Solution:

Let the two numbers be $x$ and $y$ such that $$xy = 9 \text{ and } \dfrac{1}{x} = \dfrac{4}{y}.$$

We get that $$y = \dfrac{9}{x}$$$$\dfrac{1}{x} = \dfrac{4x}{9}.$$$$x = \dfrac{3}{2},$$ since $x$ is positive.

Then $$y = 9 \div \dfrac{3}{2} = 6.$$

The desired sum is then $$6 + \dfrac{3}{2} = \dfrac{15}{2}.$$

Thus, D is the correct answer.