$1 \frac{10}{11} \times 17 \frac{2}{7} \times 36 \frac{1}{6} = x$

### Concept & Logic To mathematically multiply mixed fractions, you must first convert all of them into improper fractions. Then, aggressively cancel out common factors between the numerators and denominators before computing the final multiplication. ### Step-by-Step Solution * Given expression: $1 \frac{10}{11} \times 17 \frac{2}{7} \times 36 \frac{1}{6}$ * Convert each mixed fraction to an improper fraction: $1 \frac{10}{11} = \frac{1 \times 11 + 10}{11} = \frac{21}{11}$ $17 \frac{2}{7} = \frac{17 \times 7 + 2}{7} = \frac{119 + 2}{7} = \frac{121}{7}$ $36 \frac{1}{6} = \frac{36 \times 6 + 1}{6} = \frac{216 + 1}{6} = \frac{217}{6}$ * Substitute these fractions back into the multiplication expression: $\frac{21}{11} \times \frac{121}{7} \times \frac{217}{6}$ * Perform cross-cancellation to simplify the chain: Cancel $11$ and $121$: $\frac{121}{11} = 11$ Cancel $7$ and $21$: $\frac{21}{7} = 3$ * The resulting expression is now: $3 \times 11 \times \frac{217}{6} = 33 \times \frac{217}{6}$ * Simplify $33$ and $6$ by dividing both by $3$: $11 \times \frac{217}{2} = \frac{2387}{2}$ * Convert the final improper fraction back to a mixed number: $\frac{2387}{2} = 1193$ with a remainder of $1$, which gives $1193 \frac{1}{2}$. ### Exam Strategy & Shortcut Convert to improper fractions and immediately scan the board for multiples. The numbers $121$ and $11$ pair perfectly, as do $21$ and $7$. Grouping the $(21/7)$ visually as $3$ and $(121/11)$ as $11$ instantly reduces the messy problem to $33 \times (217/6)$. From there, simplifying by $3$ makes it a straightforward 2-digit multiplication and division by $2$. ### Common Pitfall A critical mistake students make is attempting to multiply the whole integer parts separately from the fractional parts (e.g., assuming the answer starts with $1 \times 17 \times 36$). This logic only works for addition and subtraction, not multiplication sequences. ### Final Answer **Therefore, the correct answer is $1193 \frac{1}{2}$.**