Calculating the New Profit Sharing Ratio

The initial profit sharing ratio between partners A, B, and C is 4 : 3 : 2. The total parts in the initial ratio are $4 + 3 + 2 = 9$.

A new partner, D, is admitted with a $\frac{1}{8}$ share.

The remaining profit share for the old partners (A, B, C) is calculated as: $1 - \text{D's Share} = 1 - \frac{1}{8} = \frac{8}{8} - \frac{1}{8} = \frac{7}{8}$.

This remaining share of $\frac{7}{8}$ will be distributed among A, B, and C according to their old profit sharing ratio (4 : 3 : 2).

Determining New Shares for A, B, and C

Calculate the new share for each old partner by multiplying their original fraction by the remaining profit share ($\frac{7}{8}$):

• A's New Share = Original Share $\times$ Remaining Share
  $ \text{A's New Share} = \frac{4}{9} \times \frac{7}{8} = \frac{28}{72} $
• B's New Share = Original Share $\times$ Remaining Share
  $ \text{B's New Share} = \frac{3}{9} \times \frac{7}{8} = \frac{21}{72} $
• C's New Share = Original Share $\times$ Remaining Share
  $ \text{C's New Share} = \frac{2}{9} \times \frac{7}{8} = \frac{14}{72} $

Incorporating D's Share

D's share is given as $\frac{1}{8}$. To maintain a common denominator with the other partners' shares (which is 72), we convert D's share: $ \text{D's Share} = \frac{1}{8} = \frac{1 \times 9}{8 \times 9} = \frac{9}{72} $

Final New Profit Sharing Ratio

The new profit sharing ratio is formed by combining the new shares of A, B, C, and D:
A : B : C : D = $\frac{28}{72} : \frac{21}{72} : \frac{14}{72} : \frac{9}{72}$
Removing the common denominator, the new ratio is: 28 : 21 : 14 : 9