jmc

The area of $△BFD$ is the area of square $ABCE$ subtracted by the the area of the three triangles around it. Arbitrarily assign the side length of the square to be $6$. The ratio of the area of $△BFD$ to the area of $ABCE$ is $$\frac{36-12-12-2}{36}=\frac{10}{36}=\boxed{\frac{5}{18}}$$