If x, y and y-1x are not 0, then (x-1y)/(y-1x) equals

1984 AMC 12 Problem 2

Attempt Problem 2 of the 1984 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1984 AMC 12 solutions, or check the answer key.

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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}

Write the numerator and denominator as single fractions

Both parts contain the common factor

xy-1

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.