About Partnership
A partnership question is a ratio question with one extra ingredient: time. Two partners who put in the same money for different lengths of time do not deserve the same share, so the profit is divided in proportion to capital multiplied by the period it was invested.
What you need to understand
- Profit is shared in the ratio of capital multiplied by time, not in the ratio of capital alone.
- A partner who joins part of the way through earns only for the months actually invested, not for the whole year.
- The equivalent capital of a partner is the capital multiplied by the number of months. Only this product can be compared with another partner.
- If the periods are equal the profit ratio reduces to the ratio of the capitals.
- If the capitals are equal the profit ratio reduces to the ratio of the periods.
- Read carefully whether a working partner also draws a salary, because that is settled before the profit is shared.
Formulas to remember
How to work through these questions
- Write down each partner equivalent capital as a single number before forming any ratio at all.
- Add those equivalent capitals to get the divisor.
- Multiply the total profit by the fraction belonging to the partner the question asks about.
- When the question gives one share and asks for the total, divide that share by that partner fraction of the whole.
- Check that the shares add back to the total profit, which catches nearly every slip.
Mistakes that cost marks
- Comparing the capitals and ignoring how long each was invested.
- Crediting a partner with a full twelve months when the question says the partner joined part way through.
- Dividing by the wrong equivalent capital when converting a share back into the total profit.
- Splitting the profit equally on the assumption that equal effort deserves equal reward.
- Swapping the ratio around so that the partner with the larger contribution receives the smaller share.
Worked example
A invests Rs. 20000 at the start of a year. After 4 months B joins with Rs. 15000. Find the ratio in which the year profit is shared.
- A has the money in for the whole twelve months, so the equivalent capital of A is 20000 x 12 = 240000.
- B is in for 12 - 4 = 8 months, so the equivalent capital of B is 15000 x 8 = 120000.
- The ratio is 240000 : 120000, which reduces to 2 : 1.
- Check: A put in more money and for longer, so A must take the larger share, and twice as much is reasonable.
Answer: 2 : 1
Practice questions with answers
A few Partnership questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
A puts Rs. 5000 into a business at the beginning of the year, and B puts in Rs. 4000 after 2 months. In what ratio are the profits shared?
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A
3 : 2
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B
5 : 2
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C
15 : 1
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D
2 : 3
Answer: Option A — with explanation
Profit is shared in the ratio of capital multiplied by the time it was invested.
A: 5000 x 12 = 60000
B: 4000 x 10 = 40000
So the ratio is 60000 : 40000 = 3 : 2.
Common mistakes
- gave B the whole year as well: 15 : 1
- inverted the ratio: 2 : 3
- added the two contributions when forming the first term: 5 : 2
Question 2
A invests Rs. 12000 for 11 months and B invests Rs. 12000 for 8 months in the same business. If the total profit is Rs. 228000, find A's share.
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A
96000
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B
114000
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C
165818.18
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D
132000
Answer: Option D — with explanation
Shares are in the ratio of capital multiplied by time.
A: 12000 x 11 = 132000; B: 12000 x 8 = 96000.
Total = 228000, so A's share = 228000 x 132000/228000 = 132000.
Common mistakes
- gave B's share of the profit: 96000
- split the profit equally and ignored the contributions: 114000
- divided the profit in the inverse ratio of the contributions: 165818.18
Question 3
A and B are partners with capitals of Rs. 11000 and Rs. 6000 for 3 months and 3 months respectively. A receives Rs. 66000 as his share of the profit. What was the profit of the business?
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A
187000
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B
36000
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C
132000
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D
102000
Answer: Option D — with explanation
The shares are in the ratio of capital multiplied by time.
A: 11000 x 3 = 33000; B: 6000 x 3 = 18000.
A's share is 33000/51000 of the profit, so total = 66000 x 51000/33000 = 102000.
Common mistakes
- divided by the wrong contribution when finding the total: 187000
- gave B's share instead of the total profit: 36000
- assumed the two partners took equal shares: 132000
Question 4
A puts Rs. 4000 into a business at the beginning of the year, and B puts in Rs. 18000 after 9 months. In what ratio are the profits shared?
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A
8 : 3
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B
17 : 9
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C
8 : 9
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D
9 : 8
Answer: Option C — with explanation
Profit is shared in the ratio of capital multiplied by the time it was invested.
A: 4000 x 12 = 48000
B: 18000 x 3 = 54000
So the ratio is 48000 : 54000 = 8 : 9.
Common mistakes
- inverted the ratio: 9 : 8
- added the two contributions when forming the first term: 17 : 9
- gave B the whole year as well: 8 : 3
Question 5
A and B are partners, A investing Rs. 9000 for 7 months and B investing Rs. 8000 for 9 months. Out of a profit of Rs. 135000, how much goes to A?
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A
67500
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B
63000
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C
72000
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D
154285.71
Answer: Option B — with explanation
Shares are in the ratio of capital multiplied by time.
A: 9000 x 7 = 63000; B: 8000 x 9 = 72000.
Total = 135000, so A's share = 135000 x 63000/135000 = 63000.
Common mistakes
- gave B's share of the profit: 72000
- divided the profit in the inverse ratio of the contributions: 154285.71
- split the profit equally and ignored the contributions: 67500
Frequently asked questions
Why is the capital multiplied by the time?
Because a partner with half the money invested for twice as long has contributed just as much as the other partner. The profit follows the contribution, so the two must be multiplied together.
What if two partners invest the same amount for different periods?
The profit is then shared in the ratio of the periods. The equal capitals cancel out of the comparison.
How do I find the total profit from one share?
Divide that share by that partner fraction of the total equivalent capital. If a share of Rs. 600 represents one third of the whole, the total profit is Rs. 1800.
What is a sleeping partner?
A partner who contributes capital but takes no part in running the business. The question will usually say whether such a partner also receives a salary, and that is paid before the profit is divided.
