<s>[INST] Why does $\frac{a}{\frac{b}{x}} = x \times \frac{a}{b}$? As much as it embarasses me to say it, but I always had a hard time understanding the following equality:
$$
\frac{a}{\frac{b}{x}} = x \times \frac{a}{b}
$$
I always thought that the left-hand side of the above equation was equivalent to
$$
\frac{a}{\frac{b}{x}} = \frac{a}{b} \div \frac{x}{1} = \frac{a}{b} \times \frac{1}{x}
$$
What am I doing wrong, here?
 [/INST] $\frac{a}{\frac{b}{x}}$ means a divided by $\frac{b}{x}$.
Note that 
$$\frac{b}{x}\frac{x}{b} =1$$
This means that 
$$a \frac{b}{x}\frac{x}{b} = a$$
Now divide both sides by $\frac{b}{x}$ and you get
$$a\frac{x}{b} = \frac{a}{ \frac{b}{x}}$$
The mistake you make is confusing $\frac{a}{\frac{b}{x}}$ with $\frac{\frac{a}{b}}{x}$. In general
$$\frac{a}{\frac{b}{x}}\neq \frac{\frac{a}{b}}{x}$$
as they have different meanings.
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