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Solution

(i)

	R	T
OLD RATIO	3	2

Sacrificing Ratio = Old Ratio × Surrender Ratio
R’s = 3/5 × 1/4 = 3/20
T’s = 2/5 × 1/5 = 2/25

New Ratio = Old Ratio − Sacrificing Ratio
R = 3/5-3/20 = 9/20
T = 2/5-2/25 = 8/25
S’s Share = R’s Sacrifice + S’s Sacrifice
= 3/20 + 2/25 = 23/100

	R	T	S
New profit-sharing ratio	9/20	8/25	23/100
=	45/100	32/100	23/100
=	45	32	23

(ii)

	A	B
OLD RATIO	1	1

C admits for 1/4^th share of profit
Let the combined share of A, B and C be = 1
Combined share of A and B = 1-C’s share = 1-1/4 = 3/4

New Ratio = Combined share of A and B ×2/3
A’s = 3/4 × 2/3 = 6/12
B’s = 3/4 × 1/3 = 3/12

	A	B	C
New profit-sharing ratio	6/12	3/12	1/4
=	6/12	3/12	3/12
=	2	1	1

(iii)

	A	B
OLD RATIO	3	2

  C admits for 1/5share of profit

A’s sacrifice	= C’s share × 1/5
	= 1/5 × 1/5
	= 1/25
B’s sacrifice	= C’s share × 4/5
	= 1/5 × 4/5
	= 4/25

New Ratio = Old Ratio − Sacrificing Ratio
A’s = 3/5-1/25 = 14/25
B’s = 2/5-4/25 = 6/25

	A	B	C
New profit-sharing ratio	14/25	6/25	1/5
=	14/25	6/25	5/25
=	14	6	5

(iv)

	X	Y	Z
OLD RATIO	3	2	1

W admits for 1/6 share of profit
Let combined share of all partner after W’s admission be = 1
Combined share X and Y in the new firm = 1- Z’s share- W’s share
= 1-1/6-1/6 = 4/6

New Ratio = Old Ratio × Combined share of X and Y

A’s	= 3/5×4/6
	= 12/30
B’s	= 2/5×4/6
	= 8/30

	X	Y	Z	W
New profit-sharing ratio	12/30	8/30	1/6	1/6
=	12/30	8/30	5/30	5/30
=	12	8	5	5

(v)

	A	B
OLD RATIO	1	1

C admits for 1/5 share
D admits for 1/6 share
Let combined share of all partner after C and D’s admission be = 1
Combined share of profit of A and B after C and D’s admission = 1- C’s share – Z’s share
= 1-1/5-1/6 = 19/30

New Ratio = Old Ratio × Combined share of A and B

A’s	= 1/2 × 19/30
	= 19/60
B’s	= 1/2 × 19/30
	= 19/60

	X	Y	Z	W
New profit-sharing ratio	19/60	19/60	1/5	1/6
=	19/60	19/60	12/60	10/60
=	19	19	12	10

(vi)

	A	B
OLD RATIO	3	2

C admits for 1/5 share of profit
Let the combined share of all partners after C’s admission be = 1
Combined share of A and B after C’s admission = 1 – C’s share
= 1-1/4 = 3/4

New ratio of A and B each= Combined share of A and B × 1/2
= 3/4 × 1/2 = 3/8 each                 

	X	Y	Z
New profit-sharing ratio	3/8	3/8	1/4
=	3/8	3/8	2/8
=	3	3	2