Rectangle $ABCD$, pictured below, shares $50\%$ of its area with square $EFGH$. Square $EFGH$ shares $20\%$ of its area with rectangle $ABCD$. What is $\frac{AB}{AD}$?


$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10$
If we shift $A$ to coincide with $E$, and add new horizontal lines to divide $EFGH$ into five equal parts:


This helps us to see that $AD=a/5$ and $AB=2a$, where $a=EF$.
Hence $\dfrac{AB}{AD}=\dfrac{2a}{a/5}=10$.