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Important Formulae

Ratio of Divisions of Gains:

1. 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.

2. 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.

Practice Questions

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 A, out of an annual profit of Rs. 56,700. A.16000 B.16500 C.16800 D.17000

Ratio of shares of A,B and C = 120000 : 135000 : 150000
Ratio = 8 : 9 : 10
= 8+9+10 = 27
∴ A's share = Rs. (56700 x 8/27) = 16800

2.

 Ram started a business investing Rs. 45000. After 3 months, Ramesh joined him with a capital of Rs. 60000. After another 6 months, Rakesh joined them with a capital of Rs. 90000. At the end of the year, they made a profit of Rs. 16500. Find the share Ramesh ? A.6400 B.6500 C.6600 D.6700

Ram invested his capital for 12 months, Ramesh for 9 months, rakesh for 3 months.
So, Ratio of capitals = (45000 x 12) : (60000 x 9) : (90000 x 3)
= 540000 : 540000 : 270000
= 2 : 2 : 1.
So, Ramesh share = Rs. (16500 x 2/5) = Rs. 6600

3.

 A, B and C enter into partnership. A invests 3 times as much as B invests and B invests two third of what C invests. At the end of the year, the profit earned is Rs. 6600. What is the share of B ? A.1200 B.1250 C.1300 D.1350

Let C capital = Rs. X.
Then B capital = Rs. 2/3 X
A capital = Rs. (3 x 2/3 X) = Rs. 2X
∴ Ratio of their capitals = 2X : 2/3 X : X
= 6 : 2 : 3
Hence, B share = Rs. ( 6600 x 2/11) = Rs. 1200.

4.

 Four milkmen rented a pasture. A grazed 24 cows for 3 months, B 10 cows 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? A.3100 B.3150 C.3200 D.3250

Ratio of shares of A, B, C, and D = (24x3) : (10x5) : (35x4) : (21x3)
Ratio of A : B : C : D = 72 : 50 : 140 : 63
Let Total rent be Rs. X. Then, A's share = Rs. 72X/325
∴ 72X/325 = 720
⇒ X = (720 x 325)/72 = 3250

5.

 A invested Rs. 76000 in a business. After few months, B joined him with Rs. 57000. At the end of year, the total profit was divided between them in the ratio 2 : 1. After how many months did B join ? A.3 months B.4 months C.5 months D.6 months

Suppose B joined after X months.
Then, B's money was invested for (12 – x) months.
∴ (76000 x 12)/(57000 x (12-X)) = 2/1
⇒ 912000 = 114000 (12 – X)
⇒ 114 (12 – X) = 912
⇒ 12 – X = 8
⇒ X = 4
Hence B joined after 4 months.

6.

 A, B and C enter into a partnership by investing in the ratio of 3 : 2 : 4. After one year, B invests another Rs. 270000 and C, at the end of 2 years, also invests Rs. 270000. At the end of three years, profits are shared in the ratio of 3 : 4 : 5. Find the initial investment of C ? A.3600 B.360000 C.36000 D.3600000

Let the initial investments of A, B and C be Rs. 3X, Rs. 2X and Rs. 4X respectively.
Then, (3X x 36) : [(2X x 12) + (2X + 270000) x 24] : [(4X x 24) + (4X + 270000) x 12] = 3 : 4 : 5
⇒ 108X : (72X + 6480000) : (144X + 3240000) = 3 : 4 : 5
∴ (108 X)/(72 X+6480000) = 3/4
⇒ 432 X = 216X + 19440000
⇒ 216 X = 19440000
X = 90000
C's initial investment = 4X = Rs. 360000.

7.

 P and Q started a business investing Rs. 85000 and Rs. 15000 respectively. In what ratio the profit earned after 2 years be divided between P and Q respectively. A.73:3 B.17:3 C.3:17 D.13:17

P:Q = 85000:15000 = 17:3

8.

 Kamal started a business investing Rs. 9000. After five months, Sameer joined with a capital of Rs. 8000. If at the end of year, they earn a profit of Rs. 6970, then what will be the share of Sameer in the profit? A.10000 B.10500 C.11000 D.11500

Kamal : Sameer = (9000 x 12) : (8000 x 7)
⇒ Kamal : Sameer = 108 : 56 = 27 : 14
∴ Sameer's share = Rs. ( 6970 x 14/41 ) = Rs. 10500