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I totally get what division means in the real world. "dollars / hour", well, that's the number of dollars you will make in one hour. "kilometers / gallon" is the distance you can go with a gallon of gas. Division means that given a certain amount of one thing, you'll get a certain amount of another thing.
I'm so good with division that you could give me a ratio I've never seen before and I can tell you what it means. "Burgers / McDonalds"? It's the average number of burgers a McDonalds will produce. "Dolphin / Miles"? Every mile you drive, you get this amount of dolphins.
Multiplication on the other hand makes no sense at all. What the heck is a foot-pound? And I don't really mean the definition. I mean, what does multiplication actually do for these two values? What does multiplying two values say about their relationship? What if I were to say kilometer-hours? Or dollar-kilograms? Or dolphin-miles? What would those things mean?
Bonus points go to explaining this in a very simple, clear manner.