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Partnership

Competitive Mathematics · Arithmetic

6 Part4 worked examples10 practice questions

Two friends open a shop together — one puts in ₹1,00,000 and the other ₹2,00,000. If the year’s profit is ₹90,000, how is it split? The answer is the one idea of this chapter: each partner gets profit in proportion to the money he put in and how long it stayed. The ratio of profits equals the ratio of capital × time. If time is equal only the capital ratio matters; if capital is equal only the time ratio; if both differ, their product.

The profit ratio (capital × time)
1

Simple partnership

Either equal capital or equal time

When either capital or time is the same for all.

Equal time or equal capital

Whatever is equal cancels

If one factor is equal, profit follows the other.

SituationProfit ratio
equal time, different capitals
equal capital, different times
Example 1 · Basic
A and B start a business with ₹30,000 and ₹50,000. The year’s profit is ₹72,000. Find their shares.
Solution, step by step
  1. 1Ratio ; total 8 parts, one part .
  2. 2A gets ₹27,000; B ₹45,000.
Answer ₹27,000 and ₹45,000
Example 2 · Standard
Three partners put in ₹40,000 each. A stays 8 months, B 6 and C 12. What is C’s share of a ₹39,000 profit?
Solution, step by step
  1. 1Equal capital ⟹ ratio .
  2. 2C’s share
Answer ₹18,000
Example 3 · Basic
A’s investment is three times B’s. What is A’s share in an annual profit of ₹64,000?
Solution, step by step
  1. 1Ratio ; 4 parts; one part .
  2. 2A’s share
Answer ₹48,000
BasicTry it yourself first
A invested ₹6,000, B invested ₹9,000 in a partnership and after 1 year the profit was ₹20,000. What is A's share?
BasicTry it yourself first
A invested ₹4,000, B invested ₹6,000, C invested ₹10,000 in a partnership and after 1 year the profit was ₹40,000. What is C's share?
StandardTry it yourself first
A invested ₹15,000, B invested ₹25,000, C invested ₹20,000 in a partnership and after 1 year the profit was ₹48,000. What is B's share?
AdvancedTry it yourself first
A invested ₹27,000, B invested ₹45,000, C invested ₹36,000, D invested ₹12,000 in a partnership and after 1 year the profit was ₹60,000. What is D's share?
BasicTry it yourself first
2 partners each put in ₹20,000 and stayed 3, 5 months respectively. The total profit is ₹24,000. What is B's share?
StandardTry it yourself first
3 partners each put in ₹30,000 and stayed 6, 9, 12 months respectively. The total profit is ₹54,000. What is C's share?
AdvancedTry it yourself first
3 partners each put in ₹45,000 and stayed 4, 7, 10 months respectively. The total profit is ₹63,000. What is B's share?

Sharing a loss

The same ratio, the other direction

A loss is shared in the same ratio as a profit.

In a loss every partner bears it in the ratio of capital × time. The one who put in more bears more. (Sometimes a question says a partner gives only labour — he does not bear the capital loss; read that condition carefully.)

Example 4 · Standard
A, B and C put in ₹20,000, ₹30,000 and ₹50,000. The business lost ₹15,000. What is B’s share of the loss?
Solution, step by step
  1. 1Ratio ; 10 parts; one part .
  2. 2B’s loss
Answer ₹4,500
BasicTry it yourself first
A put in ₹8,000, B put in ₹12,000. The business lost ₹5,000. What is B's share of the loss?
StandardTry it yourself first
A put in ₹20,000, B put in ₹35,000, C put in ₹45,000. The business lost ₹20,000. What is C's share of the loss?
AdvancedTry it yourself first
A put in ₹24,000, B put in ₹30,000, C put in ₹36,000, D put in ₹60,000. The business lost ₹15,000. What is D's share of the loss?
2

Compound partnership

Different capitals and different times

Profit follows the products capital × time; joining later, changing capital and reversing the ratios.

Capital × time

“Capital-months”

Multiply each partner’s capital by his months and take the ratio.

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Changing capital and reversing the ratios

Count each stretch separately

Capital changes mid-year, or profit and capital ratios are given and the times are asked.

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3

Working partners and rent

Salary first, then sharing

Profit after salary or commission, and sharing rent.

Working and sleeping partners

Take out the salary first

The working partner first gets a salary or commission; the balance goes by capital.

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Sharing rent

Number × time

Rent of a pasture or godown — in the ratio of animals / goods × months.

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4

Traps and fast tricks

Slips · tricks

Where marks go in partnership.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

Simplest ratio

Habits that shorten the work.

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5

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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6

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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