A, B, and C are partners in a business with investments of Rs. 5,000, Rs. 7,000, and Rs. 10,000 respectively. B receives 12\% of the total profit as a commission for managing the business. The remaining profit is shared among A, B, and C in proportion to their investments at the end of the year. If B's total profit is Rs. 2,500 less than the combined profit of A and C, what is the profit earned by A?

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Accepted Answer

**Solution:**

1. Let the total profit be denoted by *P*.

2. B receives 12\% of *P* as a commission for managing the business, so B's commission = $0.12P$.

3. The remaining profit after B's commission is:
   $$P - 0.12P = 0.88P$$

4. This remaining profit is divided among A, B, and C in the ratio of their investments:
   - A : B : C = 5000 : 7000 : 10000 = 5 : 7 : 10

5. The sum of the ratio parts = $5 + 7 + 10 = 22$.

6. B's share from the remaining profit = $\frac{7}{22} 	imes 0.88P$.

7. Therefore, B's total profit = commission + share =
   $$0.12P + \frac{7}{22} 	imes 0.88P = 0.12P + 0.28P = 0.4P$$

8. A's profit from the remaining profit = $\frac{5}{22} 	imes 0.88P = 0.2P$.

9. C's profit from the remaining profit = $\frac{10}{22} 	imes 0.88P = 0.4P$.

10. According to the problem, B's profit is Rs. 2,500 less than the combined profit of A and C:
    $$	ext{B's profit} = (	ext{A's profit} + 	ext{C's profit}) - 2500$$

Substitute the values:
    $$0.4P = (0.2P + 0.4P) - 2500$$
    $$0.4P = 0.6P - 2500$$

11. Rearranging:
    $$0.6P - 0.4P = 2500$$
    $$0.2P = 2500$$
    $$P = \frac{2500}{0.2} = 12500$$

12. Now, calculate A's profit:
    $$	ext{A's profit} = 0.2P = 0.2 	imes 12500 = 2500$$

**Answer:** Rs. 2,500

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