Algebraic substitution with a given ratio If x / y = 4 / 5, evaluate 4/7 + (2y − x) / (2y + x).

Introduction / Context:
This question tests the standard technique of substituting proportional values from a given ratio to simplify an algebraic expression. By expressing x and y in terms of a single parameter, you can reduce the compound fraction to a simple rational number and then add the constant 4/7.

Given Data / Assumptions:

• x / y = 4 / 5.
• Expression to evaluate: 4/7 + (2y − x) / (2y + x).
• Assume y ≠ 0 and denominators remain nonzero.

Concept / Approach:
Let x = 4k and y = 5k for some nonzero k. This preserves the ratio x/y = 4/5. Substitute into the expression and simplify the resulting rational terms. Parameter k cancels out, leaving a pure number that can be added to 4/7.

Step-by-Step Solution:
Set x = 4k and y = 5k.Compute numerator: 2y − x = 10k − 4k = 6k.Compute denominator: 2y + x = 10k + 4k = 14k.Thus (2y − x)/(2y + x) = 6k/14k = 3/7.Add: 4/7 + 3/7 = 1.

Verification / Alternative check:
Pick a concrete pair (x, y) with ratio 4:5, e.g., x = 4 and y = 5. Then (2*5 − 4)/(2*5 + 4) = 6/14 = 3/7, and 4/7 + 3/7 = 1, confirming the result.

Why Other Options Are Wrong:
3/7 is the value of just the fractional part, not the whole expression.11/7 and 2 come from arithmetic slips while adding or simplifying the ratio.

Common Pitfalls:
Forgetting to scale both x and y consistently; not canceling k; or adding 4/7 incorrectly. Keep the ratio substitution clean and perform fraction addition with like denominators.

Final Answer:
1