Problem 415

The equality of the ratios $x:4$ and $9:y$ is equivalent to the equation $\dfrac{x}{4} = \dfrac{9}{y}$. (Note that $x$ and $y$ are both positive so we are not dividing by 0.)

This equation is equivalent to the equation $xy = 4(9)= 36$.

Thus, we want to determine the number of pairs of integers $(x,y)$ for which $xy=36$.

Since the positive divisors of $36$ are 1, 2, 3, 4, 6, 9, 12, 18, 36, then 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.