Evaluate the nested fractional expression exactly: [ 5 − ( 3/4 + { 2 1/2 − ( 1/2 + 1/6 − 1/7 ) } ) ] / 2. Convert mixed numbers to improper fractions and simplify.

Introduction / Context:
Nestings with mixed numbers and multiple signs test attention to order of operations and accurate handling of fractions. This problem requires converting a mixed number to an improper fraction, simplifying the inner parentheses first, and then carefully executing additions and subtractions before the final division by 2.

Given Data / Assumptions:

• Expression: [ 5 − ( 3/4 + { 2 1/2 − ( 1/2 + 1/6 − 1/7 ) } ) ] / 2.
• “2 1/2” denotes the mixed number 5/2.
• Perform parentheses and brackets from inside out; apply division by 2 to the whole bracket.

Concept / Approach:
Work from the innermost parentheses outward. Combine the small fractions inside, subtract from the mixed number, add 3/4, subtract from 5, and finally divide by 2. Keeping everything as exact fractions avoids rounding errors and ensures a simplified result.

Step-by-Step Solution:
Innermost parentheses: 1/2 + 1/6 − 1/7 → common denominator 42 gives (21 + 7 − 6)/42 = 22/42 = 11/21.Compute the curly bracket: 2 1/2 − (11/21) = (5/2) − (11/21) = (105 − 22)/42 = 83/42.Add 3/4: 3/4 + 83/42 = (63 + 166)/84 = 229/84.Subtract from 5: 5 − 229/84 = 420/84 − 229/84 = 191/84.Divide by 2: (191/84) / 2 = 191/168.

Verification / Alternative check:
Convert to decimals as a check: 1/2 + 1/6 − 1/7 ≈ 0.523809; 2.5 − 0.523809 ≈ 1.976190; + 0.75 ≈ 2.726190; 5 − 2.726190 ≈ 2.273810; ÷ 2 ≈ 1.136905. Now 191/168 ≈ 1.136905 — exact match.

Why Other Options Are Wrong:

• 223/168, 183/168, 205/168, 171/168: Each is close but results from a minor sign error or denominator mismatch in one of the inner steps.

Common Pitfalls:
Misreading the scope of the final “/ 2” (it applies to the entire bracket); forgetting to convert 2 1/2 to 5/2; adding fractions without a common denominator.

Final Answer:
191/168