The age of a son today is $\frac{1}{4}\text{th}$ of the age of his father. After $20$ years, the age of the father will be twice the age of his son. Find the age of the father today.

Let the current age of the father be $F$ and the current age of the son be $S$.
According to the first condition:

$$S=\frac{1}{4}F⟹F=4S$$

According to the second condition (after $20$ years):

$$F+20=2(S+20)$$

Substitute $F=4S$ into the second equation:

$$4S+20=2S+40$$

$$4S-2S=40-20$$

$$2S=20⟹S=10$$

Now, find the father's current age:

$$F=4S=4\times 10=40\text{ years}$$