Find the value of $x$ in the following $$1\frac{2}{3} \div \frac{2}{7} \times \frac{x}{7} = 1\frac{1}{4} \times \frac{2}{3} \div \frac{1}{6}$$

Concept & Formula The core concept involves following the correct order of operations (BODMAS/PEMDAS) and handling operations on mixed fractions. The key rule for dividing fractions is to multiply by their reciprocal: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$. Step-by-Step Solution * First, convert the mixed fractions into improper fractions: $$1\frac{2}{3} = \frac{5}{3}$$ $$1\frac{1}{4} = \frac{5}{4}$$ * Substitute these into the original equation: $$\frac{5}{3} \div \frac{2}{7} \times \frac{x}{7} = \frac{5}{4} \times \frac{2}{3} \div \frac{1}{6}$$ * Apply the division-to-multiplication rule (invert the fraction immediately following the division sign) on both sides: LHS: $$\frac{5}{3} \times \frac{7}{2} \times \frac{x}{7} = \frac{5 \times 7 \times x}{3 \times 2 \times 7} = \frac{5x}{6}$$ RHS: $$\frac{5}{4} \times \frac{2}{3} \times \frac{6}{1} = \frac{5 \times 2 \times 6}{4 \times 3 \times 1} = \frac{60}{12} = 5$$ * Equate LHS and RHS: $$\frac{5x}{6} = 5$$ * Solve for $x$: $$5x = 30 \implies x = 6$$ Exam Strategy & Shortcut Convert mixed numbers to improper fractions immediately. Cancel common terms from the numerators and denominators *before* multiplying them out. For example, on the LHS, the $7$ in the numerator and denominator cancel out instantly, saving calculation time. Common Pitfall A frequent error is misinterpreting the order of operations by performing multiplication before division. Always process division and multiplication strictly from left to right as they hold the same mathematical precedence. Final Answer Therefore, the correct answer is **6**.