Partnership

This is the starting point. All partners keep their money in the business for the same time: the full year, 6 months or 2 years, it does not matter which. When time is the same for everybody it cancels out, and profit is shared simply in the ratio of the capitals.

So the method has three steps:

• Write the capitals as a ratio and simplify it (cut zeros, divide by the common factor).
• Add the terms to get the total parts.
• Find 1 part from whatever is given, then answer the question in parts.

Radhika and Meera start a stationery shop with ₹42,000 and ₹56,000. Ratio = 42 : 56 = 3 : 4, total 7 parts. If the profit is ₹17,500, 1 part = ₹2,500, so Radhika gets ₹7,500 and Meera gets ₹10,000.

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The exam changes only what is given:

• Total profit given → find a share. 1 part = total ÷ total parts.
• One partner's share given → find the total or another share. 1 part = that share ÷ his parts.
• Difference of two shares given → 1 part = difference ÷ difference of their parts.
• Percentage asked → a partner's percentage = his partstotal parts × 100.

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Watch one small point. If the question says at the end of 6 months the profit is ₹X and both partners invested for those same 6 months, it is still a simple partnership. The time is the same for both, so it cancels.

Sometimes the question does not give the capitals directly. It gives a condition instead: "Q's capital is three times R's", "3 times A's capital = 4 times B's", "A invested ₹5,000 more than B", or "P invested 15 of the total". Your first job is to turn the condition into a clean ratio. After that it is Type 1 again.

1. "Times" relations: start from the smallest. If B = 2C and A = 3B, take C = 1. Then B = 2 and A = 6, so A : B : C = 6 : 2 : 1. Starting from the person everyone is compared with keeps fractions away.

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2. Equal multiples: aA = bB = cC. Here the capitals are in the inverse ratio: A : B : C = 1a : 1b : 1c. Multiply by the LCM of a, b, c to remove the fractions.
If 4A = 5B = 10C, then A : B : C = 14 : 15 : 110. The LCM of 4, 5, 10 is 20, so the ratio is 5 : 4 : 2. Check: 4 × 5 = 5 × 4 = 10 × 2 = 20.

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3. Two ratios joined: A : B and B : C. Make the common term equal, exactly as in the Ratio chapter. If A : B = 2 : 5 and B : C = 3 : 4, then A : B : C = 6 : 15 : 20.

4. "More than / less than" with a total. Let the smallest capital be x, write the others as x + something, add them up to the total and solve. Then use the actual capitals as the ratio.
Take Radhika, Meera and Sana, who invest ₹31,000 in all. Meera puts in ₹2,000 more than Sana and Radhika ₹3,000 more than Meera. Sana = x, Meera = x + 2,000, Radhika = x + 5,000. So 3x + 7,000 = 31,000 and x = 8,000. Capitals: 13,000 : 10,000 : 8,000.

5. Fractions of the total. If P puts in 13 of the total and Q puts in 14, then R puts in 1 − 13 − 14 = 512. Ratio = 13 : 14 : 512 = 4 : 3 : 5.

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The same ratio thinking also handles questions where two investments are made in a given ratio and each earns a different return. The overall return is then a weighted result of the two, as the last practice questions show.

Now the times are different. One partner keeps his money for 12 months, another for 8 months, a third for 10 months. As we saw in the foundation, profit is shared in the ratio of capital × time, so each partner's money-months decide his share.

Method:

• Multiply each partner's capital by his time (the same unit for all: months with months, years with years).
• Write these products as a ratio and simplify.
• Divide the profit in that ratio, exactly as in Type 1.

Radhika puts in ₹9,000 for 8 months and Meera puts in ₹6,000 for 10 months.
Radhika = 9,000 × 8 = 72,000; Meera = 6,000 × 10 = 60,000.
Ratio = 72 : 60 = 6 : 5. If the profit is ₹4,400, then 11 parts = ₹4,400 and 1 part = ₹400, so Radhika gets ₹2,400 and Meera gets ₹2,000.

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Simplify before multiplying. Capitals 9,000 : 6,000 = 3 : 2 and times 8 : 10 = 4 : 5. Profit ratio = (3 × 4) : (2 × 5) = 12 : 10 = 6 : 5. The answer is the same, and the numbers stay tiny.

If the question gives a capital ratio and a time ratio, multiply them term by term. Capital 2 : 3 and time 5 : 4 give profit 10 : 12 = 5 : 6. This works for three partners too: first with first, second with second, third with third.

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Two quick checks come up again and again:

• Who gets more? Compare the products. Equal products mean equal profit.
• Years or months: both are fine, as long as everybody is in the same unit.

The business runs for a fixed period (usually one year), but not every partner is there for the whole period. Someone joins later, or someone leaves early. The only new skill is to read each partner's time correctly. After that it is Type 3.

Joins after x months (in a one-year business) ⇒ his money works for (12 − x) months.
Leaves after x months ⇒ his money works for x months.
A partner who is there from start to end ⇒ 12 months.

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If the business runs for 2 years, the full period is 24 months, so joins after 6 months means 24 − 6 = 18 months. Always subtract from the full period of the business, not automatically from 12.

