dimensional analysis - What allows us to treat physical units in algebra?

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).

Answer

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.