Capital, time and profit-sharing ratios

Outcome

Calculate capital-time contributions, divide profit among partners, and recover an unknown investment duration from an agreed sharing ratio.

Concept and assumptions

Use a 12-month accounting period throughout this lesson. Each partner’s stated capital remains unchanged during their active months. Assume the agreement allocates the entire stated distributable profit in proportion to capital multiplied by time, with no salary or commission. Returned capital is not part of this profit.

A larger investment supports the business more; keeping it invested longer also increases its contribution. Under this model, equal capital for equal time receives equal profit. Therefore:

Contribution weight = capital × active months. Partner’s share = total profit × partner’s weight ÷ sum of all weights.

Measure every duration in the same unit. The weights have units of rupee-months; their ratio has no units. When durations are equal, they cancel, leaving the capital ratio. When capitals are equal, only durations matter. Neither shortcut is valid merely because the business has several partners.

Simplify the weights together, add the ratio parts, and distribute the profit using that total. The shares must add back to the original profit. All answers here are exact.

Worked examples

Example 1 — Equal durations. A invests ₹45,000 and B ₹75,000, both for 12 months. Divide ₹36,000 profit.

Weights are 45,000 × 12 : 75,000 × 12 = 3:5. Total parts = 3 + 5 = 8. A receives 36,000 × 3/8 = ₹13,500. B receives 36,000 × 5/8 = ₹22,500. Check: 13,500 + 22,500 = 36,000.

Example 2 — Unequal durations. During the 12-month period, A invests ₹60,000 for 12 months and B ₹90,000 for 8 months. Profit is ₹50,400.

A’s weight = 60,000 × 12 = 720,000. B’s weight = 90,000 × 8 = 720,000. The ratio is 1:1, so each receives 50,400 ÷ 2 = ₹25,200. B’s larger capital is exactly offset by its shorter duration.

Example 3 — Three partners. A invests ₹40,000 for 12 months, B ₹60,000 for 9 months, and C ₹80,000 for 6 months. Divide ₹75,000.

Weights: A = 40,000 × 12 = 480,000; B = 60,000 × 9 = 540,000; C = 80,000 × 6 = 480,000. Dividing each by 60,000 gives 8:9:8. Total parts = 25; one part = 75,000 ÷ 25 = 3,000. A = 8 × 3,000 = ₹24,000; B = 9 × 3,000 = ₹27,000; C = 8 × 3,000 = ₹24,000. Their sum is ₹75,000.

Common mistakes

Using capital alone despite unequal durations; treating a ratio part as a fraction of another partner’s part rather than of the total; mixing months with years; adding returned investment to distributable profit.

Practice questions

1. A invests ₹28,000 and B ₹42,000, each for 12 months. Divide ₹35,000 profit.
2. A invests ₹48,000 for 10 months and B ₹40,000 for 12 months. Divide ₹54,000 profit.
3. A invests ₹30,000 for 12 months, B ₹45,000 for 8 months, and C ₹60,000 for 3 months. Divide ₹45,000 profit.
4. A invests ₹50,000 for 12 months. B invests ₹75,000 for an unknown number of months. Their profit ratio A:B is 4:3. Find B’s duration.

Worked answers

1. Equal durations cancel: 28,000:42,000 = 2:3. Total parts = 5. A = 35,000 × 2/5 = ₹14,000; B = 35,000 × 3/5 = ₹21,000.
2. Weights = 48,000 × 10 : 40,000 × 12 = 480,000:480,000 = 1:1. Each receives 54,000 ÷ 2 = ₹27,000.
3. Weights = (30,000 × 12):(45,000 × 8):(60,000 × 3) = 360,000:360,000:180,000 = 2:2:1. Total parts = 5; one part = 45,000 ÷ 5 = 9,000. A and B each receive 2 × 9,000 = ₹18,000; C receives ₹9,000.
4. Let B invest for t months. Then 600,000/(75,000t) = 4/3. Cross-multiplication gives 1,800,000 = 300,000t, hence t = 6 months. Check: 600,000:450,000 = 4:3.