$A$ and $B$ are equal partners. They admit $C$ and $D$ as partners with $\frac{1}{5}\text{th}$ and $\frac{1}{6}\text{th}$ share respectively. What is the profit sharing ratio of all the partners?

Let the total profit of the firm be $1$.
Share given to $C=\frac{1}{5}$
Share given to $D=\frac{1}{6}$

$$\text{Total share of new partners}=\frac{1}{5}+\frac{1}{6}=\frac{6+5}{30}=\frac{11}{30}$$

$$\text{Remaining share for old partners }(A\text{ and }B)=1-\frac{11}{30}=\frac{19}{30}$$

Since $A$ and $B$ are equal partners, they divide the remaining share equally:

$$\text{New share of }A=\frac{1}{2}\times \frac{19}{30}=\frac{19}{60}$$

$$\text{New share of }B=\frac{1}{2}\times \frac{19}{30}=\frac{19}{60}$$

Expressing all shares with a common denominator of $60$:

$$\text{Share of }C=\frac{1}{5}=\frac{12}{60}$$

$$\text{Share of }D=\frac{1}{6}=\frac{10}{60}$$

Thus, the new profit sharing ratio of $A:B:C:D$ is:

$$19:19:12:10$$