Compute the value of (1/4 + (1/4 ÷ 5/4)) / ((1/4 * 1/4) + 2 1/4). Express your final answer as a simplified fraction.

Introduction / Context:
This simplification problem combines division of fractions, multiplication, and addition with a mixed number. Careful stepwise evaluation avoids sign and denominator errors. The final answer should be given as a reduced fraction where possible.

Given Data / Assumptions:

• The expression is (1/4 + (1/4 ÷ 5/4)) / ((1/4 * 1/4) + 2 1/4).
• 2 1/4 denotes the mixed number 9/4.
• Follow standard operator precedence: division/multiplication before addition.

Concept / Approach:
Evaluate the numerator and denominator separately. Convert the mixed number to an improper fraction. Use reciprocal multiplication for the division 1/4 ÷ 5/4, then collect terms over common denominators to form a single fraction over a single fraction, which can be inverted and multiplied.

Step-by-Step Solution:
Compute 1/4 ÷ 5/4 = (1/4) * (4/5) = 1/5.Numerator: 1/4 + 1/5 = (5 + 4)/20 = 9/20.Denominator: (1/4 * 1/4) + 2 1/4 = 1/16 + 9/4 = 1/16 + 36/16 = 37/16.Overall: (9/20) / (37/16) = (9/20) * (16/37) = 144/740 = 36/185.

Verification / Alternative check:
Convert all terms to decimals as a cross-check: 1/4 ÷ 5/4 = 0.2; numerator 0.25 + 0.2 = 0.45; denominator 0.0625 + 2.25 = 2.3125; 0.45 / 2.3125 ≈ 0.194594…; 36/185 ≈ 0.194594… — match.

Why Other Options Are Wrong:

• 16/25 / 32/185 / 9/20: Each arises from a typical arithmetic slip (e.g., adding denominators, forgetting to invert 37/16, or stopping at the intermediate 9/20).
• None of these: Not correct because 36/185 is exactly obtained.

Common Pitfalls:
Leaving 2 1/4 as 2 + 1/4 in mixed operations; adding fractions without common denominators; failing to reduce 144/740.

Final Answer:
36/185