Q.A, B and C are partners sharing profits and losses in the ratio 3:2:1. B retires from the firm, and A and C decide to share future profits and losses in the ratio 3:2. Calculate the gaining ratio of A and C.

When a partner retires, the continuing partners' gain is measured as their new share of profits minus their old share; the shares that gain together must always add up exactly to the retiring partner's own share, which is a quick built-in check.

Step 1 — Old ratio. A : B : C = 3 : 2 : 1, so as fractions of the whole firm, A = 3/6 = 1/2, B = 2/6 = 1/3, C = 1/6.

Step 2 — New ratio. After B retires, A and C agree to share in 3 : 2, so A = 3/5 and C = 2/5.

Step 3 — Gain of each continuing partner. Gain = New Share − Old Share.

Step 4 — Express both gains over a common denominator (30) and state the ratio. A's gain = 1/10 = 3/30; C's gain = 7/30. Gaining ratio of A : C = 3 : 7.

Step 5 — Independent cross-check (dual-solve). The gains of the continuing partners must together equal exactly the share given up by the retiring partner. B's old share was 2/6 = 1/3 = 10/30. Sum of gains = 3/30 + 7/30 = 10/30 = 1/3. This matches B's share exactly, confirming the gaining ratio is correct.

Students often confuse the gaining ratio with the new profit-sharing ratio itself, or compute the gain as Old Share − New Share (the sacrificing formula) instead of New − Old. Another common slip is forgetting to convert all fractions to a common denominator before comparing/adding them, which produces an incorrect ratio.