A Pythagorean fraction is a number of the form $a/b$ or $b/a$ where $a$ and $b$ are the legs of a Pythagorean triple. Are there simple necessary and/or sufficient conditions for determining whether a rational number can be expressed as a ratio of two Pythagorean fractions? As an example, I would be interested to know if $4/9$ can be expressed as a ratio of two Pythagorean fractions. A brute force search of all primitive Pythagorean triples $(a,b,c)$ with $c < 6000$ found that $4/9$ could not be written as a ratio of two Pythagorean fractions when one of the fractions has a corresponding hypotenuse less than 6000.