Partnership

Partnership Basics: Partners, Capital and Sharing the Profit

A partnership is a business started by two or more people who each put in some money. The money a partner puts in is called his capital (or investment). At the end of the year, or whatever period the question gives, the profit is shared among the partners, and so is a loss.

Two kinds of partner.

• A working (active) partner invests money and also runs the business. He is often paid a salary or a commission for this work, on top of his share.
• A sleeping partner only invests money. He takes no part in running the business and gets only his share of the profit.

The one rule behind every question. A partner's profit depends on how much money he put in and for how long it stayed in the business:

$$\text{profit} \propto \text{capital} \times \text{time}$$

So for two partners,

$$\frac{P_1}{P_2} = \frac{C_1 \times T_1}{C_2 \times T_2}$$

Simple partnership. If all the partners keep their money in for the same time, the time cancels and the profit is shared in the ratio of the capitals.

"A, B and C start a shop with ₹1,20,000, ₹1,60,000 and ₹2,00,000 and run it for a year. The profit is ₹54,000." Take out the common factor 40,000: the capitals are in the ratio 3 : 4 : 5, which is 12 parts in all.

One part = 54,000 ÷ 12 = ₹4,500, and each share is that many parts.

One ratio, three kinds of question. Once you have the ratio, every question is about parts.

• Total profit from one share: C's ₹22,500 is 5 parts, so one part is ₹4,500 and the 12 parts make ₹54,000.
• A share from the difference of two shares: if C gets ₹9,000 more than A, those 5 − 3 = 2 parts are ₹9,000, so one part is ₹4,500 again.
• One share from another: B's share = A's share × $\frac{4}{3}$.

Loss is shared the same way. A loss is split in exactly the ratio a profit would be. If the shop above had lost ₹7,200, one part would be ₹600 and A would bear 3 parts, ₹1,800. More on loss comes later in these notes.

Capital and share are different things. The share is a part of the profit, not of the capital. A partner who put in ₹1,20,000 and gets ₹13,500 still owns his ₹1,20,000 as well.

Capital × Time: Different Durations, Joining Late and Leaving Early

When partners keep their money in for different lengths of time, multiply each capital by the number of months it stays in. This product decides the share.

Why multiply? ₹60,000 kept in for 4 months does the same work for the business as ₹20,000 kept in for 12 months, because 60,000 × 4 = 20,000 × 12. So capital × months is a fair measure of what each partner gave.

Reading the time correctly. Unless the question says otherwise, the business runs for 12 months and the profit is shared at the end of the year.

If the profit is shared at the end of 18 months or 2 years, use 18 or 24 in place of 12.

A full line-up. "A starts a business with ₹36,000. After 3 months B joins with ₹48,000, and after 6 months C joins with ₹60,000. The profit at the end of the year is ₹85,000."

The ratio is 432 : 432 : 360 = 6 : 6 : 5 (dividing by 72), which is 17 parts. One part = 85,000 ÷ 17 = ₹5,000, so A and B get ₹30,000 each and C gets ₹25,000. C put in the most money but gets the least, because his money worked for the shortest time.

Leaving early. "A and B start with ₹50,000 and ₹40,000. After 8 months A takes out all his money. The year's profit is ₹33,000." A: 50 × 8 = 400. B: 40 × 12 = 480. So A : B = 400 : 480 = 5 : 6, and A gets ₹15,000 while B gets ₹18,000.

Capitals and times given as ratios. "The capitals are in the ratio 4 : 5 : 6 and the times in the ratio 3 : 4 : 2." Multiply term by term: 12 : 20 : 12 = 3 : 5 : 3. Never add the two ratios.

Years instead of months. If every partner's time is a whole number of years, you can multiply by years. Just use the same unit for everyone.

[SSC Tip] The usual slip is to use the joining month as the time. "B joined after 3 months" means B's money worked for 9 months, not 3. Write the months column first, then multiply.

Capital Changed Once During the Year

Sometimes a partner adds money or takes some out partway through the year. Then his capital is not one number for the whole year. Break the year at the month of the change, find capital × months for each piece and add the pieces.

"After 4 months" means the old capital stays for the first 4 months and the new capital for the remaining 8.

"X and Y start a business with ₹60,000 and ₹80,000. After 4 months X adds ₹20,000 and Y takes out ₹20,000. The profit at the end of the year is ₹63,000."

Work in thousands. X has 60 for 4 months, then 80 for 8 months. Y has 80 for 4 months, then 60 for 8 months.

