Application of ratios in partnership: Definition, Questions, Examples

Application of ratios in partnerships is a very crucial chapter of Mathematics, particularly in determining the profit sharing ratio (PSR), which dictates how profits and losses are distributed among partners.

Profit sharing ratio in partnership deed is usually specified. For example, if partners contribute equally, the PSR is 1 : 1. However, if contributions are unequal, such as 2 : 3 : 5, profits are shared accordingly. The new profit sharing ratio formula in admission of a partner is recalculated to reflect the updated contributions, ensuring that profits are shared according to the revised ratios.

Profit sharing Ratio formula = $\frac{\text{Partner’s capital contribution}}{\text{Total capital contribution}}$

Profit sharing ratio questions resources like “profit sharing ratio class 12 DK goel solutions”, “profit sharing ratio class 12 solutions”, and profit sharing ratio calculators guide these calculations.

We will also discuss the profit sharing ratio in LLP (Limited Liability Partnership).

What is Partnership?

When two or more individuals formally agree to run a business or company and decide to share profits depending on the investment, it is called a Partnership.

Partnership depends on three main factors: Investment, time, and profit/loss

Investment $\propto$ Time, Time $\propto$ Profit/Loss.

So, Investment $\propto$ Profit/Loss

There are various types of partnerships.

A general partnership is a type of partnership where two or more people run a business together with equal rights and profits are distributed as per their investment.

A limited partnership is a type of partnership where there will be both limited and general partners. Limited partners will have limited responsibilities and they will not participate in daily business run in. General partners will have unlimited access and they will handle day-to-day operations.

A Limited Liability Partnership (LLP) is a type of partnership where all partners have limited liability and responsibility.

Partnership at will is a partnership where two or more partners can start a business together and other people can join in the middle. Also, anybody can leave the business at their will.

Concept of Profit Sharing Ratio

Profit sharing ratio is the most important part of a partnership. It will decide how the profits will be distributed among the partners based on the investment and the time period it was invested.

Profit sharing Ratio formula = $\frac{\text{Partner’s capital contribution}}{\text{Total capital contribution}}$

Suppose the capital invested by A and B is in the ratio x : y. That means for every x part A invested B invested y part.

Total capital is (x + y).

Now, we will divide their part by total capital to get how many parts of the profits they will each get.

Profit sharing ratio of A = $\frac{x}{x+y}$, Profit sharing ratio of B = $\frac{y}{x+y}$

So, A will get $\frac{x}{x+y}$ × Total profit

And B will get $\frac{y}{x+y}$ × Total profit

If a new partner joins or anyone withdraws money between the said time period, then we have to adjust the profit sharing ratio accordingly.

Example:

A, B, and C started a business by investing INR 9,000, INR 18,000 and INR 21,000, respectively. If B's share in the profit earned by them is INR 759, then the total profit earned by them together is:

Sol: Ratio of investment

A : B : C = 9000 : 18000 : 21000

⇒ A : B : C = 3 : 6 : 7

Let $x$ be the whole profit.

According to the question,

B's share = 759

⇒ $\frac{6x}{3+6+7} = 759$

⇒ $x = \frac{759×16}{6} = 2024$

Hence, the correct answer is INR 2,024.

How to calculate profit if the investment is made for an equal time period

When partners invest money for an equal time period, then determining the profit sharing ratio process is easy. First, calculate the total investment during that time. Then calculate the the profit sharing ratio using the capital investments.

Let’s understand this with an example.

Suppose A invests 1000, B invests 2000, and C invests 3000 for 12 months. Then how will the profit be distributed among them?

Total money A invested in 12 months = 1000 × 12 = 12000

Total money B invested in 12 months = 2000 × 12 = 24000

Total money C invested in 12 months = 3000 × 12 = 36000

Total investment = 12000 + 24000 + 36000 = 72000

Profit sharing ratio for A = $\frac{12000}{72000}=\frac{1}{6}$

Profit sharing ratio for A = $\frac{24000}{72000}=\frac{1}{3}$

Profit sharing ratio for A = $\frac{36000}{72000}=\frac{1}{2}$

A will get $\frac{1}{6}$ × Profit

B will get $\frac{1}{3}$ × Profit

C will get $\frac{1}{2}$ × Profit

Example:

Suresh, Dinesh, and Ramesh became partners in a business by investing money in the ratio of 3 : 6 : 8. If their investments are increased by 5%, 15%, and 20%, respectively, then what will be the ratio of their profits for one year?

Sol: The initial ratio of their investment = 3 : 6 : 8

Let the investment by Suresh be $3x$.

Investment by Dinesh is $6x$.

And, Investment by Ramesh is $8x$.

Now, their investments are increased by 5%, 15%, and 20% respectively.

