# Quotient of Quotients of Real Numbers

## Theorem

$\forall x \in \R, y, w, z \in \R_{\ne 0}: \dfrac {x / y} {w / z} = \dfrac {x \times z} {y \times w}$

## Proof

 $\displaystyle \frac {x / y} {w / z}$ $=$ $\displaystyle \frac x y \times \frac 1 {w / z}$ Definition of Real Division $\displaystyle$ $=$ $\displaystyle \frac x y \times \frac z w$ Reciprocal of Quotient of Real Numbers $\displaystyle$ $=$ $\displaystyle \dfrac {x \times z} {y \times w}$ Product of Quotients of Real Numbers

$\blacksquare$