For real numbers w and z , \cfrac(1)/(w) + (1)/(z)(1)/(w) - (1)/(z) = 2014. What is (w+z)/(w-z) ?

2014 AMC 10B Problem 9 — Solution

2014 AMC 10B Problem 9

Attempt Problem 9 of the 2014 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2014 AMC 10B solutions, or check the answer key.

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For real numbers $w$ and $z ,$ $\cfrac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014.$ What is $\frac{w+z}{w-z} ?$

$-2014$

$\dfrac{-1}{2014}$

$\dfrac{1}{2014}$

$1$

$2014$

Answer: A

Concepts:algebraic manipulation fraction

Difficulty rating: 1020

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 .

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