RRB JE Junior Engineer · Quantitative Aptitude
Partnership
Distribution of profit among partners based on their capital invested and time period.
Five concepts. Partnership is ratio with a clock attached: every partner's weight is capital multiplied by the months it stayed invested, and profit (or loss) divides in that ratio. Equal durations collapse the clock away; a working partner's salary is cut off the top before the ratio ever runs; and anyone who joins or leaves mid-year must be split into time-segments before the products are formed.
- RRB JE Junior Engineer
- Easy level
- 5 concepts
- 42 practice questions
1Profit is shared by capital × time
A partner who puts money in for longer earns proportionally more, so the fair weight for each partner is capital multiplied by the time it stayed invested — not capital alone. Two partners with the same capital × months product have equal claim on the profit, even when one put in more cash for fewer months.
That product is also called the equivalent capital: investing C for t months is the same claim as investing Ct for one month. Reducing every partner to a single capital × months number is the whole method; the profit ratio then falls out as the ratio of those numbers.
Figure. The two capital × months products are drawn to the same scale and come out equal, which is why A and B share the profit 1 : 1 even though B put in more cash. The figure is the products, not the capitals — comparing the cash bars instead would have shown 12000 against 18000 and taught the wrong ratio.
How it works
- Form each weightWrite capital × months (or days — any common unit) for every partner.
- Read the ratioThe profit ratio is exactly the ratio of those products; cancel a common factor if it helps.
- Hand out the profitEach share is (that partner's weight / sum of weights) × total profit.
Higher capital, shorter time
A invests ₹12000 for 12 months and B invests ₹18000 for 8 months. If the total profit is ₹2700, find B's share.
- A's weight = 12000 × 12144000
- B's weight = 18000 × 8144000
- Profit ratio A : B = 144000 : 1440001 : 1
- B's share = (1/2) × 2700₹1350
Pro tip. Higher capital for a shorter time can equal lower capital for a longer time — always multiply before you compare. Here the products matched, so the profit split evenly even though the cash put in did not.
A invests ₹6000 for 8 months and B invests ₹9000 for 4 months. Profit is ₹2100. A's share is
- ₹1200
- ₹840
- ₹1050
Weights are 6000 × 8 = 48000 and 9000 × 4 = 36000, so the ratio is 4 : 3 and A's share is (4/7) × 2100 = ₹1200. Sharing by capital alone (2 : 3) gives ₹840. Splitting the profit in half gives ₹1050. The months are different, so capital alone is the wrong ratio.
2Equal time collapses to the capital ratio
When every partner stays invested for the same duration, the common time factor cancels out of every weight, and the profit ratio is simply the capital ratio. That is a simple partnership: form C1 : C2 : C3 and share.
The trap is doing the multiplication anyway with mismatched units, or — more often — treating an equal-time problem as if the times still differed because the question mentioned a year. If the durations match, skip the time factor entirely and share by capitals alone.
Figure. With time cancelled, the profit bars are just the capital parts 5 and 7. Of 12 parts, P takes 5 — and on a ₹3600 profit that is ₹1500. The figure has no time axis because equal duration leaves nothing for one to show.
How it works
- Confirm equal durationSame start and same end for every partner — including "for the year" said of all of them.
- Write the capital ratioReduce C1 : C2 : C3 by any common factor.
- Share the profitEach share is (that capital part / sum of parts) × total profit.
Same period, different capital
P and Q start a business investing ₹5000 and ₹7000 for the same period. If the profit is ₹3600, find P's share.
- Same time, so profit ratio = capital ratio5000 : 7000
- Reduce 5000 : 70005 : 7
- Total parts = 5 + 712
- P's share = (5/12) × 3600₹1500
Pro tip. When durations match, the time factor is a common multiplier you can cancel before writing anything down. Re-introducing "× 12 months" on both sides changes no ratio and only invites an arithmetic slip.
A and B invest in the ratio 3 : 5 for the same period. Profit is ₹2400. A's share is
- ₹900
- ₹1440
- ₹1200
Equal time, so the profit ratio is 3 : 5 and A's share is (3/8) × 2400 = ₹900. Giving A the larger part (5/8) yields ₹1440. Splitting evenly yields ₹1200. The smaller capital takes the smaller share.
3Unequal durations — multiply first, then share
When the durations differ, the partnership is compound: compute each partner's capital × months first, then share the profit in the ratio of those products. This is the most tested variant, because the capital ratio and the profit ratio are no longer the same number.
