2017 AMC 12A Problem 2

Below is the professionally curated solution for Problem 2 of the 2017 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 12A solutions, or check the answer key.

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Concepts:algebraic manipulation fraction

Difficulty rating: 1020

The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers?

$1$

$2$

$4$

$8$

$12$

Solution:

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.

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