A and B are partners in a firm sharing profits in the ratio of $3:2$. They admit M as a new partner for a $\frac{1}{4}$ share. The new profit-sharing ratio between A and B is set to be $2:1$. Calculate their sacrificing ratio.

Let us calculate the sacrificing ratio step by step using the formula:
$$\text{Sacrifice}=\text{Old Share}-\text{New Share}$$
Determine Old Shares:
$$\text{Old Share of A}=\frac{3}{5},\text{Old Share of B}=\frac{2}{5}$$
Determine New Shares:M is admitted for a $\frac{1}{4}$ share. The remaining share for A and B is:
$$\text{Remaining Share}=1-\frac{1}{4}=\frac{3}{4}$$
A and B divide this remaining share in a $2:1$ ratio:
$$\text{New Share of A}=\frac{2}{3}\times \frac{3}{4}=\frac{6}{12}=\frac{2}{4}=\frac{1}{2}$$
$$\text{New Share of B}=\frac{1}{3}\times \frac{3}{4}=\frac{3}{12}=\frac{1}{4}$$
Calculate the Sacrifice Made by Each Partner:For A:
$$\text{Sacrifice of A}=\frac{3}{5}-\frac{2}{4}=\frac{12-10}{20}=\frac{2}{20}$$
For B:
$$\text{Sacrifice of B}=\frac{2}{5}-\frac{1}{4}=\frac{8-5}{20}=\frac{3}{20}$$
Determine the Sacrificing Ratio:Comparing their sacrifices ($\frac{2}{20}:\frac{3}{20}$), we get a ratio of $2:3$.