1984 AMC 12 第 2 题

Question

若

x

、

y

和

y-\frac1x

都不为

0

，则

\frac{x-\frac1y}{y-\frac1x}

等于

If

x,

y

and

y-\frac1x

are not

0,

then

\frac{x-\frac1y}{y-\frac1x}

equals

1

\frac{x}{y}

\frac{y}{x}

\frac{x}{y}-\frac{y}{x}

xy-\frac1{xy}

将分子和分母分别写成一个分式

Write the numerator and denominator as single fractions

两部分都含有公因式

xy-1

Both parts contain the common factor

xy-1

有

x-\frac1y=\frac{xy-1}{y}

和

y-\frac1x=\frac{xy-1}{x}

。已知条件保证所需各量不为零，所以它们的商为

\frac{\frac{xy-1}{y}}{\frac{xy-1}{x}}=\frac{x}{y}\text{。}

所以正确答案是 B。

We have

x-\frac1y=\frac{xy-1}{y}

and

y-\frac1x=\frac{xy-1}{x}.

The hypotheses make the needed quantities nonzero, so their quotient is

\frac{\frac{xy-1}{y}}{\frac{xy-1}{x}}=\frac{x}{y}.

Therefore, the correct answer is B.

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