$A$, $B$, and $C$ are partners sharing profits in the ratio of $3:2:1$. $D$ is admitted into the firm for a $\frac{1}{4}\text{th}$ share of profit, which he acquires as $\frac{1}{8}\text{th}$ from $A$ and $\frac{1}{8}\text{th}$ from $B$. Calculate the new profit-sharing ratio of all four partners.

Let's calculate the new profit shares by deducting each partner's sacrifice from their old share:
$$\text{New Share}=\text{Old Share}-\text{Sacrificed Share}$$
Given details:\begin{itemize}\item Old shares: $A=\frac{3}{6}$, $B=\frac{2}{6}$, $C=\frac{1}{6}$ \item Sacrifices made to $D$: $A$ gives $\frac{1}{8}$, $B$ gives $\frac{1}{8}$, $C$ gives $0$.\end{itemize}Let's compute the new values:\begin{itemize}\item \textbf{For A:}
$$\text{New Share of }A=\frac{3}{6}-\frac{1}{8}=\frac{24-6}{48}=\frac{18}{48}$$
\item \textbf{For B:}
$$\text{New Share of }B=\frac{2}{6}-\frac{1}{8}=\frac{16-6}{48}=\frac{10}{48}$$
\item \textbf{For C:}
$$\text{New Share of }C=\frac{1}{6}=\frac{1\times 8}{6\times 8}=\frac{8}{48}$$
\item \textbf{For D:}
$$\text{Share of }D=\frac{1}{4}=\frac{1\times 12}{4\times 12}=\frac{12}{48}$$
\end{itemize}Putting them together into a common denominator ratio:
$$\text{New Ratio}=18:10:8:12$$
Simplifying by dividing throughout by 2 gives:
$$\text{New Profit-Sharing Ratio}=9:5:4:6$$