Writing the capital ratio and stopping is the standard miss. The months are not decoration — they are the other half of every weight — and a partner who invested for half as long has half the claim of the same capital left in for the full term.
Figure. The bars are the capital × months products 96000 and 72000 — ratio 4 : 3 — not the capitals 8000 and 12000. Drawing capital bars instead would reverse which partner looks larger and would hand the student the wrong ratio.
How it works
- List capital and monthsOne row per partner; do not skip a partner whose capital looks "obvious".
- MultiplyForm each capital × months product before writing any ratio.
- Share by the productsReduce the product ratio, then take (part / sum) × profit.
Products that do not match
A invests ₹8000 for 12 months and B invests ₹12000 for 6 months. If the total profit is ₹4200, find each partner's share.
- A's weight = 8000 × 1296000
- B's weight = 12000 × 672000
- Ratio 96000 : 720004 : 3
- A = (4/7) × 4200; B = (3/7) × 4200₹2400 and ₹1800
Pro tip. B put in more cash and still takes the smaller share, because B's money worked for half as long. If your answer has the larger capital taking the larger share on an unequal-time question, you probably shared by capital alone.
A invests ₹6000 for 10 months and B invests ₹9000 for 5 months. Profit is ₹3500. A's share is
- ₹2000
- ₹1400
- ₹1750
Weights are 6000 × 10 = 60000 and 9000 × 5 = 45000, so the ratio is 4 : 3 and A's share is (4/7) × 3500 = ₹2000. Sharing by capital alone (2 : 3) gives A ₹1400. An even split gives ₹1750. The products, not the capitals, set the ratio.
4Working-partner salary comes off the top
A working partner may take an agreed salary or commission before profit is divided. That fixed amount is deducted from the total profit first; only the remainder is shared in the capital × time ratio. The sleeping partners never touch the salary line.
Apply the ratio to the reduced profit, then — if the question asks for the working partner's total receipt — add the salary back onto that partner's share. Forgetting the deduction shares a pot that has already been partly spent.
Figure. One bar is the whole profit. The salary segment is cut off first; only the remaining length is split 2 : 3 between A and B. A's total receipt is the salary segment plus A's share segment — ₹500 + ₹1200 — not the share alone.
How it works
- Deduct the salaryRemaining profit = total profit − working partner's fixed salary or commission.
- Share the remainderDivide what is left by the capital × time ratio (or capital ratio, if times match).
- Add salary back if askedThe working partner's total receipt is salary plus that partner's share of the remainder.
Salary, then the ratio
A (working partner) and B invest ₹10000 and ₹15000 for the same period. The year's profit is ₹3500, of which A takes ₹500 as salary before the rest is shared. Find B's share and A's total receipt.
- Remaining profit = 3500 − 500₹3000
- Capital ratio A : B (equal time)2 : 3
- B's share = (3/5) × 3000₹1800
- A's total = 500 + (2/5) × 3000₹1700
Pro tip. Deduct any fixed salary or commission before the capital-time ratio runs. Sharing ₹3500 in 2 : 3 would give B ₹2100 — ₹300 too much — because that calculation spends the salary twice, once as A's extra and once inside the pool.
A working partner takes ₹400 salary from a ₹1900 profit; capitals are 2 : 3 for equal time. The other partner's share is
- ₹900
- ₹1140
- ₹760
Remainder 1900 − 400 = 1500, shared 2 : 3, so the other partner gets (3/5) × 1500 = ₹900. Sharing the full ₹1900 in 2 : 3 gives ₹1140. Giving that partner the working partner's part of the remainder gives ₹760. Salary comes off before the ratio.
5Join or leave mid-year — segment the months
If a partner joins or leaves mid-year, that partner's money was not invested for the whole term. Split the investment into time-segments: the months before the change at one capital, the months after at the other, and add the capital × months products. Partners who stay the whole year contribute one product; partners who move contribute a sum of products.
The usual error is to give the late joiner a full-year weight, or to start the clock from January for money that arrived in May. Count only the months the cash was actually in.
Figure. A's bar runs the full year; B's bar starts when B joins and covers only the remaining eight months. The weights are capital times those drawn lengths — 12000 × 12 against 9000 × 8 — not capital times twelve for both.
How it works
- Mark the calendarNote who is in for which stretch of the year — whole year, from month k, until month m.