⇒ Investment by Suresh

= $3x+5\% \text{ of }3x=3x+\frac{5}{100}\times3x=3x+\frac{3x}{20}=\frac{63x}{20}$

Investment by Dinesh

= $6x+15\%\text{ of }6x=6x+\frac{15}{100}\times6x=6x+\frac{18x}{20}=\frac{138x}{20}$

And, Investment by Ramesh

= $8x+20\%\text{ of }8x=8x+\frac{20}{100}\times8x=8x+\frac{32x}{20}=\frac{192x}{20}$

So, Required ratio = $\frac{63x}{20}:\frac{138x}{20}:\frac{192x}{20}=63:138:192=21:46:64$

Hence, the correct answer is 21 : 46 : 64.

How to calculate profit if the investment is made for unequal time period

When partners invest money for an unequal time period, then we have to calculate the total investment each made during their own time period. Sum up all the investments to get the total investment. Then calculate the the profit sharing ratio using the capital investments.

Let’s understand this with an example.

Suppose A invests 1000 for 12 months, B invests 2000 for 9 months, and C invests 3000 for 6 months. Then how will the profit be distributed among them after 12 months?

Total money A invested in 12 months = 1000 × 12 = 12000

Total money B invested in 8 months = 2000 × 9 = 18000

Total money C invested in 6 months = 3000 × 6 = 18000

Total investment = 12000 + 18000 + 18000 = 48000

Profit sharing ratio for A = $\frac{12000}{48000}=\frac{1}{4}$

Profit sharing ratio for A = $\frac{18000}{48000}=\frac{3}{8}$

Profit sharing ratio for A = $\frac{18000}{48000}=\frac{3}{8}$

A will get $\frac{1}{4}$ × Profit

B will get $\frac{3}{8}$ × Profit

C will get $\frac{3}{8}$ × Profit

Example:

The ratio of investments of P and Q is 4 : 5. P invests for 1 year and Q invests for 2 years. What is the ratio of profit of P and Q?

Sol: The ratio of investment of P and Q = 4 : 5

Time ratio (in years) = 1 : 2

The profit ratio is equal to the product of investment and time ratio.

So, the profit ratio of P and Q = (4 × 1) : (5 × 2) = 4 : 10 = 2 : 5

Hence, the correct answer is 2 : 5.

Active Partner and Passive Partner

In partnership, partners can have different types of roles or involvements. On this basis, we can divide it into two types, Active Partner and Passive Partner.

As the name suggests, an Active Partner is someone who operates on a daily basis and maintains the business or company.

A passive Partner is someone who doesn't take part in day-to-day operations and only contributes financially.

They are also called Sleeping Partners.

Example:

A is a working partner and B is a sleeping partner in a company. Their investment ratio is 3 : 5.

A gets 20% of the profit for managing the company and the rest of the profit is shared as per the investment. If the total profit is 6400, what will A get?

Let investment by Active partner, A be 3x.

And investment by Sleeping partner, B is 5x.

Total investment = 3x + 5x = 8x

A’s profit sharing ratio = $\frac{3x}{8x}=\frac{3}{8}$

B’s profit sharing ratio = $\frac{5x}{8x}=\frac{5}{8}$

A gets 20% of 6400 for managing the company.

20% of 6400 = 1280

The rest of the money = 6400 - 1280 = 5120, which will divided as per investment.

A gets = 5120 × $\frac{3}{8}$ + 1280 = 1920 + 1280 = 3200

B gets = 5120 × $\frac{5}{8}$ = 3200

Difference between Active Partner and Passive Partner

Tips and Tricks

• Investment $\propto$ Time, Time $\propto$ Profit/Loss.

So, Investment $\propto$ Profit/Loss

• Profit sharing Ratio formula = $\frac{\text{Partner’s capital contribution}}{\text{Total capital contribution}}$

• When partners invest money for an equal time period, then multiply the money by the time period and calculate the profit-sharing ratio.

• When partners invest money for an unequal time period, then we have to calculate the total investment each made during their own time period.

Practice Questions

Q1. In a partnership business, B's capital was half of A's. If after 8 months, B withdrew half of his capital and after 2 more months, A withdrew $\frac{1}{4}$th of his capital, then the profit ratio of A to B will be:

1. 5 : 2

2. 10 : 23

3. 2 : 5

4. 23 : 10

Hint: Find the profit ratio by taking the ratio of the sum of the product of the amount invested and their corresponding time period.

Answer:

Let $x$ be B's capital.

Then, A's capital will be $2x$.

Profit ratio of A and B = (capital of A × time period of A) : (capital of B × time period of B)

= $(2x × 10 + \frac{3}{4} × 2x×2) : (x × 8 + \frac{x}{2} × 4)$

= $(20x+3x) : (8x+2x)$

= $23x:10x$

= $23:10$

Hence, the correct answer is 23 : 10.

