So, proportional reasoning works for me here. I’d sure like to have a convincing visual, though! The visuals that I commonly use for fraction division work a lot better with common denominators than common numerators.

I do like the focus on the size of the pieces, on your second page. The inverse proportionality between the size of the pieces and the denominator of the fractions is a good intuition here. I still can’t quite get it into a nice visual without making common denominators eventually, though, for instance with a rectangle divided horizontally into b pieces and vertically into c pieces so that I can see why the ratio of the piece sizes leads me to c/b as the result.