I had to remind myself exactly what the point of cross multiplying is.

…it’s essentially just a label given to a specific set of algebraic operations. That it even has a name seems stupid to me. We shouldn’t focus on memorizing specific cases like this when understanding why it works will get you there just as quickly. Heck in the case of cross multiplying, I think it works against the interests of the students’ learning. It’s a shortcut that hides the fact that you’re multiplying both sides by both denominators, when “do the same thing on both sides of the equals sign” is algebra 101.