Partnership Model 3 — The Complex Timeline

Partnerships · 25 min

Aptitude · Partnerships · Model 3

Everybody pays only for the months they stayed

Now partners both arrive late and leave early, sometimes in the same question. The equation does not change at all. What changes is that you have to draw a timeline in month numbers before you multiply anything, because the sentence will not hand you a duration.

Move the join and leave months and rebuild the timeline →
Treat the business like a hotel: you are billed for your own stay, whatever the hotel does before you arrive or after you leave.

01 The idea

Start month, end month, then multiply

A starts a business with ₹20,000. Three months in, B joins with ₹30,000. Three months after that, A leaves. B stays to the end of the year. Neither partner’s duration is stated anywhere in that sentence.

Draw it in month numbers instead. A runs from month 0 to month 6, so A stayed six months. B runs from month 3 to month 12, so B stayed nine. Once those two numbers exist on paper, the question is Model 2 with different products.

The hotel picture is the one to keep. The business is a hotel and the capital is the room rate; your bill is your rate times your own stay. Whether the hotel was open before you arrived, and whether it stays open after you leave, has nothing to do with what you owe.

What makes this model hard is language, not arithmetic. “Left after 4 months” is a stay of four. “Left 4 months before the end” is a stay of eight. “Left halfway through the year” is six, and “stayed a third of the year” is four. Each of those phrases appears in real papers, and each one needs converting before you touch a multiplication.

Write every partner as a start month and an end month. The stay is the difference, and the capital-months are the capital times the stay.
StayThe months between a partner’s start month and their end month. This is the T in the golden equation, and it is what the question makes you compute rather than telling you.
Before the endA phrase that measures backwards from month 12. “Leaves 2 months before the end” sets the end month to 10, so the stay is 10 minus the start month.
HandoverOne partner leaves and another joins at the same moment, so the stays do not overlap. It is still priced by the same rule; overlap never enters the arithmetic.

02 Worked example

A in for six months, B in for nine

This sum runs through the whole lesson. A starts a business with ₹20,000. After 3 months B joins with ₹30,000. After another 3 months A leaves. The year’s profit is ₹39,000. Find each share.

PartnerStart → endStayCapital-months
Amonth 0 → 66120,000
Bmonth 3 → 129270,000
Overlapmonths 3 → 63not used
1
Draw the timeline in month numbersEvery date phrase becomes a start month and an end month. Do this before any multiplication, on its own line.A: month 0 → 6, stay 6  ·  B: month 3 → 12, stay 9
2
A’s capital-monthsA held ₹20,000 for six months. The business ran on for another six without A, and that is not A’s concern.20,000 × 6 = 120,000
3
B’s capital-monthsB held ₹30,000 from month 3 to the close, which is nine months.30,000 × 9 = 270,000
4
Ratio, parts, one partCancel by 30,000. Thirteen parts, and the profit was chosen so a part is a round number.120,000 : 270,000 = 4 : 9 ⇒ 13 parts, 1 part = 39,000 / 13 = ₹3,000
5
Read off the sharesParts times the value of a part, and the two must return the profit.A = 4 × 3,000 = ₹12,000, B = 9 × 3,000 = ₹27,000

The two partners overlapped for three months, and that fact never entered the arithmetic. Each product is built from one partner’s own capital and one partner’s own stay, which is why a handover with no overlap at all is priced by exactly the same rule.

03 The method

The phrase-to-stay table is the method

The equation is unchanged from Model 2. This section is about the six sentences that encode a stay, because that is where the model is actually difficult.

Stay = end month − start month, then profit ratio = CA × stayA : CB × stayB. Add one product per extra partner.
“After n months” and “n months before the end” are opposites. The first counts forwards from month 0, so it gives you a start month or an end month directly. The second counts backwards from month 12, so the month you want is 12 − n. A question that uses both in one sentence — joins after 5, leaves 2 before the end — has a stay of (12 − 2) − 5 = 5 months.
PhraseStart → endStay
In for the whole year0 → 1212
Leaves after 4 months0 → 44
Leaves 4 months before the end0 → 88, not 4
Joins after 3, leaves after 3 more3 → 63
Joins after 5, leaves 2 before the end5 → 105
Stays a third of the year0 → 4fraction of 12 first

05 Cheat sheet

Model 3 on one page

The rule is one line. The rest of this table is the vocabulary that hides a stay.

