2014 AMC 10B Problem 9

Problem 9 of 25EasierAlgebra

For real numbers

w

and

z ,

\cfrac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014.

What is

\frac{w+z}{w-z} ?

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Solution

Observe that:

\begin{aligned} \dfrac{wz}{wz}\cdot \dfrac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} &= 2014\\ \dfrac{w+z}{z-w} &= 2014\\ \dfrac{w+z}{w-z} &= -1\cdot 2014\\ \dfrac{w+z}{w-z}&=-2014.\end{aligned}

Thus, the correct answer is A .

Concepts: algebraic manipulation · fraction

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