2017 AMC 12A Problem 2

Problem 2 of 25EasierAlgebra

The sum of two nonzero real numbers is

4

times their product. What is the sum of the reciprocals of the two numbers?

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Solution

Let the numbers be

x

and

y,

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.

Concepts: algebraic manipulation · fraction

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.