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1. Partnership: When two or more than two persons run a business jointly, they are called partners and the deal is known as partnership.
Partnership is an association of two or more persons who invest their money in order to carry on a certain business. a partner who manages the business is called working partner and the partner who simply invests is called the sleeping partner.
If the period of investment is same for each partner, then the profit or loss is divided in the ratio of their investments.
The simple formula involved is (if there are two partners A and B)
Investment of A x period of investment of A = profit /loss of A
Investment of B x period of investment of B profit /loss of B
OR,
Monthly equivalent investment of A= profit /loss of A
Monthly equivalent investment of B profit /loss of B
2. Ratio of Division of Gains:
i) When investments of all the partners are for the same time, the gain or loss is distributed among the partners in the ratio of their investments.
Suppose A and B invest Rs. x and Rs. y respectively for a year in a business, then at the end of the year:
(A’s share of profit) : (B's share of profit) = x : y.
ii) When investments are for different time periods, then equivalent capitals are calculated for a unit of time by taking (capital x number of units of time). Now, gain or loss is divided in the ratio of these capitals.
Suppose A invests Rs. x for p months and B invests Rs. y for q months, then
(A’s share of profit) : (B's share of profit) = xp : yq.
3. Working and Sleeping Partners: A partner who manages the business is known as a working partner and the one who simply invests the money is a sleeping partner.
EXAMPLES
1. A, B and C started a business by investing Rs. 1,20,000, Rs. 1,35,000 and ,Rs.1,50,000 respectively. Find the share of each, out of an annual profit of Rs. 56,700.
Sol. Ratio of shares of A, Band C = Ratio of their investments
= 120000 : 135000 : 150000 = 8 : 9 : 10.
A’s share = Rs. (56700 x (8/27))= Rs. 16800.
B's share = Rs. ( 56700 x (9/27)) = Rs. 18900.
C's share = Rs. ( 56700 x (10/27))=Rs. 21000.
2. Alfred started a business investing Rs. 45,000. After 3 months, Peter joined him with a capital of Rs. 60,000. After another 6 months, Ronald joined them with a capital of Rs. 90,000. At the end of the year, they made a profit of Rs. 16,500. Find the lire of each.
Sol. Clearly, Alfred invested his capital for 12 months, Peter for 9 months and Ronald
for 3 months.
So, ratio of their capitals = (45000 x 12) : (60000 x 9) : (90000 x 3)
= 540000 : 540000 : 270000 = 2 : 2 : 1.
Alfred's share = Rs. (16500 x (2/5)) = Rs. 6600
Peter's share = Rs. (16500 x (2/5)) = Rs. 6600
Ronald's share = Rs. (16500 x (1/5)) = Rs. 3300.
3. A, Band C start a business each investing Rs. 20,000. After 5 months A withdrew Rs.6000, B withdrew Rs. 4000 and C invests Rs. 6000 more. At the end of the year, a total profit of Rs. 69,900 was recorded. Find the share of each.
Sol. Ratio of the capitals of A, Band C
= 20000 x 5 + 15000 x 7 : 20000 x 5 + 16000 x 7 : 20000 x 5 + 26000 x 7
= 205000:212000 : 282000 = 205 : 212 : 282.
A’s share = Rs. 69900 x (205/699) = Rs. 20500 I
B's share = Rs. 69900 x (212/699) = Rs. 21200;
C's share = Rs. 69900 x (282/699) = Rs. 28200.
4. A, Band C enter into partnership. A invests 3 times as much as Band B invests twothird of what C invests. At the end of the year, the profit earned is Rs. 6600. What is the share of B?
Sol. Let C's capital = Rs. x. Then, B's capital = Rs. (2/3)x
A’s capital = Rs. (3 x (2/3).x) = Rs. 2x.
Ratio of their capitals = 2x : (2/3)x :x = 6 : 2 : 3.
Hence, B's share = Rs. ( 6600 x (2/11))= Rs. 1200.
5. Four milkmen rented a pasture. A grazed 24 cows for 3 months; B 10 for 5 months; C 35 cows for 4 months and D 21 cows for 3 months. If A's share of rent is Rs. 720, find the total rent of the field.
Sol. Ratio of shares of A, B, C, D = (24 x 3) : (10 x 5) : (35 x 4) : (21 x 3) = 72 : 50 : 140 : 63.
Let total rent be Rs. x. Then, A’s share = Rs. (72x)/325
(72x)/325=720 ó x=(720 x 325)/72 = 3250
Hence, total rent of the field is Rs. 3250.
6. A invested Rs. 76,000 in a business. After few months, B joined him Rs. 57,000. At the end of the year, the total profit was divided between them in ratio 2 : 1. After bow many months did B join?
Sol. Suppose B joined after x months. Then, B's money was invested for (12  x)
(76000 x 12)/(57000 x (12x) =2/1 ó 912000=114000(12x)
114 (12  x) = 912ó12x=8óx=4
Hence, B joined after 4 months.
7. Three partners A, B and C invest Rs 1600, Rs 1800 and Rs 2300 respectively in a business. How should they divide a profit of Rs 399?
sol: here profit is to divided in the ratio 16:18:23
A's share of profit=16/(16+18+23)x 399= Rs 112.
B's share of profit=18/(16+18+23)x 399= Rs 126
C's share of profit=23/(16+18+23)x 399= Rs 161
8..A. B and C enter into a partnership. A advances Rs 1200 for 4 months. B gives Rs 1400 for 8 months and C Rs 1000 for 10 months. they gain Rs 585 altogether. find the share of profit each?
sol: monthly equivalent of A = 1200 x 4= 4800
Monthly equivalent of B = 1400 x 8= 11200
Monthly equivalent of C = 1000 x 10= 10000
So, profit is to be divided in the ratio 48:112:100 i.e. 12:28:25
A's share of profit is (12/65)x 585=Rs108
B's share of profit is (28/65)x 585=Rs252
C's share of profit is (25/65)x 585=Rs225
9. Simran started a software business by investing Rs. 50,000. After six months, Nanda joined her with a capital of Rs. 80,000. After 3 years, they earned a profit of Rs. 24,500. What was Simran's share in the profit?
Sol: Simran : Nanda = (50000 x 36) : (80000 x 30) = 3 : 4.
Simran's share = Rs.

