Ratio, Proportion, Partnership & Ages

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high importance~2 Q in Tier 123 formulas⚡ 15 shortcuts5 subtopics

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Partnership

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⏱ 4 min read🧩 5 question types🎯 12 practice Q

The idea in one minute

Profit is divided in the ratio of capital × time — the capital-months.

Same period for all partners: time cancels, so the share ratio is just the capital ratio. Different joining dates or mid-year top-ups: build one capital-months row per partner.

A working partner's fee or per cent comes off the profit first; only the residue is divided.

One rule for everything

Profit share $\propto$ capital $\times$ time. Call the product capital-months.

Same period for everyone: the time cancels, and the share ratio is just the capital ratio. Different joining dates, withdrawals or top-ups: build one row per partner and add up.

Rule: Share $\propto$ capital $\times$ months. Nothing else matters in this topic.

The capital-months table

A invests Rs 12,000 for 12 months. B joins after 4 months with Rs 18,000, so B's money works only 8 months.

Partner	Capital-months
A	$12000 \times 12 = 144000$
B	$18000 \times 8 = 144000$

Ratio 1 : 1 — a Rs 9,600 profit splits 4,800 each. A late joiner can catch up only by investing more.

Tip: A late joiner's months run from the joining month to the year end, never from the start.

Mid-year capital changes

Split the year at each change and add capital × months.

A invests 15,000 for 4 months and 20,000 for 8: $60000 + 160000 = 220000$ . Against B's $18000 \times 12 = 216000$ , the ratio is $55 : 54$ .

Same engine for withdrawals: A holds 30,000 for 8 months then 20,000 for 4, giving $240000 + 80000 = 320000$ capital-months.

Watch: Using only the first capital or only the final capital is the classic error on these.

The working partner's cut

A partner who manages the business takes a fee or a per cent of the profit off the top. Only the residue is divided by capital-months, and the working partner's own capital still counts in that split.

Profit 9,000; B manages and draws 10%: cut 900, residue 8,100 split by capitals $3 : 2$ . B's split share is 3,240, and B's total receipt is 4,140 if the question asks for it.

Careful: The cut applies to the whole profit, not to the residue. And take it off before sharing, not after.

Rent of a pasture

The same engine in rural costume: animals are the capital, months are the months.

A grazes 15 cows for 8 months (120 cow-months); B grazes 25 cows for 4 months (100). Rent 2,200 splits $6 : 5$ , so A pays 1,200 and B pays 1,000.

Wages and workmen

Wages for a job split the same way: men × days. Three men working 5, 8 and 10 days share wages in $5 : 8 : 10$ . Nothing new to learn — capital-months again.

Rs 1,200 split for 2, 3 and 5 days worked: ten parts of 120, so the shares are 240, 360 and 600.

Keep the numbers small

Drop zeroes early: 12,000 × 12 is 144 thousand-months. Ratios survive any common division, and every zero you drop is one less slip. When the products look unwieldy, divide all of them by 1,000 before forming the ratio.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Simple partnership (same time, different capitals)

How to spot it:

Partners invest different amounts for the same period; a share or the total profit is asked.

P_1 : P_2 = C_1 : C_2 \quad (\text{same time})

Method

Reduce the capital ratio to small terms (divide by the GCD).
Add the parts; one part = profit ÷ total parts.
Multiply by the asked partner's parts.

Why it works:

With time equal for everyone, it cancels out of the ratio.

Try this

A and B invest Rs 45,000 and Rs 35,000 for one year. From a profit of Rs 8,000, what does A receive?

Show solution

Ratio $45000 : 35000 = 9 : 7$ (16 parts).
A $= \dfrac{9}{16} \times 8000 = 4500$ .

Answer

Rs 4,500

Type 2very common3 practice Q

Partner joins late or leaves early

How to spot it:

'A starts the business; B joins after k months' — time differs, so capitals alone mislead.

P_1 : P_2 = C_1T_1 : C_2T_2

Method

Compute capital-months for each (joiner: from joining month to year end).
The profit ratio is the ratio of these products.
Reverse version: ratio known, one capital unknown — equate the products.

Why it works:

Money working longer earns a proportionally larger share; only the product counts.

Try this

A starts a business with Rs 16,000. After 6 months, B joins with Rs 24,000. From a year-end profit of Rs 14,000, B's share is:

Show solution

A: $16000 \times 12 = 192000$ ; B: $24000 \times 6 = 144000$ .
Ratio $4 : 3$ .
B $= \dfrac{3}{7} \times 14000 = 6000$ .

Answer

Rs 6,000

Type 3common2 practice Q

Mid-year capital change

How to spot it:

'A invests X and adds or withdraws Y after k months' — one partner's capital moves mid-year.

