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

Answer: Option A

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