Partnership Concepts including solved problems with shortcut methods

Partnership: When two or more two persons run a business then they are called partners and when they agree to share profit and losses, the deal is known as partnership.

Elements of partnership

Capital: The sum of money invested by the partners to start any business is known as capital.

Equivalent capital: When capital invested by partners is multiplied by the time of investment, the product thus obtained is known as equivalent capital.

Types of partnerships.

Case 1: when the investment and the time of investment are equal, the profit and loss are distributed equally.
            m1:m2:m3 where m1,m2,m3 are the investment of respective partners

Case 2: When an investment is different but the time of investment is the same. then the profit and loss are shared in the ratio of their time invested.
           t1:t2:t3 where t1,t2,t3 are the time periods.

Case 3: When the investment in a business is different then the profit and loss are shared in the ratio of their equivalent capital. 
           m1t1:m2t2:m3t3

Case 4: Multiple investments for different time periods.

           Σ m1t1:Σ m2t2:Σ m3t3 …: Σmntn

• The formula for compound partnership can also be written as:

A's Capital × A's Time in partnership / B's capital × B's Time in partnership = A's profit/B's profit

• If the investment is in the ratio of a:b:c and time of investment is in the ratio of x:y:z

Then, 

The Profit would be distributed in the ratio of ax: by CZ

Ex.1 What will be the ratio of Profit if three partners A, B, and C are investing 20,000 Rs. , 45,000 Rs. , and Rs. 60,000 for one year.

Sol. The ratio of profits of A, B, and C will be

20000:45000:60000

4: 9: 12 Ans.

Ex.2 A began a business with 450 R.s and was joined of towards by B with 300 Rs. when did B join if profit at the end of the year is in ratio 2:1?

Sol. let B joined of x months

Then we have 450*20/300*x =1/2

300* 2x  = 450*12

x= 9 months

B joined After =12-9 = 3 Months  Ans

Ex.3 If A and B invested 8 and 10 Rs. respectively A invested for 5 months and b invested for 8 months then the ratio of their profit will be.

Sol. The ratio of their profit will be

8*5:10*8 = 40:80

1:2 Ans

Ex.4 A and B enter into a partnership with Capital in the ratio 5:6 At the end of 8 months. A withdraw his Capital. If they receive profits in ratio 5:9.Find how long B's Capital is used.

Sol. Let B's time of partnership is x

5 * A's time in partnership / 6 * B's time in partnership= 5/9

5/6 * 9/5 = B's time/A's time

3/2 = x/8 

x = 12 months Ans

Ex.5 A, B, C are partners. A receive 2/5 of the profit. B and C Share the Remaining profit equally. A's income is increased by 220 Rs. When profit rises from 8 to 10%. find the Capital invested by A, B, C.

Sol. For A's share

10% -8% = 220 Rs.

2% = 220 Rs.

1% = 110 Rs.

A's share is 100% = 11,000 Rs

For B and C's Share

2/5 = 11,000 Rs.

1 = 11,000 * 5/2

Remaining part is 3/5 thus

1*3/5 = 11,000 * 5/2 * 3*5 = 16,500 Rs

B and C has 8250 Rs. Capital each  Ans

Ex.6 Two partners invested 50,000 and 70,000 Rs. respectively in a business and agreed that 70% of the profit should be divided equally between then and the remaining profit in the ratio of investment. if one partner gets 90 Rs. more than the other. Find the total profit made in the business.

Sol. The difference comes only due to 30% of the profit

suppose the total profit is x.

Then 30% of x is distributed in the ratio

50,000:70,000= 5:7

30% of 5/5+7 = 30* 5x /12 = x/8

Share of second partner = 30*7x/5+7=7x/40

Now difference in share = 7x/40 - x/8 = 90 Rs.

7x-5x/40 = 90

x = 1800 Rs.

Second method :

90(100/30)(5+7/7-5)= 1800 Rs.  Ans

Ex.7 A and B invested in the ratio 3:2 in a business. If 5% of the total profit goes directly to the charity and A's share is 855. Find the total profit.

Sol. Suppose the total profit is Rs. 100

5 Rs. Directly goes to charity

Now 95 Rs. is divided in the ratio of 3:2

A\s share = (95/3+2)*3 = 57

But we see that A's actual profit is 855 Rs.

