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

### Concept & Proportionality When given a ratio of sums, introduce a common multiplier to represent the actual values. From there, sum the expressions to find the total in terms of the multiplier and equate it to the given total sum. ### Step-by-Step Solution 1. Let the expressions be proportional to a constant $k$. $a + b = 6k$ $b + c = 7k$ $c + a = 8k$ 2. Add all three equations together: $(a + b) + (b + c) + (c + a) = 6k + 7k + 8k$ $2(a + b + c) = 21k$ 3. We are given that $a + b + c = 14$. Substitute this into the equation: $2(14) = 21k$ $28 = 21k$ $k = \frac{28}{21} = \frac{4}{3}$ 4. We need to find the value of $c$. We know that $(a + b + c) = 14$. 5. From our first equation, we know $a + b = 6k$. Let's find its value: $a + b = 6 \times \left(\frac{4}{3}\right) = 8$ 6. Substitute $(a + b)$ into the total sum equation: $8 + c = 14$ $c = 14 - 8 = 6$ ### Exam Strategy & Shortcut Instead of solving for all variables individually, directly find the value of the missing variable by subtracting the sum of the other two variables from the total sum. Since $c$ is missing from $(a+b)$, finding $(a+b)$ gets you to the answer in one step. ### Common Pitfall A common mistake is assuming $a+b = 6, b+c = 7, c+a = 8$ literally, without using a constant $k$. This would falsely lead to $2(a+b+c) = 21 \Rightarrow a+b+c = 10.5$, which contradicts the given condition $a+b+c = 14$. ### Final Answer Therefore, the correct answer is **6**.