Q.P, Q and R share profits and losses in the ratio 5:3:2. Q retires, and P and R decide to share the future profits of the firm in the ratio 3:2. Calculate

Agree — a retiring partner leaves (goes out of) the firm, so the term 'outgoing partner' is correct.

Reasoning

Retirement means a partner ceasing to be a member of the firm while the firm continues its business with the remaining partners. Because such a partner is leaving/withdrawing from the firm, the same partner is described as an outgoing partner.

Both terms refer to the same partner, so the statement is correct.

Disagree — gain ratio arises on retirement/death of a partner; on admission of a new partner it is the sacrifice ratio that is found.

Reasoning

When the shares of continuing partners change, we compare their old and new ratios:

• Gain ratio = New Ratio − Old Ratio, and it is worked out when a partner retires or dies so that the remaining partners share the outgoing partner's goodwill.
• On admission, the old partners lose a part of their share, so the relevant figure is the sacrifice ratio (Old Ratio − New Ratio), not the gain ratio.

Hence the statement, which links gain ratio to admission, is incorrect.

The ratio obtained by deducting the Old Ratio from the New Ratio is the Gaining Ratio.

Why: On the retirement or death of a partner, the continuing partners acquire the outgoing partner's share, so their profit shares increase. The extra share each continuing partner gains is measured as:

Gaining Ratio = New Ratio − Old Ratio

It is the mirror image of the sacrificing ratio (used on admission), and it decides how the continuing partners bear the outgoing partner's goodwill.

The blank is Old Ratio: New Ratio − Old Ratio = Gain Ratio.

When a partner retires or dies, the continuing partners usually acquire (gain) his share of profit. The gain each continuing partner makes is the excess of his new share over his old share. Hence Gaining Ratio = New Ratio − Old Ratio. (This is the mirror image of the sacrificing ratio computed on admission, where Sacrifice = Old Ratio − New Ratio.)

Unless a new ratio is agreed, retiring only changes the total; the remaining partners keep their old mutual ratio, so X and Z share 5 : 2.

X, Y and Z shared in the ratio 5 : 3 : 2. When Y retires, his share is taken over by X and Z, and in the absence of any other agreement the continuing partners share future profits in their old mutual ratio.