1. **State the problem:** Simplify the expression $$\frac{2xy}{x} \div \frac{6xy}{x^2}$$. 2. **Rewrite the division as multiplication by the reciprocal:** $$\frac{2xy}{x} \times \frac{x^2}{6xy}$$ 3. **Simplify each fraction:** - In $$\frac{2xy}{x}$$, cancel the common factor $x$: $$\frac{2\cancel{x}y}{\cancel{x}} = 2y$$ - In $$\frac{x^2}{6xy}$$, cancel one $x$ from numerator and denominator: $$\frac{x\cancel{x}}{6\cancel{x}y} = \frac{x}{6y}$$ 4. **Multiply the simplified expressions:** $$2y \times \frac{x}{6y}$$ 5. **Cancel the common factor $y$:** $$2\cancel{y} \times \frac{x}{6\cancel{y}} = \frac{2x}{6}$$ 6. **Simplify the fraction $$\frac{2x}{6}$$ by dividing numerator and denominator by 2:** $$\frac{\cancel{2}x}{\cancel{6}3} = \frac{x}{3}$$ **Final answer:** $$\boxed{\frac{x}{3}}$$