A, B and C were partners in a business who shared profits and losses in the ratio of $$\frac{1}{5}:\,\frac{1}{3}:\,\frac{7}{{15}}$$ respectively. C retired and his share was purchased by A and B in the ratio of 3 : 2. The new profit sharing ratio of A and B will be:

The correct answer is $\boxed{\text{C}}$.

Initially, A, B, and C shared profits in the ratio of $\frac{1}{5}:\,\frac{1}{3}:\,\frac{7}{{15}}$, which is equivalent to $6:10:14$.

When C retires, his share is purchased by A and B in the ratio of 3 : 2. This means that A will now have a share of $\frac{6}{10} + \frac{3}{14} = \frac{12}{35}$, and B will have a share of $\frac{10}{10} + \frac{2}{14} = \frac{22}{35}$.

Therefore, the new profit sharing ratio of A and B is $\frac{12}{35} : \frac{22}{35} = \boxed{12 : 13}$.

Here is a brief explanation of each option:

• Option A: $\frac{13}{12}$. This is the ratio of A’s share to C’s share. However, C’s share is no longer relevant, as he has retired.
• Option B: $\frac{12}{15}$. This is the ratio of A’s share to B’s share before C retired. However, this ratio changes when C retires, as A and B purchase his share in different proportions.
• Option C: $\frac{12}{13}$. This is the correct answer. It is the ratio of A’s share to B’s share after C retires.
• Option D: $\frac{14}{15}$. This is the ratio of A’s share to B’s share if C had not retired. However, C has retired, so this ratio is no longer relevant.