1. **State the problem:** Simplify the expression $$\frac{5 \times 10^{7}}{2 \times 10^{5}}$$. 2. **Recall the rule for division of powers of 10:** When dividing powers of 10 with the same base, subtract the exponents: $$\frac{10^{a}}{10^{b}} = 10^{a-b}$$. 3. **Apply the rule to the expression:** $$\frac{5 \times 10^{7}}{2 \times 10^{5}} = \frac{5}{2} \times \frac{10^{7}}{10^{5}} = \frac{5}{2} \times 10^{7-5} = \frac{5}{2} \times 10^{2}$$ 4. **Simplify the fraction:** $$\frac{5}{2} = 2.5$$ 5. **Multiply by the power of 10:** $$2.5 \times 10^{2} = 2.5 \times 100 = 250$$ **Final answer:** $$250$$