Partnership for CSAT: Capital, Time and Profit Sharing

Partnership is the CSAT topic that tests how business partners split profit: each partner's share is proportional to capital invested multiplied by the time it stayed invested. This one sentence answers every partnership question UPSC has asked, because CSAT never tests legal partnership types, only this capital-time rule. A sleeping partner invests money but does not manage the business; a working partner manages it and is usually paid a commission from the profit before the capital-time split.

The capital-time rule

If two partners invest C1 and C2 for T1 and T2 months, profits divide in the ratio C1 x T1 : C2 x T2. When everyone invests for the full year, time cancels out and the ratio is just the capitals.

Worked example 1. A and B invest Rs 5,000 and Rs 4,000; the year's profit is Rs 2,250. A's share? Ratio 5:4, total 9 parts. A gets 5/9 x 2250 = Rs 1,250.

Worked example 2. Three partners invest Rs 16,000, Rs 18,000 and Rs 23,000; profit Rs 19,380. Ratio 16:18:23, total 57 parts, each part = 19380/57 = Rs 340. Shares: Rs 5,440, Rs 6,120, Rs 7,820.

Compound partnership: capital times time

When partners join at different dates or for different durations, multiply each capital by its months before forming the ratio.

Worked example 3. Ajay advances Rs 1,200 for 4 months, Vijay Rs 1,400 for 8 months, Sanjay Rs 1,000 for 10 months; profit Rs 585. Capital-time weights: 4800, 11200, 10000. Divide by 400: 12:28:25. Total 65 parts, each = 585/65 = Rs 9. Shares: Rs 108, Rs 252, Rs 225.

Worked example 4. Arun starts with Rs 2,000; Babita joins after 3 months with Rs 4,000 (9 months); Chandan puts Rs 10,000 for 2 months only; profit Rs 5,600. Weights: 2000 x 12 = 24000, 4000 x 9 = 36000, 10000 x 2 = 20000. Ratio 24:36:20 = 6:9:5. Total 20 parts, each = 280. Shares: Rs 1,680, Rs 2,520, Rs 1,400. The trap is using 12 months for a partner who joined late.

Worked example 5. A invests Rs 1,600 for 9 months, B Rs 1,200 for 6 months, C Rs 800 for 12 months; profit Rs 2,600. Weights: 14400, 7200, 9600 = 6:3:4. Total 13 parts, each = 200. A's share = Rs 1,200.

The working partner's commission

A working partner is paid for management before the capital-time division. Subtract the commission first, then split the remainder.

Worked example 6. A (sleeping) puts Rs 40,000, B (working) puts Rs 50,000. B receives 25 percent of the profit for managing; the rest is divided by capital ratio. Total profit Rs 4,800. Commission = 1200. Remainder 3600 splits 4:5, so B gets 2000 more. B's total = Rs 3,200. The classic error is splitting first and adding the commission to the wrong base.

Finding the missing capital or the missing time

The ratio equation works backwards: if the profit ratio and two of the three quantities (capital, time) are known, solve for the third.

Worked example 7. A starts with Rs 35,000; after 5 months B joins; year-end profits are in the ratio 2:3. B's capital? 35000 x 12 : B x 7 = 2:3. Cross-multiplying: 3 x 420000 = 2 x 7B, so B = Rs 90,000.

Worked example 8. Aman invests Rs 60,000; Raju joins later with Rs 40,000; profits are 2:1. When did Raju join? Let Raju invest for m months: 60000 x 12 : 40000 x m = 2:1. So 720000 = 2 x 40000m, giving m = 9 months, meaning Raju joined after 3 months.

Worked example 9. X and Y start with investments in the ratio 6:7; profits are 2:3; X's money stayed 7 months. Y's months? 6 x 7 : 7 x Ty = 2:3 gives 42 x 3 = 14 x Ty, so Ty = 9 months.

Ratio puzzles: capital, time and profit all in play

Profit ratio = capital ratio x time ratio, so any one of the three ratios is the product or quotient of the other two.

Worked example 10. Capitals are in the ratio 1:2:3 and durations in the ratio 1:2:3. Profit ratio = 1x1 : 2x2 : 3x3 = 1:4:9.

Worked example 11. Capitals 2:5:7, profits 6:25:49. Duration ratio = profit/capital = 6/2 : 25/5 : 49/7 = 3:5:7.

Worked example 12. Capitals 5:6:8; after 4 months A increases his share by 50 percent; profit Rs 15,500. A's weight = 5 x 4 + 7.5 x 8 = 20 + 60 = 80; B = 6 x 12 = 72; C = 8 x 12 = 96. Ratio 80:72:96 = 10:9:12. Total 31 parts, each = 500. B's share = Rs 4,500. Mid-year changes mean splitting that partner's weight into two phases.

PYQ walkthroughs

Worked example 13. P puts Rs 14,000 more than Q; P invests for 8 months, Q for 10 months; P's share exceeds Q's by Rs 400 out of Rs 2,000 total (CSAT 2024). P gets Rs 1,200 and Q Rs 800, ratio 3:2. Let Q = x: 8(x + 14000) : 10x = 3:2. Cross-multiplying: 16x + 224000 = 30x, so 14x = 224000, x = 16000. P's capital = Rs 30,000.

Worked example 14. A invests 1/3 of the capital for 1/3 of the duration, B invests 1/4 of the capital for 1/4 of the duration, C invests the rest for the whole duration; profit Rs 17,000 (CSAT 2026). Weights: A = 1/3 x 1/3 = 1/9, B = 1/4 x 1/4 = 1/16, C = (1 - 1/3 - 1/4) x 1 = 5/12. As fractions of 144: 16:9:60. Total 85 parts, each = 200. C's share = 60 x 200 = Rs 12,000.

Traps that cost marks

First, using 12 months for a partner who joined late or left early: the duration is the actual months the money stayed. Second, applying the working partner's commission to the wrong base: it comes off the total profit before the capital-time split. Third, adding a mid-year capital change as if it applied all year: split the partner's timeline into phases. Fourth, in reverse questions, cross-multiplying the wrong pair: the equation is (C1 x T1)/(C2 x T2) = P1/P2, keep each partner's quantities together.

Speed tips for the exam hall

Write every partnership as a single line of capital-time products, then reduce the ratio by cancelling common factors before touching the profit figure. For missing-capital or missing-time questions, set up the proportion and cross-multiply: it is one equation, one unknown. For mid-year changes, break that partner into two virtual partners (before and after the change). And in the PYQ style with fractions of capital and duration, convert to a common denominator immediately, as in example 14.

Key Terms

• Sleeping partner is a partner who invests capital but takes no part in managing the business. His profit share comes purely from the capital-time ratio.
• Working partner is a partner who manages the business day to day. He is usually paid a commission or salary from the total profit before the remaining profit is split by the capital-time ratio.
• Capital-time product is a partner's investment multiplied by the number of months it stayed in the business, for example Rs 4,000 for 9 months = 36,000. Profit shares follow the ratio of these products.
• Compound partnership is a partnership where partners invest different capitals for different durations, so the profit ratio must use capital-time products rather than capitals alone.
• Profit-sharing ratio is the ratio in which the year's profit is divided among partners. Under the capital-time rule it equals the ratio of the partners' capital-time products.
• Management commission is a fixed share of profit paid to the working partner for running the business, deducted before the capital-time division of the remainder.