CaseRuleOn the lesson sum
Stayend month − start monthA: 6 − 0 = 6
Capital-monthscapital × stay20,000 × 6 = 120,000
Leaves after n monthsstay = ncounted from the start
Leaves n months before the endstay = 12 − nnot n — the most-set trap
Joins at j, leaves n before the endstay = (12 − n) − jjoins 5, leaves 2 early → 5
Overlap between partnersnever used3 months here, and irrelevant
Handover with no overlapsame rule4 months each side
The timeline goes on paper firstStart month and end month for every partner, before any multiplication. Every error in this model is a stay computed in your head from a sentence designed to mislead.
Overlap is a distractorEach product uses only one partner’s own capital and own stay. Two partners who never coexist are priced by the identical rule, which is worth checking on a handover question.
The year is an assumptionIf the accounts close at month 8, then “leaves 2 months before the end” means month 6, not month 10. Read the period of the business before you count backwards from it.

06 Where & why

Where this shows up

Model 3 is where the chapter starts testing reading rather than arithmetic, which is why it appears in the harder sections of most papers.

Bank PO · SSC CGL
Staggered exits among three partners

“B leaves after 6 months, C after 9.” Three products, three stays, and the capitals chosen so the ratio cancels.

TCS NQT · Capgemini
The “before the end” phrasing

A single question will mix “joins after 3 months” with “leaves 2 months before the end” to see whether you count in both directions.

CAT · XAT
Reverse: when did the partner leave?

The profit ratio is given and a stay is missing. Solve for the stay, then convert it into whatever the question actually asked for.

Replacement questions
One partner hands over to another

Stays that do not overlap at all. Worth doing once, because it proves to you that overlap was never part of the method.

Model 4 leaves this arithmetic alone and changes what is in the pot: a working partner takes a salary off the top before any of this happens. Model 5 keeps the pot and changes the capital mid-year.

07 Interview questions

What gets asked

Ten, and most of them are a sentence to decode rather than a sum to do.

How do you handle a partner who joins late and leaves early?
Write their start month and end month, take the difference as their stay, and multiply by their capital. Nothing about the equation changes. Joining after 3 months and leaving 2 months before the end gives a stay of (12 − 2) − 3 = 7.
A invests ₹20,000 from the start and leaves at month 6; B invests ₹30,000 from month 3 to the end. Split a ₹39,000 profit.
₹12,000 and ₹27,000. A supplies 20,000 × 6 = 120,000 and B supplies 30,000 × 9 = 270,000, so the ratio is 4 : 9 over 13 parts, and one part is ₹3,000.
What is the difference between “left after 4 months” and “left 4 months before the end”?
The first is a stay of 4, counted forwards from the start. The second sets the end month to 12 − 4 = 8, so for a partner who started at month 0 the stay is 8. Same four in the sentence, double the stay.
Does the overlap between two partners affect the split?
No. Each product uses only that partner’s own capital and own stay, and the products never interact. Two partners who overlap for nine months and two who hand over with no overlap at all are priced by the identical rule.
A (₹4,000) and B (₹6,000) start together; A stays the full year, B leaves after some months, and the profit splits 4 : 3. When did B leave?
After 6 months. From (4,000 × 12) / (6,000 × T) = 4 / 3 you get 48 / 6T = 4 / 3, so T = 6. Reverse questions in this model are the same equation solved for a stay.
Three partners each invest ₹20,000. A stays 12 months, B leaves after 4, C joins when B leaves. Find the ratio.
3 : 1 : 2. The capitals are equal so they cancel, and the ratio is just the stays: 12 : 4 : 8. This is the cleanest demonstration in the chapter that time carries the same weight as money.
A partner stays for a third of the year. What is the stay?
Four months. Convert the fraction of the period into months before you do anything else — a third of 12. The same applies to “halfway through the year”, which is six, and to “a quarter of the term”.
How is Model 3 different from Model 2?
Model 2 has one date to read, a joining delay, so the stay is 12 − n. Model 3 has two dates, a start and an end, so the stay is a difference of month numbers and neither of them need be 0 or 12. The arithmetic after the stays is identical in both.
What is the fastest way to avoid errors here?
One line per partner reading “month a to month b, stay b − a”, written before any multiplication. Every wrong answer in this model comes from converting a phrase into a duration mentally, and the line makes that impossible.
Where does this shape actually occur?
In any venture where the funding roster changes during the accounting period — a partner exiting a firm mid-year, a short-term contract funded by different parties in sequence, a property held jointly and then bought out. Real settlements are computed exactly this way, by the months each stake was held.

