1. **State the problem:** Simplify the expression $$\frac{6x^2y^3}{2x^2y^2} \times \frac{10x^3y^4}{18y^2}$$. 2. **Rewrite the expression:** $$\left(\frac{6x^2y^3}{2x^2y^2}\right) \times \left(\frac{10x^3y^4}{18y^2}\right)$$ 3. **Simplify the first fraction:** $$\frac{6x^2y^3}{2x^2y^2} = \frac{\cancel{6}^3 \cancel{x^2} y^3}{\cancel{2}^1 \cancel{x^2} y^2} = \frac{3y^3}{y^2}$$ 4. **Simplify the second fraction:** $$\frac{10x^3y^4}{18y^2} = \frac{10x^3y^4}{18y^2} = \frac{\cancel{10}^{5} x^3 y^4}{\cancel{18}^{9} y^2} = \frac{5x^3 y^4}{9 y^2}$$ 5. **Simplify powers of y in both fractions:** $$\frac{3y^3}{y^2} = 3y^{3-2} = 3y$$ $$\frac{5x^3 y^4}{9 y^2} = \frac{5x^3 y^{4-2}}{9} = \frac{5x^3 y^2}{9}$$ 6. **Multiply the simplified fractions:** $$3y \times \frac{5x^3 y^2}{9} = \frac{3 \times 5 x^3 y \times y^2}{9} = \frac{15 x^3 y^3}{9}$$ 7. **Simplify the fraction:** $$\frac{15 x^3 y^3}{9} = \frac{\cancel{15}^5 x^3 y^3}{\cancel{9}^3} = \frac{5 x^3 y^3}{3}$$ **Final answer:** $$\frac{5 x^3 y^3}{3}$$