24500 x

3

= Rs. 10,500.
 
7
 
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?
Sol: 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

2

= Rs. 890.
 
9
 
11. Aman started a business investing Rs. 70,000. Rakhi joined him after six months with an amount of Rs.. 1,05,000 and Sagar joined them with Rs. 1.4 lakhs after another six months. The amount of profit earned should be distributed in what ratio among Aman, Rakhi and Sagar respectively, 3 years after Aman started the business?
Sol: Aman : Rakhi : Sagar = (70,000 x 36) : (1,05,000 x 30) : (1,40,000 x 24) = 12 : 15 : 16.
12. A began a business with Rs. 85,000. He was joined afterwards by B with Rs. 42,500. For how much period does B join, if the profits at the end of the year are divided in the ratio of 3: 1?
Sol: Suppose B joined f+or x months. Then,
Then,

85000 x 12

=

3
 
42500 x x

1

x =

85000 x 12

= 8.
 
42500 x 3

So, B joined for 8 months.
13. A starts business with Rs. 3500 and after 5 months, B joins with A as his partner. After a year, the profit is divided in the ratio 2 : 3. What is B's contribution in the capital?
Sol: Let B's capital be Rs. x.
Then,

3500 x 12

=

2
 
7x

3

14x = 126000
x = 9000.
14. A and B entered into partnership with capitals in the ratio 4: 5. After 3 months, A withdrew of his capital and B withdrew of his capital. The gain at the end of 10 months was Rs. 760. A's share in this profit is:
Sol: A : B =

4x x 3 +

4x 

1

x 4x

x 7

:

5x x 3 +

5x 

1

x 5x

x 7
 
4

5

= (12x + 21x) : (15x + 28x)
= 33x :43x
= 33 : 43.
A's share = Rs.

760 x

33

= Rs. 330.
 
76

15. A and B started a partnership business investing some amount in the ratio of 3: 5. C joined then after six months with an amount equal to that of B. In what proportion should the profit at the end of one year be distributed among A, B and C?
Sol: Let the initial investments of A and B be 3x and 5x.
A : B : C = (3x x 12) : (5x x 12) : (5x x 6) = 36 : 60 : 30 = 6 : 10 : 5.
16. A, B, C rent a pasture. A puts 10 oxen for 7 months, B puts 12 oxen for 5 months and C puts 15 oxen for 3 months for grazing. If the rent of the pasture is Rs. 175, how much must C pay as his share of rent?
Sol: A : B : C = (10 x 7) : (12 x 5) : (15 x 3) = 70 : 60 : 45 = 14 : 12 : 9.
C's rent = Rs.

