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I'm trying to prove this : $\frac{a}b=\frac{c}d⟺\frac{a+b}{a−b}=\frac{c+d}{c−d}$

but going from $\frac{a}b=\frac{c}d$

I already proved it backwards

$\frac{a+b}{a−b}=\frac{c+d}{c−d}$ I multiply each side by (a-b)(c-d)

$ac-ad+bc-bd = ac - bc + ad - bd$ so $2bc=2ad$ which by dividing both sides by $2bd$ gives us : $$\frac{a}b=\frac{c}d$$

any ideas how to proceed ?

Edit : this equivalence only works when $a\neq b $ and $c \neq d$.

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    You want to know how to go from $\frac ab = \frac cd$ to $\frac{a-b}{a+b} = \frac{c-d}{c+d}$? I just want to clarify what you want. – 2017-02-21
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    yes and sorry for being unclear – 2017-02-21
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    My friend it's ok to be unclear at times. – 2017-02-21
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    @1.2.3 Equivalence relations is not a correct tag. – 2017-02-21

2 Answers 2

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Suppose that $a \neq b, c\neq d$, and $b,d \neq 0$. Then, $\frac ab \neq 1$. We are asked to assume that $\frac ab = \frac cd$.

Now, we can do the following:

$$ \frac ab = \frac cd \implies \frac ab + 1 = \frac cd + 1 \implies \frac{a+b}b = \frac{c+d}d\\ \frac ab = \frac cd \implies \frac ab - 1 = \frac cd - 1 \implies \frac{a-b}{b} = \frac{c-d}{d}\\ $$

Divide the first equation by the second: $$ \frac{\frac{a+b}{b}}{\frac{a-b}b} = \frac{\frac{c+d}d}{\frac{c-d}d} \implies \frac{a+b}{a-b} = \frac{c+d}{c-d} $$

This is a proof, provided the conditions that are given in the starting hold true.

On a side note, I think this is called the componendo-dividendo equality (I may be wrong).

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I do not think the two statements are equivalent. Take $a=b=c=d=1$ then your first equality holds while the second on is undefined.

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    yes I forgot to mention that the trick fails for a=b and c=d. thanks for reminding – 2017-02-21
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    -1;This is not an answer but more of a comment. – 2017-02-21
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    @A---B Somewhat agree and certainly +1 for explaining the downvote. On the other hand the question is 'I am trying to prove this claim' and I answer 'This claim is false'. – 2017-02-21
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    Yes you are right in a way but this was pretty trivial that OP forgot to mention. – 2017-02-21