2017 AMC 12A Problems/Problem 2

Problem

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

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12$

Solution

Let $x, y$ be our two numbers. Then $x+y = 4xy$ . Thus,

$\frac{1}{x} + \frac{1}{y} = \frac{x+y}{xy} = 4$ .

$\boxed{ \textbf{C}}$ .

Solution 2. Use Simon's Favorite Factoring Trick to find x and y (4xy)=x+y 4xy-x-y=0 (4x-1)(y-1/4)=1/4 (4x-1)=1/2 x=3/8 (y-1/4)=1/2 y=3/4 add the reciprocals 8/3+4/3=4

2017 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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All AMC 10 Problems and Solutions

2017 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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2017 AMC 12A Problems/Problem 2

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