Why $\frac {a}{b} = \frac {c}{d} \implies \frac {a+c}{b+d} = \frac {a}{b} = \frac {c}{d}$?

Question

In geometry class, this property was quickly introduced:

$$\dfrac {a}{b} = \dfrac {c}{d} \implies \dfrac {a+c}{b+d} = \dfrac {a}{b} = \dfrac {c}{d} $$

It usually avoids some boring quadratic equations, so it's useful but I don't really get it. Seems trivial but I can't prove.

How to demonstrate it?

Answer 1

$$\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a+c}{b+d}=\frac{a+\frac{ad}{b}}{b+d}=\frac{ab+ad}{b(b+d)}=\frac{a}{b}=\frac{c}{d}$$

Answer 2

Hint: $$ \frac{p}{q} = \frac{r}{s} $$ if and only if $ps = rq$.

Answer 3

Assume that none of the unknowns is zero. Then, let $$\dfrac{a}{b}=\dfrac{c}{d}=k$$ This implies, $$ a=kb,\qquad c=kd$$

$$\therefore{a+c}=k(b+d)$$ $$\implies \dfrac{a+c}{b+d}=k$$