So my friend asked me why angles are dimensionless, to which I replied that it's because they can be expressed as the ratio of two quantities -- lengths.
Ok so far, so good.
Then came the question: "In that sense even length is a ratio. Of length of given thing by length of 1 metre. So are lengths dimensionless?".
This confused me a bit, I didn't really have a good answer to give to that. His argument certainly seems to be valid, although I'm pretty sure I'm missing something crucial here.
Answer
Your friend's question is perceptive but not at odds with your earlier answer.
When you compare the length of something with a unit (1 meter), the ratio is indeed a unitless number.
But then all numbers (1.5, $\pi$, 42) are unitless. When you want to determine speed you divide displacement by time - each of which has units. But what you enter into you calculator are just the numbers - you handle the units separately.
"The runner covered 100 meter in 10 seconds. What was his average speed?" Is solved by calculating the numerical ratio 100/10 and adding the dimensional ratio m/s to preserve the units. Most calculators don't have (or need) a means to enter units (some sophisticated computer programs do - to help you avoid mistakes by mixing units).
For some physical calculations you need to take the logarithm - when you do, you ALWAYS have to divide the quantity by some scale factor with the same units as it is not possible to take the $\log$ of a unit.
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