Aptitude · Partnerships · Model 5

A capital that changes is a sum, not an average

A partner adds money or takes some out part way through the year, so their capital has two levels rather than one. Their capital-months are the two segments added together. Averaging the two capitals and running the average for twelve months gives a different answer, and it is right only in one special case.

Move the change month and watch the average break →

Break the year at the change and add the pieces: C₁×m + C₂×(12−m). Never average the two capitals.

01 The idea

Two capitals, two segments

X starts with ₹5,000 and takes ₹1,000 back out after four months, so X holds ₹5,000 for four months and ₹4,000 for the remaining eight. Y puts in ₹6,000 and leaves it alone all year.

X’s capital-months are the two segments added: 5,000 × 4 = 20,000 plus 4,000 × 8 = 32,000, giving 52,000. Y’s are 6,000 × 12 = 72,000. The ratio 52,000 : 72,000 cancels to 13 : 18, and on a ₹31,000 profit that pays X ₹13,000 and Y ₹18,000.

The tempting shortcut is to average X’s two capitals — ₹5,000 and ₹4,000 give ₹4,500 — and run that for twelve months, which is 54,000. That is 2,000 too high, and it would pay X ₹13,285.71. Two things are wrong with it: the figure is not the answer, and it is not even a whole number of rupees, which is itself a signal that the method is off.

An average weights both capitals equally, and X’s two capitals did not run for equal times — four months against eight. The average is right precisely when the change lands at month six, and wrong every other time. Adding the segments is right always, so there is nothing to gain by remembering the exception.

Cut the year at every change of capital, price each segment as capital × its own months, and add. The segments are what the ratio compares.

SegmentA stretch of months over which a partner’s capital did not change. Priced as capital × months in the segment and then added to the other segments.

Equivalent capitalA partner’s total capital-months, the sum of their segments. It is what the profit ratio compares, and it is a sum — never a mean.

Change monthThe month at which the capital moves. A change at month m makes two segments of m and 12 − m months.

02 Worked example

₹5,000 dropping to ₹4,000 at month four

This sum runs through the whole lesson. X starts with ₹5,000 and withdraws ₹1,000 after 4 months. Y invests ₹6,000 for the whole year. The profit is ₹31,000. Find each share.

Phase	X’s capital	Months	Capital-months
Months 1–4	₹5,000	4	20,000
Months 5–12	₹4,000	8	32,000
X, total	—	12	52,000
Average capital × 12	₹4,500	12	54,000 — wrong

Cut X’s year at month fourOne partner, two capital levels, so two segments. Write the months for each before you multiply anything.months 1–4 at ₹5,000 · months 5–12 at ₹4,000

Price the two segments and add themEach level is charged for the months it actually ran. The two products are then added, which is where the model gets its name and its only real idea.5,000 × 4 + 4,000 × 8 = 20,000 + 32,000 = 52,000

Test the average-capital shortcutAveraging ₹5,000 and ₹4,000 gives ₹4,500, and twelve months of that is 54,000. It weights both capitals equally, and they ran for four months and eight.average route: 4,500 × 12 = 54,000, against the true 52,000

Bring in Y and form the ratioY never changed, so Y is a single segment. Then it is the usual ratio, parts and one-part sequence.6,000 × 12 = 72,000 ⇒ 52,000 : 72,000 = 13 : 18, 31 parts, 1 part = ₹1,000

Read off the sharesParts times the value of a part. Compare with what the average would have paid X.X = ₹13,000, Y = ₹18,000 (the average route says X ₹13,285.71)

The average is not a rough version of the right answer — it is the answer to a different question, the one where the capital changed at month six. Adding segments costs one extra multiplication and is correct for every change month, which is the whole argument for it. The same segment-sum handles three or four changes with no new idea at all.

03 The method

One segment per stretch, however the change is worded

The arithmetic never changes. What changes is how the question describes the new capital — a rupee amount, a fraction, a percentage, or a multiple.

Capital-months = C₁×m₁ + C₂×m₂ + …, one term per stretch of unchanged capital, with the months summing to the period of the business. Then form the ratio against the other partners as usual.

Work out the new capital before you work out any segment. “Withdraws 25% of his capital” on ₹16,000 means a new capital of ₹12,000, and “adds 25%” on ₹12,000 means ₹15,000 — two different bases, which is what the question is testing. Write the capital level for each stretch on its own line first, and the segments become mechanical.

How the change is worded	New capital	Watch for
Adds ₹10,000 to ₹40,000	`50,000`	the plain case
Withdraws ₹1,000 from ₹5,000	`4,000`	the lesson sum
Withdraws 25% of ₹16,000	`12,000`	percent of their own capital
Adds one-third of ₹30,000	`40,000`	fraction of the current level
Doubles his capital	`2 × current`	a multiple, not an addition
Average the two capitals	`only if the change is at month 6`	adds the segments instead

05 Cheat sheet

Model 5 on one page

One rule, four wordings of a change, and the one case where the shortcut is safe.

