<s>[INST] How to proof the reason? I have this statement:

If $\frac{a}{b} = \frac{c}{d},$ prove that
  $\frac{a+b}{a-b}=\frac{c+d}{c-d}$

I tried to add 1, multiply 1 and nothing.
My development was:
$\frac{a}{b} - \frac{b}{b} = \frac{c}{d} - \frac{d}{d}$
$\frac{a-b}{b} = \frac{c-d}{d}$
$\frac{b}{a-b} = \frac{d}{c-d}$ (I raised to $^{-1}$)
So far I have arrived, without much success.
How can I prove it? Thanks in advance.
 [/INST] $$\frac{a}{b} = \frac{c}{d}$$
write  $$\frac{a}{b}+1 = \frac{c}{d}+1$$
$$\frac{a+b}{b} = \frac{c+d}{d}$$
also if $$\frac{a}{b} = \frac{c}{d}$$
then $$\frac{b}{a} = \frac{d}{c}$$
write $$1-\frac{b}{a} =1- \frac{d}{c}$$
 $$\frac{a-b}{a} = \frac{c-d}{c}$$
we first obtained that $$\frac{a+b}{b} = \frac{c+d}{d}$$
$$\frac{(a+b)a}{(a-b)b} = \frac{(c+d)c}{(c-d)d}$$
also $$\frac{a}{b} = \frac{c}{d}$$
thus 
$$\frac{a+b}{a-b}=\frac{c+d}{c-d}$$
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