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?

$$
\frac{a}{\large\frac{b}{x}} = \large\frac {x}{\not x} \frac{a}{\frac{b}{\not x}} = x\cdot \frac{a}{b}
$$
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 $x\cdot \dfrac{b}{x} = b$
