Algebraic simplification (compound fraction): If a = p/(p + q) and b = q/(p − q), evaluate ab / (a + b) in simplest form.

Introduction / Context:This problem requires careful manipulation of rational expressions. By putting a and b over a common denominator, we can form a + b and ab explicitly and then simplify the compound fraction ab/(a + b). The key move is recognizing that identical factors cancel cleanly.

Given Data / Assumptions:

• a = p/(p + q)
• b = q/(p − q)
• p ≠ ±q (to avoid zero denominators)

Concept / Approach:Compute numerator and denominator separately: ab and a + b. Use the identity (p + q)(p − q) = p^2 − q^2 and simplify. The cancellation of (p^2 − q^2) between numerator and denominator will leave a compact expression in p and q.

Step-by-Step Solution:

Verification / Alternative check:Test with concrete values (e.g., p = 3, q = 1) to ensure the simplified expression matches the numeric computation of ab/(a + b). It does.

Why Other Options Are Wrong:

• Other options either invert the result or drop necessary terms, failing to account for the combination in the denominator.

Common Pitfalls:Forgetting to combine like terms p(−q) + q(p) = 0 (they cancel), leaving p^2 + q^2. Also, do not try to add numerators without establishing the common denominator first.

Final Answer:pq / (p^2 + q^2)