1. **Problem:** Multiply the rational expressions $\frac{2a^2 b}{b^2 c} \times \frac{b}{a}$. 2. **Formula and rules:** When multiplying rational expressions, multiply numerators together and denominators together: $$\frac{A}{B} \times \frac{C}{D} = \frac{A \times C}{B \times D}$$ Remember to factor and simplify common factors. 3. **Step-by-step solution:** - Multiply numerators: $2a^2 b \times b = 2a^2 b^2$ - Multiply denominators: $b^2 c \times a = a b^2 c$ So the expression is: $$\frac{2a^2 b^2}{a b^2 c}$$ 4. **Simplify by canceling common factors:** - Cancel $a$ in numerator and denominator: $$\frac{2\cancel{a} a b^2}{\cancel{a} b^2 c} = \frac{2 a b^2}{b^2 c}$$ - Cancel $b^2$ in numerator and denominator: $$\frac{2 a \cancel{b^2}}{\cancel{b^2} c} = \frac{2 a}{c}$$ 5. **Final answer:** $$\boxed{\frac{2a}{c}}$$ This means the product simplifies to $\frac{2a}{c}$.