The equality of the ratios $x: 4$ and $9: y$ is equivalent to the equation $\frac{x}{4}=\frac{9}{y}$. This equation is equivalent to the equation $xy=4(9)=36$. The positive divisors of 36 are $1,2,3,4,6,9,12,18,36$, so the desired pairs are $(x, y)=(1,36),(2,18),(3,12),(4,9),(6,6),(9,4),(12,3),(18,2),(36,1)$. There are 9 such pairs.