Componendo/Dividendo

Given:

$\begin{equation*} \frac{a}{b}=\frac{c}{d} \end{equation*}$

Componendo:

$\begin{equation*} \frac{a+b}{b}=\frac{c+d}{d} \end{equation*}$

Dividendo:

$\begin{equation*} \frac{a-b}{b}=\frac{c-d}{d} \end{equation*}$

The proofs are “obvious”. You are just adding/subtracting 1 to both sides.

What’s more interesting is combining the two. Divide the first by the second to get:

$\begin{equation*} \frac{a+b}{a-b}=\frac{c+d}{c-d} \end{equation*}$