(Partners make Sacrifice in Old Ratio)
A and B are partners sharing profits in the ratio of 3 : 1. C is admitted into partnership for 1/8th of the profits. Calculate sacrificing ratio and new profit-sharing ratio.

Explanation

Since, C's share is given without mentioning as to what C a acquires from A and B separately, C acquires his share from A and B in their old profit-sharing ratio. Therefore, the sacrifice made by A and B is in the ratio of 3 : 1.
:. A's sacrifice `=(1)/(8)xx(3)/(4)=(3)/(32)`, B's sacrifice `=(1)/(8)xx(1)/(4)=(1)/(32)`
Hence, Sacrificing Ratio of A and B is `3//32:1//32" or"3:1`. In this case, the sacrificing ratio will always be the old profit-sharing ratio. Of course, the new ratio is different and it is calculated as follows:
Suppose total profit is 1
C takes 1/8th share out of 1. Remaining share `=1-(1)/(8)=(7)/(8)`
:. A's New Share `=(3)/(4)xx(7)/(8)=(21)/(32)` B's New Share `(1)/(4)xx(7)/(8)=(7)/(32)`
Thus, New Profit-sharing Ratio of A: B : C = 21/32 : 7/32 : 4/32 : 4/32 =21 : 7 : 4.