If a : b = 3 : 8, evaluate the expression (5a − 3b) / (2a + b) exactly. Assume a and b are non-zero real numbers consistent with the given ratio.

Introduction / Context:
This question tests how to use a ratio to evaluate an algebraic expression. When a : b is given, we can represent a and b as multiples of a common factor and then substitute into the expression. In many aptitude problems, the factor cancels out, leaving a pure numerical fraction. The key is to substitute consistently and simplify carefully with signs.

Given Data / Assumptions:

• a : b = 3 : 8 • So a = 3k and b = 8k for some non-zero real k • Required: (5a - 3b) / (2a + b)

Concept / Approach:
Replace a and b by 3k and 8k. Then compute numerator and denominator separately: Numerator = 5a - 3b Denominator = 2a + b Finally, cancel the common factor k to get a simplified fraction. This method works because the expression is homogeneous (both numerator and denominator are linear in a and b).

Step-by-Step Solution:
1) Let a = 3k and b = 8k 2) Compute numerator: 5a - 3b = 5(3k) - 3(8k) = 15k - 24k = -9k 3) Compute denominator: 2a + b = 2(3k) + 8k = 6k + 8k = 14k 4) Form the ratio: (5a - 3b)/(2a + b) = (-9k)/(14k) 5) Cancel k (k ≠ 0): = -9/14

Verification / Alternative check:
Pick an easy ratio instance such as a = 3 and b = 8: (5a - 3b)/(2a + b) = (15 - 24)/(6 + 8) = (-9)/14 = -9/14. This confirms the result is consistent for the given ratio.

Why Other Options Are Wrong:
• 9/14 misses the negative sign from 15k - 24k. • ±14/9 are reciprocals and do not match the computed linear simplification. • -3/5 is a different fraction and fails substitution with a = 3, b = 8.

Common Pitfalls:
• Forgetting to multiply b by 3 in the term 3b. • Cancelling the wrong quantities or forgetting the sign in the numerator.

Final Answer:
-9/14