2014 AMC 10B Problem 9

1. Clear the small fractions. The fractions $\frac1w$ and $\frac1z$ make the expression messy. Multiplying the top and the bottom of a fraction by the same number does not change it, so multiply both by $wz$:
- top: $\left(\frac1w + \frac1z\right) \cdot wz = z + w$,
- bottom: $\left(\frac1w - \frac1z\right) \cdot wz = z - w$.
2. So the given equation says
$$\frac{z + w}{z - w} = 2014.$$
3. Compare with the question. We want $\frac{w + z}{w - z}$. The top $w + z$ is the same as $z + w$. The bottom $w - z$ is the negative of $z - w$ (for example, $5 - 3 = 2$ but $3 - 5 = -2$).
4. Flipping the sign of the bottom flips the sign of the whole fraction, so
$$\frac{w + z}{w - z} = -2014.$$

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

A fraction built out of $\frac1w$ and $\frac1z$ becomes simple as soon as you multiply top and bottom by $wz$. The problem is built so that the simple form is almost the target. The only real work is noticing that the subtraction on the bottom is in the other order, and keeping track of the minus sign.

Find actual numbers that work. We need $\frac{z + w}{z - w} = 2014$. Try $z = 2015$ and $w = 2013$: then $z + w = 4028$ and $z - w = 2$, and $4028 \div 2 = 2014$. Now $\frac{w + z}{w - z} = \frac{4028}{-2} = -2014$.