So X : Y = 880 : 800 = 11 : 10. One part = 63,000 ÷ 21 = ₹3,000, so X gets ₹33,000 and Y gets ₹30,000.

Both partners held the same two amounts, just at different times, and still X gets more. X had the larger amount for the longer stretch.

Fractions of the capital. "A starts with ₹90,000 and after 6 months takes out half of it." A has 90 for 6 months and 45 for 6 months: 540 + 270 = 810. A quick way to see it: half the money for half the year counts the same as the full ₹90,000 for 9 months, and 90 × 9 = 810.

"Takes out $\frac{1}{3}$ of his capital after 3 months" on ₹60,000 leaves ₹40,000, so the partner has 60 × 3 + 40 × 9 = 180 + 360 = 540.

Two common misreadings.

• "Withdraws ₹20,000" means the capital drops by 20,000. It does not become 20,000.
• "Adds ₹20,000 after 4 months" counts for 8 months, not 4.

When everyone changes in the same way. If A, B and C each add 50% to their capital after 6 months, the profit ratio is still the ratio of their starting capitals. Every partner's money is scaled by the same factor in the same months, so nothing changes between them. Spotting this saves you a whole table.

Draw the table even for one change. Two rows and three columns take ten seconds and stop you from using only the starting capital or only the final capital.

Several Changes and Three Partners: The Timeline Table

With several changes, mark every month where something happens and treat the year as a set of periods. Inside each period every partner's capital is fixed, so capital × months for that period is quick. Then add along each row.

"A, B and C start a business with ₹30,000, ₹40,000 and ₹50,000. After 3 months A adds ₹10,000. After 6 months B takes out ₹20,000. After 9 months C takes out ₹20,000. The profit at the end of the year is ₹75,000."

The changes come after months 3, 6 and 9, so the year breaks into four periods of 3 months each. In thousands:

A : B : C = 450 : 360 : 540 = 5 : 4 : 6 (dividing by 90), which is 15 parts. One part = ₹5,000, so the shares are ₹25,000, ₹20,000 and ₹30,000.

Each row on its own. You do not need a column for every period. For each partner, cut his own year only where his capital changes. A has 30 for 3 months and 40 for 9 months, so 90 + 360 = 450. This is quicker when each partner changes only once.

Counting each amount for the months it stays. Another way is to follow the money. A's first ₹30,000 stays all 12 months and his extra ₹10,000 stays 9 months: 30 × 12 + 10 × 9 = 450. A withdrawal works the other way round. B's ₹40,000 would count 40 × 12 = 480, but ₹20,000 of it was missing for the last 6 months, so take away 20 × 6 = 120 and get 360.

One partner fixed, the other in instalments. "A puts in ₹90,000 for the whole year. B puts in ₹30,000 at the start, ₹30,000 more after 4 months and ₹30,000 more after 8 months." Each instalment counts for the months it is in:

A has 90 × 12 = 1,080. So A : B = 1,080 : 720 = 3 : 2, even though both have put in ₹90,000 by the end.

Finding the Unknown: Capital, Time or the Month of a Change

Many questions give the profit split and ask for something that went into it: a partner's capital, how long his money stayed, or when he made a change. Write profit ratio = capital × time ratio and solve for the unknown.

When did B join? "A starts a business with ₹45,000. Some months later B joins with ₹60,000. At the end of the year the profit is shared equally. After how many months did B join?"

Say B's money stayed t months. Equal shares mean 45 × 12 = 60 × t, so t = 9. B's money was in for 9 months, so B joined after 3 months. The options nearly always include 9 as well, because it answers a different question: how many months B's money was in.

What was B's capital? "A puts in ₹40,000 for the whole year and B joins after 4 months. Of a profit of ₹36,000, B gets ₹16,000. Find B's capital."

A gets ₹20,000, so A : B = 20,000 : 16,000 = 5 : 4. With B's capital as x thousand:

$$\frac{40 \times 12}{x \times 8} = \frac{5}{4} \Rightarrow 40x = 1920 \Rightarrow x = 48$$

B put in ₹48,000.

When was the change made? "A and B start with ₹50,000 and ₹60,000. After some months A takes out ₹20,000. At the end of the year the profit is shared in the ratio 2 : 3. When did A take the money out?"

