$15 \cdot 34$ For positive real numbers $x$ and $y$ define $x * y=\frac{x \cdot y}{x+y}$, then
(A) “*” is commutative, but not associative.
(B) “*” is associative, but not commutative.
(C) “*” is neither commutative nor associative.
(D) “*” is both commutative and associative.
(E) None of these.
(25th American High School Mathematics Examination, 1974)
[Solution] From $x * y=\frac{x y}{x+y}=\frac{y x}{y+x}=y * x$, and
$$
(x * y) * z=\frac{x y}{x+y} * z=\frac{\frac{x y z}{x+y}}{\frac{x y}{x+y}+z}=\frac{x y z}{x y+x z+y z} \text {, }
$$

Similarly, $\quad x *(y * z)=\frac{x y z}{x y+x z+y z}$.
Therefore, “*” is not only commutative but also associative.
Hence, the answer is $(D)$.