Aptitude · Partnerships · Model 2

“Joins after 4 months” means eight, not four

One partner starts the business and another arrives part way through. The arithmetic is the golden equation unchanged; the entire difficulty is translating a joining date into a number of months invested, and the sentence is built to make you get it wrong.

Slide the joining month and watch the trap open up →

A delay of n months leaves T = 12 − n months invested. The number in the sentence is the delay, not the time.

01 The idea

The delay is not the duration

A starts a business with ₹21,000 in January. B joins after four months, in May, with ₹12,000. At the end of December the profit is ₹14,500. B’s money was inside the business from May to December, which is eight months.

Four is the delay. Eight is the time. The question hands you the first and needs the second, and it is the only real obstacle in the model. Everything after this one subtraction is Model 1 with different products.

Get it wrong and the numbers still look plausible. Using four gives the ratio 21 : 4 and pays B ₹2,320; using eight gives 21 : 8 and pays B ₹4,000. Both add up to ₹14,500, neither is obviously absurd, and one of them is worth no marks.

The fix is mechanical: before you multiply anything, write down each partner’s months invested on its own line. A phrase like “joins after 4 months” becomes “T = 8” on paper, and once it is on paper you cannot use the 4 by accident.

Convert every date phrase into months invested first, on its own line. Then form capital × time and finish as in Model 1.

Joining delayThe months that pass before a partner arrives. Written n, it is the number the question gives you and the number you must not use directly.

Months investedHow long the money was actually inside the business, T = 12 − n for a one-year business. This is the T in the golden equation.

Equal-profit conditionWhen two partners take equal shares their products match: C₁T₁ = C₂T₂. It turns most reverse questions in this model into a single division.

02 Worked example

₹21,000 from January, ₹12,000 from May

This sum runs through the whole lesson. A starts a business with ₹21,000. After 4 months B joins with ₹12,000. At the end of the year the profit is ₹14,500. Find each share.

Reading of “after 4 months”	B’s T	Ratio	A’s share
Correct: T = 12 − 4	8	21 : 8	₹10,500
Trap: T = 4	4	21 : 4	₹12,180

Translate the delay into months investedB arrived four months in, so B’s money was inside for the remaining eight. Write this line before you write anything else.A: T = 12 · B: T = 12 − 4 = 8

A’s capital-monthsA started the business and stayed to the close, so A supplies the full twelve months.21,000 × 12 = 252,000

B’s capital-monthsB’s capital times the eight months it was in. The 12,000 is larger than A’s per-month contribution but runs for less of the year.12,000 × 8 = 96,000

Ratio, parts, one partCancel by 12,000. Twenty-nine parts is not a friendly number, which is exactly why the profit was chosen as ₹14,500.252,000 : 96,000 = 21 : 8 ⇒ 29 parts, 1 part = 14,500 / 29 = ₹500

Read off the sharesParts times the value of a part. The two shares must return the profit exactly.A = 21 × 500 = ₹10,500, B = 8 × 500 = ₹4,000

The whole model is step one. Steps two to five are the procedure from Model 1, applied to products instead of capitals. Put the trap version beside the right one, as below, and you can see that nothing about the wrong answer looks wrong.

03 The method

The equation, and running it backwards

Half the questions in this model give you the ratio and withhold a joining month. That is the same equation solved for T.

A’s profit : B’s profit = C_A × 12 : C_B × (12 − n), where n is B’s joining delay. For three partners, one product each and one delay each.

Equal profits means equal products, so the times are inversely proportional to the capitals. If A has ₹10,000 for 12 months and B has ₹30,000, then B needs 120,000 / 30,000 = 4 months to draw level — and therefore joined after eight. One division, then one subtraction, and mixing up which of the two the question asked for is the second trap in this model.

Question shape	Given	The move
Find a share	capitals and one delay	`C×12 : C×(12−n)`
Find the months invested	the profit ratio	`T = C1×T1 / (C2 × ratio)`
Find the joining delay	the profit ratio	`n = 12 − T`
Equal profits	two capitals	`C1×T1 = C2×T2`
Three partners joining in turn	“after 2 more months”	`delays accumulate: 2, then 2+2`
Durations given directly	“invests for 5 months”	`no subtraction at all`

05 Cheat sheet

Model 2 on one page

Six lines of translation and one equation. The translation is where the marks are lost.

