2014 AMC 10B Problem 9

Question

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

Clear the tiny fractions by multiplying numerator and denominator by

wz

Watch the sign change between

z-w

and

w-z

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 .

Problem 9 in other years