1. **State the problem:** Solve for $\Delta x$ given the equation $$\frac{\sqrt{45}}{\Delta x} = \frac{6x}{2\sqrt{5}}.$$\n\n2. **Simplify square roots and fractions:** \n$\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$. \nRewrite the equation as $$\frac{3\sqrt{5}}{\Delta x} = \frac{6x}{2\sqrt{5}}.$$\n\n3. **Simplify the right side fraction:** \n$$\frac{6x}{2\sqrt{5}} = \frac{6x}{2 \cdot \sqrt{5}} = \frac{3x}{\sqrt{5}}.$$\nNow the equation is $$\frac{3\sqrt{5}}{\Delta x} = \frac{3x}{\sqrt{5}}.$$\n\n4. **Cross multiply:** \n$$3\sqrt{5} \times \sqrt{5} = 3x \times \Delta x.$$\n\n5. **Simplify the left side:** \n$\sqrt{5} \times \sqrt{5} = 5$, so $$3 \times 5 = 3x \Delta x.$$\nThis gives $$15 = 3x \Delta x.$$\n\n6. **Solve for $\Delta x$:** \nDivide both sides by $3x$ to isolate $\Delta x$: $$\Delta x = \frac{15}{3x} = \frac{5}{x}.$$\n\n**Final answer:** $$\boxed{\Delta x = \frac{5}{x}}.$$