2.10 Partnership: Profit Sharing Basics (Time-Weighted)

Figure: Profit splits by Capital × Time, not capital alone.

In partnership problems that intersect with Profit & Loss, when partners invest different amounts for different time periods, profit is shared in the ratio of (Investment × Time), not investment alone.

Profit-sharing ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂) : …

Worked Example 24: A invests Rs 5000 for 12 months, and B invests Rs 6000 for 8 months in a business. If the total profit at the end of the year is Rs 3600, find each partner’s share.

Solution: A’s equivalent capital = 5000 × 12 = 60,000. B’s equivalent capital = 6000 × 8 = 48,000. Ratio A : B = 60,000 : 48,000 = 5 : 4 (dividing by 12,000). Sum of ratio parts = 9. Each part = 3600/9 = 400. A’s share = 5 × 400 = Rs 2000. B’s share = 4 × 400 = Rs 1600. (Check: 2000 + 1600 = 3600 ✓)

Worked Example 25 (no-profit-no-loss with unequal CPs): A man bought two horses for a total of Rs 15,000. He sold one at a profit of 20% and the other at a loss of 10%, and on the whole transaction he neither gained nor lost. Find the cost price of each horse.

Solution: Let CP of first horse (sold at 20% profit) = C₁, and CP of second horse (sold at 10% loss) = C₂. C₁ + C₂ = 15,000 … (i) Net profit = 0 ⇒ Profit from first = Loss from second ⇒ 0.20 C₁ = 0.10 C₂ ⇒ C₂ = 2C₁ … (ii) Substitute in (i): C₁ + 2C₁ = 15,000 ⇒ 3C₁ = 15,000 ⇒ C₁ = Rs 5000, C₂ = Rs 10,000. Verification: Profit on horse 1 = 0.2 × 5000 = 1000. Loss on horse 2 = 0.1 × 10000 = 1000. Net = 0 ✓.

Worked Example 24A (equal profit-sharing despite unequal capital and time): Three partners A, B and C invest Rs 4000 for 6 months, Rs 6000 for 4 months, and Rs 8000 for 3 months respectively in a joint venture. If the total profit at the end is Rs 9000, find each partner’s share.

Solution: Equivalent capitals: A = 4000×6 = 24,000; B = 6000×4 = 24,000; C = 8000×3 = 24,000. Ratio A:B:C = 24000:24000:24000 = 1:1:1. So the profit is split equally: each partner gets 9000/3 = Rs 3000. This example is a useful reminder that “who invested the most money” is irrelevant by itself — it is the product of capital and time that decides the share, and three very different-looking investments can still produce an identical ratio.

Worked Example 25A (mid-year capital withdrawal AND a late-joining partner — combined edge case): A starts a business with Rs 8000. After 4 months, B joins with Rs 12,000. Then, 2 months later (i.e., 6 months from the start), A withdraws Rs 2000 of his own capital. At the end of the year (12 months total), the total profit is Rs 7500. Find each partner’s share.

Solution: Break A’s investment into two time-slices: - Months 0–6 (6 months): A’s capital = Rs 8000. - Months 6–12 (6 months, after withdrawal): A’s capital = 8000 − 2000 = Rs 6000.

A’s equivalent capital = (8000 × 6) + (6000 × 6) = 48,000 + 36,000 = 84,000. B invests for the remaining 8 months of the year (12 − 4) at a constant Rs 12,000: B’s equivalent capital = 12,000 × 8 = 96,000. Ratio A : B = 84,000 : 96,000 = 7 : 8 (dividing by 12,000). Sum of parts = 15. Each part = 7500/15 = 500. A’s share = 7 × 500 = Rs 3500. B’s share = 8 × 500 = Rs 4000. Check: 3500 + 4000 = 7500 ✓. This is the hardest standard variant of the topic: a partner’s capital changes mid-year and a second partner joins late — always split the changing partner’s investment into separate time-slices and add the equivalent capitals before forming the ratio.

Worked Example 25B (a partner leaves early and a new partner joins — edge case): A and B start a business with capitals of Rs 15,000 and Rs 20,000 respectively. After 5 months, A leaves the business, and at that very point C joins with Rs 25,000. At the end of the year, the total profit is Rs 14,700. Find each partner’s (A, B, C) share.

Solution: A invested for only the first 5 months: A’s equivalent capital = 15,000 × 5 = 75,000. B invested for the full 12 months: B’s equivalent capital = 20,000 × 12 = 2,40,000. C invested for the remaining 12 − 5 = 7 months: C’s equivalent capital = 25,000 × 7 = 1,75,000. Ratio A : B : C = 75,000 : 2,40,000 : 1,75,000. Dividing throughout by 5,000: 15 : 48 : 35. Sum of parts = 15 + 48 + 35 = 98. Each part = 14,700/98 = 150. A’s share = 15 × 150 = Rs 2250. B’s share = 48 × 150 = Rs 7200. C’s share = 35 × 150 = Rs 5250. Check: 2250 + 7200 + 5250 = 14,700 ✓.