Problem 367

If $x:4$ is equal to $9:y$, then $\dfrac{x}{4}=\dfrac{9}{y}$ or $xy=36$.

Since $x$ and $y$ are positive integers, then $(x,y)$ is a positive divisor pair of $36$.

The positive divisor pairs of $36$
with $x<y$ are $(1,36)$, $(2,18)$, $(3,12)$, and $(4,9)$.

The reversals of these $4$ ordered
pairs (with $x>y$) are also
positive divisor pairs.

In addition to these $4\times2=8$
ordered pairs, $(6,6)$ is also a
positive divisor pair of $36$, and so
there are $9$ such ordered pairs.