A, B and C enter into a partnership and their shares are in the ratio 1/2 :…

Concept: In a partnership, profit is shared in the ratio of each partner's capital-months — that is, capital invested multiplied by the time for which it remains invested. When a partner changes their capital partway through the period, compute the capital-months for each sub-period separately and add them before forming the ratio.

Application:

1. The capital ratio 1/2 : 1/3 : 1/4 is converted to whole numbers by multiplying by the LCM of 2, 3, 4, which is 12: this gives 6 : 4 : 3. Let A, B and C's capitals be 6x, 4x and 3x.

2. A invests 6x for the first 2 months, then withdraws half his capital (leaving 3x) for the remaining 10 months, so the business runs for 2 + 10 = 12 months in total.

3. A's capital-months = (6x × 2) + (3x × 10) = 12x + 30x = 42x.

4. B's capital-months = 4x × 12 = 48x.

5. C's capital-months = 3x × 12 = 36x.

6. The profit ratio A : B : C = 42 : 48 : 36, which simplifies (divide by 6) to 7 : 8 : 6, a total of 7 + 8 + 6 = 21 parts.

7. B's share = (8/21) × 378 = 8 × 18 = Rs. 144.

Cross-check: A's share = (7/21) × 378 = 126 and C's share = (6/21) × 378 = 108. Adding all three shares: 126 + 144 + 108 = 378, which equals the total profit, confirming B's share of Rs. 144 is correct.