**Partnership :**Whenever at least two people take part in a business as co-proprietors, the association is known as an Partnership. A partnership includes at least two individuals (partners).**Partners :**The people who take part in a business are Called partners.**Profit :**Positive Difference between the amount spent and sum earned in Business when earning is more than Capital**Loss :**Negative Difference between the amount spent and sum earned in Business when earning is less than Capital**Profit/ Loss Share :**Its a Profit or loss of individuals who are partner in Business which depends on Capital and Investment Time.**Investment Time**: Period of times capital is invested in business.**Capitals**: Amount of sum invested in business by individuals or partners.**RATIO OF DIVISIONS OF GAINS**: When investments of all the prawn are for the same time, the gain or loss is distributed among the partners .**WORKING PARTNERS**: A partner who manages the business.**SLEEPING PARTNERS**: A partner who simply invests the money in business.

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__Solution / Short-Tricks / Formula for solving Partnership Problems__

**If**

__Case 1:__**capitals of two partners be Rs. C1 and Rs. C2 for the same period**and the total profit be Rs. P, then shares of the partners in the profit are

If the

**capitals of three partners be Rs. C1, Rs C2 and Rs. C3 for the same period**, and the total profit be Rs. P, then shares of the partners in the profit are

**Example 1:**

*A, B and C invested Rs. 20,000, Rs. 50,000 and Rs. 40,000, in a business. The net profit for the year was Rs. 12,100. This was divided in proportion to their investments. Find out the amount of profit each partner has earned.*

**Solution:**We have,

- CI. = 20,000,
- C2 = 50,000,
- C3 = 40,000 and
- P = 12,100.

__Profit share of A:__

=(C1 x P )/( C1+ C2 + C3)

=(20,000 x 12,1000/( 20,000 + 50,000 + 40,000)

= 2/ 11 x 12,100

= Rs. 2,200.

__Profit share of B:__

= (C2 x P) / (C1+ C2 + C3)

= (50,000x12,100) / (20,000 + 50,000 + 40,000)

= 5/11 x 12,100

= Rs. 5,500.

__Profit share of C:__

= (C3 x P) / (C1+ C2 + C3)

= (40,000x12,100) / (20,000 + 50,000 + 40,000)

=4/11 x 12' 100

= Rs. 4,400.

**If the capitals of two partners be Rs. C1 and Rs. C2 for the periods t1 and t2, and the total profit be Rs. P, then shares of the partners in the profits are:**

__Case 2:__If the capitals of three partners be Rs. C1, Rs. C2 and Rs. C3 for the periods t1, t2 and t3, respectively, and the total profit be Rs. P, then shares of the partners in the profit are

__Example 2:__*A and B are two partners in a business*. A contributes Rs. 1,200 for 5 months and B contributes Rs. 750 for 4 months. If total profit is Rs. 450,

*find out their respective shares*.

**Solution:**We have,

- CI. = 1200,
- C2 = 750,
- t1= 5, t2 = 4 and
- P = 450.

__Profit share of A__

= (c1 x t1 x P) / ( C1t1 + C2t2)

= (1,200x5x450 ) / (1,200x5 + 750x4)

= Rs' 27,00,000 / 9000

= Rs. 300

__Profit share of B__

= ( C2 x t2 x P) / ( C1t1 + C2t2)

=(750x4x450) / (1,200x5 + 750x4)

= 13,50,000 / 9.000

= Rs. 150

**If the capitals of two partners be Rs. C1 and Rs. C2 for the periods t1 and t2 respectively, then**

__Case 3 (A):__**(Profit of A / Profit of B) = (c1 x t1) / (c2 x t2)**

**It there is a loss in the business, then**

*Note:***Loss of (A: B) = (Cl X t1) : (C2 x t2)**

**If the capitals of three partners be Rs. C1, Rs. C2 and Rs. C3 for the periods t1, t2 and t3 respectively, then Loss / Profit Share are:**

__Case 3 (B)__:**Profit Share of (A : B : C) = (Cl X t1) : (C2 x t2) : (C3 x t3)**

*It there is a loss in the business, then*

**Note:****Loss of (A: B: C) = (Cl X t1) : (C2 x t2) : (C3 x t3)**

Example 3: There are three partners A, B and C in a business. A invests Rs. 2000 for 5 months, B invests Rs. 1200 for 6 months and C invests Rs. 2500 for 3 months. Find out the ratio of their shares in the profit?

**Solution:**Here,

C1 x t1 = 2,000 x 5 = 10,000,

C2 x t2 = 1,200 x 6 = 7,200 and

C3 s t3 = 2,500 x 3 = 7,500.

Profit of A: Profit of B: Profit of C = C1xt1 : C2xt2 : C3xt3

= 10,000: 7,200: 7,500 or 100:72:75.

**If the capitals of three partners in a business invested in the ratio of CI: C2: C3 and their profits are in the ratio P1: P2: P3, If the timing of their investments is in the ratio of t1: t2: t3 then**

__Case 4 (A):__

__ratio of timing of their investments__- (t1 : t2 : t3) = P1/C1 : P2/C2 : P3/C3

__ratio of their capitals invested is__- (C1 : C2 : C3) = P1/t1 : P2/t2 : P3/t3

**Example 4:**Anuu, Manu and Tanu invested capitals in a business the ratio 4:6:9. At the end of the business, they received their shares of profits in the ratio 2:3:5. Find the ratio of time for which they invested their capitals.

**Solution:**We have

- C1: C2: C3 = 4:6:9
- P1: P2: P3 = 2:3:5

the ratio of time for which Anuu, Manu and Tanu invested their capitals

= (P1 /C1) : (P2 /C2) : (P3 / C3)

= 2/4 : 3/6 : 5/9

= 1/2 : 1/2 : 5/9

= 9:9:10.

**Example 5:**Gupta, Singhal and Kansal starts a business. If the ratio of their periods of investments are 1:2:5 and their profits are in the ratio of 3:4:5, find the ratio of capitals of Gupta, Singhal and Kansal.

**Solution:**We have,

- P1: P2: P3 = 3:4:5
- t1: t2: t3 = 1:2:5

= P1/t1 : P2/t2 : P3/t3

= 3/1 : 4/2 : 5/5

= 3 : 2 : 1

Thus, Gupta, Singhal and Kansal invested their capitals in the ratio of 3:2:1.

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