'A', 'B' and 'C' are partners sharing profits in the ratio of 3 : 2 : 1. 'D' is admitted for \(\frac{{{1^{{\rm{th}}}}}}{6}\)share and 'C' should retain his original share.

The sacrificing ratio of A and B will be:

The correct answer is 3 : 2

Key PointsA's old share= 3 /6
B's old share= 2/6
C's old share= 1/6

Share for D - 1/6

Remaining share = 1 - 1/6 = 5/6

Share retained by C = 1/ 6

Remaining Share = 5 / 6 - 1/ 6 = 4 / 6

This remaining share will be shared by A and B in their old ratio, i.e, 3 : 2

A's new share =  \({4\over6}\times {3\over5} = {12\over 30}\) 

B's new share = \({4\over6}\times {2\over5} = {8\over 30}\)

New Ratio of A : B : C : D = 12 : 8 : 5 : 5

Sacrificing ratio = old Ratio - New Ratio

Sacrifice by A = 3/5 - 12/30 = 6/30

Sacrifice by B = 2/5 - 8/30 = 4/30

Sacrificing Ratio = 6 : 4 = 3 : 2

Alternate Method

When the new ratio is not given in the question, the old ratio itself is the sacrificing ratio.

Hence, the correct answer is 3 : 2