New Profit-Sharing Ratio and Sacrificing Ratio

Whenever a new partner is admitted, two ratios must be worked out before any other adjustment can be made: the New Profit-Sharing Ratio (NPSR) — the ratio in which all partners, old and new, will share profits going forward — and the Sacrificing Ratio — the ratio in which the old partners give up a part of their old share in favour of the new partner. Almost every subsequent step (sharing goodwill, sharing the revaluation profit/loss, distributing reserves) depends on getting one or both of these ratios right first.

Determining the New Profit-Sharing Ratio

The new partner's share is fixed in one of two ways, and the method used decides how the calculation proceeds.

Case (a): The new partner's share is given directly, and nothing is said about how the old partners will share the remainder. In this case, the general rule is that the old partners continue to share the remaining profit in their old profit-sharing ratio, unless the question states otherwise.

Method:

1. New partner's share = the share stated in the question.
2. Remaining share (for the old partners together) = 1 − new partner's share.
3. Each old partner's new share = old partner's own old ratio share × remaining share.

Case (b): The new partner acquires their share from the old partners in a specified ratio (for example, "the new partner acquires 3/20th from A and 1/10th from B", or "in the ratio of 2:1 from the old partners"). Here the new partner's total share is the sum of what is taken from each old partner, and:

Each old partner's new share = old partner's old share − the portion of their share taken by the new partner.

In both cases, once every partner's new share is found, they are expressed as a ratio (over a common denominator) to state the New Profit-Sharing Ratio.

Sacrificing Ratio

The Sacrificing Ratio is the ratio in which the old partners surrender, or "sacrifice", part of their former share of profit in favour of the incoming partner. It matters because the new partner must compensate the old partners for the share of goodwill (and, effectively, of future profits) they are giving up — and that compensation is always shared among the old partners in the sacrificing ratio, not in the old profit-sharing ratio (though the two ratios can turn out to be the same in some cases).

Sacrificing Share (for each old partner) = Old Share − New Share

The Sacrificing Ratio is simply the ratio of the old partners' sacrificing shares to one another. …