I have a vague intuition as to why speed is distance / time, but mathematically speaking I don't understand the division aspect of it.
One of the first things I learned in arithmetic was that you can't add oranges to apples (not unless you abstract them both to "fruit"); that essentially, numbers in an operation need to be of the same unit.
How can you divide say, $x$ kilometers by $y$ hours? Based on my notion of division, it would mean "how many times do $y$ hours fit into $x$ kilometers?", which doesn't make any sense to me. Of course I understand the idea of $x$ kilometers per $y$ hours, but I don't see how per turns into division.
How do you make sense out of this?