### Partnership

Partnership is an important chapter in Quantitative section of IBPS and other competitive examination.
A Partnership is an association of two or more persons who put their money together in order to carry on a certain business. It is of two kinds:
1). Simple 2). Compound
Simple partnership: If the capital of the partners is invested for the same period, the partnership is called simple.
Compound partnership: If the capitals of the partners are invested for different lengths of time, the partnership is called compound.
In a group of n persons invested the different amount for a different period then their profit ratio is :

At1 : Bt2 : Ct3 : Dt4 …….. : Xt4

{ Here first-person invested amount A for t1 period, Second persons invested amount B for t2 period, and so on….}

### Problem Related to This Topic

(1). Rajesh invested Rs. 76000 in a business. After few months Moni joined him and invests Rs. 57000. At the end of the year both of them share the profits at the ratio of 2:1. After how many months Moni joined Rajesh?
Solution:
We can simply calculate per month investment of both partnership:
Rajesh invested Rs. 76,000 for 12 months and, Moni invested Rs. 57,000 for X months.
Now,
(76000 x 12)( 57000 × X) = 21
⇒ (76 × 12) / 2 = 57x
⇒ x = 8
So, Moni invested his money for 8 months and he joined after 4 months.

(2). X and Y started a business by investing money in the ratio of 5:6. Z joined them after a few months by sharing an amount equal to Y’s share. At the end of the year 20% profit was earned which was equal to Rs. 98,000. How much money was invested by Z?
Solution:
First of all,
here we will calculate the ratios
X = 5 × 12 = 60
Y = 6 × 12 = 72
Z = 6 × 6 = 36
Total investment at the end of year = 98000 × 10020 = Rs. 4,90,000

⇒ Investment by Z = 490000 × 36168 × 2
= Rs. 210000

(3). A,B and C enter into partnership. A advances Rs. 1200 for 4 months, B Rs. 1400 for 8 months, and C Rs. 1000 for 10 months. They gain Rs. 585 altogether. Find the share of each.
Solution:
Rs. 1200 in 4 months earns as much profit as Rs. 1200*4 or Rs. 4800 in 1 month.
Rs. 1400 in 8 months earns as much profit as Rs. 1400*8 or Rs. 11200 in 1 month.
Rs. 1000 in 10 months earns as much profit as Rs. 1000*10 or Rs. 10,000 in 1 month.
Therefore, the profit should be divided in the ratios of 44800, 11,200 and 10,000 i.e. in the ratios of 12,28 and 25.
Now, 12+28+25 = 65
A’s share = 1265 x 585 = Rs. 108
B’s share = 2865 x 585 = Rs. 252
C’s share = 2565 x 585 = Rs. 225
Note: In compound partnership, the ratio of profits is directly proportional to both money and time, so they are multiplied together to get the corresponding shares in the ratio of profits.

(4). A and B enter into a partnership for a year. A contributes Rs. 1500 and B Rs. 2000. After 4 months they admit C, who contributes Rs. 2250. If B withdraws his contribution after 9 months, how would they share a profit of Rs. 900 at the end of the year?
Solution:
A’s share : B’s share : C’s share
= 1500 x 12 : 2000 x 9 : 2250 x 8
= 15 x 12 : 20 x 9 : 22.5 x 8
= 180 : 180 : 180 = 1 : 1: 1
Therefore, each of them gets Rs. 900/3 = Rs. 300.

(5). Two partners invest Rs. 125,000 and Rs. 8500 respectively in a business and agree that 60% of the profit should be divided equally between them and the remaining profit is to be treated as interest on capital. If one partner gets Rs.300 more then the other, find the total profit made in the business.
Solution:
The difference counts only due to 40 % of the profit which was distributed according to their investments.
Let the total profit be Rs. X
Then 40% of X is distributed in the ratio
125,000 : 85,000 = 25 : 17
Therefore, the share of the first partner = 40% of X (25/{25+17})
= 40% of X (25/40) = 40X/100[25/42] = 5X/21
Now, from the questions,
The difference in shares = (5X/21) – (17X/105) = 300
Hence, (300 x 105)/8 = Rs. 3937.50

(6). A and B entered into a partnership, investing Rs. 16,000 and Rs. 12,000 respectively. After 3 months, ‘A’ withdrew Rs.5000, while B invested Rs. 5000 more. After 3 more months, C joins the business with a capital of Rs. 21,000. After a year, they obtained a profit of Rs. 26,400. By what value does the share of B exceed the share of C?
Solution:
A invested Rs. 16,000 for 3 months and Rs.(16,000 – 5000) for 9 months.
B invested Rs. 12,000 for 3 months and Rs.(12,000 + 5000) for 9 months.
C invested in Rs. 21,000 for 6 months.
Now, A’s share : B’s share : C’s share
= (16 x 3 + 11 x 9) : (12 x 3 + 17 x 9) : (21 x 6)
= 147 : 189 : 126 = 7 : 9:6
Therefore, B’s share exceeds that of C by
(26400/7+9+6) x (9-6) = (26400 x 3)/22 = Rs. 3600

(7). X, Y and Z jointly thought of engaging themselves in a business venture. It was agreed that X would invest Rs. 6500 for 6 months, Y Rs. 8400 for 5 months and Z, Rs. 10,000 for 3 months. X wants to be the working member for which, he was to receive 5% of the profits. The profit earned was Rs. 7400. Calculate the share of Y in the profit.
Solution:
For managing,
X received = 5% of Rs. 7400 = Rs. 370.
Balance = Rs. (7400 – 370) = Rs. 7030.
Ratio of their investments = X : Y : Z = (6500 x 6) : (8400 x 5) : (10000 x 3)
= 39000 : 42000 : 30000
= 13 : 14 : 10
Hence, Y’s share = Rs. 7030 x 1437 = Rs. 2660

(8). Vijay started a business investing Rs. 90000. After 3 months, Pooja joined him with a capital of Rs. 120000. After another 6 months, Aman joined them with a capital of Rs. 180000. At the end of the year, they made a profit of Rs. 40000. What would be Aman’s share in it?
Solution:
Ratio of investments of Vijay, Pooja and Aman
Vijay : Pooja : Aman = 90000 x 12 : 120000 x 9 : 180000 x 3 = 2 : 2 : 1
Aman’s share = Rs. 40000 x 15 = Rs. 8000

(9). A, B and C enter into a partnership in the ratio 7/2: 4/3 :6/5 . After 4 months, A increases his share 50%. If the total profit at the end of one year be Rs. 21,600, What B’s share in the profit?
Solution:
Ratio of initial investments = 72 :43 : 65 = 105 : 40 : 36.

Let the initial investments be 105X, 40X and 36X.

A : B : C = (105X x 4 + 150100x 105X x 8) : (40X x 12) : (36X x 12)
100
= 1680X : 480X : 432X = 35 : 10 : 9.

Hence, B’s share = Rs.21600 x 1054= Rs. 4000.

(10). Arun, Kamal and Vinay invested Rs. 8000, Rs. 4000 and Rs. 8000 respectively in a business. Arun left after six months. If after eight months, there was a gain of Rs. 4005, then what will be the share of Kamal?
Solution:
Arun : Kamal : Vinay = (8,000 x 6) : (4,000 x 8) : (8,000 x 8)
= 48 : 32 : 64
= 3 : 2 : 4.
Kamal’s share = Rs.4005 x 29 = Rs. 890.