Case	Rule	On the lesson sum
Capital changes once	`C1×m + C2×(12−m)`	5,000×4 + 4,000×8 = 52,000
Capital changes twice	`three segments, still added`	months split 3, 3 and 6
Partner never changes	`capital × 12`	6,000 × 12 = 72,000
Percentage change	`percent of their own capital`	25% off 16,000 → 12,000
Average of the two capitals	`wrong unless m = 6`	54,000 instead of 52,000
Change at month 6 exactly	`average happens to work`	both halves are 6 months
Withdraws the whole capital	`that is Model 3`	a partner leaving, not a shift

Segments add, capitals do not averageThe mean weights both levels equally and they almost never ran for equal times. The segment sum needs one extra multiplication and is correct for every change month.

An awkward share is a warningThe average route pays X ₹13,285.71 on the lesson sum — not a whole rupee. Exam sums are built so the correct method lands on round figures.

Find the new capital first“Withdraws 25%” and “adds 25%” use the partner’s own capital as the base, so the two changes are different rupee amounts. Write the level for each stretch before you price any segment.

06 Where & why

Where this shows up

Model 5 is the most computational model in the chapter and the one that combines most freely with the others.

Bank PO · SSC CGL

One add and one withdrawal in the same question

Two partners, each with two segments. Four multiplications and a ratio, and the numbers are chosen so it cancels.

TCS NQT · Infosys

Percentage and fraction changes

“A withdraws 25% while B adds 25%.” The two 25% figures are different rupee amounts because the bases differ, which is the whole question.

CAT · XAT

Unknown capital with a shift

“B starts with x and adds ₹2,000 after 6 months; the ratio is 6 : 7 — find x.” The segment sum becomes a linear expression in x.

The full-chapter question

Joining, leaving and shifting in one timeline

Papers that carry a five-mark partnership question usually build it here, because a segment table absorbs every other model as a special case.

With this model the chapter closes, and the whole of it fits on one line: count the capital-months. Model 1 counts one segment each, Models 2 and 3 change the months, Model 4 changes the pot before the split, and Model 5 splits a partner’s own year into pieces.

07 Interview questions

What gets asked

Ten, and the third one is the reason this model is set as often as it is.

A partner adds capital part way through the year. How do you handle it?

Break their year at the change and price each stretch separately, then add. A partner who holds C₁ for m months and C₂ for the rest supplies C₁×m + C₂×(12−m) capital-months, and that sum is what the ratio compares.

X holds ₹5,000 for 4 months and ₹4,000 for 8; Y holds ₹6,000 all year. Split a ₹31,000 profit.

₹13,000 and ₹18,000. X supplies 20,000 + 32,000 = 52,000 capital-months and Y supplies 72,000, so the ratio is 13 : 18 over 31 parts, and one part is ₹1,000.

Why can you not just average the two capitals?

Because an average weights both levels equally and they ran for unequal times — four months against eight here. Averaging ₹5,000 and ₹4,000 gives 54,000 capital-months against the true 52,000, and pays X ₹13,285.71 instead of ₹13,000.

Is the average ever right?

Yes, when the change falls exactly at month six, because then each capital genuinely ran for half the year and the mean is the correct weighted mean. It is not worth remembering as a shortcut, since adding the segments is one multiplication more and always correct.

What if the capital changes twice?

Three segments instead of two, added the same way. A partner who holds ₹10,000 for 3 months, ₹15,000 for 3 and ₹20,000 for 6 supplies 30 + 45 + 120 = 195 thousand-months. The method does not grow in difficulty, only in length.

“A withdraws 25% and B adds 25%” — are those the same amount?

Only if their capitals are equal. Each percentage is of that partner’s own capital, so 25% off ₹16,000 is ₹4,000 while 25% added to ₹12,000 is ₹3,000. Work out both new capital levels before you touch a segment.

How do you deal with an unknown starting capital?

Write the segment sum as an expression in x and set the ratio up as an equation. If B starts with x and adds ₹4,000 after 6 months, B supplies 6x + 6(x + 4,000) = 12x + 24,000, and the given ratio turns that into one linear equation.

What is the difference between a withdrawal and leaving the business?

A withdrawal leaves a smaller positive capital in place, so the partner still has a second segment. Withdrawing everything means the partner has left, which is Model 3 — one segment and then nothing. If a withdrawal would take the capital to zero or below, you have misread the question or it is a different model.

How does this model relate to the other four?

Hard

A starts on 1 January with ₹10,000. After 3 months B joins with ₹20,000. After 2 more months A adds ₹5,000 and B withdraws ₹5,000. At the end of month 9 A leaves entirely. The profit for the year is ₹1,02,000. (a) Find each partner’s share. (b) State which of the five models each feature of the timeline belongs to, and say why one segment table is enough to handle all of them at once.

Follow-up

This is Models 2, 3 and 5 in a single question, and the only way through it is a phase table with months and a capital level on every row. Part (b) is the connection worth writing down: every model in the chapter is a special case of a list of segments.

Show the hint

Build a two-column table of phases for each partner. A’s rows cover months 1–5 and 6–9 and stop; B’s cover 4–5 and 6–12.