1. **State the problem:** Simplify the expression $6a \times \frac{5b^2}{3a^3}$. 2. **Write the expression:** $$6a \times \frac{5b^2}{3a^3}$$ 3. **Multiply the numerators and denominators:** $$\frac{6a \times 5b^2}{3a^3} = \frac{30ab^2}{3a^3}$$ 4. **Simplify the fraction by dividing numerator and denominator by 3:** $$\frac{\cancel{3}0ab^2}{\cancel{3}a^3} = \frac{10ab^2}{a^3}$$ 5. **Simplify the powers of $a$ using the rule $\frac{a^m}{a^n} = a^{m-n}$:** $$\frac{10ab^2}{a^3} = 10b^2a^{1-3} = 10b^2a^{-2}$$ 6. **Rewrite with positive exponents:** $$10b^2a^{-2} = \frac{10b^2}{a^2}$$ **Final answer:** $$\frac{10b^2}{a^2}$$