In $\triangle ABC$ we have $AB=13$, $BC=15$, and $AC=14$. Points $D$ and $E$ are on $AB$ and $AC$ respectively, with $AD=6$ and $AE=7$. What is the ratio of the area of triangle $ADE$ to the area of the quadrilateral $BCED$?

1. $[ADE]:[ABC]=(AD\cdot AE):(AB\cdot AC)=(6\cdot7):(13\cdot14)=42:182=3:13$.
2. So $[ADE]:[BCED]=\dfrac3{13}:\left(1-\dfrac3{13}\right)=\dfrac3{13}:\dfrac{10}{13}=\boxed{\dfrac3{10}}$.