Partnership problems are among the most scoring and formula-driven topics in Arithmetic. Yet many students get confused because profit sharing depends not only on capital but also on time, and sometimes on work done by partners. A small difference in investment duration or contribution can completely change the final profit distribution.
If you understand these three ideas:
✔ Capital invested
✔ Time for which the capital is used
✔ Working vs. Sleeping partners
Then you can solve ANY partnership question effortlessly.
This Partnership Problems covers formulas, concepts, shortcuts, cases, examples, FAQs, and exam tips, making it a complete resource for competitive exams.
Quick Overview: Partnership Formulas
| Concept / Situation | Equivalent Value Considered | Used For Calculation | Formula (With Meaning of Symbols Inside Row) |
|---|---|---|---|
| Partners investing for same time | Capital only | Profit ratio | Profit share = x : y (x = A’s capital, y = B’s capital) |
| Partners investing for different time | Capital × Time | Profit ratio | Profit share = xp : yq (p = time of A, q = time of B) |
| Working partner included | Capital + Work contribution | Salary + Profit share | Profit = Salary to working partner + Remaining profit in capital ratio |
| Sleeping partner | Capital only | Profit share | Sleeping partner gets profit only in investment ratio |
| Equivalent capital method | Capital × Time | For multiple partners | Equivalent capital of each = Capital × Time |
| Total profit distribution | Equivalent capital of all partners | Individual share | Share = (Individual Equivalent Capital ÷ Total) × Profit |
Formulas for Partnership
Understanding how profit is shared in a partnership requires only one idea profit is always divided in the ratio of (capital × time). But many students get confused because partners may invest at different times, invest different amounts, or take salaries as working partners.
So before using partnership-specific formulas, you must clearly understand when to use capital, when to include time, and when salary must be deducted.
Capital–Time Formulas (Foundation of Partnership Problems)
Every partnership problem simple or complex, is based on these three core relationships:
1. Profit depends on Capital
Higher the capital → higher the profit share.
2. Profit depends on Time
Longer the investment duration → larger the share.
3. Profit share ∝ (Capital × Time)
This is the universal rule.
Basic Profit Share Formula
Partner’s Profit Share = (Capital × Time/Total of all (Capital × Time))×Total Profit
This applies everywhere in simple partnership, compound partnership, working partner, or entry/exit type problems.
1. Simple Partnership (Equal Time Investment)
When all partners invest for the same duration, profit is divided based only on capital.
Formula: Profit Ratio (Same Time)
A′s Profit : B′s Profit = x : y
Where:
x = capital of A
y = capital of B
Why this formula works?
Because time is equal, only capital varies.
Profit depends on “who invested more money.”
Common mistakes students make:
- Using time even when time is same
- Forgetting to simplify capital ratio
- Assuming a larger time gives more profit (not applicable here)
Key tip:
Use this only when all partners invest for same time period.
2. Partnership With Different Time Periods (Capital–Time Method)
If partners invest for different durations, time affects profit directly.
So profits depend on Equivalent Capital, calculated as:
Equivalent Capital = Capital×Time
Formula: Profit Ratio (Different Time)
A′s Profit : B′s Profit = xp : yq
Where:
x = capital of A
p = months invested by A
y = capital of B
q = months invested by B
Why do we multiply (capital × time)?
Because a partner investing less capital for a longer duration may still contribute more overall.
Common errors
- Not converting years to months
- Using time in years for one partner and months for another
- Forgetting that profit depends on BOTH factors
Key tip:
Whenever partners join/leave at different times → Always use capital × time.
3. Working Partner Formula (Salary + Profit Share)
A working partner takes an active role, and receives a salary, commission, or bonus for managing the business.
Formula for Working Partner Profit Calculation
- Subtract salary from total profit
- Divide remaining profit in the ratio of (capital × time)
Why salary is subtracted first?
Because salary is a fixed payment for managing work.
Profit sharing happens after salary is paid.
Common mistakes
- Adding salary inside the profit ratio
- Splitting salary between partners
- Dividing salary + profit together
Key tip:
Working partner ALWAYS gets salary separately.
4. Sleeping Partner Formula
A sleeping partner only invests money. He does not manage the business.
Formula:
Sleeping partner gets:
Profit Share = Capital-Time Ratio Only
Why no salary?
Because they only invest, not manage.
Common mistakes
- Thinking sleeping partner also gets salary
- Dividing salary among both partners
- Ignoring time contribution
Key tip:
Sleeping partner = investment only, no extra income.
5. Equivalent Capital Method (For Any Number of Partners)
This method is ideal for 3 or more partners or when multiple time entries exist.
Formula:
Equivalent Capital = Capital×Time
Then:
Profit Share = (Equivalent Capital/Total)×Profit
Why use this method?
It standardises all investments into a single comparable unit.
Common errors
- Forgetting to convert all time into same unit
- Missing individual entry/exit times
- Treating time as irrelevant
Key tip:
Use this method for 3+, entry/exit, or time-changed investments.
6. General Profit Distribution Formula (Universal Formula)
This is the ultimate formula used in all partnership questions.
Formula:
Partner’s Share=(Capital × Time/Sum of all (Capital × Time))×Total Profit
Why this formula works?
