RMO ( 2016 ) Triangle Vs Quadrilateral

Let $ABC$ be a triangle. Let $D, E$ be a points on the segment $BC$ such that $BD = DE = EC$ . Let $F$ be the midpoint of $AC$ . Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ , respectively.

Determine the ratio of the area of the triangle $APQ$ to that of the quadrilateral $PDEQ$ .

If this ratio can be expressed as $\dfrac pq$ , where $p$ and $q$ are coprime positive integers, submit your answer as $p + q$ .