Given 4x = 18y, determine the exact value of (x / y) − 1.

Introduction / Context:
This quick algebra question checks proportional reasoning and manipulation of ratios. From a simple linear relation between x and y, you are asked to find (x / y) − 1 exactly as a rational number.

Given Data / Assumptions:

• Equation: 4x = 18y.
• Both x and y are real numbers (y ≠ 0 for x/y to be defined).

Concept / Approach:
Transform the linear relation into a ratio for x/y. Divide both sides by 4y to isolate x/y directly, then subtract 1. Keep fractions in simplest terms to avoid arithmetic errors.

Step-by-Step Solution:
Start with 4x = 18y.Divide both sides by 4y: (4x)/(4y) = (18y)/(4y) ⇒ x/y = 18/4.Reduce the fraction: x/y = 9/2.Compute (x/y) − 1 = (9/2) − 1 = (9/2) − (2/2) = 7/2.

Verification / Alternative check:
Pick numbers satisfying 4x = 18y. For example, let y = 2 ⇒ 18y = 36 ⇒ 4x = 36 ⇒ x = 9. Then x/y = 9/2 and (x/y) − 1 = 7/2, confirming the result.

Why Other Options Are Wrong:
5/2, 3/2, 2, and 9/2 do not represent (x/y) − 1. Note 9/2 is x/y itself, not the requested expression. The others are arbitrary mis-subtractions or reductions.

Common Pitfalls:
Forgetting to reduce 18/4 to 9/2; subtracting 1 incorrectly; or computing y/x by mistake. Always check which ratio is asked for and simplify carefully.

Final Answer:
7/2