1. **State the problem:** We want to express $\frac{1}{20\sqrt{20}}$ in the form $\frac{\sqrt{5}}{n}$, where $n$ is a positive integer. 2. **Rationalise the denominator:** Start with the original expression: $$\frac{1}{20\sqrt{20}}$$ Multiply numerator and denominator by $\sqrt{20}$ to remove the square root from the denominator: $$\frac{1}{20\sqrt{20}} \times \frac{\sqrt{20}}{\sqrt{20}} = \frac{\sqrt{20}}{20 \times 20} = \frac{\sqrt{20}}{400}$$ 3. **Simplify the square root:** Note that $\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$, so: $$\frac{\sqrt{20}}{400} = \frac{2\sqrt{5}}{400} = \frac{\sqrt{5}}{200}$$ 4. **Compare with the desired form:** We have expressed the original fraction as $\frac{\sqrt{5}}{200}$, so $n = 200$. **Final answer:** $n = 200$