If (a+b)/(3) =(b+c)/(6) =(c+a)/(5) then prove that: (a+b+c)/(a) =7

Question

If
`(a+b)/(3) =(b+c)/(6) =(c+a)/(5)` then prove that:
`(a+b+c)/(a) =7`

Answer 1

`(a+b+c)/(a) =7`

Explanation:

Given :-

• 
  `(a+b)/(3) =(b+c)/(6) =(c+a)/(5)`

And we need to prove that ,

• 
  `(a+b+c)/(a) =7`

So let us assume that ,

`(a+b)/(3) =(b+c)/(6) =(c+a)/(5)=k`

Where k is a constant . Now equate each of the three terms separately to k . Therefore we have ,

• 
  `(a+b)/(3)=k`

`\implies a + b = 3k \quad \dots (i)`

Similarly we can say that ,

`\implies c + b = 6k \quad \dots (ii)`

`\implies a + c = 5k\quad \dots (iii)`

Subtracting (i) and (ii) :-

• 
  `\implies a - c = -3k \quad \dots (iv)`

Adding (iv) and (iii) :-

`\implies 2c = 4k`

`\implies \boxed{c = 4k }`

Put this is (ii) :-

`\implies b +4k = 6k`

`\implies \boxed{b= 2k }`

Similarly we will get ,

`\implies \boxed{a = k }`

Proving the given equation :-

`\implies (a+b+c)/(a) \\\\\implies (k+2k+4k)/(k) \\\\\implies (7k)/(k)=\boxed{\red{ 7}}`

Hence proved !