1. **Stating the problem:** Simplify the expression $$\frac{4b^{2}c}{12bc^{3}}$$ and verify if the simplification is correct. 2. **Formula and rules:** When simplifying fractions with variables, divide coefficients and subtract exponents of like bases using the rule $$\frac{a^{m}}{a^{n}} = a^{m-n}$$. 3. **Simplify coefficients:** $$\frac{4}{12} = \frac{\cancel{4}^{1}}{\cancel{12}^{3}} = \frac{1}{3}$$. 4. **Simplify variables:** - For $b$: $$\frac{b^{2}}{b^{1}} = b^{2-1} = b^{1} = b$$ - For $c$: $$\frac{c^{1}}{c^{3}} = c^{1-3} = c^{-2} = \frac{1}{c^{2}}$$ 5. **Combine all:** $$\frac{4b^{2}c}{12bc^{3}} = \frac{1}{3} \times b \times \frac{1}{c^{2}} = \frac{b}{3c^{2}}$$ 6. **Conclusion:** Your simplification should be $$\frac{b}{3c^{2}}$$. If your answer differs, it is incorrect. **Final answer:** $$\boxed{\frac{b}{3c^{2}}}$$