Radhika starts a bakery with ₹16,000. After 4 months Meera joins with ₹18,000. At the end of the year the profit is ₹10,500.
Radhika = 16,000 × 12 = 1,92,000; Meera = 18,000 × 8 = 1,44,000.
Ratio = 192 : 144 = 4 : 3. 7 parts = ₹10,500, 1 part = ₹1,500, so Radhika gets ₹6,000 and Meera gets ₹4,500. Meera put in more money but for fewer months, so she gets less.

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When dates are given (like "started in March, joined in November, accounts closed after 2 years"), count the months carefully: from March to November is 8 months. Draw a small timeline like the one below and write each partner's months on it.

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A partner who withdraws all his capital after some months is the same as one who leaves. His time is only the months for which his money stayed in the business.

Here the partners stay for the whole period, but somebody changes his capital in between. He adds money, withdraws some, doubles it, halves it, or changes it by some percentage. The rule stays the same: profit ∝ money-months. We only have to work out the money-months of each period separately and add them.

Effective capital = (capital in period 1 × its months) + (capital in period 2 × its months) + …

Radhika starts with ₹20,000 and after 3 months adds ₹4,000. Meera starts with ₹30,000 and after 3 months withdraws ₹6,000. The business runs for one year.
Radhika = 20,000 × 3 + 24,000 × 9 = 60,000 + 2,16,000 = 2,76,000
Meera = 30,000 × 3 + 24,000 × 9 = 90,000 + 2,16,000 = 3,06,000
Ratio = 276 : 306 = 46 : 51.

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How to read the usual changes:

• "adds ₹x" ⇒ new capital = old + x. "withdraws ₹x" ⇒ new capital = old − x.
• "doubles" ⇒ × 2. "halves" or "withdraws half" ⇒ × 12.
• "increases by 25%" ⇒ × 54. "increases by 50%" ⇒ × 32. "decreases by 10%" ⇒ × 910. "decreases by 13" ⇒ × 23. Your percentage–fraction table from the Percentage chapter does this work.

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When only a ratio of capitals is given (like 3 : 2 : 4), treat the ratio terms as the capitals: A = 3 units, B = 2 units, C = 4 units. Doubling A makes it 6 units; halving B makes it 1 unit. The real rupee values are never needed.

Some questions mix this type with Type 4: one partner joins midway while another changes his capital. Keep one row per partner and fill in the months period by period. Each partner's months must add up to the months he was actually in the business.

So far the capitals and times were given and we found the profit. Now the question is turned around: the profit ratio (or a share) is given, and you have to find a capital, a time, or the month when someone joined. Nothing new is needed. It is the same rule read backwards.

Since Profit ∝ Capital × Time:

• Capital ∝ ProfitTime, so divide each profit term by its time.
• Time ∝ ProfitCapital, so divide each profit term by its capital.

Radhika and Meera share a profit in 5 : 6. Radhika's money stayed for 10 months and Meera's for 8 months. Capitals = 510 : 68 = 12 : 34 = 2 : 3.

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Unknown months: make an equation. Set the ratio of money-months equal to the profit ratio, with the unknown inside it.
Radhika puts in ₹10,000 for the whole year. Meera joins later with ₹16,000, and the profit is shared 3 : 2. Let Meera's time be t months.
10,000 × 1216,000 × t = 32 ⇒ 2,40,000 = 48,000 t ⇒ t = 5.
So Meera's money worked for 5 months, which means she joined after 7 months. Always check what is asked: for how many months or after how many months.

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Unknown amount added. Put the unknown amount x into the period after the change, build both partners' money-months, and set their ratio equal to the profit ratio.

When only the time ratio is asked and the capitals and the profit ratio are given, use Time ∝ ProfitCapital directly. No equation is needed.

In a real business the whole profit is not always divided straight away. Some money is taken out first: a salary or commission for the partner who runs the business, a donation to charity, rent, money kept back for reinvestment, or interest on capital. Only the remaining profit is divided in the capital (× time) ratio. Careless mistakes happen most in this type, so read every line slowly.

Step 1: take out what goes first.

• Monthly salary ⇒ salary × 12 for a year.
• "x% of the profit" for managing ⇒ x% of the total profit (unless the question clearly says x% of the remaining profit).
• Charity, rent or reinvestment of r% ⇒ only (100 − r)% of the profit is left to share.

Step 2: divide the rest in the ratio of capital × time.

Step 3: add back for the working partner. His total = salary or commission + his share of the rest. A sleeping partner gets only his share of the rest.

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Take a salary case first. Radhika and Meera invest in the ratio 2 : 3. Radhika runs the shop and takes ₹1,500 a month from the profit. The year's profit is ₹48,000.
Salary for the year = 1,500 × 12 = ₹18,000. Remaining = ₹30,000, shared 2 : 3 ⇒ ₹12,000 and ₹18,000.
Radhika's total = 18,000 + 12,000 = ₹30,000; Meera's total = ₹18,000.

