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 total volume of the mixture be $4$ units.
* Glycerin $=3$ units
* Water $=1$ unit

Let a fraction $x$ of the total mixture be drawn off and replaced with water.

* Volume of mixture removed $=4x$ units
* Glycerin removed $=3x$ units, Water removed $=1x$ units

Remaining quantities after removing the mixture and adding $4x$ units of water:

$$\text{Remaining Glycerin}=3-3x$$

$$\text{New Water Amount}=1-x+4x=1+3x$$

Since the final ratio of glycerin and water is $1:1$, the quantities must be equal:

$$3-3x=1+3x$$

$$3-1=3x+3x$$

$$2=6x⟹x=\frac{2}{6}=\frac{1}{3}$$

Hence, $\frac{1}{3}$ fraction of the mixture must be drawn off and substituted.