1. **State the problem:** Tiana has $2 \frac{2}{3}$ gallons of paint. She uses half of it to paint her room, then gives $\frac{3}{8}$ of the remaining paint to her brother. We need to find how many gallons she gives to her brother. 2. **Convert mixed number to improper fraction:** $$2 \frac{2}{3} = \frac{3 \times 2 + 2}{3} = \frac{8}{3}$$ 3. **Calculate paint used for the room:** She uses half of the total paint: $$\frac{1}{2} \times \frac{8}{3} = \frac{8}{6} = \frac{4}{3}$$ gallons. 4. **Calculate remaining paint:** $$\frac{8}{3} - \frac{4}{3} = \frac{4}{3}$$ gallons remain. 5. **Calculate paint given to brother:** She gives $\frac{3}{8}$ of the remaining paint: $$\frac{3}{8} \times \frac{4}{3} = \frac{3 \times 4}{8 \times 3} = \frac{12}{24}$$ 6. **Simplify the fraction:** $$\frac{12}{24} = \frac{\cancel{12}}{\cancel{24}} = \frac{1}{2}$$ gallons. **Final answer:** Tiana gives her brother $\frac{1}{2}$ gallons of paint.