08 Practice problems

Six on the timeline

Write the start and end month for every partner before you multiply anything. Three of these hide a stay behind a phrase rather than stating it.

The plain quitter

Easy
Ramesh and Suresh start a business with ₹50,000 and ₹60,000. Ramesh leaves after 5 months and Suresh stays for the full year. The annual profit is ₹38,800. Find Suresh’s share.
Follow-up
The parts do not cancel to anything friendly, which is deliberate — the profit was chosen to fit them. Trust the ratio and let the odd-looking part count stand.
Show the hint
250 against 720 in thousands. The parts total 97, and ₹38,800 divides by it.

Staggered exits

Easy
X, Y and Z start a business with ₹20,000, ₹30,000 and ₹40,000. X stays the full year, Y leaves after 6 months and Z leaves after 9. The total profit is ₹39,000. Find Z’s share.
Follow-up
Three products, and the largest capital does not produce the largest claim by as much as you would expect. Compute all three before forming any ratio.
Show the hint
240, 180 and 360 in thousands. Cancel by 60.

Before the end

Medium
X and Y start a business with ₹50,000 and ₹70,000. X leaves 3 months before the end of the year and Y stays throughout. The profit is ₹43,000. Find X’s share.
Follow-up
The phrase counts backwards from month 12, so X’s stay is 9 and not 3. That single reading is the whole question, and the parts total 43 to confirm you read it right.
Show the hint
X’s end month is 12 − 3 = 9, and X started at 0.

Late in and early out

Medium
Amit starts a project with ₹20,000. Sumit joins after 3 months with ₹40,000, then leaves 4 months before the end of the year. Find their profit ratio.
Follow-up
Both directions of counting appear in one sentence: a start measured forwards and an end measured backwards. Sumit’s stay is neither 3 nor 4 nor 8.
Show the hint
Sumit runs from month 3 to month 8. That is a stay of 5.

Backwards to a leaving month

Medium
A invests ₹45,000 and B invests ₹60,000; B stays the full year and A leaves after some months. Out of a total profit of ₹66,000, A receives ₹18,000. After how many months did A leave?
Follow-up
You are given a share rather than a ratio, so the first move is to find B’s share by subtraction and reduce the two to a ratio. Only then does the equation have one unknown.
Show the hint
B gets ₹48,000, so the ratio is 3 : 8. Now solve 45T : 60 × 12 = 3 : 8.

Two stays, one equation

Hard
(a) Ravi and Kiran start with ₹50,000 and ₹70,000 and both leave before the year ends. Ravi’s money was invested 2 months longer than Kiran’s and their profits are equal; find Kiran’s stay. (b) A invests ₹80,000 and withdraws completely after x months, at which point B replaces him with ₹1,20,000 for the rest of the year; if their profits are equal, find x. (c) Part (b) does not give a whole number of months — explain why that is a legitimate answer and not a sign of an error.
Follow-up
Part (a) has two unknown stays linked by a difference, so the equal-profit condition becomes a linear equation rather than a division. Part (c) is the marks: capital-months is a continuous quantity, and nothing in the model requires a stay to be an integer.
Show the hint
For (a) let Kiran’s stay be T and Ravi’s be T + 2. For (b) A’s stay is x and B’s is 12 − x.