Aptitude · Partnerships

Half the money for twice as long is the same claim

Profit is not shared in the ratio of what the partners put in. It is shared in the ratio of capital multiplied by the months that capital stayed in. Every one of the five models in this chapter is that single product with a different timeline attached.

Set two capitals and two timelines, then walk the split →

Profit ratio = capital × time, never capital alone. The unit is the capital-month, and it behaves exactly like the person-day in Time and Work.

01 The idea

Rent for a shared room

You and a friend take one room together. You move in on 1 January and stay all twelve months. Your friend moves in on 1 July and stays six. When the bill comes, nobody argues for a 50–50 split. Your friend used the room for half as long, so your friend owes half as much.

A business is that arrangement in reverse. Partners put money in instead of moving in, and the bill becomes the profit to be handed out. What each partner earns a claim on is not the money they contributed but the money multiplied by the number of months it stayed inside the business.

So ₹6,000 left in for twelve months and ₹12,000 left in for six months are equal claims. Both products come to 72,000. On an ₹18,000 profit each partner takes ₹9,000, even though one of them put in twice as much money as the other. Read the capitals alone and you would have said ₹6,000 and ₹12,000, which is a different answer to a different question.

That product deserves a name you say out loud: the capital-month. One rupee inside the business for one month is one capital-month, and a partner’s claim is the count of them. If you have done Time and Work this is familiar furniture — there, five people for eight days and ten people for four days are the same forty person-days. Here, ₹6,000 for twelve months and ₹12,000 for six are the same 72,000 capital-months. Same structure, different nouns.

Multiply each partner’s capital by the months it was inside the business, and share the profit in the ratio of those products.

Capital-monthOne rupee inside the business for one month. A partner supplies capital × months of them, and that count is the claim. It is the partnership twin of the person-day.

Profit ratioThe ratio the profit is divided in. It always equals the ratio of capital-months: C₁T₁ : C₂T₂ — never the ratio of the capitals, unless the times happen to be equal.

One partThe profit divided by the total number of ratio parts. Every share is a whole number of parts, which is why the parts method beats taking fractions of the profit.

02 Worked example

₹6,000 for a year against ₹12,000 for six months

This sum runs through the whole lesson. A puts in ₹6,000 and leaves it for all twelve months. B puts in ₹12,000 and leaves it for six. The year’s profit is ₹18,000. Find each share.

Partner	Capital	Months in	Capital-months
A	₹6,000	12	72,000
B	₹12,000	6	72,000
Total	—	—	144,000
Capitals alone	6,000 : 12,000	ignored	gives 1 : 2, not 1 : 1

Count A’s capital-monthsCapital times months in. A left ₹6,000 for the full year.6,000 × 12 = 72,000 capital-months

Count B’s capital-monthsTwice the money for half the time. Watch what the product does.12,000 × 6 = 72,000 capital-months

Form the ratio and count the partsThe two products are identical, so the claims are identical. Cancel to whole numbers and add them.72,000 : 72,000 = 1 : 1 ⇒ 2 parts

Resist the ratio of the capitalsThe capitals read 6,000 : 12,000, which is 1 : 2. That is the answer to the question where both partners stayed the same length of time — and they did not.1 : 2 would pay A 6,000, short by ₹3,000

Divide the profit once, then multiply outOne part is the profit over the parts. Both shares come from it, and they must add back to ₹18,000.1 part = 18,000 / 2 = 9,000 ⇒ A = ₹9,000, B = ₹9,000

The capital-month is the quantity the whole chapter conserves. Once you have counted them, the rest is a ratio and a division, and it does not matter whether the difference in months came from joining late, leaving early, or adding money halfway through. Every model from here changes only how you count.

03 The method

The golden equation, and the parts shortcut

One equation, and one habit of arithmetic that keeps every question in the chapter to two divisions or fewer.

A’s profit : B’s profit = (A’s capital × A’s months) : (B’s capital × B’s months). Extend it partner by partner for three or more — the products never interact, so adding a partner adds one product.

Cancel before you multiply, then divide the profit exactly once. Strip the shared zeros from the capitals, form the ratio in small whole numbers, add the parts, and compute one part = profit / total parts. Every share is then a multiplication. Taking fractions of the profit partner by partner is the same answer with four times the arithmetic and four times the rounding risk.

What the question says	Months in (T)	Watch for
In for the whole year	`12`	the default when nothing is said
Joins after 4 months	`12 − 4 = 8`	not 4
Leaves after 4 months	`4`	counted from the start, so no subtraction
Leaves 4 months before the end	`12 − 4 = 8`	not 4
Invests for 5 months	`5`	already a duration, use it as given
Business ran only 8 months	`total is 8`	the year is not always 12

05 Cheat sheet

The golden equation on one page

The rule, the shortcut, and the two readings that look right and are not.