Say A withdrew after x months. A has 50 for x months and 30 for (12 − x) months, which is 50x + 30(12 − x) = 360 + 20x. B has 60 × 12 = 720. So (360 + 20x) : 720 = 2 : 3, which gives 360 + 20x = 480 and x = 6. A took the money out after 6 months.

Capital ratio from the profit and time ratios. Since profit ∝ capital × time, capital ∝ profit ÷ time. "Profits are in the ratio 2 : 3 : 5 and the times in the ratio 4 : 3 : 2. Find the ratio of the capitals." Capitals = $\frac{2}{4} : \frac{3}{3} : \frac{5}{2}$ = $\frac{1}{2} : 1 : \frac{5}{2}$. Multiply by 2 to get 1 : 2 : 5.

Time ratio from the profit and capital ratios. "Profits are in the ratio 3 : 4 and capitals in the ratio 5 : 8." Times = $\frac{3}{5} : \frac{4}{8}$ = $\frac{3}{5} : \frac{1}{2}$ = 6 : 5.

Splitting a total capital. "A and B put in ₹1,50,000 between them. A's money is in for 8 months and B's for 12 months, and they get equal profits." Equal shares mean A × 8 = B × 12, so A : B = 12 : 8 = 3 : 2, the times turned upside down. A put in ₹90,000 and B ₹60,000.

Checking an answer. Put your answer back in. With B's ₹48,000 for 8 months, B has 48 × 8 = 384 against A's 40 × 12 = 480, and 480 : 384 = 5 : 4, as the question said.

Capitals Given as Percentages, Fractions or Relations

Sometimes the capitals are not given in rupees but described in terms of each other. Turn the description into a ratio first; the rest is the usual work.

Percentages. "A's capital is 25% more than B's, and C's capital is 20% less than B's. The year's profit is ₹1,22,000." Take B's capital as 100. Then A = 125 and C = 80, so A : B : C = 125 : 100 : 80 = 25 : 20 : 16, which is 61 parts. One part = 1,22,000 ÷ 61 = ₹2,000, so the shares are ₹50,000, ₹40,000 and ₹32,000.

Choose 100 for the partner the others are compared with. If C were compared with A ("C's capital is 20% less than A's"), take A as 100.

One capital as a fraction of another. "A's capital is $\frac{3}{5}$ of B's" means A : B = 3 : 5. "A's capital is $1\frac{1}{2}$ times B's" means 3 : 2.

Equal multiples. "3 times A's capital = 4 times B's capital = 6 times C's capital." Then A : B : C = $\frac{1}{3} : \frac{1}{4} : \frac{1}{6}$. Multiply by 12 to get 4 : 3 : 2. Writing 3 : 4 : 6 is the trap.

Capitals as multiples, with different joining times. "A starts a business with some money. After 4 months B joins with twice A's amount, and after 6 months C joins with three times A's amount." Take A's capital as 1. Then A = 1 × 12 = 12, B = 2 × 8 = 16 and C = 3 × 6 = 18, so the profit goes 12 : 16 : 18 = 6 : 8 : 9. You never need the actual amount.

Two ratios linked by one partner. "A : B capitals are 2 : 3 and B : C capitals are 4 : 5." Make B's term the same: 8 : 12 and 12 : 15, so A : B : C = 8 : 12 : 15.

Fractions of the capital and of the time. "A puts in $\frac{1}{3}$ of the total capital for $\frac{1}{4}$ of the time, B puts in $\frac{1}{4}$ of the capital for $\frac{1}{2}$ of the time, and C puts in the rest of the capital for the whole time. The profit is ₹45,000."

C's part of the capital is $1 - \frac{1}{3} - \frac{1}{4} = \frac{5}{12}$, and C's time is the whole time, 1. Multiply the capital fraction by the time fraction for each partner:

So the profit goes 2 : 3 : 10, which is 15 parts of ₹3,000: A gets ₹6,000, B ₹9,000 and C ₹30,000.

Profit shares given as fractions. "A gets $\frac{2}{5}$ of the profit and B and C share the rest equally" is already a ratio: A : B : C = $\frac{2}{5} : \frac{3}{10} : \frac{3}{10}$ = 4 : 3 : 3. If the capitals or times are also given, find the missing one as in the previous part of these notes.

The Working Partner: Salary or Commission Comes First

A working partner is paid for running the business before the profit is shared. The order never changes:

1. Take the salary or commission out of the profit.
2. Share what is left in the ratio of capital × time.
3. The working partner's total = his salary or commission + his share.

