In a partnership, A invests 1/6 of the capital for 1/6 of the time, B invests…

Concept:

In a compound partnership, when partners invest different fractions of the capital for different fractions of the time, the profit-sharing ratio equals the ratio of (capital fraction invested) × (time fraction invested) for each partner — not the capital fraction alone. Each partner's share of the total profit is (that partner's ratio part ÷ sum of all ratio parts) × total profit.

Application:

1. Let the total capital be 6 units and the total time also be 6 units (a convenient common multiple of the denominators 6 and 3).

2. A invests 1/6 of the capital (= 1 unit) for 1/6 of the time (= 1 unit), so A's investment-time product = 1 × 1 = 1.

3. B invests 1/3 of the capital (= 2 units) for 1/3 of the time (= 2 units), so B's investment-time product = 2 × 2 = 4.

4. C invests the remaining capital = 6 − (1 + 2) = 3 units, for the whole time = 6 units, so C's investment-time product = 3 × 6 = 18.

5. The profit-sharing ratio A : B : C = 1 : 4 : 18, whose parts sum to 1 + 4 + 18 = 23.

6. A's share of the total profit = (A's ratio part ÷ sum of ratio parts) × total profit = (1/23) × 2300 = 100.

Cross-check:

• Using B's ratio part: B's share = (4/23) × 2300 = 400.

• Using C's ratio part: C's share = (18/23) × 2300 = 1800.

• The three shares add up to 100 + 400 + 1800 = 2300, matching the total profit exactly, confirming the ratio 1 : 4 : 18.

• Independently, working directly in fractions (without assuming a base of 6): A's combined fraction = (1/6) × (1/6) = 1/36; B's combined fraction = (1/3) × (1/3) = 1/9 = 4/36; C's combined fraction = (1/2) × 1 = 1/2 = 18/36 — giving the same 1 : 4 : 18 ratio.