The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12$
Let $x, y$ be our two numbers. Then $x+y = 4xy$. Thus, 
$\frac{1}{x} + \frac{1}{y} = \frac{x+y}{xy} = 4$. 
$\boxed{ \textbf{C}}$.