Simplify: $10 \frac{1}{8} \text{ of } \frac{12}{15} \div \frac{35}{36} \text{ of } \frac{20}{49}$.

### Concept & Rule This problem tests your exact understanding of the BODMAS hierarchy. The word 'of' translates to multiplication, but critically, operations connected by 'of' must be calculated **before** division. $$A \text{ of } B \div C \text{ of } D = (A \times B) \div (C \times D)$$ ### Step-by-Step Solution * **Given:** $10 \frac{1}{8} \text{ of } \frac{12}{15} \div \frac{35}{36} \text{ of } \frac{20}{49}$ * **Step 1: Convert mixed numbers and evaluate the 'of' terms on both sides** * Left side: $10 \frac{1}{8} = \frac{81}{8}$ * Left 'of' evaluation: $\frac{81}{8} \times \frac{12}{15}$ * Simplify by cross-canceling ($12$ and $15$ by $3$ to get $\frac{4}{5}$): $\frac{81}{8} \times \frac{4}{5} = \frac{81}{2} \times \frac{1}{5} = \frac{81}{10}$ * Right side 'of' evaluation: $\frac{35}{36} \times \frac{20}{49}$ * Simplify by cross-canceling (divide $35$ and $49$ by $7$; divide $20$ and $36$ by $4$): * $\frac{5}{9} \times \frac{5}{7} = \frac{25}{63}$ * **Step 2: Perform the central division** * Now substitute the simplified blocks back into the equation: $\frac{81}{10} \div \frac{25}{63}$ * Convert division to multiplication by flipping the second fraction: $\frac{81}{10} \times \frac{63}{25}$ * Multiply numerators and denominators: $\frac{81 \times 63}{10 \times 25} = \frac{5103}{250}$ * **Step 3: Convert the improper fraction back to a mixed number** * $5103 \div 250$: We know $250 \times 20 = 5000$. * The remainder is $103$. * Result: $20 \frac{103}{250}$ ### Exam Strategy & Shortcut Treat everything before the $\div$ sign as one massive bracket, and everything after as another massive bracket. Simplify each block independently through aggressive cross-cancellation before attempting to divide. Never multiply large numerators (like $35 \times 20$) without checking for common factors first. ### Common Pitfall The most frequent error is treating 'of' identically to a standard multiplication sign ($\times$) and processing strictly left-to-right: $(10 \frac{1}{8} \times \frac{12}{15}) \div (\frac{35}{36}) \times (\frac{20}{49})$. This completely changes the mathematical meaning and will result in a totally incorrect fraction. ### Final Answer **Therefore, the correct answer is $20 \frac{103}{250}$.**