The value of 6⅔ ÷ 2½ × 3¾ - 5½ × 4¼ + 1⅔ (⅞ + ¾ × ⅔) is

In an expression combining brackets, division, multiplication, addition and subtraction, apply the order of operations strictly — resolve brackets first, then carry out multiplication and division from left to right, and finally addition and subtraction from left to right. Every mixed number must first be rewritten as an improper fraction before any operation is applied to it.

1. Convert every mixed number to an improper fraction: 6⅔ = 20/3, 2½ = 5/2, 3¾ = 15/4, 5½ = 11/2, 4¼ = 17/4, 1⅔ = 5/3, ⅞ = 7/8, ¾ = 3/4, ⅔ = 2/3.

2. Resolve the innermost bracket first: ¾ × ⅔ = 1/2, so ⅞ + 1/2 = 7/8 + 4/8 = 11/8.

3. Evaluate the division-multiplication chain from left to right: 6⅔ ÷ 2½ × 3¾ = 20/3 ÷ 5/2 × 15/4 = 20/3 × 2/5 × 15/4 = 10.

4. Evaluate the second product: 5½ × 4¼ = 11/2 × 17/4 = 187/8.

5. Evaluate the third term using the bracket's value: 1⅔ × 11/8 = 5/3 × 11/8 = 55/24.

6. Combine the three terms over the common denominator 24: 10 − 187/8 + 55/24 = 240/24 − 561/24 + 55/24 = −266/24 = −133/12 = −11 ¹/₁₂.

Cross-check with decimals: 6.667 ÷ 2.5 × 3.75 = 10; 5.5 × 4.25 = 23.375; 1.667 × (0.875 + 0.5) = 1.667 × 1.375 ≈ 2.292. So 10 − 23.375 + 2.292 = −11.083, which equals −11 ¹/₁₂ — confirming the fraction-based result.

The value of the expression is −11 ¹/₁₂.