Solution:

Let $p$ and $q$ be practical. For any $k \leq p q$, we can write
$$k = a q + b \text{ with } 0 \leq a \leq p,\ 0 \leq b < q.$$
Since $p$ and $q$ are practical, we can write
$$a = c_1 + \ldots + c_m, \quad b = d_1 + \ldots + d_n$$
where the $c_i$'s are distinct divisors of $p$ and the $d_j$'s are distinct divisors of $q$. Now
$$\begin{aligned} k &= (c_1 + \ldots + c_m) q + (d_1 + \ldots + d_n) \\ &= c_1 q + \ldots + c_m q + d_1 + \ldots + d_n. \end{aligned}$$
Each of $c_i q$ and $d_j$ divides $p q$. Since $d_j < q \leq c_i q$ for any $i, j$, the $c_i q$'s and $d_j$'s are all distinct, and we conclude that $p q$ is practical.