1. **State the problem:** Simplify the expression $$10 \sqrt{20} \times \frac{1}{2 \sqrt{5}}$$. 2. **Recall the formula and rules:** When multiplying and dividing expressions with square roots, use the property $$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$$ and simplify fractions by canceling common factors. 3. **Rewrite the expression:** $$10 \sqrt{20} \times \frac{1}{2 \sqrt{5}} = \frac{10 \sqrt{20}}{2 \sqrt{5}}$$ 4. **Simplify the fraction:** $$\frac{10}{2} = \cancel{\frac{10}{2}} 5$$ So the expression becomes: $$5 \times \frac{\sqrt{20}}{\sqrt{5}}$$ 5. **Use the property of square roots division:** $$\frac{\sqrt{20}}{\sqrt{5}} = \sqrt{\frac{20}{5}}$$ 6. **Simplify inside the square root:** $$\sqrt{\frac{20}{5}} = \sqrt{4}$$ 7. **Calculate the square root:** $$\sqrt{4} = 2$$ 8. **Multiply the simplified terms:** $$5 \times 2 = 10$$ **Final answer:** $$10$$