A, B and C are partners. They share profits and losses in $$\frac{1}{2}:\,\frac{1}{3}:\,\frac{1}{6}.$$   C died what will be the gaining ratio of A and B?

Gaining Ratio = New Ratio − Old Ratio
Step 1: Find the old profit-sharing ratio
A : B : C = 12 : 13 : 16
Total = 12 + 13 + 16 = 41
So,
A's old share = 12/41
B's old share = 13/41
C's old share = 16/41
Step 2: Determine the new ratio
Since no new ratio is given, we assume A and B continue in their old ratio (12:13).
Thus,
A's new share = 12/25
B's new share = 13/25
Step 3: Calculate gain of each partner
A's gain:
12/25 - 12/41 = 192/1025 =12
13/25 - 13/41 = 208/1025 =13