Case	Rule	On the lesson sum
Everyone in the same length of time	`ratio = C1 : C2`	the months cancel out
Times differ at all	`ratio = C1×T1 : C2×T2`	72,000 : 72,000 = 1 : 1
One part	`profit / total parts`	18,000 / 2 = 9,000
A partner’s share	`their parts × one part`	1 × 9,000 = 9,000
Capital ratio when the times differ	`wrong answer`	1 : 2 would pay 6,000 / 12,000
Averaging a capital that changed	`only if it changed at month 6`	see Model 5
Unit to think in	`capital-month`	6,000 × 12 = 72,000

Capital-months is the invariantIt plays the role person-days plays in Time and Work. Count them first and every question in the chapter turns into a ratio you already know how to finish.

Time means months inside, not months elapsed“Joins after 4 months” gives T = 8. “Leaves after 4 months” gives T = 4. The same four in the sentence, two different values of T.

The shares must add to the profitOne addition catches almost every slip. If A and B come to more or less than the profit you printed, the ratio or the parts are wrong, not the division.

06 Where & why

Where this shows up

Partnerships is a small chapter that carries reliable marks, because the questions are short and the method never changes.

TCS NQT · Infosys

Two partners, unequal months

Almost always one product each and a ratio. The numbers are built so the parts divide the profit exactly, which is itself a check on your ratio.

Bank PO · SSC CGL

Three partners with staggered dates

The load is in reading the timeline, not in the arithmetic. Write each partner’s start and end month down before you multiply anything.

CAT · XAT

Reverse questions

The profit ratio is given and a capital or a joining month is missing. The same equation, solved for a different letter.

Real firms

Profit-sharing clauses in a partnership deed

Deeds price capital by the time it was employed, and pay a working partner a salary before the split — which is Model 4 written in legal language.

One sentence to carry into the rest of the module: count the capital-months. Models 2 to 5 differ only in how the months are counted, and Model 4 in what comes out of the profit before you start.

07 Interview questions

What gets asked

Ten questions, from the definition to the honest comparison an interviewer uses to check you understand why time enters at all.

How is profit shared in a partnership?

In the ratio of capital multiplied by time, not capital alone. Each partner supplies capital × months invested capital-months, and the profit is divided in the ratio of those products. When every partner is in for the same period the time cancels and the capitals alone are enough.

Why does time enter the ratio at all?

Because a rupee that sat in the business for twelve months did twelve months of work and a rupee that sat for six did six. The business is paying for capital employed, and capital employed is a money-times-duration quantity. Ignoring the duration prices two unequal contributions as if they were equal.

A puts in ₹6,000 for 12 months, B puts in ₹12,000 for 6 months. Split an ₹18,000 profit.

₹9,000 each. A supplies 6,000 × 12 = 72,000 capital-months and B supplies 12,000 × 6 = 72,000, so the ratio is 1 : 1. Reading the capitals alone would have given 1 : 2 and paid A only ₹6,000, which is the standard error in this chapter.

What exactly is a capital-month?

One rupee inside the business for one month. It is a unit of capital employed, so a partner’s claim is a count of them. It works the same way person-days work in Time and Work — a single product that absorbs both givens and lets you compare partners who share neither.

How does this compare with Time and Work?

Structurally they are the same chapter. Time and Work conserves person-days: 5 people for 8 days and 10 people for 4 days both do 40. Partnerships conserves capital-months: ₹6,000 for 12 months and ₹12,000 for 6 both supply 72,000. In both, the product is the invariant and the two factors are interchangeable.

Someone joins after 4 months. What is their time?

Eight months, not four. The four counts the delay before they arrived; the time you want is the months their money was actually inside the business, which is 12 − 4. This single substitution is the most-set trap in the chapter and it is why Model 2 exists as a named model.

Why use the parts method rather than fractions of the profit?

It replaces one division per partner with one division in total. Find the value of one part once, then every share is a multiplication. It also gives you a free check: if the value of one part comes out to an awkward number, the ratio is usually wrong, because exam sums are built so it does not.

Can two partners with very different capitals get equal shares?

Yes, whenever their capital-month products match. ₹6,000 for 12 months and ₹12,000 for 6 both come to 72,000 and split the profit evenly. Halving the time and doubling the money leaves the claim untouched, which is the sentence to remember from this lesson.

Does the order in which partners joined change the answer?

Hard

A contractor pays ₹3,000 for a day’s work. (a) You send 10 workers for 12 hours and Raju sends 10 workers for 6 hours — split the fee. (b) On a second ₹3,000 job you send 15 workers for 4 hours and Raju sends 6 workers for 10 hours — split that too. (c) Name the quantity you conserved in both parts, state its partnership equivalent, and say in one sentence why a partner who doubles their capital and halves their time has an unchanged claim.

Follow-up

Parts (a) and (b) are Time and Work problems, not partnership problems, and they still yield to the same product. Part (c) is the point: recognising one invariant behind two chapters is what stops you memorising two methods.

Show the hint

Man-hours in (a) and (b); capital-months in the answer to (c). Both are a rate times a duration.