A, B, C were partners in a partnership firm their profitsharing ratio was 5:3:2. B retires and the new profitsharing ratio between A and C was 3:2. Calculate gaining ratio.

The correct option is 1:2.

Step-by-Step Explanation:

To find the gaining ratio, we first identify the old profit-sharing ratio and the new profit-sharing ratio of the remaining partners.

The old profit-sharing ratio among partners A, B, and C is 5:3:2.
This means:
A's old share = $\frac{5}{10}$
B's old share = $\frac{3}{10}$
C's old share = $\frac{2}{10}$

After B retires, the new profit-sharing ratio between A and C is 3:2.
This means:
A's new share = $\frac{3}{5}$
C's new share = $\frac{2}{5}$

The formula to calculate the gaining share of a partner is:
Gaining Share = New Share - Old Share

Let's calculate the gaining share for each remaining partner:

For Partner A:
A's Gaining Share = A's New Share - A's Old Share
A's Gaining Share = $\frac{3}{5}-\frac{5}{10}$
To subtract these fractions, we find a common denominator, which is 10:
A's Gaining Share = $\frac{6}{10}-\frac{5}{10}=\frac{1}{10}$

For Partner C:
C's Gaining Share = C's New Share - C's Old Share
C's Gaining Share = $\frac{2}{5}-\frac{2}{10}$
Again, converting to the common denominator of 10:
C's Gaining Share = $\frac{4}{10}-\frac{2}{10}=\frac{2}{10}$

Thus, the gaining ratio between A and C is the ratio of their gaining shares:
Gaining Ratio = A's Gain : C's Gain
Gaining Ratio = $\frac{1}{10}:\frac{2}{10}$
Gaining Ratio = 1:2