Componendo and dividendo

Componendo and dividendo is a method of simplification in mathematics.

According to it,

$\text{if } \frac{a}{b} = \frac{c}{d} \text{ then } \frac{a+b}{a-b} = \frac{c+d}{c-d}.$

[edit] Proof

Add one to both sides of the equality a / b = c / d to get

$\frac{a}{b} + 1 = \frac{c}{d} + 1 \implies \frac{a+b}{b} = \frac{c+d}{d}. \qquad (1)$

Now subtract one to get

$\frac{a}{b} - 1 = \frac{c}{d} - 1 \implies \frac{a-b}{b} = \frac{c-d}{d}. \qquad (2)$

The desired result follows after dividing (1) by (2).

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