MHB Can You Solve This Unique Fraction Challenge?

[anemone]

Here is this week's POTW:

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Evaluate $\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}$, given that $\dfrac{(a-b)(b-c)(c-a)}{(a+b)(b+c)(c+a)}=\dfrac{1}{11}$.

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[anemone]

Congratulations to the following members for their correct solution! (Cool)
1. maxkor
2. DaalChawal

Solution from DaalChawal:

Spoiler

We have
$a \over a+b$ + $b \over b+c$ +$c \over c+a$

= $1 \over 2$ $[3 + \sum\limits_{cyc} \frac{(a-b)}{(a+b)} ]$

= $\frac{1}{2}$ $[3 - \prod\limits_{cyc} \frac{(a-b)}{(a+b)}]$

= $\frac{1}{2} [3 - \frac{1}{11}]$

= $\frac{16}{11}$