1. **State the problem:** Simplify the expression $5 \frac{1}{3} \div 3 \frac{3}{5} + 2 \frac{1}{3}$. 2. **Convert mixed numbers to improper fractions:** $5 \frac{1}{3} = \frac{16}{3}$ $3 \frac{3}{5} = \frac{18}{5}$ $2 \frac{1}{3} = \frac{7}{3}$ 3. **Rewrite the expression with improper fractions:** $$\frac{16}{3} \div \frac{18}{5} + \frac{7}{3}$$ 4. **Division of fractions rule:** To divide by a fraction, multiply by its reciprocal. $$\frac{16}{3} \div \frac{18}{5} = \frac{16}{3} \times \frac{5}{18}$$ 5. **Multiply fractions:** $$\frac{16}{3} \times \frac{5}{18} = \frac{16 \times 5}{3 \times 18} = \frac{80}{54}$$ 6. **Simplify the fraction by canceling common factors:** $$\frac{80}{54} = \frac{\cancel{2} \times 40}{\cancel{2} \times 27} = \frac{40}{27}$$ 7. **Add the fractions:** $$\frac{40}{27} + \frac{7}{3}$$ Convert $\frac{7}{3}$ to denominator 27: $$\frac{7}{3} = \frac{7 \times 9}{3 \times 9} = \frac{63}{27}$$ 8. **Add the numerators:** $$\frac{40}{27} + \frac{63}{27} = \frac{40 + 63}{27} = \frac{103}{27}$$ 9. **Convert the improper fraction to a mixed number:** $$103 \div 27 = 3 \text{ remainder } 22$$ So, $$\frac{103}{27} = 3 \frac{22}{27}$$ **Final answer:** $3 \frac{22}{27}$