Ratio, Proportion, Partnership & Ages

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high importance~2 Q in Tier 122 formulas⚡ 15 shortcuts5 subtopics

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Subtopic 3 of 5·← Proportion & proportional division of terms·Problems on ages →

Partnership

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Profit is divided in the ratio of capital × time invested.

Same period: share ratio = capital ratio $C_1 : C_2 : C_3$ .
Different periods (or a partner joins later / withdraws early): share ratio $= C_1 T_1 : C_2 T_2 : C_3 T_3$ (capital-months).
A working partner's fee/salary/commission is taken OUT of the profit first; the rest is divided by capital-time ratios.

For a mid-year capital change, split the year into stretches and add capital × months over the stretches.

📖 Notes 🧩 4 question types ⚡ Formulas 🎯 12 practice Q

Detailed notes

One rule runs the whole subtopic

Profit is divided in the ratio of capital × time (capital-months). Same period → the time cancels and only the capital ratio matters. Different joining dates, withdrawals or top-ups → build one row per partner: $\text{share}_i \propto C_1t_1 + C_2t_2 + \cdots$ Then each partner's money = fraction × profit. Nothing else exists in this topic.

The capital-months table

Partner	Working
A: ₹12,000 for 12 months	$12000 \times 12 = 144000$
B: ₹18,000 for 8 months (joined after 4)	$18000 \times 8 = 144000$
Ratio 144 : 144 = 1 : 1. Months for a late joiner run from the JOINING month to the end of the year, never from the start.

Mid-year capital changes

Split the year into stretches and add capital × months across them: A holds ₹30,000 for 8 months, then ₹20,000 for 4 → $30000 \times 8 + 20000 \times 4 = 320000$ capital-months. Using only the final capital (or only the first) is the classic error. The inverse question appears too: "they share profits equally, B joined after 6 months — find B's capital": equate the capital-month products.

Working partners

A partner who works takes a salary/fee/commission out of the profit first; only the residue is divided by capital-months (which still include the working partner's own capital-months). Example: profit ₹9,000; B manages and draws 10% → cut ₹900 → residue ₹8,100 split by capitals. The cut percentage applies to the whole profit, not to the residue.

Comparing two claims

"Who got more?" and "difference between shares" need only the capital-month ratio: difference = gap parts × value of one part. Reduce the capital-month numbers by their GCD early — the arithmetic stays tiny (₹'000 units help).

How the questions vary the story

Same capital-months engine, four costumes:

Rent of a pasture: grazers pay in ratio of (animals × months) — cows are the "capital".
Wages of a contractor: workers' wages split by (men × days) — identical arithmetic.
Consumption of a granary: provisions last longer or shorter as (men × days) changes — the inverse uses the same product.
Sleeping vs working partner: the sleeping partner invests only; the working one takes the fee — combine patterns rp-n4 with any of the above.

Unit discipline

Drop the zeroes early: 12,000 × 12 = 144 'thousand-months' — write 144, not 144000. Ratios are what survive, and every zero you cancel is one less chance of an arithmetic slip. If the products look unwieldy, divide all of them by 1000 (or 10000) before forming the ratio; the ratio is unchanged.

Quick revision

Share ∝ capital × months. Same period → capital ratio.
Late joiner: months from joining to year-end.
Capital change: add capital × months per stretch.
Working partner: cut from the top, share the residue.

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Simple partnership (same time, different capitals)very common2 practice Q

How to spot it:

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

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

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

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

Ratio $45000 : 35000 = 9 : 7$ (16 parts) → A $= \frac{9}{16} \times 8000 = ₹4{,}500$ .

Type 2: Partner joins late / leaves early (capital × time)very common3 practice Q

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

Compute capital-months: investment × months in business (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: money working for longer deserves a proportionally larger share — only the product counts.

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

A: $16000 \times 12 = 192000$ ; B: $24000 \times 6 = 144000$ → ratio 4 : 3 → B $= \frac{3}{7} \times 14000 = ₹6{,}000$ .

Type 3: Mid-year capital changecommon2 practice Q

How to spot it:

'A invests X and adds/withdraws Y after k months' — or 'invested a for 4 months and b for the rest'.

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

Split the year at every change.
Add capital × months over the stretches.
Take the ratio across partners; a single partner with changes just sums their own stretches.

Why: each stretch of capital earns its own capital-months, and they add up.

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

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

Type 4: Working partner (salary / commission off the top)common2 practice Q

How to spot it:

'B 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

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

Why: the fee is for work done, so it must not be distorted by the profit ratio.

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

Cut $= ₹1{,}440$ → residue $= ₹12{,}960$ ; ratio $5 : 3$ → B $= \frac{3}{8} \times 12960 = ₹4{,}860$ .

Formulas

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

Shortcut tricks

⚡ Capital-months table

One row per partner: capital × months. The profit ratio is the row ratio — no other step.

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

A: 12,000 × 12 = 144,000; B: 18,000 × 8 = 144,000 ⇒ 1 : 1 ⇒ ₹4,800 each.

⚡ Deduct the working partner's cut first

Salary/commission % applies to the whole profit; only the residue is shared by capital-time.

Example: A and B invest ₹6,000 and ₹4,000. B manages the business and gets 10% of the profit. On a ₹9,000 profit, find A's share.

Cut = ₹900; residue ₹8,100 split 3 : 2 ⇒ A = $\frac{3}{5} \times 8100$ = ₹4,860.

⚡ Split the year for capital changes

Add capital × months across the stretches of the year.

Example: A starts with ₹20,000 and adds ₹5,000 after 6 months. B invests ₹22,500 throughout. Profit ratio?

A: $20000 \times 6 + 25000 \times 6 = 270000$ ; B: $22500 \times 12 = 270000$ ⇒ 1 : 1.

Where students lose marks

Ignoring time when partners join at different dates.
Applying the working partner's % after sharing instead of before.
For months: counting the joiner's period from the start of the year instead of from the joining month.
For mid-year capital changes, using only the final capital.

Practice sets — 12 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 12 questions

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