I would really appreciate help with this because it's been driving me insane for a while now...

I understand what "per" means in "$x$ kilometers *per* $y$ hours". What I don't understand is how to make the use of the division bar in rates coherent with the definition of division.

Based on my understanding of division, it would mean in this case: "how many times do $y$ hours fit into $x$ kilometers?", which doesn't make any sense to me.

How do you make sense of dividing a certain kind of unit by a completely different kind? Do you think of it as a different kind of division?