175 x

9

= Rs. 45.
 
35
 
17. A and B started a business in partnership investing Rs. 20,000 and Rs. 15,000 respectively. After six months, C joined them with Rs. 20,000. What will be B's share in total profit of Rs. 25,000 earned at the end of 2 years from the starting of the business?
Sol: A : B : C = (20,000 x 24) : (15,000 x 24) : (20,000 x 18) = 4 : 3 : 3.
B's share = Rs.

25000 x

3

= Rs. 7,500.
 
10
 
18. Three partners shared the profit in a business in the ratio 5 : 7 : 8. They had partnered for 14 months, 8 months and 7 months respectively. What was the ratio of their investments?
Sol: Let their investments be Rs. x for 14 months, Rs. y for 8 months and Rs. z for 7 months respectively.
Then, 14 xs: 8y: 7z = 5: 7: 8.
Now,

14x

=

5

98x = 40y y =

49

x

8y

7

20

And,

14x

=

5

112x = 35z z =

112

x =

16

x.

7z

8

35

5

x : y : z = x :

49

x

:

16

x

= 20 : 49 : 64.

20

5
 
19. A, B, C subscribe Rs. 50,000 for a business. A subscribes Rs. 4000 more than B and B Rs. 5000 more than C. Out of a total profit of Rs. 35,000, A receives:
Sol: Let C = x.
Then, B = x + 5000 and A = x + 5000 + 4000 = x + 9000.
So, x + x + 5000 + x + 9000 = 50000
3x = 36000
x = 12000
A : B : C = 21000 : 17000 : 12000 = 21 : 17 : 12.
A's share = Rs.

35000 x

21

= Rs. 14,700.
 
50

20. A, B and C enter into a partnership in the ratio : : . After 4 months, A increases his share 50%. If the total profit at the end of one year be Rs. 21,600, then B's share in the profit is:
Sol: Ratio of initial investments =

7

:

4

:

6

= 105: 40: 36.
 
2

3

5

Let the initial investments be 105x, 40x and 36x.
A : B : C =

105x x 4 +

150

x 105x x 8

: (40x x 12) : (36x x 12)
 
100

= 1680x : 480x : 432x = 35 : 10 : 9.
Hence, B's share = Rs.

21600 x

10

= Rs. 4000.
 
54

21. Reena and Shaloo are partners in a business. Reena invests Rs. 35,000 for 8 months and shaloo invests Rs.42,000 for 10 months. Out of a profit of Rs.31, 570. Reena's share is
Ratio of their shares

=(35000 ×8):(42000 ×10)
 
=2 : 3.
 
Reena's share

=Rs.(31570 ×2/5)
 
=Rs.12628.

22. P and Q started a business investing Rs.85,000 and Rs.15,000 respectively. In what ratio the profit earned after 2 years be divided between P and Q respectively?
Sol: P : Q

= 85000 : 15000

=85 : 15
 
=17 : 3

23.Anand and Deepak started a business investing Rs.22,500 and Rs.35,000 respectively. Out of a total profit of Rs. 13,800. Deepak's share is
Sol: Ratio of their shares

= 22500 : 35000

=9 : 14.
 
Deepak's share

= Rs.(13800×14/23)

= Rs. 8400.

25.Kamal started a business investing Rs.9000. After five months, Sameer joined with a capital of Rs.8000. If at the end of the year, they earn a profit of Rs.6970, then what will be the shares of sameer in the profit?
Kamal : Sameer

=(9000×12):(8000 ×7)

= 108 : 56
 
= 27 : 14.
 
Sameer's share

= Rs.(6970 ×14/41)

=Rs.2380.

26. A, B and C enter into a partnership. They invest Rs. 40,000 , Rs. 80,000 and Rs.1,20,000 respectively. At the end of the first year, B withdraws Rs. 40,000 while at the end of the second year, C withdraws Rs.80,000. In what ratio will the profit be shared at the end of 3 years?
A : B: C

=(40000×36) : (80000×12+40000×24) : (120000×24+40000×12)

=144 :192 :336
 
= 3 : 4 : 7.

27. A started a business with Rs.21,000 and is joined afterwards by B with Rs.36,000. After how many months did B join if the profits at the end of the year are divided equally?
Sol: Suppose B joined after x months.
 
