Q.X, Y and Z share profits in the ratio 3 : 2 : 1. Y retires, and it is agreed that in future X and Z will share profits in the ratio 5 : 3. Calculate the new profit-sharing ratio and the gaining ratio.

Here the new ratio is given directly, so the gaining ratio has to be worked backward as New Share minus Old Share.

Expressing X and Z's old shares (3/6 and 1/6) and new shares (5/8 and 3/8) over a common denominator of 24 gives old shares of 12/24 and 4/24, and new shares of 15/24 and 9/24. X's gain = 15/24 − 12/24 = 3/24; Z's gain = 9/24 − 4/24 = 5/24. This gives a gaining ratio of 3 : 5.

Old ratio of X, Y, Z = 3 : 2 : 1 (out of a total of 6 parts), so:

• X's old share = 3/6
• Y's old share (retiring) = 2/6
• Z's old share = 1/6

The question states directly that after Y's retirement, X and Z will share future profits in the ratio 5 : 3, so:

New ratio, X : Z = 5 : 3.

To find the gaining ratio, express every share over a common denominator. The old shares use a denominator of 6, and the new shares use a denominator of 8 (5 + 3); the LCM of 6 and 8 is 24.

Gaining ratio, X : Z = 3/24 : 5/24 = 3 : 5.

As a check, the total of the two gains (3/24 + 5/24 = 8/24 = 1/3) exactly equals Y's old share (2/6 = 1/3) — confirming that X and Z between them have picked up exactly the share Y gave up, and no more.

A quick check that always works: the sum of the gaining-ratio fractions (here 3/24 + 5/24) must equal the outgoing partner's old share (here Y's 2/6 = 8/24) — if it does not, a calculation error has been made somewhere.

A very common error is to assume the gaining ratio is automatically the same as the new ratio (5 : 3) — that is only true when the outgoing partner's old share happens to have been zero, which is never the case. The gaining ratio must always be derived as New Share minus Old Share, over a common denominator, not read off directly from the new ratio.