A fixed salary. "A and B start a business with ₹1,50,000 and ₹1,00,000. A runs it and gets ₹2,000 a month as salary. The profit for the year is ₹69,000."

• Salary for 12 months = ₹24,000.
• Left to share = 69,000 − 24,000 = ₹45,000, in the ratio 3 : 2.
• A's share = ₹27,000 and B's share = ₹18,000.
• A gets 24,000 + 27,000 = ₹51,000 in all.

Two wrong answers turn up often. Sharing the full ₹69,000 in the ratio 3 : 2 gives A ₹41,400 (salary forgotten), and adding ₹24,000 to that gives ₹65,400 (salary paid twice). The salary is never shared in the capital ratio.

A percentage of the profit. "A and B invest ₹70,000 and ₹50,000. A, the working partner, gets 10% of the profit for running the business, and the rest is shared in the ratio of capitals. The profit is ₹72,000."

• A's commission = 10% of 72,000 = ₹7,200.
• Left = ₹64,800, shared 7 : 5, so A gets ₹37,800 and B gets ₹27,000.
• A gets 7,200 + 37,800 = ₹45,000 in all.

Working back to the profit. If only A's total is given, write it as a part of the profit P. Above, A gets 0.1P + 0.9P × $\frac{7}{12}$ = 0.1P + 0.525P = 0.625P. So if you were told A got ₹45,000 in all, P = 45,000 ÷ 0.625 = ₹72,000.

The same works with a monthly salary. "A gets ₹2,000 a month and ₹51,000 in all for the year" means A's share of the rest is 51,000 − 24,000 = ₹27,000. That is 3 parts of 5, so the rest is ₹45,000 and the profit is ₹69,000.

[Tier 2] Commission on the profit after commission. Some questions say the commission is r% of the profit left after paying the commission. Then the commission C satisfies $C = \frac{r}{100}(P - C)$, which gives

$$C = P \times \frac{r}{100 + r}$$

With r = 10 and P = ₹66,000, the commission is 66,000 × $\frac{10}{110}$ = ₹6,000. Check: 10% of the remaining ₹60,000 is ₹6,000.

[SSC Tip] Underline the words "for managing", "as salary" or "as commission". Whatever follows them is paid first and goes to one partner only. The words "of the profit" or "of the remaining profit" tell you which amount the percentage is taken on.

Interest on Capital and Profit Kept Aside

Interest on capital. Partners sometimes agree that each one first gets simple interest on his capital at a fixed rate, and only the rest of the profit is shared. For one year, interest = capital × rate ÷ 100.

"A and B invest ₹80,000 and ₹50,000. Each gets 10% a year interest on his capital, and the rest of the profit is shared equally. The year's profit is ₹40,000."

The gap between their totals, ₹3,000, comes only from the interest, since the rest was shared equally.

If the rest is shared in the ratio of the capitals instead, the interest step changes nothing: the interest is 8 : 5 and the rest is 8 : 5, so the whole profit ends up 8 : 5. Interest on capital matters only when the rest is shared in some other ratio, usually equally.

For part of a year, use months: interest on ₹60,000 at 12% a year for 6 months is 60,000 × 12 × $\frac{6}{12}$ ÷ 100 = ₹3,600.

Profit kept aside. A part of the profit may go to charity, be saved as a reserve or be put back into the business. Take it out first, then share only the rest.

"The year's profit is ₹75,000. 10% is given to charity and the rest is shared between A and B in the ratio 4 : 5." The amount to share is 90% of 75,000 = ₹67,500, so A gets 67,500 × $\frac{4}{9}$ = ₹30,000 and B gets ₹37,500.

To go backwards, undo the steps in reverse order. If A's ₹30,000 is 4 parts out of 9, the amount shared was ₹67,500. That was 90% of the profit, so the profit was 67,500 ÷ 0.9 = ₹75,000.

Several steps stacked. A question may stack them: first a reserve, then a working partner's commission, then the shares. Follow the order the question gives.

"Of a profit of ₹1,50,000, 20% is kept as a reserve. A gets 10% of the remaining profit for managing, and the rest is shared equally by A and B." The reserve of ₹30,000 leaves ₹1,20,000. The commission of ₹12,000 leaves ₹1,08,000, so each gets ₹54,000, and A gets ₹66,000 in all. If the commission had been 10% of the whole profit, it would be ₹15,000 instead, so read "of the profit" and "of the remaining profit" carefully.

For a final check, add every amount paid out: it must come back to the total profit.

