If the fraction a/b is positive, which of the following statements must always be true?

Introduction / Context:This question tests sign analysis for rational numbers. A fraction a/b is positive when its numerator and denominator have the same sign: both positive or both negative. Recognizing the necessary and sufficient condition prevents incorrect assumptions about individual signs.

Given Data / Assumptions:

• a/b > 0.
• b ≠ 0 (division by zero is undefined).

Concept / Approach:For any real numbers a and b (b ≠ 0), a/b > 0 if and only if a and b have the same sign. Algebraically, that means the product ab is positive, i.e., ab > 0. This holds regardless of the individual signs of a or b as long as both are not zero simultaneously and b ≠ 0.

Step-by-Step Solution:Case 1: a > 0 and b > 0 → a/b > 0 and ab > 0.Case 2: a < 0 and b < 0 → a/b > 0 and ab > 0.Both cases share ab > 0 as the must-true condition.

Verification / Alternative check:Counterexamples show why individual sign claims fail: If a = -2 and b = -3, then a/b = 2/3 (positive), yet a > 0 and b > 0 are both false. However, ab = 6 > 0 remains true.

Why Other Options Are Wrong:a > 0: Could be negative if b is also negative.b > 0: Could be negative if a is also negative.a + b > 0: Sum can be any sign even if a/b is positive.a and b are both even: Parity is irrelevant to sign.

Common Pitfalls:Assuming positivity forces both numerator and denominator individually positive; ignoring the valid both-negative case.

Final Answer:ab > 0