A fraction of a fraction

Statement

Simplify the complex fraction $\dfrac{\ \dfrac{2x}{6x - 6}\ }{\ \dfrac{8x^2}{x - 1}\ }.$

Solution

Dividing by a fraction is the same as multiplying by its reciprocal: $\frac{2x}{6x - 6} \cdot \frac{x - 1}{8x^2}.$ Pull out $6x - 6 = 6(x - 1)$: $\frac{2x}{6(x - 1)} \cdot \frac{x - 1}{8x^2} = \frac{2x\,(x - 1)}{6(x - 1)\cdot 8x^2} = \frac{2x}{48x^2} = \frac{1}{24x}.$ (The cancellations hold for $x \neq 0,\ x \neq 1$.) $\boxed{\dfrac{1}{24x}}$