Loss Sharing and Profit Over Several Years

Loss. A loss is shared in exactly the same ratio as a profit would be, capital × time. Each partner's part of the loss comes out of his capital.

"A and B invest ₹60,000 and ₹90,000 for a year and lose ₹20,000." The ratio is 2 : 3, so A bears ₹8,000 and B bears ₹12,000. When the business closes, A gets back 60,000 − 8,000 = ₹52,000 and B gets back ₹78,000.

Profit in one year, loss in the next. Find each year's share with that year's ratio. Add the profit share and subtract the loss share to get a partner's overall gain or loss.

Profit given year by year. If the question gives a separate profit for each year, share each year's profit in that year's ratio.

"A and B start a business with ₹50,000 and ₹30,000. At the start of the second year C joins with ₹40,000 and A takes out ₹10,000. The profit is ₹40,000 in the first year and ₹33,000 in the second."

One profit for the whole period. If the question gives only one profit at the end of two years, add capital × months over all 24 months.

With the same capitals and a single two-year profit of ₹1,90,000: A has 50 × 12 + 40 × 12 = 1,080, B has 30 × 24 = 720 and C has 40 × 12 = 480. The ratio 1,080 : 720 : 480 = 9 : 6 : 4, so A gets ₹90,000, B ₹60,000 and C ₹40,000.

Do not mix the two methods. With the yearly profits above, sharing the total ₹73,000 in the ratio 9 : 6 : 4 would give A about ₹34,579 in place of the correct ₹37,000, because the two years earned different amounts per rupee invested.

A partner who leaves after some years. "A, B and C run a business for 3 years. A leaves at the end of the first year and C joins at the start of the third year" is just joining and leaving with years as the unit: count each partner's capital × years over the 3 years, or convert to months if any period is not a whole year.

Grazing Rent: Animals × Months

When several people rent a pasture (a field for grazing) together, the rent is shared just like a profit. The number of animals plays the part of capital and the number of months plays the part of time:

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

"A, B and C rent a pasture for ₹9,500. A grazes 12 cows for 5 months, B grazes 15 cows for 4 months and C grazes 18 cows for 6 months. How much does each pay?"

The ratio 60 : 60 : 108 = 5 : 5 : 9 is 19 parts. One part = 9,500 ÷ 19 = ₹500, so A and B pay ₹2,500 each and C pays ₹4,500.

Different animals. If horses, cows and sheep graze together, first turn every animal into one kind, using the equivalence the question gives. "2 horses graze as much as 3 cows" means 1 horse = $\frac{3}{2}$ cows.

"A grazes 10 horses for 6 months and B grazes 18 cows for 5 months. 2 horses eat as much as 3 cows. The rent is ₹12,000." A's 10 horses are worth 15 cows, so A uses 15 × 6 = 90 cow-months. B uses 18 × 5 = 90 cow-months. They pay equally, ₹6,000 each. Counting each horse as one cow would give 60 : 90 and make A pay only ₹4,800, which is wrong.

With three kinds, convert all of them to the middle one. If 2 horses = 3 cows and 4 sheep = 1 cow, then a horse counts as $\frac{3}{2}$ cows and a sheep as $\frac{1}{4}$ of a cow.

Finding the number of animals or months. Set up the ratio as before. "A grazes 20 sheep for 6 months and B grazes some sheep for 4 months. Of a rent of ₹3,600, B pays ₹1,600. How many sheep did B graze?" A pays ₹2,000, so 120 : 4n = 2,000 : 1,600 = 5 : 4. Then 5 × 4n = 4 × 120, so 20n = 480 and n = 24.

Days instead of months. Some questions give days. Nothing changes: use animals × days for everyone.

[SSC Tip] Grazing questions hide the same two traps as partnership: forgetting the months, and forgetting to turn different animals into one kind. Add an "in one kind" column to the table whenever more than one animal is named.

Solved Examples

Question 1

P, Q and R start a business with ₹75,000, ₹1,05,000 and ₹1,20,000. At the end of the year the profit is ₹70,000. Find R's share.

Answer. All three kept their money in for the whole year, so the profit goes in the ratio of the capitals. Dividing by 15,000, the ratio is 5 : 7 : 8, which is 20 parts. One part = 70,000 ÷ 20 = ₹3,500, so R gets 8 × 3,500 = ₹28,000. (P gets ₹17,500 and Q gets ₹24,500.)

