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

Answer: 2.

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About mvtrinh

Retired high school math teacher.
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