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Now a case where part of the profit is shared equally. Radhika and Meera invest ₹36,000 and ₹24,000. They agree that 40% of the profit will be shared equally (for their equal work) and the rest in the ratio of capitals. Profit = ₹15,000.
Equal part = 40% of 15,000 = ₹6,000 ⇒ ₹3,000 each. Rest = ₹9,000 in 3 : 2 ⇒ ₹5,400 and ₹3,600.
Radhika = 3,000 + 5,400 = ₹8,400; Meera = 3,000 + 3,600 = ₹6,600.

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Going backwards. If a share is given and the total is asked, undo the steps. A partner's share = his partstotal parts × (100 − r)% of the total profit, so total profit = share × total partshis parts × 100100 − r.

Interest on capital. If every partner first gets interest at the same rate on his capital for the same time, that interest is itself in the ratio of capitals. The rest is also shared in the ratio of capitals. So the final shares are simply in the ratio of capitals; the interest does not change who gets what.

A sleeping partner with a reduced share. If a silent partner is to get only half of what he would get as an active partner, halve his ratio term and then divide: 5 : 4 : 6 becomes 5 : 4 : 3.

In the last type, interest at the same rate on every capital left the final shares in the capital ratio. That happens only when the rest of the profit is also shared in the capital ratio. Many deeds say something different: each partner first gets interest on his capital, and the remaining profit is shared in an agreed ratio, very often equally. Now the capital ratio decides only the interest part.

Work in four steps:

• Interest on each capital = capital × rate × time100
• Divisible profit = total profit − total interest
• Share the divisible profit in the agreed ratio
• Partner's total = his interest + his share of the divisible profit

Radhika puts in ₹70,000 and Meena ₹30,000. Each first gets 8% a year on her capital, and the rest is shared equally. The year's profit is ₹20,000. Interest: Radhika ₹5,600 and Meena ₹2,400, together ₹8,000. Divisible profit = 20,000 − 8,000 = ₹12,000, so ₹6,000 each. Radhika's total is 5,600 + 6,000 = ₹11,600 and Meena's is 2,400 + 6,000 = ₹8,400.

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When the balance is shared equally, two partners' totals differ only by the interest on the difference of their capitals. Here 8% of (70,000 − 30,000) = ₹3,200, and indeed 11,600 − 8,400 = 3,200. The profit figure does not matter for this difference.

Time matters. If a capital stays in the firm for only part of the year, the interest is for that period only: capital × rate × months12. A partner who joins after 6 months gets half a year's interest.

A partner's loan is not his capital. Sometimes a partner lends extra money to the firm. That money does not enter the capital ratio and buys no extra share of the profit. The firm owes him interest on it, and this interest is paid before the profit is shared, like a salary.

Radhika and Meena have ₹50,000 each as capital and are equal partners. Meena also lends ₹20,000 to the firm at 6% a year, and the profit before this interest is ₹14,000. Loan interest = ₹1,200 goes to Meena first. The remaining ₹12,800 is shared equally, ₹6,400 each. Radhika gets ₹6,400 and Meena gets 6,400 + 1,200 = ₹7,600.

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Some deeds make the sharing depend on how big the profit is. For example: "the first ₹40,000 of profit is shared equally, and whatever is above that is shared in the ratio 3 : 2." Or a partner who manages the business gets a bonus only on the profit above a certain figure. This works like income-tax slabs: every slab has its own rule, and each rupee of profit falls in exactly one slab.

Follow three steps:

• Cut the profit at the limit: slab 1 is the profit up to the limit, slab 2 is the amount above it.
• Share each slab by its own rule.
• Add up what each partner gets from all the slabs.

Radhika and Meena share the first ₹25,000 of profit equally and the rest in the ratio 2 : 3. The profit is ₹75,000. Slab 1 = ₹25,000, so ₹12,500 each. Slab 2 = 75,000 − 25,000 = ₹50,000, and in 2 : 3 that is ₹20,000 and ₹30,000. Radhika's total is 12,500 + 20,000 = ₹32,500 and Meena's is 12,500 + 30,000 = ₹42,500.

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Check whether the limit is crossed. If the profit is smaller than the limit, the second rule never applies. Under the same deed a profit of ₹18,000 is simply shared equally, ₹9,000 each.

Going backwards. When a partner's share is given and the profit is asked, first find what that partner would get at the limit itself. If the given share is bigger, the profit has crossed the limit, and the extra comes from slab 2. Suppose Meena gets ₹39,500 under the deed above. At the limit she has ₹12,500, so slab 2 gave her 39,500 − 12,500 = ₹27,000. That is 35 of slab 2, so slab 2 = ₹45,000 and the profit is 25,000 + 45,000 = ₹70,000. If the given share is smaller than the share at the limit, the profit never reached the limit.

Bonus on the excess. "A gets 5% of the profit above ₹80,000 as a bonus and the rest is shared equally." With a profit of ₹1,00,000 the bonus is 5% of 20,000 = ₹1,000. The remaining ₹99,000 is shared equally, ₹49,500 each. A's total is ₹50,500 and B's is ₹49,500.