Question 2

A, B and C invest ₹36,000, ₹54,000 and ₹72,000 in a business for one year. C's share of the profit is ₹9,000 more than A's. Find the total profit and B's share.

Answer. Dividing by 18,000, the capitals are 2 : 3 : 4. C has 2 parts more than A, and those 2 parts are ₹9,000, so one part is ₹4,500. The total is 9 parts = ₹40,500, and B's share is 3 parts = ₹13,500.

Question 3

A invests ₹54,000 for 8 months, B invests ₹72,000 for 5 months and C invests ₹48,000 for 9 months. The total profit is ₹51,000. Find B's share.

Answer. Multiply each capital by its months (in thousands): A = 54 × 8 = 432, B = 72 × 5 = 360, C = 48 × 9 = 432. Dividing by 72, the ratio is 6 : 5 : 6, which is 17 parts. One part = 51,000 ÷ 17 = ₹3,000, so B gets ₹15,000.

Question 4

Rakesh starts a business with ₹42,000. After 5 months Suresh joins him with ₹63,000. At the end of the year the profit is ₹45,000. What is Suresh's share? (a) ₹27,000 (b) ₹24,000 (c) ₹21,000 (d) ₹18,000

Answer. Rakesh's money is in for 12 months, but Suresh's is in for only 12 − 5 = 7 months. In thousands, Rakesh = 42 × 12 = 504 and Suresh = 63 × 7 = 441. Dividing by 63, the ratio is 8 : 7, which is 15 parts of ₹3,000. Suresh gets 7 × 3,000 = ₹21,000. Option (a) is what you get if you share only by capital (2 : 3) and forget the months. The answer is (c) ₹21,000.

Question 5

A, B and C start a business with ₹28,000, ₹42,000 and ₹56,000. A leaves after 6 months, and B leaves 3 months before the end of the year. The profit at the end of the year is ₹87,000. Find each person's share.

Answer. A stays 6 months, B stays 12 − 3 = 9 months and C stays all 12. In thousands: A = 28 × 6 = 168, B = 42 × 9 = 378, C = 56 × 12 = 672. Dividing by 42, the ratio is 4 : 9 : 16, which is 29 parts. One part = 87,000 ÷ 29 = ₹3,000, so A gets ₹12,000, B ₹27,000 and C ₹48,000.

Question 6

X and Y start a business with ₹80,000 and ₹60,000. After 5 months X takes out half of his capital. The profit at the end of the year is ₹70,000. How should it be shared?

Answer. X has 80 (thousand) for 5 months and 40 for the other 7 months: 400 + 280 = 680. Y has 60 for 12 months = 720. So X : Y = 680 : 720 = 17 : 18, which is 35 parts of ₹2,000. X gets ₹34,000 and Y gets ₹36,000.

Question 7

A starts a business with ₹50,000 and B with ₹40,000. After 3 months A takes out ₹10,000. After 6 months B puts in ₹20,000 more. After 9 months A puts in ₹20,000 more. The profit at the end of the year is ₹78,000. Find each share.

Answer. Make a timeline for each person (in thousands).

A : B = 570 : 600 = 19 : 20, which is 39 parts of ₹2,000. A gets ₹38,000 and B gets ₹40,000.

Question 8

A, B and C start a business with ₹60,000, ₹45,000 and ₹75,000. After 4 months A adds ₹15,000 and C takes out ₹15,000. After 8 months B adds ₹30,000. The year's profit is ₹1,14,000. Find each share.

Answer. The changes come after months 4 and 8, so split the year into three periods of 4 months (in thousands).

Dividing by 60, the ratio is 14 : 11 : 13, which is 38 parts. One part = 1,14,000 ÷ 38 = ₹3,000, so A gets ₹42,000, B ₹33,000 and C ₹39,000.

Question 9

A puts ₹1,20,000 into a business for the whole year. B puts in ₹40,000 at the start, another ₹40,000 after 3 months and another ₹40,000 after 6 months. The year's profit is ₹84,000. Find B's share.

Answer. Count each of B's instalments for the months it stays: 40 × 12 + 40 × 9 + 40 × 6 = 480 + 360 + 240 = 1,080. A has 120 × 12 = 1,440. So A : B = 1,440 : 1,080 = 4 : 3, and B gets 84,000 × $\frac{3}{7}$ = ₹36,000. Both put in ₹1,20,000 in the end, but B's money came in bit by bit, so B gets less than half.

