Aptitude · Partnerships · Model 1

When the clock is the same for everyone, drop it

In Model 1 every partner is in for the whole period, so the months are a common factor in every capital-month product and cancel out of the ratio. What is left is the ratio of the capitals, and the whole model is the parts method applied to it.

Set the capitals and watch the parts do the work →

Equal times means the months cancel, so profit ratio = capital ratio. This is the only model where you may ignore time, and only because it is identical for all.

01 The idea

The 12 that cancels

Ram puts in ₹30,000 and Shyam puts in ₹50,000. Both stay for the full year. The capital-month products are 30,000 × 12 and 50,000 × 12, and the 12 appears in both.

A factor common to both sides of a ratio cancels. So 360,000 : 600,000 is the same ratio as 30,000 : 50,000, which is 3 : 5. The time did not disappear from the problem; it dropped out of the ratio because it was equal. Change Shyam’s months to ten and it comes straight back.

That is worth being precise about, because “in Model 1 you ignore time’’ is a sentence students carry into Model 2 and lose marks with. The honest version is: when every partner supplies the same number of months, the ratio of the products equals the ratio of the capitals.

What is left is the parts method. Reduce 30,000 : 50,000 to 3 : 5, add the parts to get 8, divide the ₹16,000 profit once to get ₹2,000 a part, and read off ₹6,000 and ₹10,000. Two divisions in the whole question, and the second one is the check.

Reduce the capitals to a small whole-number ratio, add the parts, divide the profit once, then multiply each partner’s parts.

Simple partnershipEvery partner is in for the same period, usually the full year. Nobody joins late, leaves early, adds capital or withdraws it, so T is a common factor.

PartsThe sum of the reduced ratio terms. In 3 : 5 there are 8 parts, so the profit is cut into eight equal pieces and each partner takes a whole number of them.

Reverse questionA question that hands you one share, or the gap between two shares, instead of the total. Both give the value of one part directly, which is all you need.

02 Worked example

Ram ₹30,000, Shyam ₹50,000, profit ₹16,000

This sum runs through the whole lesson. Ram and Shyam start a business with ₹30,000 and ₹50,000. At the end of the year the profit is ₹16,000. Find each share.

Partner	Capital	Parts	Share
Ram	₹30,000	3	₹6,000
Shyam	₹50,000	5	₹10,000
Total	₹80,000	8	₹16,000

Check the timeline before anything elseNothing in the question says either partner joined late or left early, so both supplied twelve months. That is what licenses the next step.(30,000 × 12) : (50,000 × 12) = 30,000 : 50,000

Strip the ratio to whole numbersCancel the three shared zeros, then divide by the 10 still in common. Small numbers are the point.30,000 : 50,000 → 30 : 50 → 3 : 5

Add the partsThe parts are the denominator for the entire question, so count them carefully once rather than twice.3 + 5 = 8 parts

Value one partOne division. If this lands on an awkward number, suspect the ratio before you suspect the division.16,000 / 8 = ₹2,000 per part

Multiply each partner’s partsRam has 3 parts and Shyam has 5. Add the shares back and they must return ₹16,000.Ram = 3 × 2,000 = ₹6,000, Shyam = 5 × 2,000 = ₹10,000

Notice what never appeared: the number 12. It cancelled at step one and the rest of the sum was a ratio and a division. Model 2 is this exact procedure with one partner’s 12 replaced by a smaller number, so the ratio no longer reduces to the capitals.

03 The method

One rule, five question shapes

The rule is one line. What varies is which of the four quantities — capitals, ratio, one share, total profit — the question withholds.

Equal times ⇒ profit ratio = capital ratio. Then one part = total profit / total parts and share = parts × one part.

Whatever you are given, convert it to the value of one part. A total profit divides by all the parts. One partner’s share divides by their own parts. The difference between two shares divides by the gap in their parts. All three roads reach the same number, and from there every other figure is a multiplication.

Question shape	What you are given	The move
Find a share	capitals and the total profit	`parts × profit / total parts`
Find the total profit	one partner’s share	`share / its parts × total parts`
Find the total profit	the gap between two shares	`gap / gap in parts × total parts`
Find a missing capital	one capital and the profit ratio	`capital ratio = profit ratio`
Fractions of the capital	A has 1/3, B has 1/4, C the rest	`take the LCM as the total`
Percentage of the remaining	X takes 50%, Y takes 30% of the rest	`30% of 50, not of 100`

05 Cheat sheet

Model 1 on one page

The rule, the three roads to the value of one part, and the two readings that get marked wrong.

