How does this separation of the variable from the fraction result in $3\cdot \frac{1}{x}$ & not $\frac{3}{x}$ I wish to write the following expression as the product of a whole number or a fraction and variable expression. The answer given in my textbook is as follows:
$\frac{3}{x} = \frac{3\cdot 1}{1\cdot x} = \frac{3}{1} \cdot \frac{1}{x} = 3\cdot \frac{1}{x} $
However, I am not sure how to get to $3\cdot \frac{1}{x} $. When I complete the multiplication I get to $\frac{3}{1} \cdot \frac{1}{x} = \frac{3}{x}$, which of course loops me back round!

As you stated we have
$\frac{3}{x} = \frac{3\cdot 1}{1\cdot x} = \frac{3}{1} \cdot \frac{1}{x}$
Now we know that  $\frac{3}{1}=3$ (dividing by 1 does not change a number).
Hence, we have
$ \frac{3}{1} \cdot \frac{1}{x}=3\cdot \frac{1}{x}$
