The ratio of glycerin and water in a mixture is $3:1$. How much fraction of the mixture must be drawn off and substituted by water so that the ratio of glycerin and water becomes $1:1$?

Let the initial quantity of the mixture be $4x$ liters.
Glycerin = $3x$, Water = $x$.

Let $y$ liters of the mixture be drawn off and replaced with $y$ liters of water.
Quantity of glycerin removed = $\frac{3}{4}y$
Quantity of water removed = $\frac{1}{4}y$

New quantity of Glycerin = $3x-\frac{3}{4}y$
New quantity of Water = $x-\frac{1}{4}y+y=x+\frac{3}{4}y$

Since the final ratio is $1:1$:

$$3x-\frac{3}{4}y=x+\frac{3}{4}y$$

$$2x=\frac{6}{4}y$$

$$2x=\frac{3}{2}y⟹y=\frac{4}{3}x$$

Fraction of the total mixture removed = $\frac{y}{4x}=\frac{\frac{4}{3}x}{4x}=\frac{1}{3}$.