Three-sum ratio system: If (a + b) : (b + c) : (c + a) = 6 : 7 : 8 and also a + b + c = 14, find the value of c. Show how to convert the chained ratio into individual variables.

Introduction / Context: Ratios involving sums of variables can be converted into linear equations by introducing a common factor. Summing the three pairwise sums cleverly yields a direct relationship to a + b + c, unlocking individual values.

Given Data / Assumptions:

• (a + b) : (b + c) : (c + a) = 6 : 7 : 8.
• a + b + c = 14.

Concept / Approach: Let a + b = 6k, b + c = 7k, c + a = 8k. Then (a + b) + (b + c) + (c + a) = 2(a + b + c) = (6 + 7 + 8)k = 21k. This gives k and then c from a simple combination of the three sums.

Step-by-Step Solution:

Verification / Alternative check: Compute a + b = 6*(4/3) = 8, b + c = 28/3, c + a = 32/3; sum/2 gives 14. c = 6 satisfies all relations.

Why Other Options Are Wrong: 7, 8, 10, 5 do not satisfy the three simultaneous relations when checked against a + b + c = 14.

Common Pitfalls: Forgetting that the sum of the three pairwise sums equals 2(a + b + c), or misapplying the half step when isolating a single variable.

Final Answer: 6