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I have been thinking about this problem:

$$Speed = \frac{Distance}{Time}$$

Following this, is makes sense that the units of speed is m/s. However, I do not follow why we are able to divide units to derive a new unit, which turns out to be intuitive later on (e.g. meter/second gives you 'meter per second' which makes intuitive sense as a unit of speed).

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  • $\begingroup$ You are over-thinking this. The unit follows from the definition of speed. They don't "turn out to be intuitive later on", they start out as being just as intuitive as the definition. $\endgroup$ – StephenG Feb 3 '18 at 5:23
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I'm a bit confused about the exact nature of your question. When you divide units, you are forming a ratio of two quantities. Often, these quantities are things we know how to or would like to measure. So when you divide distance by time, you get a ratio of one quantity you can measure (distance) and another quantity you can measure (time) which is meaningful.

This can actually end up being a useful thing to do in physics, and is known as Dimensional Analysis.

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You are confusing dimensions and units. Dimensions group together co-measurable units. For example 5 seconds from now (second is a unit of measure) is $\frac{1}{12}$ minutes from now where the conversion made is multiplication by dimensionless quantity.

Another example: In Special relativity due to nature of $c$ and/or the identification (up to an important sign) of spacial and temporal units having the same dimension renders velocity dimensionless:

If I can measure potential gravitational energy with a ladder and a ruler and convert it to a mass using division by a velocity, then these are co measurable as well, etc.

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