Case	Rule	On the lesson sum
Equal times	`profit ratio = capital ratio`	3 : 5
One part	`profit / total parts`	16,000 / 8 = 2,000
Given one share	`share / its parts`	6,000 / 3 = 2,000
Given the gap between shares	`gap / gap in parts`	4,000 / 2 = 2,000
Fractions of the total capital	`LCM of the denominators`	1/3 and 1/4 → total 12
Percent of the remaining capital	`read the base carefully`	30% of 50 is 15, not 30
Using unequal times here	`wrong model`	that is Model 2 onward

Time did not vanish, it cancelledIt is a common factor of every product, which is a different thing from being absent. Say it the second way and Model 2 will not catch you out.

One part is the whole questionTotal profit, a single share, or the difference between two shares — each one gives you the value of a part in one division. Everything else follows by multiplication.

An awkward part value is a warningExam sums are built so the profit divides by the total parts exactly. If 1 part comes out to a recurring decimal, re-check the cancelling before you re-check the division.

06 Where & why

Where this shows up

Model 1 is the easy mark in the chapter and it appears in every paper that carries partnerships at all.

TCS NQT · Cognizant

Direct three-capital splits

Capitals chosen so the ratio cancels to single digits. The work is in the cancelling and nothing else.

Bank PO · RRB

Difference-of-shares questions

“A gets ₹3,200 more than C, find the total profit.” The gap in parts is the only thing you need to spot.

SSC CGL

Fraction and percentage capitals

“A invests one-third, B invests one-quarter, C the rest.” Take the LCM as the total capital and the fractions become whole parts.

Option elimination

Divisibility as a shortcut

If A holds 7 parts out of a whole-number split, A’s share is a multiple of 7. On a multiple-choice paper that sometimes settles the question without any arithmetic.

Model 1 is not a separate technique. It is the golden equation with the times equal, and the moment one partner’s months differ you are in Model 2 with the same five steps.

07 Interview questions

What gets asked

Ten, running from the definition to the reverse questions that make up most of the marks.

What is a simple partnership?

One where every partner’s capital is in for the same period. Because the time is a common factor of every capital-month product it cancels out of the ratio, so the profit is shared in the ratio of the capitals alone.

Ram invests ₹30,000 and Shyam ₹50,000 for a year. Split a ₹16,000 profit.

₹6,000 and ₹10,000. The ratio is 30,000 : 50,000 = 3 : 5, which is 8 parts, so one part is 16,000 / 8 = ₹2,000. Ram takes 3 parts and Shyam takes 5.

Is it correct to say time is irrelevant in Model 1?

No, and the distinction matters. Time is equal, so it cancels out of the ratio — it is a common factor, not an absent quantity. Students who learn it as “ignore time” carry that into Model 2, where one partner’s months differ and the cancelling no longer happens.

A and B invest in the ratio 5 : 8 and A’s share is ₹15,000. Find the total profit.

₹39,000. A’s 5 parts are worth ₹15,000, so one part is ₹3,000, and the total is 13 parts — 13 × 3,000. Being handed one share is as good as being handed the total, because either one gives you the value of a part.

Two partners invest in the ratio 4 : 9 and the difference in their shares is ₹15,000. Find the total profit.

₹39,000. The gap is 9 − 4 = 5 parts, so one part is 15,000 / 5 = ₹3,000, and 13 parts is ₹39,000. Reach for the gap in parts whenever the question gives you a difference rather than a total.

A invests one-third of the capital, B one-quarter, and C the rest. How do you set that up?

Take the total capital as the LCM of the denominators, here 12. Then A is 4, B is 3 and C is 12 − 7 = 5, so the ratio is 4 : 3 : 5. Choosing the LCM as the total is what keeps every term a whole number.

X invests 50% of the capital, Y invests 30% of the remaining, and Z the rest. Find the ratio.

10 : 3 : 7. Take the total as 100: X is 50, the remainder is 50, and Y is 30% of that 50 = 15, so Z is 35. The trap is reading Y as 30% of 100 — the base is the remainder, and the question says so.

If A invests twice as much as B and B thrice as much as C, what is the ratio?

6 : 3 : 1. Set the smallest to 1: C = 1, B = 3, A = 6. Working from the smallest end keeps you out of fractions, and it also removes the risk of inverting one of the two statements.

How is Model 1 different from Model 2?

Hard

(a) A, B and C invest such that 2 × A’s capital = 3 × B’s = 4 × C’s. The profit is ₹52,000; find the difference between A’s and C’s shares. (b) Instead, A contributes one-third of the total capital and B contributes as much as A and C together; the profit is ₹84,000, so find C’s share. (c) Explain in one sentence why the ratio in (a) is 6 : 4 : 3 and not 2 : 3 : 4.

Follow-up

Both halves hide the ratio behind a relation rather than stating it, and the two relations invert in opposite directions. Part (c) is the marks: a multiplier on a capital divides the ratio term, which is the single most common inversion error in the chapter.

Show the hint

For (a) set all three products to the LCM of 2, 3 and 4. For (b), B equalling A + C forces B to be exactly half the total.