Why does $\frac{4}{2} = \frac{2}{1}$? I take for granted that $\frac{4}{2} = \frac{2}{1}$. 
Today, I thought about why it must be the case. My best answers amounted to $\frac{4}{2}=2$ and $\frac{2}{1}=2$; therefore $\frac{4}{2}=\frac{2}{1}$.  However, that explanation seems circular:


*

*one can express $2$ as $\frac{2}{1}$. 

*As such, to say $\frac{4}{2}$ equals $\frac{2}{1}$ because both equal $2$, is nearly saying $\frac{4}{2} = \frac{2}{1}$ (the question) and $\frac{2}{1}=\frac{2}{1}$ (trivial, at best). 


So why does $\frac{4}{2} = \frac{2}{1}$?

If $c \neq 0$, do you agree that $\frac{c}{c} = 1$? Do you also agree that 1 is a unit and a multiplicative identity element?
Given any rational number $\frac{a}{b}$, if we multiply both the numerator and the denominator by the same number, we're computing $\frac{a}{b} \times \frac{c}{c} = \frac{a}{b} \times 1$. So, if we have $c = 2$, and we compute $\frac{2}{1} \times \frac{c}{c}$, we have $\frac{2}{1} \times \frac{2}{2} = \frac{4}{2} = \frac{2}{1} \times 1$.
