1. **State the problem:** Multiply the mixed number $2 \frac{2}{8}$ by the fraction $\frac{2}{9}$. 2. **Convert the mixed number to an improper fraction:** $$2 \frac{2}{8} = 2 + \frac{2}{8} = \frac{2 \times 8}{8} + \frac{2}{8} = \frac{16}{8} + \frac{2}{8} = \frac{18}{8}$$ 3. **Simplify the improper fraction:** $$\frac{18}{8} = \frac{\cancel{18}}{\cancel{8}} = \frac{9}{4}$$ 4. **Multiply the fractions:** $$\frac{9}{4} \times \frac{2}{9} = \frac{9 \times 2}{4 \times 9} = \frac{\cancel{9} \times 2}{4 \times \cancel{9}} = \frac{2}{4}$$ 5. **Simplify the result:** $$\frac{2}{4} = \frac{\cancel{2}}{\cancel{4}} = \frac{1}{2}$$ 6. **Final answer:** $$\boxed{\frac{1}{2}}$$