X, Y, Z are sharing profits in the ratio of 6 : 5 : 3. A is admitted into partnership for 1/8th share. The sacrificing ratio of X, Y and Z is
6 : 5 : 3
The sacrificing ratio is the ratio in which the old partners give up their shares in favour of the incoming partner. When the question does not say how the new partner's share is taken, the old partners are assumed to give it up in their old profit-sharing ratio.
A's share of \( \dfrac{1}{8} \) is contributed in the ratio 6 : 5 : 3: X gives \( \dfrac{1}{8} \times \dfrac{6}{14} = \dfrac{6}{112} \), Y gives \( \dfrac{1}{8} \times \dfrac{5}{14} = \dfrac{5}{112} \) and Z gives \( \dfrac{1}{8} \times \dfrac{3}{14} = \dfrac{3}{112} \). Total \( = \dfrac{14}{112} = \dfrac{1}{8} \).
Sacrificing ratio \( = 6 : 5 : 3 \), the same as the old ratio. The remaining \( \dfrac{7}{8} \) is shared 6 : 5 : 3, so the new ratio of X, Y, Z and A is \( 42 : 35 : 21 : 14 = 6 : 5 : 3 : 2 \).
Hence, 6 : 5 : 3.
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