1. **State the problem:** Simplify the complex fraction $$\frac{2 \text{ of } \frac{2}{3} \div 1 \frac{2}{3} + 3}{\frac{1}{2} + 3 \frac{1}{2} \text{ of } 2 - 1 \frac{1}{2}}$$. 2. **Convert mixed numbers to improper fractions:** - $1 \frac{2}{3} = \frac{5}{3}$ - $3 \frac{1}{2} = \frac{7}{2}$ - $1 \frac{1}{2} = \frac{3}{2}$ 3. **Rewrite 'of' as multiplication and substitute the values:** Numerator: $$2 \times \frac{2}{3} \div \frac{5}{3} + 3$$ Denominator: $$\frac{1}{2} + \frac{7}{2} \times 2 - \frac{3}{2}$$ 4. **Calculate the numerator step-by-step:** - Multiply: $2 \times \frac{2}{3} = \frac{4}{3}$ - Divide: $\frac{4}{3} \div \frac{5}{3} = \frac{4}{3} \times \frac{3}{5} = \frac{12}{15} = \frac{4}{5}$ - Add 3: $\frac{4}{5} + 3 = \frac{4}{5} + \frac{15}{5} = \frac{19}{5}$ 5. **Calculate the denominator step-by-step:** - Multiply: $\frac{7}{2} \times 2 = 7$ - Sum: $\frac{1}{2} + 7 - \frac{3}{2} = \frac{1}{2} - \frac{3}{2} + 7 = -1 + 7 = 6$ 6. **Divide numerator by denominator:** $$\frac{19}{5} \div 6 = \frac{19}{5} \times \frac{1}{6} = \frac{19}{30}$$ **Final answer:** $$\boxed{\frac{19}{30}}$$