jmc

The coordinates of $P$ can be written as $\left(a,\frac{15a}{8}\right)$ and the coordinates of point $Q$ can be written as $\left(b,\frac{3b}{10}\right)$. By the midpoint formula, we have $\frac{a+b}{2}=8$ and $\frac{15a}{16}+\frac{3b}{20}=6$. Solving for $b$ gives $b=\frac{80}{7}$, so the point $Q$ is $\left(\frac{80}{7},\frac{24}{7}\right)$. The answer is twice the distance from $Q$ to $(8,6)$, which by the distance formula is $\frac{60}{7}$. Thus, the answer is $\boxed{67}$.