Partnership Problems: Method with Worked Examples

Partnership problems ask how to split a profit (or loss) between people who invested in the same business. The problems below apply one conditional rule under three stated assumptions: the agreement says profit is shared in the ratio capital x time for each partner, no partner draws a salary or interest unless the problem says so, and amounts and months are counted exactly as stated. Real agreements can set different terms; this article covers only the stated model.

The partnership rule

When the agreement says profit is shared by capital x time, each partner's share is proportional to the product of their capital and the time it stayed in the business. For partners 1 and 2, share 1 : share 2 = (capital 1 x time 1) : (capital 2 x time 2). When all partners invest for the same duration, the times cancel and the capitals alone set the ratio.

capital

the money a partner puts into the business, in rupees

time

how long that capital stays invested, in months here; use one time unit throughout a problem

share ratio

capital x time for each partner, reduced to simplest terms

Worked examples

Example 1: same duration, different capitals

A invests Rs 12,000 and B invests Rs 18,000 in a business for one year. The year's profit is Rs 7,500. Find each partner's share.

1. Same time, so the ratio is the capitals: 12,000 : 18,000 = 2 : 3.
2. Total parts: 2 + 3 = 5, so each part is 7,500/5 = Rs 1,500.
3. A gets 2 x 1,500 = Rs 3,000; B gets 3 x 1,500 = Rs 4,500.
4. Check: 3,000 + 4,500 = 7,500.

A receives Rs 3,000 and B receives Rs 4,500.

Example 2: different durations

A invests Rs 10,000 for 12 months and B invests Rs 15,000 for 8 months. The profit is Rs 9,000. Find the shares.

1. A's capital x time: 10,000 x 12 = 1,20,000.
2. B's capital x time: 15,000 x 8 = 1,20,000.
3. Ratio 1,20,000 : 1,20,000 = 1 : 1, so each gets 9,000/2 = Rs 4,500.
4. Check: 4,500 + 4,500 = 9,000.

Each partner receives Rs 4,500.

Example 3: three partners

A invests Rs 5,000 for 12 months, B invests Rs 8,000 for 9 months, and C invests Rs 10,000 for 6 months. The profit is Rs 8,000. Find the shares.

1. Capital x time: A 5,000 x 12 = 60,000; B 8,000 x 9 = 72,000; C 10,000 x 6 = 60,000.
2. Ratio 60,000 : 72,000 : 60,000 = 5 : 6 : 5; total parts 16, so each part is 8,000/16 = Rs 500.
3. A gets 5 x 500 = Rs 2,500; B gets 6 x 500 = Rs 3,000; C gets 5 x 500 = Rs 2,500.
4. Check: 2,500 + 3,000 + 2,500 = 8,000.

A receives Rs 2,500, B Rs 3,000 and C Rs 2,500.

Example 4: a partner joins late

A starts a business with Rs 20,000. After 4 months B joins with Rs 25,000. The year's profit is Rs 11,000. Find the shares.

1. A's capital works all 12 months: 20,000 x 12 = 2,40,000.
2. B's capital works 12 - 4 = 8 months: 25,000 x 8 = 2,00,000.
3. Ratio 2,40,000 : 2,00,000 = 6 : 5; total parts 11, so each part is 11,000/11 = Rs 1,000.
4. A gets 6 x 1,000 = Rs 6,000; B gets 5 x 1,000 = Rs 5,000.
5. Check: 6,000 + 5,000 = 11,000.

A receives Rs 6,000 and B receives Rs 5,000.

Example 5: a partner withdraws capital midway

A invests Rs 15,000 and withdraws Rs 5,000 after 6 months. B invests Rs 12,000 for the whole year. The year's profit is Rs 9,800. Find the shares.

1. A's capital: 15,000 for 6 months, then 10,000 for 6 months: 15,000 x 6 + 10,000 x 6 = 1,50,000.
2. B's capital x time: 12,000 x 12 = 1,44,000.
3. Ratio 1,50,000 : 1,44,000 = 25 : 24; total parts 49, so each part is 9,800/49 = Rs 200.
4. A gets 25 x 200 = Rs 5,000; B gets 24 x 200 = Rs 4,800.
5. Check: 5,000 + 4,800 = 9,800.

A receives Rs 5,000 and B receives Rs 4,800.

Summary

Rule

when the agreement says so, share ratio = capital x time for each partner

Same duration

the times cancel; capitals alone give the ratio

Joining late

count only the months the capital was actually invested

Withdrawing

split the year at the withdrawal and add capital x time for each stretch

Check

the shares must add back to the stated profit

Frequently asked questions

How is profit shared in a partnership?

When the agreement says profit is shared by capital x time, the shares follow that product for each partner, with no salary or interest drawn. The larger capital x time product takes the larger share.

What if all partners invest for the same time?

Under the capital x time agreement, the times cancel, so the profit is shared in the ratio of the capitals alone. For example, Rs 12,000 and Rs 18,000 invested for the same duration share profit in the ratio 2 : 3.

What if a partner joins late?

Count only the months that partner's capital was in the business. A capital invested 4 months after the start works for 8 months of a 12-month year.

What if a partner withdraws money midway?

Split the time at the withdrawal. Compute capital x months for the amount before the withdrawal and for the reduced amount after it, then add the two products.

Do these results decide real business payouts?

No. They are exact arithmetic under the stated assumptions. Real partnership deeds can set salaries, interest on capital, or fixed ratios, which change the shares; check the actual agreement.

Sources and scope

Method checked against CareerRide, Partnership, and LearnFrenzy, Partnership Important Formulas, both retrieved 15 November 2026. This article teaches arithmetic under stated assumptions only and makes no claim about any syllabus, board, class or real partnership agreement.

Related reading on this site: how to solve maths word problems, the Working with Fractions chapter page (shares are fractions of the profit), and maths tuition at EZ Tuitions.