Question : If $\left (a+b \right):\left (b+c \right):\left (c+a \right)= 6:7:8$ and $\left (a+b+c \right) = 14,$ then value of $c$ is:
Option 1: 6
Option 2: 7
Option 3: 8
Option 4: 14
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Correct Answer: 6
Solution :
Given:
$(a + b) : (b + c) : (c + a) = 6 : 7 : 8$
$a+b+c = 14$
Let be $a+b = 6x$ , $b+c = 7x$ and $c+a = 8x$
$\therefore 2(a+b+c)=6x+7x+8x$ ----------------------------(1)
⇒ $(a+b+c)=\frac{6x+7x+8x}{2}$
⇒ $6x+c = 10.5x$
$\therefore c = 10.5x-6x=4.5x$
From equation 1 we get,
$2×14=6x+7x+8x$ [as $a+b+c = 14$ ]
$\therefore x = \frac{28}{21}=\frac{4}{3}$
$\therefore c = 4.5\times \frac{4}{3}= 6$
Hence, the correct answer is 6.
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