$$2\frac{3}{4} \div 2\frac{2}{3} \div 1\frac{1}{12} = x$$

Concept & Formula When handling a chain of continuous division operations with mixed numbers, the standard procedure involves two steps: first, convert every mixed number into an improper fraction. Second, process the chain strictly from left to right, which is mathematically achieved by changing every division sign into a multiplication sign while simultaneously taking the reciprocal of the fraction immediately following it. Step-by-Step Solution * **Step 1: Convert all mixed numbers to improper fractions.** $$2\frac{3}{4} = \frac{4 \times 2 + 3}{4} = \frac{11}{4}$$ $$2\frac{2}{3} = \frac{3 \times 2 + 2}{3} = \frac{8}{3}$$ $$1\frac{1}{12} = \frac{12 \times 1 + 1}{12} = \frac{13}{12}$$ * **Step 2: Rewrite the original equation.** $$\frac{11}{4} \div \frac{8}{3} \div \frac{13}{12} = x$$ * **Step 3: Convert divisions to multiplications.** Change each $\div$ to $\times$ and flip the following fraction: $$x = \frac{11}{4} \times \frac{3}{8} \times \frac{12}{13}$$ * **Step 4: Simplify and Calculate.** Multiply the numerators together and the denominators together: $$x = \frac{11 \times 3 \times 12}{4 \times 8 \times 13}$$ Cancel the common factor of $4$ between the numerator ($12$) and denominator ($4$): $$x = \frac{11 \times 3 \times 3}{1 \times 8 \times 13}$$ $$x = \frac{99}{104}$$ * **Step 5: Verify against options.** The calculated value is $\frac{99}{104}$. None of the specific options (a, b, c) match this value or can be simplified to this value. Exam Strategy & Shortcut Write the expression as a single large fraction immediately: $\frac{11 \times 3 \times 12}{4 \times 8 \times 13}$. Perform cross-cancellations before executing any multiplication. This prevents you from dealing with unnecessarily large numbers like $\frac{396}{416}$, which are highly prone to calculation errors during simplification. Common Pitfall The most dangerous pitfall is applying division associatively. If a student calculates the right side first ($2\frac{2}{3} \div 1\frac{1}{12}$) and then divides $2\frac{3}{4}$ by that result, they will arrive at an entirely wrong answer. Continuous division must always be executed from left to right. Final Answer Therefore, the correct answer is **None of these**.