A and B enter into a partnership and invest Rs. 16,000 and Rs. 15,000…

Concept: When partners invest different amounts for different time periods, the profit is divided in the ratio of each partner's (capital invested) x (time period invested), not simply in the ratio of capitals. This capital-time (capital-months) product captures both how much money was put in and for how long, and is the basis for splitting any partnership profit when investments start or end at different times.

1. A invests Rs. 16,000 as a founding partner, so A's money works for the full two-year period, i.e. 24 months.

2. B invests Rs. 15,000, also as a founding partner, so B's money is also invested for the full 24 months.

3. C joins only after 9 months, contributing Rs. 20,000. Since the total period under consideration is 24 months, C's capital works for only 24 - 9 = 15 months.

4. Find each partner's capital-time product: A = 16,000 x 24 = 3,84,000; B = 15,000 x 24 = 3,60,000; C = 20,000 x 15 = 3,00,000.

5. Divide all three products by their common factor 12,000 to get the simplified profit-sharing ratio A : B : C = 32 : 30 : 25.

6. Add the ratio parts: 32 + 30 + 25 = 87. So the total profit of Rs. 36,540 is divided into 87 equal parts, each worth 36,540 / 87 = Rs. 420.

7. C's share corresponds to 25 parts, so C receives 25 x 420 = Rs. 10,500.

Cross-check: Using the same per-part value, A's share is 32 x 420 = Rs. 13,440 and B's share is 30 x 420 = Rs. 12,600. Adding all three shares: 13,440 + 12,600 + 10,500 = Rs. 36,540, which matches the total profit exactly, confirming the ratio and the division are consistent.