- Build each productWhole-year partners: capital × 12. Late joiners: capital × months present. Add segments if capital itself changes.
- Share by the totalsSum every partner's products and divide the profit in that ratio.
B joins after four months
A invests ₹12000 for the whole year. B joins after 4 months with ₹9000. Year-end profit is ₹5400. Find each partner's share.
- A's weight = 12000 × 12144000
- B's months present = 12 − 4; weight = 9000 × 872000
- Ratio 144000 : 720002 : 1
- A = (2/3) × 5400; B = (1/3) × 5400₹3600 and ₹1800
Pro tip. "Joins after 4 months" means 8 months of investment, not 4. Counting the gap instead of the presence is a reliable way to invert the ratio.
A invests ₹8000 for the whole year. B joins after 6 months with ₹8000. Profit is ₹3600. B's share is
- ₹1200
- ₹1800
- ₹2400
Weights are 8000 × 12 and 8000 × 6, so the ratio is 2 : 1 and B takes (1/3) × 3600 = ₹1200. An even split gives ₹1800. Giving B the larger share (2/3) gives ₹2400 — the result of counting B's absence as presence, or of swapping the ratio.
Notes
- Profit-Sharing Principle: Profit is shared in the ratio of each partner's 'capital \times time' contribution, since money invested longer earns proportionally more.
- Simple Partnership: When all partners invest for the same duration, profit divides simply in the ratio of their capitals.
- Compound Partnership: When durations differ, first compute each partner's capital \times months, then share profit in that ratio — this is the most tested variant.
- Working vs Sleeping Partner: A working partner may take an extra salary/commission off the top before the remaining profit is divided by capital ratio.
- Equivalent Capital: A partner who invests C for t months is equivalent to one investing Ct for one month, letting you reduce unequal-time problems to a single ratio.
Formulas
- Profit ratio (equal time): C_1 : C_2 : C_3
- Profit ratio (unequal time): C_1 t_1 : C_2 t_2 : C_3 t_3
- Partner's share = \frac{\text{his capital}\times\text{time}}{\text{total of all such products}}\times\text{Total Profit}
- After working-partner salary s: remaining (P-s) shared by capital-time ratio
Exam traps & shortcuts
- Reduce each partner to a single 'capital × months' number first; the profit ratio then falls out immediately.
- If someone joins or leaves mid-year, split their investment into time-segments and add the products.
- Deduct any fixed salary/commission of a working partner before applying the capital-time ratio to the rest.
Reference tables
The same capitals produce different profit ratios once the months stop matching. Read across a row: equal time keeps the capital ratio; unequal time replaces it with the product ratio.
| Situation | Weights | Profit ratio | Watch for |
|---|---|---|---|
| Equal time (simple) | C1, C2, C3 | C1 : C2 : C3 | Cancel the common duration; do not re-multiply |
| Unequal time (compound) | C1 t1, C2 t2, C3 t3 | C1 t1 : C2 t2 : C3 t3 | Capital ratio ≠ profit ratio |
| Working-partner salary s | ratio on (P − s) | same capital-time ratio on the remainder | Add s back only to the working partner's total |
| Join / leave mid-term | sum of segment products | ratio of those totals | Count months present, not months absent |
Every line reconstructs from the concepts above; the sheet is for the night before, not a substitute for the method.
| Quantity | Relation | Watch for |
|---|---|---|
| Profit ratio (equal time) | C1 : C2 : C3 | Time cancels; capitals alone |
| Profit ratio (unequal time) | C1 t1 : C2 t2 : C3 t3 | Multiply before comparing |
| Partner's share | (Ci ti / Σ Cj tj) × Total profit | Same form for loss |
| After working-partner salary s | share from (P − s) by the capital-time ratio | Deduct s before the ratio runs |
| Equivalent capital | C for t months ≡ Ct for 1 month | Reduce every partner to one product first |
Recap
Read only this the night before.
- Weight
- Every partner's claim is capital × months. Equal products mean equal shares, even when the cash put in differs.
- Simple
- Same duration for all → profit ratio = capital ratio. Skip the time factor.
- Compound
- Different durations → multiply first. The capital ratio is not the answer.
- Salary
- Working-partner salary or commission comes off the top; share only the remainder, then add salary back to that partner's total if asked.
- Mid-year
- Joining after k months means (12 − k) months invested. Segment, multiply, add, then share.
Practise Partnership
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