A, B and C are partners in a business. A, whose capital has been used for 5 months, claims `""_(7)^(1)` of the profit. B whose capital has been used for 7 months, claims `""_(5)^(1)` of the profit. C has invested Rs. 4,600 for 9 months. How much capital did A contribute?

Options

A. Rs. 1,800
B. Rs. 1,650
C. Rs. 1,600
D. Rs. 1,850

Answer

To solve the problem, we need to determine the capital contributed by partner A based on the given claims of profit and the capital invested by partner C.

### Step-by-Step Solution:

1. **Understanding the Profit Shares**:
- A claims \( \frac{1}{7} \) of the profit.
- B claims \( \frac{1}{5} \) of the profit.
- Let the total profit be \( P \).

2. **Finding the Remaining Profit for C**:
- The total profit can be represented as \( P = 35 \) (a common multiple of 7 and 5).
- A's share: \( \frac{1}{7} \times 35 = 5 \)
- B's share: \( \frac{1}{5} \times 35 = 7 \)
- Total profit claimed by A and B: \( 5 + 7 = 12 \)
- Therefore, C's share: \( 35 - 12 = 23 \)

3. **Setting Up the Capital Contribution**:
- Let A's capital be \( x \).
- B's capital is unknown, let it be \( y \).
- C has invested Rs. 4,600 for 9 months.
- A's capital is used for 5 months, and B's capital is used for 7 months.

4. **Using the Capital and Time Relationship**:
- The profit share is proportional to the product of capital and time.
- Thus, we can set up the equation based on the profit shares:
\[ \frac{5x}{9 \times 4600} = \frac{5}{23} \]

5. **Cross-Multiplying to Solve for x**:
- Cross-multiplying gives:
\[ 5x \times 23 = 5 \times (9 \times 4600) \]
- Simplifying:
\[ 115x = 207000 \]
- Dividing both sides by 115:
\[ x = \frac{207000}{115} = 1800 \]

6. **Conclusion**:
- Therefore, A contributed Rs. 1,800.

### Final Answer:
A's capital contribution is Rs. 1,800.