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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?

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This because we are not simplifying: at least not Ina direct sense. Take $2\dfrac{m}{s}$ this is somewhat like saying $\dfrac{2 oranges}{3 apples}$ It doesn't simplify, but, in the m/s case, it has meaning.

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  • $\begingroup$ Are you saying that the division bar in rates isn't to be interpreted as division in the arithmetic sense of the word? We use a division bar but we could've just as well used a wavy line or a dotted one? My problem originated from assuming that there is a connection between divison and the "per" in rates. $\endgroup$ – jeremy radcliff Apr 21 '15 at 23:23
  • $\begingroup$ @jeremyradcliff while they are the same, it helps to think more along the lines of per $\endgroup$ – Jimmy360 Apr 21 '15 at 23:24

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