Number System — If a/b = 4/3, evaluate (3a + 2b) / (3a − 2b) exactly by substituting proportional values and simplifying.

Introduction / Context:
Given a ratio between a and b, it is often simplest to set a and b to proportional values and evaluate the target expression directly. This prevents algebraic complexity and ensures exact simplification without solving separate equations.

Given Data / Assumptions:

• a/b = 4/3 with a, b ≠ 0.
• Expression to evaluate: (3a + 2b) / (3a − 2b).
• Proportional substitution is valid for ratios.

Concept / Approach:
Let a = 4k and b = 3k. Substitute into the expression and simplify. Because k cancels, the result is independent of the particular proportional values chosen, giving an exact rational value.

Step-by-Step Solution:
1) Put a = 4k, b = 3k.2) Compute numerator: 3a + 2b = 3*4k + 2*3k = 12k + 6k = 18k.3) Compute denominator: 3a − 2b = 12k − 6k = 6k.4) Divide: (18k)/(6k) = 3.

Verification / Alternative check:
Pick a concrete pair, e.g., a = 4, b = 3: (3*4 + 2*3)/(3*4 − 2*3) = (12 + 6)/(12 − 6) = 18/6 = 3. Matches the general result.

Why Other Options Are Wrong:
6 and 5 come from arithmetic slips in combining terms; −1 requires numerator and denominator to be negatives of each other, which they are not; 2 would require 18k/9k, not present here.

Common Pitfalls:
Failing to distribute coefficients correctly; forgetting that the common factor k cancels; sign errors in the denominator.

Final Answer:
3