New ratio and gaining ratio

When a partner retires or dies, his share of profit does not disappear: the continuing partners take it over. So their shares grow, and we must find the new ratio. Three cases:

• Nothing is said: the continuing partners take the outgoing partner's share in their old ratio. Their new ratio is then simply their old ratio between themselves. If A, B and C share 4:3:2 and C retires, A and B continue at 4:3.
• They take it in a given ratio: each continuing partner's new share = old share + share acquired.
• The new ratio is given directly.

Example: Alka (3/6) retires, and Harpreet (2/6) and Shreya (1/6) take her share in the ratio 3:2. Harpreet gets 3/5 of 1/2 = 9/30, so her new share is 2/6 + 9/30 = 19/30; Shreya gets 2/5 of 1/2 = 6/30, so hers is 1/6 + 6/30 = 11/30. New ratio 19:11.

The gaining ratio is the ratio in which the continuing partners gain: for each, gain = new share − old share. It is the mirror of the sacrificing ratio on admission, and it decides who pays the outgoing partner for his share of goodwill. (A continuing partner may also gain nothing, or even sacrifice, if the new ratio says so.)