Three boys, on an average, are $25$ years old, and their ages are in the ratio of $3:5:7$. What is the age of the youngest boy?

Given:
Number of boys = $3$

Average age = $25\text{years}$

Therefore, the sum of the ages of the three boys:

$$\text{Sum of ages}=25\times 3=75\text{years}$$

The ratio of their ages is given as $3:5:7$.

Let their ages be $3x$, $5x$, and $7x$.

According to the question:

$$3x+5x+7x=75$$

$$15x=75$$

$$x=\frac{75}{15}=5$$

The age of the youngest boy corresponds to the smallest ratio value, which is $3x$:

$$\text{Age of youngest boy}=3\times 5=15\text{years}$$