Given x / y = 3 / 4 with x and y positive, evaluate the expression: 6/7 + (y − x) / (y + x). Choose the exact value.

Introduction / Context:This problem blends a fixed ratio with a rational expression. By parameterizing x and y with a common factor, the complex-looking fraction simplifies neatly. This technique is standard in ratio problems and avoids cumbersome algebra.

Given Data / Assumptions:

• x / y = 3 / 4 with x, y > 0.
• We must evaluate 6/7 + (y − x) / (y + x).
• Arithmetic should be exact and simplified.

Concept / Approach:Let x = 3k and y = 4k for some positive k to satisfy the ratio x / y = 3 / 4. Then substitute these into the expression to simplify. The common factor k cancels cleanly in both numerator and denominator for the second term.

Step-by-Step Solution:

Verification / Alternative check:Choose a specific k, for example k = 1, giving x = 3 and y = 4. Substitute: 6/7 + (4 − 3) / (4 + 3) = 6/7 + 1/7 = 1. This confirms the general result.

Why Other Options Are Wrong:5/7 and 6/7 omit the contribution from the second term. 11/7 and 2 overcount the total by miscomputing (y − x) / (y + x).

Common Pitfalls:Plugging numbers without maintaining the 3:4 ratio, or mistakenly computing (y − x) / (y + x) as (y / x) or (x / y). Keeping the ratio structure avoids such errors.

Final Answer:1