Question 10

A starts a business with ₹64,000. Some months later B joins with ₹96,000. At the end of the year the profit is shared between A and B in the ratio 2 : 1. After how many months did B join? (a) 4 months (b) 6 months (c) 8 months (d) 9 months

Answer. Say B's money was in for t months. Then 64 × 12 : 96 × t = 2 : 1, so 96t × 2 = 768 and t = 4. That is how long B's money stayed, so B joined after 12 − 4 = 8 months. Option (a) is the trap. The answer is (c) 8 months.

Question 11

A invests ₹45,000 in a business for a full year. B joins after 3 months. Out of a profit of ₹55,000, B gets ₹30,000. How much did B invest?

Answer. A gets 55,000 − 30,000 = ₹25,000, so A : B = 25 : 30 = 5 : 6. A's capital × months is 45 × 12 = 540, and B's is x × 9 (x in thousands). So $\frac{540}{9x} = \frac{5}{6}$, which gives 45x = 3,240 and x = 72. B invested ₹72,000. Check: 540 : 648 = 5 : 6.

Question 12

A and B start a business with ₹40,000 and ₹50,000. After some months A takes out ₹15,000. Out of a year-end profit of ₹51,000, A gets ₹21,000. After how many months did A take out the money?

Answer. B gets ₹30,000, so A : B = 21 : 30 = 7 : 10. B has 50 × 12 = 600. If A withdrew after x months, A has 40x + 25(12 − x) = 300 + 15x. So (300 + 15x) : 600 = 7 : 10, which gives 300 + 15x = 420 and x = 8. A took out the money after 8 months.

Question 13

A's capital is 20% more than B's, and C's capital is 25% less than A's. A and B invest for the whole year, but C joins after 4 months. Out of a profit of ₹70,000, how much does C get?

Answer. Take B's capital as 100. Then A = 120 and C = 75% of 120 = 90. Now bring in the months: A = 120 × 12 = 1,440, B = 100 × 12 = 1,200 and C = 90 × 8 = 720. Dividing by 240, the ratio is 6 : 5 : 3, which is 14 parts of ₹5,000. C gets ₹15,000.

Question 14

A puts in $\frac{1}{5}$ of the capital for $\frac{1}{4}$ of the time, B puts in $\frac{1}{4}$ of the capital for $\frac{1}{2}$ of the time, and C puts in the rest of the capital for the whole time. Out of a profit of ₹58,000, find each share.

Answer. C's part of the capital is $1 - \frac{1}{5} - \frac{1}{4} = \frac{11}{20}$. Multiply capital by time: A = $\frac{1}{20}$, B = $\frac{1}{8}$, C = $\frac{11}{20}$. Multiplying by 40 gives 2 : 5 : 22, which is 29 parts of ₹2,000. A gets ₹4,000, B ₹10,000 and C ₹44,000.

Question 15

The profits of A, B and C are in the ratio 5 : 3 : 4, and the periods of their investments are in the ratio 2 : 3 : 1. What is the ratio of their capitals? (a) 5 : 2 : 8 (b) 10 : 9 : 4 (c) 5 : 3 : 4 (d) 2 : 1 : 4

Answer. Capital = profit ÷ time, so the capitals are $\frac{5}{2} : \frac{3}{3} : \frac{4}{1}$ = $\frac{5}{2} : 1 : 4$. Multiplying by 2 gives 5 : 2 : 8. Multiplying the profits by the times gives option (b), which is the wrong way round. The answer is (a) 5 : 2 : 8.

Question 16

A and B start a business with ₹1,60,000 and ₹2,40,000. A manages the business and gets ₹3,000 a month for it. The profit for the year is ₹1,16,000. How much does A get in all? (a) ₹46,400 (b) ₹68,000 (c) ₹82,400 (d) ₹32,000

Answer. First pay the salary: 3,000 × 12 = ₹36,000. That leaves 1,16,000 − 36,000 = ₹80,000 to share in the ratio of capitals, 2 : 3. A's share is ₹32,000, so A gets 36,000 + 32,000 = ₹68,000. Option (d) is only A's share of the rest, before the salary is added back. The answer is (b) ₹68,000.

Question 17

A and B invest ₹50,000 and ₹70,000. A is the working partner and gets 20% of the profit for managing the business. The rest is shared in the ratio of their capitals. If A gets ₹40,000 in all, find the total profit.

