If (a + b) : (b + c) : (c + a) = 6 : 7 : 8 and (a + b + c) = 14, then the value of c is = ?

Solution(By Examveda Team)

(a + b) : (b + c) : (c + a)
6 : 7 : 8
Le t 6x : 7x : 8x
∴ a + b + b + c + a = 6x + 7x + 8x
$$\eqalign{ & \Rightarrow 2a + 2b + 2c = 21x \cr & \Rightarrow 2\left( {a + b + c} \right) = 21x \cr & \Rightarrow a + b + c = \frac{{21}}{2}x \cr & \Rightarrow a + b + c = 14{\text{ given}} \cr & \Rightarrow \frac{{21x}}{2} = 14 \cr & \Rightarrow x = \frac{{28}}{{21}} = \frac{4}{3} \cr & \therefore a + b = 6 \times \frac{4}{3} = 8 \cr & \therefore a + b + c = 14 \cr & \Rightarrow 8 + c = 14 \cr & \Rightarrow c = 14 - 8 = 6 \cr} $$