2017 AMC 12A Problem 2

Let the numbers be $x$ and $y,$ so $x+y=4xy.$ Dividing both sides by $xy$ gives $$\dfrac{x+y}{xy}=4,$$ and the left side is exactly $\dfrac{1}{y}+\dfrac{1}{x}.$ So the sum of the reciprocals is $4.$ Thus, the correct answer is C.