## Inverts

Find the sum of the reciprocals of two real numbers, given that these two numbers have a sum of 50 and a product of 25.
Source: mathcontest.olemiss.edu 5/7/2007

SOLUTION

The above figure shows the graph of the line $x+y=50$ and the graph of the hyperbola $xy=25$ . The two intersection points of the two graphs are two symmetrical points $\left (49.49490,0.50510\right )$ and $\left (0.50510,49.49490\right )$. The sum of the reciprocals is
$\frac{1}{49.49490}+\frac{1}{0.50510}=2.00001$.

An alternative way to find the sum of the reciprocals without knowing the values of $x\textup{ and }y$ is:
$\frac{1}{x}+\frac{1}{y}=\frac{x+y}{xy}=\frac{50}{25}=2$