Points $B$ and $C$ lie on $\overline{AD}$. The length of $\overline{AB}$ is $2$ times the length of $\overline{BD}$, and the length of $\overline{AC}$ is $5$ times the length of $\overline{CD}$. The length of $\overline{BC}$ is what fraction of the length of $\overline{AD}$?