Partnership Problems: Sharing Profit by Capital × Time
Profit ratio = capital × time • Partner joins late • Capital withdrawn midway • Finding the unknown time • Working partner's share
SSCBankingRailway📅 Last updated: 2026-10-20
In a partnership, two or more people put money into a business and share the profit or loss. The fair share depends on how much each partner invested and for how long that money was working. That gives the rule behind every question on this topic:
Profit ratio = capital × time, worked out for each partner.
If every partner invests for the same period, time cancels out and the profit is simply shared in the ratio of capitals. If the periods differ, multiply each capital by its number of months first. Use months throughout, because most questions give joining or leaving dates in months.
Worked example 1: same period
A and B invest ₹30,000 and ₹45,000 for a year, and the profit is ₹25,000. The ratio is 30 : 45 = 2 : 3, so A gets 2/5 × 25,000 = ₹10,000 and B gets ₹15,000.
When the periods differ, a smaller sum kept in for longer can earn as much as a larger sum kept in briefly. Multiply first, then simplify the ratio.
Worked example 2: more money, less time
A invests ₹40,000 for 12 months and B invests ₹60,000 for 8 months. 40,000 × 12 = 4,80,000 and 60,000 × 8 = 4,80,000, so the ratio is 1 : 1 and they share the profit equally.
A partner who joins after k months has money in the business for only (12 − k) months of the year. This is the most common pattern in exams, and the most common error is to multiply by k instead of 12 − k.
Worked example 3: a late joiner
A starts with ₹20,000. B joins after 3 months with ₹30,000. At the year end the profit is ₹17,000. A: 20,000 × 12 = 2,40,000. B: 30,000 × 9 = 2,70,000. The ratio is 8 : 9, so B gets 9/17 × 17,000 = ₹9,000.
When a partner adds or withdraws money partway, split the year into periods with constant capital. Multiply each period's capital by its months and add the results.
Worked example 4: a withdrawal
A invests ₹50,000 and withdraws ₹20,000 after 6 months. B invests ₹60,000 for the whole year. The profit is ₹30,000.
A: 50,000 × 6 + 30,000 × 6 = 4,80,000. B: 60,000 × 12 = 7,20,000. The ratio is 2 : 3, so A gets ₹12,000 and B gets ₹18,000.
Reverse questions give the profit ratio and ask for a missing time or capital. Set capital × time for each partner equal to the given ratio and solve for the unknown.
Worked example 5: how long was B in?
A invests ₹12,000 for the year, B invests ₹16,000, and they share profit 3 : 2. So 12,000 × 12 : 16,000 × t = 3 : 2. Cross-multiplying, 1,44,000 × 2 = 3 × 16,000 × t, so t = 6 months.
When capitals are given as fractions such as 1/2 : 1/3 : 1/4, multiply each by the LCM of the denominators to get whole numbers before sharing.
Worked example 6: fractional capitals
Capitals in the ratio 1/2 : 1/3 : 1/4 become 6 : 4 : 3 after multiplying by 12. Out of a profit of ₹26,000, the first partner gets 6/13 × 26,000 = ₹12,000.
A working partner manages the business and is often paid a salary or a fixed percentage of the profit before the rest is shared. A sleeping partner only invests. Always take out the working partner's amount first, then divide what is left in the capital × time ratio.
Worked example 7: salary first
A is a working partner and gets 10% of the profit for managing the business. The rest is shared 3 : 2. Out of ₹50,000, A's salary is ₹5,000. The remaining ₹45,000 gives A ₹27,000 and B ₹18,000, so A receives ₹32,000 in all.
✘ Multiplying a late joiner's capital by the months they missed instead of the months they stayed.
✘ Sharing the whole profit by ratio before taking out a working partner's salary.
✘ Mixing years and months for different partners.
✘ Forgetting to clear fractions before adding the parts of a ratio.
Q1. A and B invest ₹24,000 and ₹36,000 for the same period. Out of a profit of ₹15,000, what is B's share? easy
✅ Answer: (d) ₹9,000
💡 Explanation: For equal periods the profit is shared in the ratio of capitals, 24,000 : 36,000 = 2 : 3. B's share is 3/5 × 15,000 = ₹9,000.
Q2. A invests ₹50,000 for 12 months and B invests ₹75,000 for 6 months. What is A's share of a profit of ₹28,000? easy
✅ Answer: (b) ₹16,000
💡 Explanation: Multiply capital by time: 50,000 × 12 = 6,00,000 and 75,000 × 6 = 4,50,000, a ratio of 4 : 3. A's share is 4/7 × 28,000 = ₹16,000.
Q3. A starts a business with ₹18,000. B joins 4 months later with ₹24,000. What is B's share of the year's profit of ₹34,000? medium
✅ Answer: (a) ₹16,000
💡 Explanation: A's capital works for 12 months and B's for 8. 18,000 × 12 : 24,000 × 8 = 2,16,000 : 1,92,000 = 9 : 8. B gets 8/17 × 34,000 = ₹16,000.
Q4. A invests ₹15,000 for a year and B invests ₹20,000. The year's profit is shared 9 : 8. For how many months was B's money in the business? medium
✅ Answer: (c) 8 months
💡 Explanation: 15,000 × 12 : 20,000 × t = 9 : 8. So 1,80,000 × 8 = 9 × 20,000 × t, which gives t = 8 months.
Q5. A, B and C invest capitals in the ratio 1/3 : 1/4 : 1/5 for the same period. What is C's share of a profit of ₹9,400? hard
✅ Answer: (d) ₹2,400
💡 Explanation: Multiply by the LCM 60 to clear the fractions: 20 : 15 : 12, a total of 47 parts. C's share is 12/47 × 9,400 = ₹2,400.
Q6. A is a working partner and takes 20% of the profit as salary. The rest is shared equally with B. What does A receive in total from a profit of ₹40,000? medium
✅ Answer: (b) ₹24,000
💡 Explanation: A's salary is 20% of 40,000 = ₹8,000. The remaining ₹32,000 is split equally, giving A another ₹16,000, a total of ₹24,000.
Why do we multiply capital by time?
Profit is earned by money that is working in the business. ₹10,000 for 6 months does the same work as ₹5,000 for 12 months, so capital × time measures each partner's real contribution, and profit is shared in that ratio.
What if a partner's capital changes several times?
Split that partner's period into pieces with constant capital. Multiply each piece's capital by its months and add the products. That total is the partner's figure in the profit ratio.
All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.
How this page was prepared. Drafted with AI assistance (Claude (Anthropic AI model)) from the sources listed above. 13 calculations on this page were recomputed by software and matched. Automated checks also covered structure, internal links and duplicate text. Software checks cannot confirm facts. Reviewed by praveenkumar on 2026-10-20 against the sources listed above.
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