If x, y and 2x+ y2 are not zero, then (2x+ y2)^-1[(2x)^-1+( y2)^-1] equals

1977 AMC 12 Problem 6

Attempt Problem 6 of the 1977 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 1977 AMC 12 solutions, or check the answer key.

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x,

y

and

2x+\frac y2

are not zero, then

\left(2x+\frac y2\right)^{-1}\left[(2x)^{-1}+\left(\frac y2\right)^{-1}\right]

equals

1

xy^{-1}

x^{-1}y

(xy)^{-1}

none of these

Rewrite the outside reciprocal using a single fraction

Combine the two reciprocals inside the brackets over a common denominator

We have

\begin{aligned} \left(2x+\frac y2\right)^{-1} &=\frac2{4x+y},\\ (2x)^{-1}+\left(\frac y2\right)^{-1} &=\frac1{2x}+\frac2y\\ &=\frac{y+4x}{2xy}. \end{aligned}

Their product is

\frac{1}{xy}=(xy)^{-1}.

Therefore, the correct answer is D.