CEMC Pascal

Algebra Difficulty 1.8 Junior Find the answer

How many pairs of positive integers $(x, y)$ have the property that the ratio $x: 4$ equals the ratio $9: y$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

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.

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