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.
1. 

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 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. 

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. 

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. 

Suppose B joined after X months.
Then, B's money was invested for (12 – x) months.
∴ (76000 x 12)/(57000 x (12X)) = 2/1
⇒ 912000 = 114000 (12 – X)
⇒ 114 (12 – X) = 912
⇒ 12 – X = 8
⇒ X = 4
Hence B joined after 4 months.
6. 

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:Q = 85000:15000 = 17:3
8. 

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