$6 \frac{5}{6} \times 5 \frac{1}{3} + 17 \frac{2}{3} \times 4 \frac{1}{2} = x$

### Concept & Strategy When evaluating complex expressions with mixed fractions and multiple operations, strictly follow the BODMAS rule (Bracket, Order, Division, Multiplication, Addition, Subtraction). Before executing operations, convert all mixed fractions into improper fractions for easier arithmetic. $$ \text{Mixed to Improper: } a \frac{b}{c} = \frac{(a \times c) + b}{c} $$ ### Step-by-Step Solution * First, convert all the mixed numbers in the equation to improper fractions: $6 \frac{5}{6} = \frac{41}{6}$ $5 \frac{1}{3} = \frac{16}{3}$ $17 \frac{2}{3} = \frac{53}{3}$ $4 \frac{1}{2} = \frac{9}{2}$ * Substitute these into the original expression: $(\frac{41}{6} \times \frac{16}{3}) + (\frac{53}{3} \times \frac{9}{2})$ * Calculate the first multiplication block: $\frac{41 \times 16}{6 \times 3} = \frac{656}{18} = \frac{328}{9}$ * Calculate the second multiplication block: $\frac{53 \times 9}{3 \times 2} = \frac{477}{6} = \frac{159}{2}$ * Add the two resulting fractions by finding the LCM of the denominators ($9$ and $2$), which is $18$: $\frac{328 \times 2}{18} + \frac{159 \times 9}{18}$ $\frac{656}{18} + \frac{1431}{18} = \frac{2087}{18}$ * Convert the final improper fraction back to a mixed number: $2087 \div 18 = 115$ with a remainder of $17$. The exact result is $115 \frac{17}{18}$. * Since $115 \frac{17}{18}$ does not match any of the given options, the answer is "None of these". ### Exam Strategy & Shortcut Use rough approximation to verify your path. Evaluate the whole numbers: $7 \times 5 = 35$, and $18 \times 4.5 = 81$. Adding them yields $116$. This tells you the exact answer is very close to $116$. Because the exact fraction is $115 \frac{17}{18}$, you can confidently select "None of these" knowing it slightly misses option (b) which is $116 \frac{2}{3}$. ### Common Pitfall Students often incorrectly multiply the whole numbers together and the fractions together separately (e.g., $6 \times 5$ and $\frac{5}{6} \times \frac{1}{3}$). This violates the distributive property. Always convert mixed numbers to improper fractions before multiplying. ### Final Answer **Therefore, the correct answer is None of these.**