jmc

Let $a$ be the smaller of the two numbers and let $b$ be the larger of the two. Then $\frac{a}{b}=\frac{3}{5}$, so $5a=3b$. Additionally, $\frac{a-4}{b+8}=\frac{2}{7}$, so cross multiplying gives $7(a-4)=2(b+8)$. We now have a system of two linear equations; solving gives $a=12$, $b=20$. Since the question asks us for the value of $b$, our answer is $\boxed{20}$.