imc

Let the two real numbers be $x,y$. We are given that $x+y=4xy,$ and dividing both sides by $xy$, $\frac{x}{xy}+\frac{y}{xy}=4.$ $$\frac{1}{y}+\frac{1}{x}=\boxed{4}.$$ Note: we can easily verify that this is the correct answer; for example, $\left(\frac{1}{2},\frac{1}{2}\right)$ works, and the sum of their reciprocals is $4$.