1984 AHSME Problems/Problem 2

Problem

If $x, y$ , and $y-\frac{1}{x}$ are not $0$ , then

$\frac{x-\frac{1}{y}}{y-\frac{1}{x}}$ equals

$\mathrm{(A) \ }1 \qquad \mathrm{(B) \ }\frac{x}{y} \qquad \mathrm{(C) \ } \frac{y}{x}\qquad \mathrm{(D) \ }\frac{x}{y}-\frac{y}{x} \qquad \mathrm{(E) \ } xy-\frac{1}{xy}$

Solution

Multiply the expression by $\frac{xy}{xy}$ to get rid of the fractional numerator and denominator: $\frac{x^2y-x}{xy^2-y}$ . This can be factored as $\frac{x(xy-1)}{y(xy-1)}$ . The $xy-1$ terms cancel out, leaving $\frac{x}{y}, \boxed{\text{B}}$ .

1984 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
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1984 AHSME Problems/Problem 2

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