Q2. A, B, and C together start a business. Three times the investment of A equals four times the investment of B and the capital of B is twice that of C. The ratio of the shares in the profit is:

1. 8 : 3 : 6

2. 3 : 8 : 6

3. 3 : 6 : 8

4. 8 : 6 : 3

Hint: Calculate the ratio of three numbers from two separate ratios by equalizing the common term.

Answer:

Given: Three times the investment of A equals four times the investment of B.

So, 3A = 4B

⇒ A : B = 4 : 3

The capital of B is twice that of C

So, B = 2C

⇒ B : C = 2 : 1

Now, A : B = 4 : 3 = 8 : 6

Also, B : C = 2 : 1 = 6 : 3

⇒ A : B : C = 8 : 6 : 3

Hence, the correct answer is 8 : 6 : 3.

Q3. A sum of Rs. 15525 is divided among Sunil, Anil, and Jamil such that if Rs. 22, Rs. 35, and Rs. 48 are diminished from their shares, respectively, their remaining sums shall be in the ratio 7 : 10 : 13. What would have been the ratio of their sums if Rs. 16, Rs. 77, and Rs. 37, respectively, were added to their original shares?

1. 9 : 13 : 17

2. 18 : 26 : 35

3. 36 : 52 : 67

4. None of these

Hint: First, find the distributed sum by subtracting the amount diminished from each share from the total sum, and then calculate the new shares of each one of them to find the new ratio.

Answer:

Total sum = Rs.15525

Distributed sum = 15525 – 22 – 35 – 48 = Rs.15420

The ratio of remaining share = 7 : 10 : 13

New share of Sunil = ($\frac{7}{7+10+13}$ × 15420) + 22 + 16

= ($\frac{7}{30}$ × 15420) + 22 + 16 = Rs. 3636

New share of Anil = ($\frac{10}{7+10+13}$ × 15420) + 35 + 77

= ($\frac{10}{30}$ × 15420) + 35 + 77 = Rs. 5252

New share of Jamil = ($\frac{13}{7+10+13}$ × 15420) + 48 + 37

= ($\frac{13}{30}$ × 15420) + 48 + 37 = Rs. 6767

So, the new ratio = 3636 : 5252 : 6767 = 36 : 52 : 67

Hence, the correct answer is 36 : 52 : 67.

Q4. A, B, and C invest to start a restaurant. The total investment is Rs. 3 lakhs. B invested Rs. 50,000 more than A and C invested Rs. 25,000 less than B. If the profit at the end of the year was Rs.14,400, then what is C's share of the profit (in Rs.)?

1. 3600

2. 4800

3. 5200

4. 4200

Hint: The ratio of profit is the same as the ratio of investment.

Answer:

Let a be the investment of A.

Investment of B = a + 50000

Investment of C = (a + 50000) – 25000

Total investment = a + (a + 50000) + [(a + 50000) – 25000]

⇒ 300000 = 3a + 75000

⇒ Investment of A = $\frac{300000-75000}{3}$ = $\frac{225000}{3}$ = Rs. 75000

⇒ Investment of B = 75000 + 50000 = Rs. 125000

⇒ Investment of C = 125000 – 25000 = Rs. 100000

Ratio of investment = 75000 : 125000 : 100000 = 3 : 5 : 4

⇒ Ratio of profit = 3 : 5 : 4

Since total profit = Rs.14400,

So, the profit received by C = $\frac{4}{3+5+4}$ × 14400

= $\frac{4}{12}$ × 14400

= Rs. 4800

Hence, the correct answer is 4800.

Q5. Two businessmen A and B, invest in a business in the ratio of 5 : 8. They decided to reinvest 30% of the profit they earned back into the business. The remaining profit they distributed amongst themselves. If A's share of the profit was Rs. 87,500, then how much profit (in Rs.) did the business make?

1. 2,27,000

2. 2,50,000

3. 3,75,000

4. 3,25,000

Hint: Find the profit distributed by subtracting the reinvested amount from the total profit and the ratio of profits is the same as the ratio of the investment.

Answer:

Reinvested amount = 30% of total profit

⇒ Remaining profit to be distributed = 70% of total profit

The ratio of investment of A and B = 5 : 8

So, the profit ratio of A and B = 5 : 8

A's share in the profit = Rs. 87500

⇒ 87500 = profit ratio of A × remaining profit

= $\frac{5}{5+8}$ × 70% of total profit

= $\frac{5×70 }{13×100}$ × total profit

⇒ Total profit = $\frac{87500×13×100 }{5×70}$ = 12500 × 26 = Rs. 325000

Hence, the correct answer is 3,25,000.

Q6. A is a working partner and B is a sleeping partner in a business. A invested Rs. 30,000 and B invested Rs. 50,000. A receives 20% of the profit for managing the business and the rest is divided in proportion to their capital. What does B get out of the profit of Rs. 10,000?

