1. **State the problem:** We are given a proportion: $3\sqrt{3}$ is to 9 as $9\sqrt{2}$ is to some unknown number $x$. We need to find $x$. 2. **Write the proportion:** $$\frac{3\sqrt{3}}{9} = \frac{9\sqrt{2}}{x}$$ 3. **Simplify the left side:** $$\frac{3\sqrt{3}}{9} = \frac{\sqrt{3}}{3}$$ 4. **Set up the equation:** $$\frac{\sqrt{3}}{3} = \frac{9\sqrt{2}}{x}$$ 5. **Cross multiply:** $$\sqrt{3} \cdot x = 3 \cdot 9 \sqrt{2}$$ $$\sqrt{3} x = 27 \sqrt{2}$$ 6. **Solve for $x$:** $$x = \frac{27 \sqrt{2}}{\sqrt{3}}$$ 7. **Simplify the radical:** $$x = 27 \frac{\sqrt{2}}{\sqrt{3}} = 27 \sqrt{\frac{2}{3}} = 27 \frac{\sqrt{6}}{3} = 9 \sqrt{6}$$ **Final answer:** $$x = 9 \sqrt{6}$$