A, B and C are partners sharing profits in the ratio of 3:2:1. D is admitted into firm for 1/4 th share of profit, which he gets 1/8 from A and 1/8 from B. Calculated new sharing ratio of all the partners

The correct option is 9:5:4:6.

Here is the step-by-step calculation to find the new profit-sharing ratio of all partners:

1. Identify the Old Profit-Sharing Ratio:
The old profit-sharing ratio of partners A, B, and C is given as 3:2:1.
Therefore, their individual old shares are:
Share of A = $\frac{3}{6}$
Share of B = $\frac{2}{6}$
Share of C = $\frac{1}{6}$

2. Calculate the Share Sacrificed by Old Partners:
D is admitted for a $\frac{1}{4}$th share, which he acquires:
From A = $\frac{1}{8}$
From B = $\frac{1}{8}$
(Note: C does not sacrifice any portion of his share for D.)

3. Calculate the New Shares of Partners:
New Share = Old Share - Share Sacrificed

For A:
New Share of A = $\frac{3}{6}-\frac{1}{8}$
To subtract these, we find the Least Common Multiple (LCM) of 6 and 8, which is 24.
New Share of A = $\frac{12-3}{24}=\frac{9}{24}$

For B:
New Share of B = $\frac{2}{6}-\frac{1}{8}$
Using the LCM of 24:
New Share of B = $\frac{8-3}{24}=\frac{5}{24}$

For C:
C did not make any sacrifice, so his share remains unchanged:
Share of C = $\frac{1}{6}$
To express C's share with the common denominator of 24, we multiply the numerator and denominator by 4:
Share of C = $\frac{1\times 4}{6\times 4}=\frac{4}{24}$

For D:
Share of D = $\frac{1}{4}$
To express D's share with the common denominator of 24, we multiply the numerator and denominator by 6:
Share of D = $\frac{1\times 6}{4\times 6}=\frac{6}{24}$

4. Determine the New Profit-Sharing Ratio:
Comparing the new shares of all partners (A, B, C, and D):
A : B : C : D = $\frac{9}{24}:\frac{5}{24}:\frac{4}{24}:\frac{6}{24}$
Thus, the new profit-sharing ratio is 9:5:4:6.