Compute the value of ( (1/2 ÷ 4) + 20 ) / ( (1/2 * 4) + 20 ). Give the result in simplest fractional form.

Introduction / Context:
Here you must apply operator precedence carefully inside numerator and denominator before forming the final ratio. Quick simplifications are possible because the fractional operations are small. The answer is best left as a simplified fraction to avoid rounding issues.

Given Data / Assumptions:

• Expression: ((1/2 ÷ 4) + 20) / ((1/2 * 4) + 20).
• Division and multiplication come before addition.
• Reduce the final fraction if possible.

Concept / Approach:
Compute each inner operation first: evaluate 1/2 ÷ 4 and 1/2 * 4. Convert the resulting mixed numbers to improper fractions to combine accurately, then divide the two totals by multiplying with the reciprocal of the denominator.

Step-by-Step Solution:
Numerator: 1/2 ÷ 4 = 1/8; so numerator total = 20 + 1/8 = 161/8.Denominator: 1/2 * 4 = 2; so denominator total = 20 + 2 = 22.Overall value = (161/8) / 22 = (161/8) * (1/22) = 161/176.

Verification / Alternative check:
As decimals: numerator ≈ 20.125; denominator = 22; ratio ≈ 0.91534…; 161/176 ≈ 0.91534… — match.

Why Other Options Are Wrong:

• 81/88 and 41/44: Close approximations but not exact; they come from halving 161/176 erroneously.
• 23/11: Results from adding before multiplying/dividing.
• 1: Occurs if you mistakenly evaluate both inner parts as 20.

Common Pitfalls:
Adding 20 before finishing ÷ and *; forgetting to convert 20 + 1/8 into an improper fraction before the final division.

Final Answer:
161/176