Olympiad Geometry · Challenge problem

In triangle $ABC$, point $D$ lies on $\overline{BC}$ with $BD:DC = 3:4$, point $E$ lies on $\overline{CA}$ with $CE:EA = 4:5$, and point $F$ lies on $\overline{AB}$ with $AF:FB = 5:3$. What is the ratio of the area of triangle $DEF$ to the area of triangle $ABC$?