The expression $$\frac{5\frac{5}{8}}{6\frac{3}{7}} \text{ of } \frac{6\frac{7}{11}}{9\frac{1}{8}} \div \frac{8}{9}\left(2\frac{3}{11} + \frac{13}{22}\right) \text{ of } \frac{3}{5}$$ equals

### Concept & Formula Convert mixed fractions to improper fractions and solve according to the order of operations under **BODMAS** rules. ### Step-by-Step Solution * **Step 1: Simplify the Complex Fractions on the Left** $$\text{First Fraction} = \frac{\frac{45}{8}}{\frac{45}{7}} = \frac{7}{8}$$ $$\text{Second Fraction} = \frac{\frac{73}{11}}{\frac{73}{8}} = \frac{8}{11}$$ Evaluate the 'of' between them: $$\frac{7}{8} \text{ of } \frac{8}{11} = \frac{7}{11}$$ * **Step 2: Simplify the Expression Inside the Parentheses** $$2\frac{3}{11} + \frac{13}{22} = \frac{25}{11} + \frac{13}{22} = \frac{50 + 13}{22} = \frac{63}{22}$$ * **Step 3: Evaluate Remaining Operators from Right to Left (BODMAS)** The expression is now: $$\frac{7}{11} \div \frac{8}{9} \times \frac{63}{22} \text{ of } \frac{3}{5}$$ Evaluate the 'of' operator first: $$\frac{63}{22} \times \frac{3}{5} = \frac{189}{110}$$ Now execute multiplication and division sequentially: $$\frac{8}{9} \times \frac{189}{110} = \frac{8 \times 21}{110} = \frac{168}{110} = \frac{84}{55}$$ Finally, execute division: $$\frac{7}{11} \div \frac{84}{55} = \frac{7}{11} \times \frac{55}{84} = \frac{5}{12}$$ ### Exam Strategy & Shortcut Notice the components cancel beautifully: $\frac{45}{8}/\frac{45}{7}$ instantly becomes $\frac{7}{8}$, and the next term becomes $\frac{8}{11}$. Their product simplifies directly to $\frac{7}{11}$, minimizing long-form multiplication overhead. ### Common Pitfall Performing the division before completing the trailing 'of' operator will produce an incorrect answer. Always treat the 'of' connection as higher priority than division. ### Final Answer **Therefore, the correct answer is 5/12.**