Answer. Say the profit is P. A gets 0.2P as commission. The other 0.8P is shared 5 : 7, so A gets 0.8P × $\frac{5}{12}$ = $\frac{P}{3}$. In all, A gets 0.2P + $\frac{P}{3}$ = $\frac{8P}{15}$. So $\frac{8P}{15}$ = 40,000 and P = ₹75,000. Check: commission ₹15,000, then A gets $\frac{5}{12}$ of ₹60,000 = ₹25,000, total ₹40,000.

Question 18

[Tier 2] A and B invest ₹90,000 and ₹1,50,000. A runs the business and gets a commission of 12.5% of the profit left after paying his commission. The rest is shared in the ratio of capitals. The total profit is ₹1,44,000. How much does A get in all?

Answer. The commission is 12.5% of what is left, so the commission is $\frac{12.5}{112.5} = \frac{1}{9}$ of the profit: 1,44,000 ÷ 9 = ₹16,000. That leaves ₹1,28,000 to share in 3 : 5, so A's share is ₹48,000. A gets 16,000 + 48,000 = ₹64,000. Check: 12.5% of ₹1,28,000 is ₹16,000.

Question 19

A and B invest ₹1,20,000 and ₹80,000. Each gets interest at 8% a year on his capital, and the rest of the profit is shared equally. The year's profit is ₹56,000. How much does each get?

Answer. Interest first: A gets 8% of 1,20,000 = ₹9,600 and B gets 8% of 80,000 = ₹6,400, which is ₹16,000 in all. The rest, ₹40,000, is split equally, ₹20,000 each. A gets ₹29,600 and B gets ₹26,400. The ₹3,200 gap is exactly the gap in their interest.

Question 20

A business makes a profit of ₹2,00,000. First 15% of the profit is kept as a reserve. Out of what is left, A gets 10% for managing the business, and the rest is shared by A and B in the ratio 3 : 2. How much does A get in all?

Answer. Reserve = ₹30,000, which leaves ₹1,70,000. A's commission is 10% of that, ₹17,000, which leaves ₹1,53,000. A's 3 parts of 5 are ₹91,800 (B gets ₹61,200). A gets 17,000 + 91,800 = ₹1,08,800.

Question 21

A invests ₹50,000 for 12 months, B invests ₹60,000 for 9 months and C invests ₹80,000 for 6 months. The business makes a loss of ₹13,500. How much of his capital does B get back?

Answer. A loss is shared like a profit. In thousands: A = 50 × 12 = 600, B = 60 × 9 = 540, C = 80 × 6 = 480. Dividing by 60, the ratio is 10 : 9 : 8, which is 27 parts of ₹500. B's part of the loss is 9 × 500 = ₹4,500, so B gets back 60,000 − 4,500 = ₹55,500.

Question 22

A and B start a business with ₹60,000 and ₹40,000. At the start of the second year A takes out ₹20,000 and C joins with ₹50,000. The profit is ₹30,000 in the first year and ₹52,000 in the second year. How much does A get over the two years?

Answer. Each year has its own profit, so share each year separately. Year 1: A : B = 3 : 2, so A gets $\frac{3}{5}$ of 30,000 = ₹18,000. Year 2: A : B : C = 40 : 40 : 50 = 4 : 4 : 5, so A gets $\frac{4}{13}$ of 52,000 = ₹16,000. A's total is ₹34,000.

Question 23

A, B and C rent a pasture for ₹9,000. A grazes 16 cows for 3 months, B grazes 12 cows for 5 months and C grazes 18 cows for 4 months. How much rent should C pay?

Answer. Share by cows × months: A = 48, B = 60, C = 72. Dividing by 12, the ratio is 4 : 5 : 6, which is 15 parts of ₹600. C pays 6 × 600 = ₹3,600.

Question 24

A grazes 16 horses for 4 months, B grazes 30 cows for 3 months and C grazes 48 sheep for 5 months in a rented field. 2 horses eat as much as 3 cows, and 4 sheep eat as much as 1 cow. The rent is ₹12,300. How much does each pay?

Answer. Turn everything into cows. 16 horses = 16 × $\frac{3}{2}$ = 24 cows, and 48 sheep = 12 cows. Cow-months: A = 24 × 4 = 96, B = 30 × 3 = 90, C = 12 × 5 = 60. Dividing by 6, the ratio is 16 : 15 : 10, which is 41 parts. One part = 12,300 ÷ 41 = ₹300, so A pays ₹4,800, B ₹4,500 and C ₹3,000.