1. Rs. 5,500

2. Rs. 5,000

3. Rs. 4,500

4. Rs. 4,000

Hint: First, subtract the managing partner’s share from the total profit. Then, divide the remaining profit in proportion to the capital invested by each partner. Calculate B’s share based on his investment.

Answer:

A receives 20% of the total profit for managing the business.

A gets 20% of Rs. 10,000 = Rs. 2,000

The remaining profit = Rs. 10,000 – Rs. 2,000 = Rs. 8,000

This remaining profit is divided in proportion to their capital.

The total capital invested = Rs. 30,000 + 50,000 = Rs. 80,000

So, B's share of the remaining profit = $\frac{50,000}{80,000}$ × 8,000 = Rs. 5,000

Hence, the correct answer is Rs. 5,000.

Q7. Anuj started a business by investing Rs. 30,000, and after 6 months Ajay joined him with a capital of Rs. 45,000. After another 3 months, Ankit joined him with a capital of Rs, 60,000. At the end of the year, profit is Rs.18,000. Find the share of Anuj in the profit.

1. Rs. 6,000

2. Rs. 12,000

3. Rs. 8,000

4. Rs. 10,000

Hint: The profit is directly proportional to the capital invested and the time duration of the investment.

Answer:

According to the question,

Total profit = Rs. 18000

Anuj invested Rs. 30,000 for 12 months.

Ajay invested Rs. 45,000 for 6 months.

And, Ankit invested Rs, 60,000 for 3 months.

Since profit is directly proportional to capital invested and the time duration of investment,

The ratio of profits of Anuj, Ajay, and Ankit respectively = 30000 × 12 : 45000 × 6 : 60000 × 3 = 4 : 3 : 2

So, the share of Anuj in profit = $18000\times \frac{4}{9}=8000$

Hence, the correct answer is Rs. 8,000.

Q8. Rohit started a business with Rs. 75000 and after some months Simaran joined him with Rs. 60000. If the profit at the end of the year is divided in the ratio 3 : 1, then after how many months did Simran join Rohit?

1. 7

2. 6

3. 8

4. 4

Hint: The share contribution is directly proportional to investment and period and the ratio is calculated per their share contribution.

Answer:

Given: The investment made by Rohit = Rs. 75,000

And investment made by Simran = Rs. 60,000

And the ratio of profits = 3 : 1

Let the time of investment Simran be $t$ months respectively.

So, $75000\times 12:60000\times t = 3:1$

⇒ $\frac{5\times 12}{4t}=\frac{3}{1}$

⇒ $t$ = 5 months.

Since we are talking about investment in a year only, so, after 12 - 5 = 7 months Simran joins Rohit.

Hence, the correct answer is 7.

Q9. S, T, and U start a business and their capitals are in the ratio of $3:4:6$. At the end, they receive the profit in the ratio of $1:2:3$. What will be the respective ratio of time for which they contribute their capital?

1. $3:2:2$

2. $2:3:3$

3. $2:2:3$

4. $4:5:3$

Hint: The ratio of share in profit = (Amount of A $\times$ time of A) : (Amount of B $\times$ time of B):(Amount of C $\times$ time of C)

Answer:

Given: The ratio of profit of S, T, and U = $1:2:3$

Let the capital of S, T, and U be $3x$, $4x$ and $6x$ respectively.

Let the time of investment of S, T, and U be $a$, $b$, and $c$ respectively.

$3ax:4bx:6cx = 1:2:3$

⇒ $\frac{3a}{4b}=\frac{1}{2}$ and $\frac{4b}{6c}=\frac{2}{3}$

⇒ $\frac{a}{b}=\frac{2}{3}$ and $\frac{b}{c}=\frac{1}{1}=\frac{3}{3}$

So, $a:b:c = 2:3:3$

Hence, the correct answer is $2:3:3$

Q10. Rs. 13000 is divided among X, Y, and Z such that 2 times X's share is equal to 3 times Y's share which is equal to 4 times Z's share. What is the share of Y?

1. Rs. 3200

2. Rs. 4800

3. Rs. 5600

4. Rs. 4000

Hint: By finding the ratio of $X, Y,$ and $Z$ to get the share from the given value.

Answer:

Let $2X = 3Y = 4Z = k$

$X =\frac{k}{2}, Y = \frac{k}{3}, Z = \frac{k}{4}$

The sum of shares of all of them together = 13000

⇒ $\frac{k}{2} + \frac{k}{3} + \frac{k}{4} = 13000$

⇒ $\frac{6k+4k+3k}{12}=13000$

⇒ $\frac{13k}{12}=13000$

⇒ $k = 12000$

$\therefore$ Share of $Y = \frac{1}{3}\times 12000=4000$

Hence, the correct answer is Rs. 4000.