Then,

21000×12=36000×(12  x)

‹=›36x=180
 
= x= 5.

28. X and Y invested in a business. They earned some profit which they divided in the ratio of 2 : 3. If X invested Rs.40,000. the amount invested by Y is
Sol: Suppose Y invested Rs.y
 
Then,

40000 /y = 2 / 3

‹=› y=(40000×3 / 2).
 
‹=› y=60000.

29. A, B, C hired a car for Rs.520 and used it for 7,8 and 11 hours respectively. Hire charges paid by B were
A : B : C

= 7 : 8 : 11.

Hire charges paid by B

= Rs(520×8/26)

= Rs.160.

30. If 4 (A's capital) = 6 (B's capital) = 10 (C's capital), then out of a profit of Rs. 4650, C will receive:
 
 
 
31.A and B entered into a partnership investing Rs. 16,000 and Rs. 12,000 respectively. After 3 months, A withdraw Rs. 5000 while B invested Rs. 5000 more. After 3 more months, C joins the business with a capital of Rs. 21,000. The share of B exceeds that of C, out of a total profit of Rs. 26,400 after one year by:
A : B : C = (16000 x 3 + 11000 x 9) : (12000 x 3 + 17000 x 9) : (21000 x 6)
 
 
 
32. In a business, A and C invested amounts in the ratio 2 :1, whereas the ratio between amounts invested by A and B was 3 : 2. If Rs. 1,57,300 was their profit, how much amount did B receive?
 
So, B : A : C = 4 : 6 : 3 or A : B : C = 6 : 4 : 3.
 

33.A and B started a business jointly. A’s investment was thrice the investment of B and the period of his investment was two times the period of investment of B. If B received Rs. 4000 as profit, then their total profit is:
Suppose B invested Rs. x for y months. Then, A invested Rs. 3x for 2y months.
 
So, A : B = (3x x 2y) : (x x y) = 6xy = 6 ; 1.
 
 
 
34. A, B and C enter into partnership. A invests some money at the beginning, B invests double the amount after 6 months and C invests thrice the amount after 8 months. If the annual profit be Rs. 27,000, C’s share is :
 
Let A's investment be Rs. x.
 
Then, Ratio of capitals = (x x 12) : (2x x 6) : (3x x 4) = 12x : 12x : 12x = 1 : 1 : 1.
 

35.A, B and C started a shop by investing Rs. 27,000, Rs. 72,000 and Rs. 81,000 respectively. At the end of the year, the profits were distributed among them. If C’s share of profit be Rs. 36,000, then the total profit was:
37. A and B invest in a business in the ratio 3 : 2. If 5% of the total profit goes to charity and A’s share is Rs. 855, the total profit is :
38. A, B, C started a business with their investments in the ratio 1 : 3 : 5, After 4 months, A invested the same amount as before and B as well as C withdraw half of their investments. The ratio of their profits at the end of the year is :
39.Simran started a software business by investing Rs. 50,000. After six months, Nanda joined her with a capital of Rs. 80, 000. After 3 years, they earned a profit of Rs. 24, 500. What was Simran’s share in the profit?
40.A and B started a business with initial investments in the ratio 14 : 15 and their annual profits were in the ratio 7 : 6. If A invested the money for 10 months, for how many months did B invest his money?
41. A, B, C hired a car for Rs. 520 and used it for 7, 8 and 11 hours respectively. Hire charges paid by B were :
42. A, B and C enter into a partnership with a capital in which A’s contribution is Rs. 10,000. If out of a total profit of Rs. 1000, A gets Rs.500 and B gets Rs. 300, then C’s capital is:
43.Reena and shaloo are partners in a business. Reena invests Rs. 35,000 for 8months and shaloo invests Rs. 42,000 for 10 months. Out of a profit of Rs. 31,570, Reena’s share is:
44.A starts business with Rs. 3500 and after 5 months, B joins with A as his partner. After a year, the profit is divided in the ratio 2 : 3. What is B’s contribution in the capital?
45. Three partners A, B, C start business. Twice A’s capital is equal to thrice B’s capital and B’s capital is four times C’s capital. Out of a total profit of Rs. 16,500 at the end of the year. B’s share is :
46.A started a business with Rs. 21,000 and is joined afterwards by B with Rs. 36,000. After how many months did B join if the profits at the end of the year are divided equally?

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