The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the two numbers?
$\textbf{(A)}\ \frac{10}{3}\qquad\textbf{(B)}\ \frac{20}{3}\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ \frac{15}{2}\qquad\textbf{(E)}\ 8$
Let the two numbers equal $x$ and $y$. From the information given in the problem, two equations can be written:
$xy=9$
$\frac{1}{x}=4 \left( \frac{1}{y} \right)$
Therefore, $4x=y$
Replacing $y$ with $4x$ in the equation,
$4x^2=9$
So $x=\frac{3}{2}$ and $y$ would then be $4 \times$ $\frac{3}{2}=6$
The sum would be $\frac{3}{2}+6$ = $\boxed{\textbf{(D)}\ \frac{15}{2}}$