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Ratio & Partnership

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CAT 2025 Lesson : Ratio & Partnership - Partnership

7. Partnership

Partnership is a common form of business which is owned by two or more partners (or owners of the business).

These partners might be working partners, who work for the business besides investing in it, or might be sleeping partners, who only invest in the business and not work for it. Therefore, working partners might draw a salary from the business while sleeping partners might not.

In any business, the excess of revenue over expenses is profit. In partnership, the profits (after paying partners' salaries) are distributed to the partners in a defined ratio.

Unless specified otherwise, in such questions, profits distributed to a partner will be directly proportional to the investment amount and the duration of the investment.

\dfrac{\text{Profit of A}}{\text{Profit of B}}

= \dfrac{\text{Investment of A} \times \text{Duration of A's Investment}}{\text{Investment of B} \times \text{Duration of B's Investment}}

Where

\text{I}_{\text{X}}

is the investment of partner X,

\text{D}_{\text{X}}

is the duration of X's investment and

\text{P}_{\text{X}}

is the Profit of partner X, the ratio of profits of 4 partners A, B, C and D is

\text{P}_{\text{A}} : \text{P}_{\text{B}} : \text{P}_{\text{C}} : \text{P}_{\text{D}} = \text{I}_{\text{A}}\text{D}_{\text{A}} : \text{I}_{\text{B}}\text{D}_{\text{B}} : \text{I}_{\text{C}}\text{D}_{\text{C}} : \text{I}_{\text{D}}\text{D}_{\text{D}}

Example 12

W, X, Y and Z invested Rs.

1,20,000

, Rs.

1,00,000

, Rs.

90,000

and Rs.

70,000

for

4, 6, 8

and

10

months respectively. If the profits at the end of the year was Rs.

75,000

, then the difference between the profits (in Rupees) that X and Y received was

Solution

Although not explicitly stated, profits are implied to be distributed in the ratio of the investments.

Substituting the investment and duration in the formula given above

\text{P}_{\text{W}} : \text{P}_{\text{X}} : \text{P}_{\text{Y}} : \text{P}_{\text{Z}}

= 120000 \times 4 : 100000 \times 6 : 90000 \times 8 : 70000 \times 10

= 24 : 30 : 36 : 35

X's share in the profit

= \dfrac{30}{24 + 30 + 36 + 35} = \dfrac{30}{125}

Similarly, Y's share in profit

= \dfrac{36}{125}

Difference between X's share and Y's share

= \left(\dfrac{36}{125} - \dfrac{30}{125} \right) \times 75000

= \dfrac{6}{125} \times 75000 =

Rs.

3600

Answer: Rs.

3600

Example 13

Larry and Sergey brought Rs.

25,000

and Rs.

40,000

as capital and started a company at the beginning of a year.

6

months after the business commenced, the company was short of funds and Larry brought in additional capital of Rs.

30,000

. Two months later, when the company needed further capital, Sergey brought in Rs.

20,000

. It was decided that the profits would be distributed in the ratio of the average capital invested during this period. If Larry received Rs.

24,000

of profit at the end of the year, then how much profit did Sergey receive?

Solution

As the capital amounts are in the multiples of

1000

and we are concerned about ratios, we can remove 3 zeroes in each of the amounts.

Larry brought in capitals of Rs.

25

and Rs.

30

which stayed in the business for

12

months and

6

months respectively.

Sergey brought in capitals of Rs.

40

and Rs.

20

which stayed in the business for

12

months and

4

months respectively.

Ratio of Profits = Ratio of Capital Invested

= 25 \times 12 + 30 \times 6 : 40 \times 12 + 20 \times 4

= 480 : 560

= 6 : 7

Let the total profit be

x

.

Larry's share in profit

= \dfrac{6}{(6 + 7)} \times x = 24000

⇒

x = 52000

Sergey's share in profit

= 52000 - 24000 = 28000

Answer: Rs.

28,000

Example 14

If the ratio of amount invested by A, B and C is

1 : 2 : 3

and the profits were split in the ratio of

3 : 4 : 5

, then what was the ratio of time period of the investments?

Solution

\text{I}_{\text{A}}\text{D}_{\text{A}} : \text{I}_{\text{B}}\text{D}_{\text{B}} : \text{I}_{\text{C}}\text{D}_{\text{C}} = \text{P}_{\text{A}} : \text{P}_{\text{B}} : \text{P}_{\text{C}}

⇒

1 \times \text{D}_{\text{A}} : 2 \times \text{D}_{\text{B}} : 3 \times \text{D}_{\text{C}} = 3 : 4 : 5

⇒

\text{D}_{\text{A}} : \text{D}_{\text{B}} : \text{D}_{\text{C}} = \dfrac{3}{1} : \dfrac{4}{2} : \dfrac{5}{3} = 9 : 6 : 5

Answer:

9 : 6 : 5

Example 15

A, B and C started a business at the beginning of a financial year with capitals of Rs.

20,000,

Rs.

30,000

and Rs.

50,000

respectively. If B and C withdrew their investments and exited the business sometime during the year and the ratio in which profits were divided was

4 : 2 : 5

, then after how many months did B and C quit?

(1)

4

and

6

(2)

8

and

6

(3)

6

and

4

(4)

6

and

8

Solution

Ratio of Capitals invested

= 20000 : 30000 : 50000 = 2 : 3 : 5

Let

b

and

c

be the number of months for which the investment of B and C remained in the business. A's investment stayed in the business for the entire duration of

12

months.

Ratio of time periods

= 12 : b : c

Ratio of Capitals Invested

=

Ratio of Profits

⇒

2 \times 12 : 3 \times b : 5 \times c = 4 : 2 : 5

Scaling up the first term in the RHS to

24

, we get

⇒

24 : 3b : 5c = 24 : 12 : 30

∴

3b = 12

and

5c = 30

⇒

b = 4

and

c = 6

Answer:

(1) 4

and

6

Example 16

Viji started a business at the beginning of a year with a capital of Rs.

1,04,000

. Three months later, Raji joined the business and brought in a capital of Rs.

91,000

. The monthly salaries that Raji and Viji were to draw was Rs.

3,000

and Rs.

2,000

respectively. These salaries were, however, to be paid as a lump sum at the end of the year. If the company's total profit (before salary payments to partners) was Rs.

2,63,000

, then the amount due to Raji (Salary + Profits) is

Solution

Viji worked for

12

months and earned a salary of

2000 \times 12 =

Rs.

24,000.

Raji worked for

9

months and earned a salary of

3000 \times 9 =

Rs.

27,000.

Profit after deducting partners' salaries

= 2,63,000 - 24,000 - 27,000 =

Rs.

2,12,000

Profits are distributed to the partners only after paying their salaries.

Viji and Raji invested Rs.

1,04,000

and Rs.

91,000

for

12

and

9

months respectively.

Ratio of Profits

= 104 \times 12 : 91 \times 9 = 32 : 21

Raji's share in profits

= \dfrac{21}{53} \times 212000 = 21 \times 4000 =

Rs.

84,000

Total amount due to Raji

=

Profit + Salary

= 84,000 + 27,000 =

Rs.

1,11,000

Answer: Rs.

1,11,000

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