Why does the following method work in isolating pro-numerals?

Question

A question simply asks to isolate (x) from 5xy=c, which is simple, right? Naturally, you'd divide both sides by (5y). But, I was wondering why does that work?

5xy = xy + xy + xy + xy + xy and 
5y = y + y + y + y + y? 

Thus, if you were to divide those two you'd remain with a 5x? not just x?

I am aware that if you add them they will equal \frac{5xy}{5y} which equals to (x), but I don't quite understand why that works, as I stated above.

Thanks in advance!

P.S I am sorry if this has the wrong tag, I really didn't know what else to tag it.

Answer 1

If you write $$\frac {5xy}{5x}=\frac{xy+xy+xy+xy+xy}{x+x+x+x+x}$$ it is clear that each $xy$ is divided by all the $x$'s, not just one of them. When you divide them each contributes $\frac y5$ and the sum is $y$ as expected.

Answer 2

Consider this: $$ \frac{(2\times 1) + (2\times 1) + (2\times 1) + (2\times 1) + (2\times 1)} {1 + 1 + 1 + 1 + 1} = \text{____}. $$

Is the answer $5,$ or is it $2$?