I understand the relationship between base lengths and area.
Letting $$AB = 4x, BC=5x, ED = y, DC = 7y$$
Since the ratio of the bases is $$5x:9x $$ the ratio of areas would be $$25x^2:81x^2$$ therefore $$25:81, 25+81=106.$$ Yet the answer is wrong (by one)! Where is my thought process incorrect??
Let area of $\triangle AED = x$. Then area of $\triangle ADC = 7x$ (same height, consider the ratio of bases).
By the same reasoning (considering that the ratio of areas of $\triangle BDC$ to $\triangle ADC$ is $5:9$), area of $\triangle BDC = (\frac 59) \cdot (7x) = \frac{35x}{9}$
So the ratio between areas of $\triangle BDC$ and $\triangle AED = \frac{35}{9}x: 8x = 35:72$. Since it's in the lowest terms, the required answer is $35 + 72 = 107$.