Age problem

Let $B$ and $A$ be the present ages of Boomer and Ana, respectively. Then

Ten years ago: $\dfrac{B-10}{A-10}=\dfrac{1}{5}$ $\implies$ $5B-50=A-10$ $\implies$ $A=5B-40$ $\color{#D61F06}(1)$

Twenty years from now: $\dfrac{B+20}{A+20}=\dfrac{1}{2}$ $\implies$ $2B+40=A+20$ $\implies$ $A=2B+20$ $\color{#D61F06}(2)$

Substituting $\color{#D61F06}(1)$ in $\color{#D61F06}(2)$ , we get

$5B-40=2B+20$ $\implies$ $5B-2B=60$ $\implies$ $5B-2B=60$ $\implies$ $3B=60$ $\implies$ $B=20$

It follows that $A=5(20)-40=100-40=60$ .

Finally, the positive difference of their ages is $60-20=$ $\color{plum}\boxed{40}$