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?
 A: $\frac{a}{b}\times\frac{1}{x}=\frac{a}{b}\div x=\frac{\frac{a}{b}}{x}\neq\frac{a}{\frac{b}{x}}=a\div\frac{b}{x}=a\times \frac{x}{b}$
A: You’re treating the fraction $$\frac{a}{b/x}$$ as if it were $$\frac{a/b}{1/x}\;.$$ If you apply the rule invert the denominator and multiply to $$\frac{a}{\frac{b}x}=\frac{a}{b/x}\;,$$ you get $$a\cdot\frac{x}b=\frac{ax}b\;.$$
To see that this really is correct, remember that the statement that $\dfrac{p}q=r$ means that $p=qr$. Thus, if $\dfrac{a}{b/x}$ really is $\dfrac{ax}b$, we should find that 
$$a=\frac{b}x\cdot\frac{ax}b\;,$$
which you can check is indeed the case.
A: The problem is that $\frac{a}{\frac{b}{x}}=\frac{\frac{a}{b}}{\frac{1}{x}}=x\frac{a}{b}$ because $\frac{1}{\frac{1}{x}}=x$.
A: $\frac{a}{\frac{b}{x}}$ 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.
A: It might help to use the definition of equality of fractions, which says that two fractions $a/b$ and $c/d$ are equal if and only if $ad = bc$.
EDIT: I think that it is always a bad idea to use the notation $\frac{a}{\frac{b}{x}}$, even though the relative length of the bars supposedly makes it unambiguous.  In my experience there is a strong correlation between people who use this notation and people who are confused.  So it may help to avoid writing such things.
A: $$
\frac{a}{\large\frac{b}{x}} = \large\frac {x}{\not x} \frac{a}{\frac{b}{\not x}} = x\cdot \frac{a}{b}
$$
Multiplying by $\dfrac xx = 1$ does not change the expression; but by multiplying numerator and denominator by $x$, the numerator becomes $ax$ and the denominator becomes $x\cdot \dfrac{b}{x} = b$
A: It seems that your problem is mostly notational. For example,
$\dfrac{\big(\frac{a}{b}\big)}{c} = \dfrac{a}{bc}$, but $\dfrac{a}{\big( \frac{b}{c} \big)} = \dfrac{ac}{b}$, and these mean different things.
As for the intuition, I have always found that in doubt, you should try with 'natural' feeling numbers. For example, try $a =1, b = c = 2$, and  you should be convinced of the differences.
A: It's really a question of which fraction is "inside" another fraction. Another way of looking at your top equation is:
$$\frac{a}{\frac{b}{x}}=a \div \frac{b}{x} = a \times \frac{x}{b} = \frac{ax}{b}$$
Hope that helps.
