If a : b :: c : d then prove that:

a : (a - b) :: c : (c - d).

Given,

a : b :: c : d

$\Rightarrow \dfrac{a}{b} = \dfrac{c}{d}$

If the original ratios are equal, their reciprocals are also equal (this is known as Invertendo) :

$\Rightarrow \dfrac{b}{a} = \dfrac{d}{c}$

Applying dividendo, we get:

$\Rightarrow \dfrac{b - a}{a} = \dfrac{d - c}{c} \\[1em] \Rightarrow -\dfrac{a - b}{a} = -\dfrac{c - d}{c} \\[1em] \Rightarrow \dfrac{a - b}{a} = \dfrac{c - d}{c} \\[1em] \text{ Now, take the reciprocal} \\[1em] \therefore \dfrac{a}{a - b} = \dfrac{c}{c - d}$

Hence, proved that a : (a - b) :: c : (c - d).