Reconstructing a bracketed expression with “of” and ratios Evaluate 3/4 ÷ (2 1/4 of 2/3) − [ (1/2 − 1/3) / (1/2 + 1/3) × 3 1/3 ] + 5/6.

Introduction / Context:
Complex fractional expressions often compress multiple ideas: “of” as multiplication, ratios of sums/differences, and mixed numbers. Interpreting the brackets sensibly and proceeding stepwise yields a clean exact fraction at the end.

Given Data / Assumptions:

• Interpret “of” as multiplication and use left-to-right evaluation for × and ÷.
• Assumed grouping: 3/4 ÷ (2 1/4 of 2/3) − [ (1/2 − 1/3) / (1/2 + 1/3) × 3 1/3 ] + 5/6.
• 2 1/4 = 9/4, 3 1/3 = 10/3.

Concept / Approach:
Compute the “of” products first inside their brackets, then resolve divisions and multiplications from left to right. Reduce intermediate fractions whenever possible to keep numbers small and accurate.

Step-by-Step Solution:
Term A: 2 1/4 of 2/3 = (9/4) × (2/3) = 18/12 = 3/2.Then 3/4 ÷ (3/2) = 3/4 × 2/3 = 1/2.Term B ratio: (1/2 − 1/3) / (1/2 + 1/3) = (1/6) / (5/6) = 1/5.Multiply by 3 1/3: (1/5) × (10/3) = 2/3.Assemble expression: 1/2 − 2/3 + 5/6.Convert to sixths: 1/2 = 3/6; −2/3 = −4/6; +5/6 remains 5/6.Sum: (3 − 4 + 5)/6 = 4/6 = 2/3.

Verification / Alternative check:
Compute decimals: 1/2 = 0.5; 2/3 ≈ 0.6667; 5/6 ≈ 0.8333 → 0.5 − 0.6667 + 0.8333 ≈ 0.6666…, confirming 2/3.

Why Other Options Are Wrong:
7/18, 49/54, 1/6, and 5/9 arise from mis-grouping or mishandling division by a fraction.

Common Pitfalls:
Dropping parentheses; interpreting “of” incorrectly; forgetting that dividing by a fraction multiplies by its reciprocal.

Final Answer:
2/3