AMC10 2014 B

AMC10 2014 B · Q9

AMC10 2014 B · Q9. It mainly tests Manipulating equations.

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

对于实数 $w$ 和 $z$，$\frac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014$。求 $\frac{w+z}{w-z}$ 的值。

(A) -2014 -2014

(B) $\frac{-1}{2014}$ $\frac{-1}{2014}$

(D) 1 1

(E) 2014 2014

Answer

Correct choice: (A)

正确答案：（A）

Solution

English (OCR): Answer (A): Note that $2014=\dfrac{\dfrac{1}{w}+\dfrac{1}{z}}{\dfrac{1}{w}-\dfrac{1}{z}}=\dfrac{\dfrac{w+z}{wz}}{\dfrac{z-w}{wz}}=\dfrac{w+z}{z-w}.$ Because $\dfrac{w+z}{z-w}=-\dfrac{w+z}{w-z}$, the requested value is $-2014$.

中文翻译：答案（A）：注意 $2014=\dfrac{\dfrac{1}{w}+\dfrac{1}{z}}{\dfrac{1}{w}-\dfrac{1}{z}}=\dfrac{\dfrac{w+z}{wz}}{\dfrac{z-w}{wz}}=\dfrac{w+z}{z-w}.$ 因为 $\dfrac{w+z}{z-w}=-\dfrac{w+z}{w-z}$，所以所求的值是 $-2014$。

Topics

AL.016 Manipulating equations