Evaluate a rational expression from a given ratio If a / b = 1 / 3, compute (3a + 2b) / (3a − 2b).

Introduction / Context:
This item uses a basic ratio to enable substitution and simplification of a rational expression. It checks whether you can correctly manipulate signs in the denominator to obtain the final value.

Given Data / Assumptions:

• a / b = 1 / 3 ⇒ a = b/3.
• Evaluate (3a + 2b) / (3a − 2b).
• Assume b ≠ 0 and denominator ≠ 0.

Concept / Approach:
Substitute a = b/3 into numerator and denominator. Factor out b to see immediate cancellation, leaving a ratio of small integers. Pay attention to the negative sign produced in the denominator.

Step-by-Step Solution:
Set a = b/3.Numerator: 3a + 2b = 3*(b/3) + 2b = b + 2b = 3b.Denominator: 3a − 2b = 3*(b/3) − 2b = b − 2b = −b.Therefore (3a + 2b)/(3a − 2b) = 3b/(−b) = −3.

Verification / Alternative check:
Choose b = 3 ⇒ a = 1. Then (3*1 + 2*3)/(3*1 − 2*3) = (3 + 6)/(3 − 6) = 9/(−3) = −3, confirming the result.

Why Other Options Are Wrong:
3 is the sign-ignored result; a common error is missing the negative in the denominator.−5 and −1 arise from arithmetic slips when combining like terms.

Common Pitfalls:
Dropping the negative sign in the denominator or substituting the ratio incorrectly (e.g., setting a = 3b instead of a = b/3). Keep track of signs carefully.

Final Answer:
−3