Question

If a/b+c=b/c+a=c/a+b and a+b+c is not equal to zero then what is the value of each ratio

Answer 1

Answer is -1 or 1/2

Explanation :

Since in the question it is asked that  "What is each fraction equal to" I assumed the fractions are (a/(b+c)),(b/(c+a)),(c/(a+b))(a/(b+c)),(b/(c+a)),(c/(a+b)).

ab+c=ba+c=ca+bab+c=ba+c=ca+b

Now,

ab+c=ba+cab+c=ba+c

⟹a(a+c)=b(b+c)⟹a(a+c)=b(b+c)

⟹a2+ac=b2+bc⟹a2+ac=b2+bc

⟹a2−b2=c(b−a)⟹a2−b2=c(b−a)

⟹(a−b)(a+b)=c(b−a)⟹(a−b)(a+b)=c(b−a)

⟹(a+b)=−c⟹(a+b)=−c or a=ba=b

Case 1 :

If (a+b)=−c(a+b)=−c

Then,

c(a+b)=−1c(a+b)=−1

∴ab+c=ba+c=ca+b=−1∴ab+c=ba+c=ca+b=−1

Case 2 :

If a=ba=b  

Then

ab+c=ca+bab+c=ca+b 

⟹aa+c=ca+a⟹aa+c=ca+a 

⟹aa+c=c2a⟹aa+c=c2a 

Solving the quadratic equation we get c=−2a,ac=−2a,a.

So, we have again two cases, either a=b=ca=b=c or  a=b=−c/2a=b=−c/2

Case 2(a):

If  a=b=ca=b=c,

then 

ab+c=ba+c=ca+b=12ab+c=ba+c=ca+b=12

Case 2(b):

If a=b=−c/2a=b=−c/2

ab+c=ba+c=ca+b=−1ab+c=ba+c=ca+b=−1

Conclusion:

There are two possibilities, fraction can be either -1 or 1/2.

OR

Let each ratio equal k. So,
a=bk+ck b+ak+ck c=ak+bk

Add the 3 equations,a+b+c= 2k(a+b+c),

If a+b+c is non zero,
k=0.5  
     If a+b+c=0
     a=-(b+c)
     And each ratio equals -1

Answer 2

Answer: 1/2

Explanation:a/(b+c) = b/(c+a) = c/(a+b)

On using 》》》》 sum of antecedents /sum of consequents

》(a+b+c)/(a+b+c+a+b+c)

》(a+b+c)/(2a+2b+2c)

》(a+b+c)/(2(a+b+c))

》cutting (a+b+c)

WE GET 1/2 or 0.5

Hope it helps you

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