Question

If a/b+c=b/c+a=c/a+b,prove that each fraction is equal to 1/2 or -1

Answer 1

$Given \: ratios : \dfrac{a}{b + c} = \dfrac{b}{a + c} = \dfrac{c}{a + b}$


From the properties of ratio and proportion, we know : -

$\mathsf{Each\:Ratio=\dfrac{Sum\:of\: antecedents}{Sum\:of\: consequents}}$



Therefore,
If a + b + c ≠ 0, so the solution will be : -

$\implies Each\:Ratio = \dfrac{a + b + c}{b + c + a + c + a + b} \\ \\ \\ \implies Each\:Ratio = \frac{a + b + c}{2a + 2b + 2c} \\ \\ \\ \implies Each\:Ratio = \dfrac{a + b +c }{2(a + b + c)} \\ \\ \\ \implies Each\:Ratio = \dfrac{1}{2}$



If a + b + c = 0, then,

• b + c = - a

• a + b = - c

• a + c = - b


Therefore,

$\implies Each\:Ratio = \dfrac{a}{b + c} = \dfrac{b}{a + c} = \dfrac{c}{a + b}$


From above, values of b + c, a + b and a + c are - a , - b and - c respectively.
Hence,

$\implies Each\:Ratio = \dfrac{a}{ - a} = \dfrac{b}{ - b} = \dfrac{c}{ - c} \\ \\ \\ \implies Each\:Ratio = - 1 = - 1 = - 1$


Hence, proved that each ratio ( fraction ) is equal to $\dfrac{1}{2}$ or - 1 .

Answer 2

Ans :

What is the value of each ratio: a/b+c=b/c+a=c/a+b?

Know Assume

a/(b+c)=b/(c+a)=c/(a+b)=k

a=k(b+c)

b=k(c+a)

c=k(a+b)

by property sum of numerator/sum of denominator is also = k

a+b+c/2(a+b+c)=k

k=1/2

If a+b+c=0

a+b=−corb+c=−a

a/b+c=a/−a=−1=k

So k=−1or1/2.