Number System — Given p/q = 4/5, evaluate the expression: 4/7 + (2q - p) / (2q + p). Provide the exact simplified value.

Introduction / Context:
This problem tests substitution with ratios and algebraic simplification. Rather than solving for explicit p and q, we can set p and q to proportional values consistent with p/q = 4/5 and then simplify the target expression directly.

Given Data / Assumptions:

• p/q = 4/5 with p, q nonzero and of the same sign.
• Expression: 4/7 + (2q - p) / (2q + p).
• We seek an exact rational number.

Concept / Approach:
Let p = 4k and q = 5k for some nonzero k. This preserves p/q = 4/5 while keeping algebra simple. Substitute into (2q - p)/(2q + p) and simplify before adding 4/7. Work with common denominators carefully to avoid mistakes.

Step-by-Step Solution:
1) Put p = 4k and q = 5k.2) Compute numerator: 2q - p = 10k - 4k = 6k.3) Compute denominator: 2q + p = 10k + 4k = 14k.4) Thus (2q - p)/(2q + p) = 6k/14k = 3/7.5) Add 4/7: 4/7 + 3/7 = 7/7 = 1.

Verification / Alternative check:
Choose a concrete pair (p, q) such as (4, 5). Then (2*5 - 4)/(2*5 + 4) = (10 - 4)/(10 + 4) = 6/14 = 3/7. Adding 4/7 again yields 1, confirming the result.

Why Other Options Are Wrong:
3/7 omits adding 4/7; 34 and 2 are not reasonable with a small rational expression; 11/7 arises if someone adds 4/7 to 1 incorrectly or mishandles denominators.

Common Pitfalls:
Forgetting to maintain proportionality (using arbitrary p, q that do not satisfy 4/5); mismanaging the common factor k; arithmetic slips when adding rational numbers.

Final Answer:
1