Concept 1
Profit follows the investment agreement
A partnership question usually asks how a common profit or loss is divided. When no special condition is stated, each partner's share is proportional to the capital invested and the length of time that capital remained in the business.
Core rule: effective investment = capital × time. Profit ratio = ratio of effective investments.
Concept 2
Equal investment periods reduce to a capital ratio
If A and B invest ₹30,000 and ₹45,000 for the same period, their profit ratio is 30000:45000 = 2:3. From a profit of ₹25,000, A receives ₹10,000 and B receives ₹15,000.
Use share = total distributable profit × partner's ratio part ÷ sum of all ratio parts.
Concept 3
Different periods require capital × time
A invests ₹40,000 for 12 months and B invests ₹60,000 for 8 months. Their effective investments are 480000 and 480000, so profit is shared equally.
| Partner | Capital | Months | Capital × months |
|---|---|---|---|
| A | ₹40,000 | 12 | 480,000 |
| B | ₹60,000 | 8 | 480,000 |
Concept 4
Break changing capital into time blocks
If capital is added or withdrawn, calculate each block separately and add the products. A invests ₹20,000 for 5 months, then ₹30,000 for 7 months: effective investment = 20000×5 + 30000×7 = 310,000 capital-months.
A partner joining after four months of a one-year business invests for 8 months, not 4. A partner leaving after seven months contributes for 7 months.
Concept 5
Salary or commission comes before the residual ratio
If a working partner receives a salary or commission, first deduct that amount from total profit unless the wording says it is included in the ratio. Divide the remaining profit by the capital-time ratio, then add the special payment to that partner.
Example: Profit ₹42,000; A receives salary ₹6,000; residual ratio A:B = 2:1. Remaining profit is ₹36,000, so A gets ₹24,000 + ₹6,000 = ₹30,000 and B gets ₹12,000.
Concept 6
Use share ratios to recover an unknown
If profit shares are known, set the corresponding capital-time products in that ratio. If A invests ₹24,000 for 12 months and B invests x for 8 months while profits are 3:2, then 24000×12 : 8x = 3:2, giving x = ₹24,000.
When one share and total profit are known, find the ratio-unit value first; when one investment or duration is unknown, form a proportion from the effective investments.
Worked methods
Examples with reasoning
Equal periods
A and B invest ₹36,000 and ₹48,000. Ratio = 3:4. From ₹28,000, their shares are ₹12,000 and ₹16,000.
Partner joins later
A invests ₹50,000 for 12 months; B invests ₹75,000 after 4 months, so for 8 months. Ratio = 600000:600000 = 1:1.
Capital withdrawn
A invests ₹40,000 for 6 months, then ₹20,000 for 6 months: 360000. B invests ₹30,000 for 12 months: 360000. They share equally.
Three partners
A, B and C invest ₹20,000, ₹30,000 and ₹40,000 for 12, 8 and 6 months. Products are all 240000, so the ratio is 1:1:1.
Working-partner commission
Profit ₹55,000; A's commission ₹5,000; remaining ratio 3:2. Residual ₹50,000 gives ₹30,000 and ₹20,000; final shares are A ₹35,000 and B ₹20,000.
Exam traps
Errors that change the profit ratio
Using only capital
Capital alone works only when investment periods are equal.
Reversing elapsed time
Joining after 3 months in a 12-month business means investing for 9 months.
Ignoring capital changes
Split the year at every addition or withdrawal and sum all capital-time blocks.
Dividing gross profit too early
Deduct salary or commission before dividing the residual profit when required.
Mixing time units
Use the same unit—normally months—for every partner.
Forgetting total ratio parts
A 2:3 share divides profit into five parts, not three.
Original practice
Check understanding after learning
Original syllabus-aligned questions—not claimed previous-year questions.
Rapid revision
The one-minute sheet
- Equal time: profit ratio = capital ratio.
- Different time: profit ratio = capital × time ratio.
- Joining after m months in a year means investment for 12−m months.
- Split changing capital into blocks and add the products.
- Handle salary or commission before dividing residual profit.
- Partner's share = distributable profit × ratio part ÷ total parts.
Source ledger
Official foundations
- SSC CGL official syllabus: Partnership Business under Mathematical Abilities
- NCERT official mathematics exemplar: ratio and proportion foundation
- NCERT official textbook portal
Explanations, worked examples and practice are original. Official references are credited and not reproduced. Drafted 25 September 2026.
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