Carefully simplify the nested fractional expression Compute the exact value of: 7 1/2 − [ 2 1/4 ÷ { 1 1/4 − 1/2(1 1/2 − 1/3 − 1/6) } ].

Introduction / Context:
This problem assesses careful simplification of a layered fractional expression containing mixed numbers, brackets, and the factor “1/2” multiplying a parenthetical group. It is a classic test of order of operations, conversion between mixed and improper fractions, and accurate arithmetic with rational numbers.

Given Data / Assumptions:

• Target expression: 7 1/2 − [ 2 1/4 ÷ { 1 1/4 − 1/2(1 1/2 − 1/3 − 1/6) } ].
• Mixed numbers should be converted to improper fractions before operations.
• Multiplication inside braces must be carried out before the outer subtraction and division.

Concept / Approach:
Work from the innermost parentheses outward. First simplify the small difference (1 1/2 − 1/3 − 1/6). Then apply the factor 1/2 to that result. Use the result to evaluate the expression inside braces. Next carry out the division in the bracket. Finally, subtract this bracket from 7 1/2. Keep all steps in exact fractional form to avoid rounding errors.

Step-by-Step Solution:
Convert: 7 1/2 = 15/2, 2 1/4 = 9/4, 1 1/4 = 5/4, 1 1/2 = 3/2.Innermost: 3/2 − 1/3 − 1/6. Since 1/3 + 1/6 = 1/2, this becomes 3/2 − 1/2 = 1.Apply 1/2: 1/2 * 1 = 1/2.Braces: 5/4 − 1/2 = 5/4 − 2/4 = 3/4.Bracket: 9/4 ÷ 3/4 = (9/4) * (4/3) = 9/3 = 3.Total: 15/2 − 3 = 15/2 − 6/2 = 9/2.

Verification / Alternative check:
Each stage can be double-checked by converting the mixed numbers first and verifying cancellations (for example, dividing by 3/4 equals multiplying by 4/3). The arithmetic aligns perfectly to give 9/2 = 4.5.

Why Other Options Are Wrong:
1: Ignores most of the interior arithmetic and the division by 3/4.4: Likely from subtracting 3 from 7 without converting to halves correctly.177/288: An unrelated small fraction produced by mishandling multiple denominators at once.

Common Pitfalls:
Forgetting to distribute the 1/2 across the inner parentheses; mixing up order of operations; or converting mixed numbers inconsistently, which can lead to sign or denominator errors. Always reduce the innermost group first and proceed outward in a structured manner.

Final Answer:
9/2