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

See Also

2017 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
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 10 Problems and Solutions
2017 AMC 12A (ProblemsAnswer KeyResources)
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

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