57 = 855

1 = 855/57

100= (855/57)*10 = 1500 Rs.

Second Method:

855(100/95)*(5/3)

1500 Rs. Ans.

Ex.8 If three partners A, B, and C are investing  30000,  40000, and 50,000 Rs. each for period of 1 year, 2 years, and 3 years respectively then find the ratio of their profits.

Sol. Ratio of their profit will be

m1t1:m2t2 :m3t3

30,000*1:40,000*2:50,000*3

30,000:80,000:150,000

3:8:15 Ans

Ex.9 Three partners undergo a partnership with an initial investment of Rs. 120000, Rs. 80000, and Rs. 150000. Next year A and B invested Rs. 80000 and Rs. 20000 respectively whereas C withdraws Rs.50000 from the business. In the third year of their partnership A, B, and  C invested Rs, 10000, Rs. 50000 and Rs, 10000 respectively. Find the ratio of their profit.

Sol. The ratio of their profit will be

                                     A                    B                    C

first-year            120000                80000            150000  

2nd year            +80000                +20000          -50000 

3rd  year           +100000               +50000        +100000

Effective Investment:

                                   A                    B                    C

1st year               120000            80000              150000

2nd year              200000           100000             100000

3rd year              300000            150000             200000

                            620000            330000            450000

The ratio of their profit will be 62:33:45 Ans.

Ex.10 A, B, C entered into partnership and their capital is in the proportion 1/3:1/4:1/5 A withdraw half his capital at the end of 4 months. out of  a total annual profit of rs. 8470, A's share is?

Sol. Ratio of their profit will be

[(x/3) *4 + (x/6) *8] : [(x/4)*12]:[(x/5)*12]

8x/3: 3x : 12x/5

40:45:36

Share of A = 8470/121 * 40 = 2800 Rs. Ans

Ex.11 A, B, C started a business each investing Rs.20000 after 5 months A withdraw Rs. 5000, B Rs.4000, and C invested Rs.6000 more. At the end of the year the total profit of Rs. 69900 was recorded what is the share of b?

Sol. The ratio of their profit will be

20000*5+15000*7:20000*5+16000*7:20000*5+26000*77

205:212:282

Share of B = (69900/699)* 212 =  Rs. 21200 Ans

Ex.12 P, Q, R started a business by investing Rs 200000, 250000,400000 respectively. They decided to receive a 10% interest in their capital and the balance of the profit to be divided equally. if they got Rs. 20500 as the annual profit, find the share of P including the interest?

Sol. 10% of interest on the capitals of P, Q, R respectively be

= 20000, 25000, 40000

Now, the total of interest = 20000+25000+40000 = 85000

Remaining of the profit after interest on capitals = 205000-85000=120000

But the remaining of the profit has to be divided equally

So, P's share of profit = 120000/3 = 40000

Total profit = interest on capital + share of principal after interest on capital

= 20000+40000 = 60000 Rs. Ans

Ex.13 A, B, and C hired a house for 1 year in Rs. 13825. They lived together for four months after that C left the house. Next five months B also left the house. Find the share of C in total rent?

Sol. Ratio of their profit will be

t₁:t₂:t₃

12:9:4

C's rent = (4/25)*13825 = Rs. 2212 Ans

Ex.14 A, B, and C are partners in a company. A gets 1/3 of total profit and B gets 1/4 of total profit. At the end of the year, C gets Rs. 5000 of profit. find the profit of A?

Sol. share of C = 1-(1/3+1/4)

so 5/12 of profit = 5000

Total profit = 5000 * 12/5 = Rs. 12000

A's share = (1/3)*12000 = Rs. 4000 Ans

Ex.15 Some amounts distributed among A, B, and C. A received 3/16 of the total amount and B received 1/4 of the total amount. If C gets Rs. 81 then B gets amount is?

Sol.C's part= 1-3/16-1/4

(16-3-4)/16 = 9/16

9/16 Part = Rs. 81

so total amount = (81*16)/9 = 144 Rs.

B's Part = (1/4) * 144 = Rs. 36 Ans