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# X and Y are partners who share net income in the ratio 4:3 respectively and have capital balances of Rs.10,000 and 8,000 respectively. Z is admitted for $\dfrac{1}{8}$ share and brings Rs.6,000 for capita and Rs.4200 for goodwill. The new capital balances of X, Y and Z. Verified
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Hint: For calculating new capital balances we have to find new capital ratios of X, Y and Z . We subtract the share fraction of Z from 1 to calculate the fraction of X and Y . We find the contribution of X and Y by New ratio = Old ratio x Remaining share of A and B
Now we take the ratio of X,Y and Z . Then we will find the distribution of goodwill amount of Z to X and Y . Then sum up the capital balances and the amount added to find the new capital income.

Complete step-by-step solution:
Old ratio = 4 : 3
The shares of X and Y are $\dfrac{4}{7}$, $\dfrac{3}{7}$ of the total net income.
The portion of X and y in new share = 1- Share of Z in new share of net income
$1 - \dfrac{1}{8} \\ = \dfrac{7}{8}$
New ratio = Old ratio x Remaining share of A and B
$X = \dfrac{4}{7} \times \dfrac{7}{8} \\ = \dfrac{4}{8} \\$
$Y = \dfrac{3}{7} \times \dfrac{7}{8} \\ = \dfrac{3}{8} \\$
The new ratio of share of net income among X , Y and Z is
$\dfrac{4}{8}:\dfrac{3}{8}:\dfrac{1}{8} \\ = 4:3:1$
Hence the new ratio of share =4:3:1
Z brings Rs.4200 for goodwill.
The sacrificing ratio of Z to X and Y is 4:3
For X the amount added from Z is
$\dfrac{4}{7} \times 4200 = 4 \times 600 \\ = 2400 \\$
For Y the amount added from Z is
$\dfrac{3}{7} \times 4200 = 3 \times 600 \\ = 1800$
So the amount shared to X and Y is Rs.2,400 and Rs.1,800 respectively.
The new capital balance are for A is $10,000 + 2,400 = 12,400$ rupees
The new capital balance of Y is $8,000 + 1800 = 9800$rupees
The new capital balance of Z is 6,000 rupees

Note: : It is necessary to ascertain the new profit sharing ratio even for the old partners when the new partner is admitted to the firm because with the admission of the new partner in the firm the new profit sharing ratios are introduced as the new partner gains the profit sharing ratio from the old partners of the firm.