For positive real numbers x and y define x*y=(x× y)/(x+y); then

(D) “ * ” is commutative and associative

(A) “ * ” is commutative but not associative

(B) “ * ” is associative but not commutative

(C) “ * ” is neither commutative nor associative

1974 AMC 12 Problem 6

Attempt Problem 6 of the 1974 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 1974 AMC 12 solutions, or check the answer key.

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For positive real numbers

x

and

y

define

x*y=\frac{x\cdot y}{x+y};

then

“

*

” is commutative but not associative

“

*

” is associative but not commutative

“

*

” is neither commutative nor associative

“

*

” is commutative and associative

none of these

The formula is visibly unchanged when

x

and

y

are interchanged

Rewrite the operation through its reciprocal:

\frac1{x*y}=\frac1x+\frac1y

Symmetry in

x,

y

makes the operation commutative. Also,

\frac1{x*y}=\frac1x+\frac1y.

Thus both

(x*y)*z

and

x*(y*z)

have reciprocal

\frac1x+\frac1y+\frac1z,

so they are equal. The operation is associative too.

Therefore, the correct answer is D.