$$9 - 1\frac{2}{9} \text{ of } 3\frac{3}{11} \div 5\frac{1}{7} \text{ of } \frac{7}{9} = x$$

Concept & Logic This simplification problem hinges on the strict hierarchy of mathematical operations. According to BODMAS (Brackets, Of, Division, Multiplication, Addition, Subtraction), the word 'of' holds a higher precedence than standard division. You must evaluate the terms connected by 'of' completely before processing the central division sign. Step-by-Step Solution * **Step 1: Convert mixed fractions to improper fractions** $$1\frac{2}{9} = \frac{11}{9}$$ $$3\frac{3}{11} = \frac{36}{11}$$ $$5\frac{1}{7} = \frac{36}{7}$$ Rewrite the equation: $$9 - \frac{11}{9} \text{ of } \frac{36}{11} \div \frac{36}{7} \text{ of } \frac{7}{9} = x$$ * **Step 2: Evaluate both 'of' operations first** Left 'of' block: $$\frac{11}{9} \text{ of } \frac{36}{11} = \frac{11}{9} \times \frac{36}{11}$$ Cancel the $11$s and divide $36$ by $9$: $= 4$ Right 'of' block: $$\frac{36}{7} \text{ of } \frac{7}{9} = \frac{36}{7} \times \frac{7}{9}$$ Cancel the $7$s and divide $36$ by $9$: $= 4$ * **Step 3: Substitute the simplified blocks back into the equation** $$9 - 4 \div 4 = x$$ * **Step 4: Execute division before subtraction** $$4 \div 4 = 1$$ $$9 - 1 = x$$ $$x = 8$$ Exam Strategy & Shortcut Mentally bracket the 'of' sections to protect them from the division operator: $9 - (A) \div (B)$. Because the fractions chosen by examiners are usually designed to cancel perfectly, focus purely on cross-canceling numerators and denominators within those brackets. Observing that $\frac{11}{9} \times \frac{36}{11}$ instantly yields $4$ saves precious time. Common Pitfall The most frequent and fatal error is evaluating the equation strictly left-to-right or prioritizing division over 'of'. If a student calculates $\frac{36}{11} \div \frac{36}{7}$ first, the entire mathematical structure collapses, resulting in a complex, messy fraction that leads to an incorrect option like $8\frac{32}{81}$. Final Answer Therefore, the correct answer is **8**.