1. **State the problem:** We need to divide the mixed numbers $1 \frac{1}{3}$ by $1 \frac{2}{3}$. 2. **Convert mixed numbers to improper fractions:** $1 \frac{1}{3} = \frac{3 \times 1 + 1}{3} = \frac{4}{3}$ $1 \frac{2}{3} = \frac{3 \times 1 + 2}{3} = \frac{5}{3}$ 3. **Recall the division rule for fractions:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $$\frac{4}{3} \div \frac{5}{3} = \frac{4}{3} \times \frac{3}{5}$$ 4. **Multiply the fractions:** $$\frac{4}{3} \times \frac{3}{5} = \frac{4 \times 3}{3 \times 5} = \frac{12}{15}$$ 5. **Simplify the fraction:** The greatest common divisor of 12 and 15 is 3, so $$\frac{12}{15} = \frac{12 \div 3}{15 \div 3} = \frac{4}{5}$$ 6. **Final answer:** $$1 \frac{1}{3} \div 1 \frac{2}{3} = \frac{4}{5}$$