investment ratio of A to B is given as 5:7.
- The profit ratio of A to B is given as 1:2.

2. **Let the Investments be Represented**:
- Let A's investment be \(5x\) and B's investment be \(7x\), where \(x\) is a common multiplier.

3. **Given Time for A**:
- A invested his money for 7 months.

4. **Let B's Investment Time be \(y\) Months**:
- We need to find \(y\), the time for which B invested his money.

5. **Using the Profit Formula**:
- The profit earned is proportional to the product of investment and time.
- Therefore, we can write the equation based on the profits:
\[ \frac{\text{Profit of A}}{\text{Profit of B}} = \frac{\text{Investment of A} \times \text{Time of A}}{\text{Investment of B} \times \text{Time of B}} \]
- Substituting the known values:
\[ \frac{1}{2} = \frac{5x \times 7}{7x \times y} \]

6. **Simplifying the Equation**:
- The \(x\) cancels out:
\[ \frac{1}{2} = \frac{5 \times 7}{7 \times y} \]
- This simplifies to:
\[ \frac{1}{2} = \frac{5}{y} \]

7. **Cross-Multiplying**:
- Cross-multiplying gives:
\[ 1 \cdot y = 2 \cdot 5 \]
- Thus:
\[ y = 10 \]

8. **Conclusion**:
- B invested his money for **10 months**.

### Final Answer:
B invested the money for **10 months**.