1. **State the problem:** Simplify the expression $$\frac{12a^3b}{9b^2} \times \frac{3}{4a}$$. 2. **Write the expression as a single fraction:** $$\frac{12a^3b}{9b^2} \times \frac{3}{4a} = \frac{12a^3b \times 3}{9b^2 \times 4a}$$ 3. **Multiply the numerators and denominators:** $$= \frac{36a^3b}{36ab^2}$$ 4. **Simplify the fraction by canceling common factors:** $$= \frac{\cancel{36}a^{3}b}{\cancel{36}a^{1}b^{2}} = \frac{a^{3}b}{a^{1}b^{2}}$$ 5. **Apply the laws of exponents:** $$= a^{3-1} b^{1-2} = a^{2} b^{-1}$$ 6. **Rewrite with positive exponents:** $$= \frac{a^{2}}{b}$$ **Final answer:** $$\frac{a^{2}}{b}$$