1. **State the problem:** Simplify the expression $$5 \times \frac{4 \times 10^2}{4 \times 10^1}$$. 2. **Write the formula and rules:** When dividing terms with the same base, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. 3. **Simplify the fraction inside:** $$\frac{4 \times 10^2}{4 \times 10^1} = \frac{4}{4} \times \frac{10^2}{10^1}$$ 4. **Cancel common factors:** $$\frac{\cancel{4}}{\cancel{4}} \times 10^{2-1} = 1 \times 10^1 = 10$$ 5. **Multiply by 5:** $$5 \times 10 = 50$$ **Final answer:** $$50$$