Phrase in the question	Months invested	On the lesson sum
Starts the business	`12`	A: 21,000 × 12 = 252,000
Joins after n months	`12 − n`	B: 12,000 × 8 = 96,000
Joins after 4 months	`8, not 4`	21 : 8, not 21 : 4
Invests for 5 months	`5`	a duration, so use it as given
Joins 2 months after B	`delays add up`	B late 2, C late 4
Equal profits	`C1T1 = C2T2`	one division for the missing T
Asked when they joined	`answer 12 − T`	find T first, then subtract

Write the T line before you multiplyOne line per partner, converting every date phrase into months invested. The trap only lands when the subtraction happens in your head.

The trap answer is not obviously wrongUsing 4 instead of 8 gives A ₹12,180 and B ₹2,320. They still add to ₹14,500, so no sum check will catch it. Only the T line catches it.

Two questions hide behind one number“How many months did B invest?” and “after how many months did B join?” differ by a subtraction from 12. Read which one was asked before you write the answer down.

06 Where & why

Where this shows up

Model 2 is the most-set model in the chapter, because one subtraction separates the students who read carefully from the students who do not.

TCS NQT · Wipro

Two partners, one joining late

Nearly always a direct question with the delay stated in months. The capitals are chosen so the products cancel to a tidy ratio.

Bank PO · SSC CGL

Reverse: after how many months did B join?

The profit ratio is given and the delay is missing. Solve for T, then remember to subtract from 12 before you answer.

Infosys · Accenture

Three partners joining in sequence

“After 2 months B joins, after 2 more months C joins.” The delays accumulate, so C is four months late, not two.

Equal-share questions

“They receive equal profits”

That single phrase is the equation C₁T₁ = C₂T₂. It usually reduces the whole question to one division.

Model 3 keeps this arithmetic and adds a leaving date, so partners can start late and finish early. If you can build the T line reliably here, that model costs you nothing new.

07 Interview questions

What gets asked

Ten, and four of them are the same subtraction asked in four different voices.

A partner joins after 4 months. What is their time?

Eight months. The four is the delay before they arrived; the time in the golden equation is the months their money was actually inside the business, which is 12 − 4. Using the 4 directly is the defining error of this model.

A starts with ₹21,000, B joins after 4 months with ₹12,000, profit ₹14,500. Find A’s share.

₹10,500. A supplies 21,000 × 12 = 252,000 and B supplies 12,000 × 8 = 96,000, so the ratio is 21 : 8 over 29 parts. One part is 14,500 / 29 = ₹500, and A takes 21 of them.

What would the trap answer have been in that question?

₹12,180 to A and ₹2,320 to B. Using B’s time as 4 gives the ratio 21 : 4 over 25 parts and a part value of ₹580. Both figures still add to ₹14,500, which is why checking the sum does not save you here.

A invests ₹10,000 for a year and B joins with ₹30,000. They receive equal profits. How long was B’s money in?

Four months. Equal profits means equal products, so 10,000 × 12 = 30,000 × T, giving T = 4. Note the question asked for the time, so four is the answer; had it asked when B joined, the answer would be eight.

A invests ₹45,000 and B joins later with ₹30,000; the profit splits 2 : 1. When did B join?

After 3 months. From (45,000 × 12) / (30,000 × T) = 2 / 1 you get T = 9, so B was in for nine months and therefore joined after 12 − 9 = 3. The two-step answer is the point of the question.

A starts, B joins after 2 months, C joins 2 months after B. What are the three times?

12, 10 and 8. B’s delay is 2 and C’s is 2 + 2 = 4, so the times are 12, 12 − 2 and 12 − 4. Delays measured from the previous joiner accumulate, and treating C’s delay as 2 is the third-partner version of the main trap.

The question says a partner invested for 5 months. Do you subtract from 12?

No. Five is already a duration, so it is the T you want. Subtract only when the sentence gives a delay or a date — “joins after”, “joins in May”. Reading a duration as a delay is the mirror image of the main error.

Does it matter whether the business ran for twelve months?

Yes, and the 12 is an assumption, not a law. If the accounts close after eight months, then a partner who joined after three was in for five, not nine. Read the period of the business before you subtract anything from it.

How does Model 2 differ from Model 1?

Hard

P invests ₹50,000 and Q joins later with ₹50,000. (a) The profit is ₹25,000 and P’s share is ₹5,000 more than Q’s; find how many months Q’s money was invested. (b) Now keep the same capitals and the same ₹5,000 gap but make the profit ₹30,000, and find Q’s months again. (c) Explain why the answer moved even though neither capital nor the gap changed, and why part (b) does not land on a whole month.

Follow-up

Part (b) breaks the habit of treating a stated difference as if it fixed the ratio. It does not — a gap only becomes a ratio once you know the total, and a fractional month is a legitimate answer rather than a sign of an error.

Show the hint

Turn the profit and the gap into two shares, reduce those to a ratio, and only then use 50 × 12 : 50 × T.