2010 AMC 12A Problem 3

Let $s$ be the side length of square $EFGH.$ The shaded overlap has width $s$ and height $AD,$ so its area is $s\cdot AD.$ Because the overlap is $50\%$ of the rectangle, $s\cdot AD=\tfrac12\,AB\cdot AD,$ so $AB=2s.$ Because it is $20\%$ of the square, $s\cdot AD=\tfrac15 s^2,$ so $AD=\tfrac{s}{5}.$ Therefore $$\dfrac{AB}{AD}=\dfrac{2s}{\frac{s}{5}}=10.$$ Thus, E is the correct answer.