10 :- Find the New Profit Sharing Ratio:
(i) R and T are partners in a firm sharing profits in the ratio of 3 : 2. S joins the firm, R surrenders 1/4th of his share and T 1/5th of his share in favour of S.
(ii) A and B are partners. They admit C for 1/4th share. In future, the ratio between A and B would be 2 : 1.
(iii) A and B are partners sharing profits and losses in the ratio of 3 : 2. They admit C for 1/5th share in the profit. C acquires 1/5th of his share from A and 4/5th share from B.
(iv) A, B and C are partners in the ratio of 1/2 : 1/3 : 1/6. D joins the firm as a new partner for 1/6th share in profits. C would retain his original share.
(v) A and B are equal partners. They admit C and D as partners with 1/5th and 1/6th share respectively.
(vi) A and B are partners sharing profits in the ratio of 5 : 3. C is admitted for 3/10th share of profit half of which was gifted by A and the remaining share was taken by C equally from A and B.

Solution :-

CASE 1 – Calculation of new profit sharing ratio
Old ratio = 3:2
R’s new share = 3/5 – (3/5 x ¼) = 3/5 – 3/20 = 9/20 x 5/5 = 45/100
T’s new share = 2/5 – (2/5 x 1/5) = 2/5 – 2/25 = 8/25 x 4/4 = 32/100
S’s share = 3/20 + 2/25 = 15 + 8/100 = 23/100
Thus, new profit sharing ratio of R,T and S is 45:32:23

CASE 2 :- Let the total profit sharing ratio of the firm be 1
C admitted for 1/4th share
Remaining share for A and B is 3/4 will they will distribute in 2:1
A’s new share = 3/4 x 2/3 = 6/12
B’s new share = 3/4 x 1/3 = 3/12
C’s share = 1/4 x 3/3 = 3/12
Thus, New profit share of A, B and C is 6:3:3 or 2:1:1

CASE 3 :- Old ratio = 3:2
C is admitted for 1/5th share
A’s new share = 3/5 – (1/5 x 1/5) = 3/5 – 1/25 = 14/25
B’s new share = 2/5 – (1/5 x 4/5) = 2/5 – 4/25 = 6/25
C’s share = 1/5 x 5/5 = 5/25
Thus, New profit sharing ratio of A, B and C is 14:6:5

CASE 4 :- Let the total profit share of the firm be 1
Old ratio = 3:2:1
D was admitted for 1/6th share
Remaining share of the firm = 1 – 1/6 – 1/6 = 4/6 = 2/3
A and B will distribute the remaining share in their old ratio i.e, 3:2
A’s new share = 2/3 x 3/5 = 6/15 x 6/6 = 36/90
B’s new share = 2/3 x 2/5 = 4/15 x 6/6 = 24/90
C’s share = 1/6 x 15/15 = 15/90
D’s share = 1/6 x 15/15 = 15/90
Thus, New profit share of A, B, C and D is 36:24:15:15 or 12:8:5:5

CASE 5:- Old ratio = 1:1
Let the total share of firm be 1
C and D are admitted for 1/5th and 1/6th share respectively.
Remaining share = 1 – 1/5 – 1/6 = 30 – 6 – 5/30 = 19/30
A’s new share = 19/30 x 1/2 = 19/60
B’s new share = 19/30 x 1/2 = 19/60
C’s share = 1/5 x 12/12 = 12/60
D’s share = 1/6 x 10/10 = 10/60
Thus, New profit sharing ratio of A, B, C and D is 19:19:12:10

CASE 6 :- Old profit sharing ratio of A and B is 5:3
C was admitted for 3/10th share
A’s new share = 5/8 – (3/10 x 1/2) – (3/20 x 1/2 ) = 5/8 – 3/20 – 3/40 = 25 – 6 – 3/40 = 16/40
B’s new share = 3/8 – (3/20 x 1/2 ) = 3/8 – 3/40 = 15 – 3/40 = 12/40
C’s share = 3/10 x 4/4 = 12/40
Thus, New profit sharing ratio of A, B and C is 16:12:12 or 4:3:3