given (((a−b)(b−c)(c−a))/((a+b)(b+c)(c+a)))=−(1/(30)) what the value of (b/(a+b))+(c/(b+c))+(a/(c+a)) ?


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Question Number 77604 by jagoll last updated on 08/Jan/20

$g i v e n$ $\frac{(a - b) (b - c) (c - a)}{(a + b) (b + c) (c + a)} = - \frac{1}{30}$ $w h a t t h e v a l u e o f$ $\frac{b}{a + b} + \frac{c}{b + c} + \frac{a}{c + a} ?$
	Answered by key of knowledge last updated on 08/Jan/20

	$not have constat answer .$ $if : \frac{a - b}{a + b} = k (and \frac{b}{a + b} = \frac{1 - k}{2})$ $\frac{b - c}{b + c} = j (\frac{c}{b + c} = \frac{1 - j}{2})$ $\Rightarrow \frac{c - a}{c + a} = - \frac{j + k}{1 + jk}$ $\frac{(a - b) (b - c) (c - a)}{(a + b) (b + c) (c + a)} = - \frac{1}{30} \Rightarrow k = \frac{j - {30 j}^{2} \mp \sqrt{j ({900 j}^{3} - {60 j}^{2} + j + 120}}{60 j}$ $\frac{b}{a + b} + \frac{c}{b + c} + \frac{a}{c + a} = ? = \frac{1}{30} (\frac{91}{2} + \frac{1}{kj})$
	Answered by john santu last updated on 09/Jan/20

	Commented by mr W last updated on 09/Jan/20

	$t h e r e i s n o u n i q u e s o l u t i o n!$ $w h a t y o u d i d i s o n l y o n e p o s s i b l e v a l u e .$ $u s i n g t h e s a m e p a t h a s y o u w e c a n$ $a l s o g e t f o r e x a m p l e$ $1 - \frac{2 b}{a + b} = - 1 \Rightarrow \frac{b}{a + b} = 1$ $1 - \frac{2 c}{b + c} = - \frac{1}{5} \Rightarrow \frac{c}{b + c} = \frac{3}{5}$ $1 - \frac{2 a}{c + a} = - \frac{1}{6} \Rightarrow \frac{c}{b + c} = \frac{7}{12}$ $\Rightarrow \frac{b}{a + b} + \frac{c}{b + c} + \frac{a}{c + a} = 1 + \frac{3}{5} + \frac{7}{12} = \frac{131}{60}$ $i f a, b, c \in R, t h e r e a r e i n f i n i t e s o l u t i o n s .$ $t h i s i s c l e a r, s i n c e w e h a v e t h r e e$ $v a r i a b l e s (= u n k n o w n s), b u t o n l y o n e$ $c o n d i t i o n (= e q u a t i o n), y o u c a n n o t$ $g e t a n u n i q u e v a l u e f o r t h e f u n c t i o n .$
	Commented by john santu last updated on 09/Jan/20

	$y e s, i a g r e e w i t h y o u r o p i n i o n s i r$ $t h e a n s w e r i p o s t e d i s n o t a s i n g l e$ $a n s w e r . c o n s i d e r i n g t h e p r o b l e m i s$ $n o t d i s p l a y e d, t h e n i t a k e o n e$ $p o s s i b l e a n s w e r$
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