$A$, $B$, and $C$ were partners in a firm with a profit-sharing ratio of $5:3:2$. $B$ retires and the new profit-sharing ratio between $A$ and $C$ is settled at $3:2$. Calculate the gaining ratio of the remaining partners.

The formula for calculating the Gaining Share of a partner is:
$$\text{Gaining Share}=\text{New Share}-\text{Old Share}$$
Given data:\begin{itemize}\item Old Ratio of $A,B,C=5:3:2$, meaning:
$$\text{Old Share of }A=\frac{5}{10},\text{Old Share of }C=\frac{2}{10}$$
\item New Ratio of $A$ and $C=3:2$, meaning:
$$\text{New Share of }A=\frac{3}{5},\text{New Share of }C=\frac{2}{5}$$
\end{itemize}Let's compute the gaining shares:\begin{itemize}\item \textbf{For A:}
$$\text{Gaining Share of }A=\frac{3}{5}-\frac{5}{10}=\frac{6-5}{10}=\frac{1}{10}$$
\item \textbf{For C:}
$$\text{Gaining Share of }C=\frac{2}{5}-\frac{2}{10}=\frac{4-2}{10}=\frac{2}{10}$$
\end{itemize}Comparing their gains:
$$\text{Gaining Ratio}=\frac{1}{10}:\frac{2}{10}=1:2$$