1. The problem is to solve the ratio $3\sqrt{3}$ is to $9\sqrt{2}$ is to what number? 2. We interpret this as a proportion: $\frac{3\sqrt{3}}{9\sqrt{2}} = \frac{9\sqrt{2}}{x}$ where $x$ is the unknown number. 3. Cross-multiply to solve for $x$: $$3\sqrt{3} \times x = 9\sqrt{2} \times 9\sqrt{2}$$ 4. Simplify the right side: $$9\sqrt{2} \times 9\sqrt{2} = 81 \times (\sqrt{2} \times \sqrt{2}) = 81 \times 2 = 162$$ 5. So the equation becomes: $$3\sqrt{3} \times x = 162$$ 6. Solve for $x$: $$x = \frac{162}{3\sqrt{3}} = \frac{162}{3} \times \frac{1}{\sqrt{3}} = 54 \times \frac{1}{\sqrt{3}}$$ 7. Rationalize the denominator: $$x = 54 \times \frac{\sqrt{3}}{3} = 18\sqrt{3}$$ Final answer: $x = 18\sqrt{3}$