The average age of three boys is $15$ years and their ages are in the proportion $3:5:7$. What is the age in years of the youngest boy?

Given that the average age of $3$ boys is $15$ years:
$$\text{Sum of their ages}=15\times 3=45\text{ years}$$

The ratio of their ages is $3:5:7$. Let their ages be $3k$, $5k$, and $7k$.

$$3k+5k+7k=45$$

$$15k=45$$

$$k=3$$

The age of the youngest boy corresponds to the smallest ratio part ($3k$):

$$\text{Age of youngest boy}=3\times 3=9\text{;}years$$