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2014 AMC 10B Problem 9

2014 AMC 10B Problem 9AMC 10 Step by Step

Name: Date: September 13, 2026

Algebraic Manipulation · difficulty 2 of 5 · about 2 min · position P1-10

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

A)
$-2014$
B)
$\frac{-1}{2014}$
C)
$\frac{1}{2014}$
D)
$1$
E)
$2014$

Answer

Key insight

Multiply top and bottom by wz: the given fraction becomes (w+z)/(z-w), which is the negative of what is asked.

Solution

Multiply the numerator and denominator of the left side by $wz$ :
$\frac{\frac1w + \frac1z}{\frac1w - \frac1z} = \frac{z + w}{z - w}.$

So $\frac{w + z}{z - w} = 2014$ . The requested expression has denominator $w - z = -(z - w)$ , hence
$\frac{w + z}{w - z} = -\frac{w + z}{z - w} = -2014.$

The answer is $\boxed{\textbf{(A)}\ -2014}$ .

Why this works

A compound fraction built from $\frac1w$ and $\frac1z$ simplifies instantly by multiplying through by $wz$ . The problem is designed so that the simplified form is almost the target; the whole difficulty is noticing the reversed subtraction in the denominator and tracking the sign.

Alternative approach

Pick values: the condition forces $\frac{z + w}{z - w} = 2014$ ; try $w = 2013$ , $z = 2015$ , which gives $\frac{4028}{2} = 2014$ . Then $\frac{w + z}{w - z} = \frac{4028}{-2} = -2014$ .

The trap

Losing the sign: after clearing denominators the bottom is z - w, not w - z, so the answer is -2014, not 2014.

Common mistakes

Losing the sign: after clearing denominators the bottom is z - w, not w - z, so the answer is -2014, not 2014.
Assuming the reciprocal relation gives $\frac{1}{2014}$ , choice (C), without actually simplifying.

Techniques

Set up the equation/formula and compute; no special trick needed · Test small/specific values or special cases to find or verify the answer