A container contains two liquids, X and Y, in the ratio $5:3$. When $8\text{liters}$ of the mixture is drawn off, and the container is filled with liquid Y, the ratio of X to Y becomes $2:3$. How many liters of liquid X was in the container initially?

Let the initial quantities of X and Y be $5k$ and $3k$ liters respectively. Total initial volume = $8k$.
When $8\text{liters}$ of the mixture is drawn off:
Amount of X removed = $8\times \frac{5}{8}=5\text{liters}$
Amount of Y removed = $8\times \frac{3}{8}=3\text{liters}$

After replacing with $8\text{liters}$ of liquid Y:
New amount of X = $5k-5$
New amount of Y = $3k-3+8=3k+5$

Given new ratio is $2:3$:

$$\frac{5k-5}{3k+5}=\frac{2}{3}$$

$$3(5k-5)=2(3k+5)$$

$$15k-15=6k+10$$

$$9k=25⟹k=\frac{25}{9}$$

Initial quantity of liquid X = $5k=5\times \frac{25}{9}=\frac{125}{9}=13\frac{8}{9}\text{liters}$.