\text{effective} = C_a t_1 + C_b t_2 + \cdots

Method

Split the year at every change.
Add capital × months over the stretches.
Take the ratio across partners.

Why it works:

Each stretch of capital earns its own capital-months, and they add up.

Try this

A invests Rs 15,000 for the first 4 months of a year and Rs 20,000 for the remaining 8 months. B invests Rs 18,000 for the whole year. The profit-sharing ratio is:

Show solution

A: $15000 \times 4 + 20000 \times 8 = 220000$ .
B: $18000 \times 12 = 216000$ .
Ratio $= 55 : 54$ .

Answer

55 : 54

Type 4common2 practice Q

Working partner (cut off the top)

How to spot it:

'A manages the business and gets x% of the profit' — or a fixed salary — before the rest is shared.

\text{residue} = \text{profit} - \text{cut}, \quad \text{residue split by } C_iT_i

Method

Take the working partner's cut out of the profit first.
Divide the residue by the capital ratio (the working partner's capital still counts).
Add the cut back if that partner's total receipt is asked.

Why it works:

The fee pays for work done, so it must not pass through the profit ratio.

Try this

A and B invest Rs 50,000 and Rs 30,000. A manages the business and receives 10% of the profit for it. From a profit of Rs 14,400, B's share is:

Show solution

Cut $= 1440$ ; residue $= 12960$ .
Ratio $5 : 3$ .
B $= \dfrac{3}{8} \times 12960 = 4860$ .

Answer

Rs 4,860

Type 5common

Rent of a pasture (animals × months)

How to spot it:

Grazing rent divided among farmers whose herds graze for different periods.

\text{rent share} \propto \text{animals} \times \text{months}

Method

Count cow-months for each farmer: animals × months.
The rent ratio is the ratio of these products.
Multiply each share of the ratio by the total rent.

Why it works:

The pasture is the partnership: animals are the capital, months are the months.

Try this

The rent of a pasture is Rs 2,200. A grazes 15 cows for 8 months and B grazes 25 cows for 4 months. A's share of the rent is:

Show solution

A: $15 \times 8 = 120$ cow-months; B: $25 \times 4 = 100$ .
Ratio $6 : 5$ .
A $= \dfrac{6}{11} \times 2200 = 1200$ .

Answer

Rs 1,200

Formula sheet

Simple partnership

P_1 : P_2 = C_1 : C_2 \quad (\text{same time})

Compound partnership

P_1 : P_2 : P_3 = C_1T_1 : C_2T_2 : C_3T_3

Working partner

\text{profit} = \text{manager's cut} + \text{residual split by } C_iT_i

Capital change mid-year

\text{effective capital} = C_a t_1 + C_b t_2 + \cdots

Shortcuts that save time

⚡ One row per partner

Write capital × months for each partner; the profit ratio is the row ratio.

Example

A invests Rs 12,000 for the full year; B joins after 4 months with Rs 18,000. Divide a Rs 9,600 profit.

Show solution

A: $12000 \times 12 = 144000$ ; B: $18000 \times 8 = 144000$ .
Ratio $1 : 1$ , so Rs 4,800 each.

Answer

Rs 4,800 each

⚡ Cut comes off the top

Salary or a per cent for the manager is removed from the whole profit before the capital-months split.

Example

A and B invest Rs 6,000 and Rs 4,000. B manages and takes 10% of a Rs 9,000 profit. Find A's share.

Show solution

Cut $= 900$ ; residue $= 8100$ .
Capitals $3 : 2$ ; A $= \dfrac{3}{5} \times 8100 = 4860$ .

Answer

Rs 4,860

⚡ Equal shares, find the capital

Equal shares mean equal capital-months. Equate the products and solve.

Example

A invests Rs 12,000 for 12 months. B joins after 6 months and they share the profit equally. Find B's capital.

Show solution

$12000 \times 12 = C \times 6$ .
$C = 24000$ .

Answer

Rs 24,000

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Ignoring time when partners join on different dates.

Share ∝ capital × months; compute capital-months for everyone.

Mistake 02

Counting a late joiner's months from the year's start.

Months run from the joining month to the year end.

Mistake 03

Sharing the profit first and taking the manager's cut after.

The cut comes off the top; only the residue is split.

Mistake 04

Using only the final capital after a mid-year change.

Add capital × months across each stretch of the year.

Mistake 05

Answering the working partner's split share when the total receipt is asked.

Add the cut back to that partner's share when 'in all' is asked.

Quick revision

Read this the night before the exam.

Share $\propto$ capital $\times$ months.
Same time for all: pure capital ratio.
Late joiner: months from joining to year end.
Mid-year change: add capital-months per stretch.
Working partner: cut off the top, split the residue.
Pasture rent and wages: the same capital-months engine.

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.

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