It combines both factors fairly:
- More capital → more profit
- More time → more profit
Common mistakes
- Forgetting to multiply capital and time
- Dividing profit only by capital
- Mixing months and years
- Ignoring working partner salary
7. Entry and Exit Formula (When Partners Join or Leave Mid-Year)
If someone joins late or leaves early, break the timeline and calculate separately.
Formula Pattern
Capital × Active Months
Sum each segment to get total equivalent capital.
Why needed?
Because profit is based only on the time a partner was in the business.
Common mistakes
- Using total year for all partners
- Ignoring mid-year entries
- Not splitting time segments correctly
Smart Tips and Practical Tricks for Solving Partnership Problems
Mastering Partnership becomes simple when you understand how capital, time, and role of partners work together. Most students make mistakes not because formulas are difficult, but because they apply them without understanding which factor actually affects profit. This section breaks down the most important concepts into clear, actionable tips so you can solve partnership questions faster and more accurately.
1. Always Use Capital × Time to Find Equivalent Investment
Many questions involve partners investing for different durations. If you don’t adjust for time, your profit ratio will always be wrong.
Equivalent Capital Formula
Equivalent Capital = Capital×Time
Example:
A invests ₹10,000 for 12 months → 120,000
B invests ₹20,000 for 6 months → 120,000
Profit ratio = 1 : 1
Even though B invested double the capital, shorter duration makes them equal.
This one step improves accuracy in almost every partnership question.
2. Identify if Duration is Same or Different
Students often ignore time completely.
- Same time → Use only capital ratio
- Different time → Use capital × time
Common cases where time differs:
✔ One partner joins later
✔ One partner withdraws capital early
✔ Partners invest in multiple phases
✔ Capital is increased mid-year
Recognizing the situation instantly helps you choose the correct formula.
3. Separate Salary Before Dividing Profit (Working Partner Rule)
If working partner gets salary/commission:
- Subtract salary from total profit
- Divide remaining profit using capital/time ratio
Example:
Total profit = ₹1,00,000
Salary to A = ₹10,000
Remaining = ₹90,000
Profit then divided according to investment.
Many students incorrectly add salary inside the ratio; do NOT do this.
4. Draw a Simple Timeline Diagram
A very underrated but powerful trick.
- Mark months: 1 to 12
- Note when each partner enters or exits
- Multiply capital × time in each segment
This visual approach prevents most mistakes related to time.
5. Focus on Patterns Instead of Memorizing Formulas
Partnership questions follow repeated patterns:
- Same investment time → capital ratio
- Different time → capital × time
- Working partner → salary first
- Sleeping partner → profit only
- Multiple partners → equivalent capital method
- Entry/exit → break into time segments
All questions come from ONE base idea:
Profit share ∝ Capital×Time
If you identify the pattern correctly, solving becomes automatic.
6. Double-Check Time Units Before Calculating
Many errors occur because students mix:
- Years with months
- Months with days
Always convert everything to ONE unit.
Most commonly: months.
Checklist:
✔ Did you convert years to months?
✔ Did you multiply capital by correct time?
✔ Did you adjust for entry/exit time?
7. Practice Typical Exam Patterns
Partnership questions appear frequently in:
- SSC (CGL, CHSL, GD, CPO)
- Banking (IBPS, SBI, RBI)
- Railway RRB
- UPSC CSAT
- State PSC
- Campus aptitude tests
The more patterns you practice, the faster you identify the formula needed.
FAQs About Partnership
Q1. Why do we use capital × time in partnership problems?
Because profit depends on both how much money is invested and for how long.
Example:
A: 10,000 for 12 months → 120,000
B: 20,000 for 3 months → 60,000
A gets double B’s profit even with smaller capital.
Q2. Why is time converted into months in partnership questions?
Months give more precise calculation. Years, days, and months mixed together often create errors.
Example:
2 years = 24 months
6 months = 6 months
Ratio = easy to handle.
Q3. What is the role of a working partner?
A working partner manages business operations and gets salary.
A sleeping partner only invests and gets profit based on capital.
Example:
A (working) invests ₹50,000
B (sleeping) invests ₹50,000
A gets salary + profit.
B gets only profit.
Q4. Why do we subtract salary before dividing profit?
Salary is a fixed expense. Profit sharing happens on the remaining amount.
Example:
Profit = 1,00,000
Salary = 20,000
Remaining = 80,000
→ Divide 80,000 as per ratio.
Q5. Does investing more capital always give more profit?
Only when time is same. If time differs, smaller investment for long duration may win.
Q6. Can a sleeping partner ever get more profit than a working partner?
Yes, if the sleeping partner invests significantly higher capital. Working partner salary does not automatically give higher profit.
Q7. Why do students get partnership ratios wrong?
Main mistakes:
✔ Not multiplying capital × time
✔ Not converting years to months
✔ Adding salary inside ratio
✔ Ignoring entry/exit time
✔ Using capital ratio instead of equivalent capital
Q8. How do partnership questions help in competitive exams?
They test ratio, time, percentage, and logical application core aptitude skills. They also improve understanding of proportional distribution.
Q9. What is the simplest way to master partnership?
Follow these 3 steps:
- Convert all time into months
- Multiply capital × time
- Form the ratio and divide profit
Everything else becomes easy.
Q10. Why is regular practice important for partnership?
Because mistakes come from small oversight time units, salary, entry/exit. Practice helps you instantly pick correct method and avoid confusion.