1. **State the problem:** Simplify the expression $1 \frac{3}{5} - 2 \frac{1}{3} + \frac{2}{5} \div 3$. 2. **Convert mixed numbers to improper fractions:** $$1 \frac{3}{5} = \frac{8}{5}, \quad 2 \frac{1}{3} = \frac{7}{3}$$ 3. **Rewrite the expression:** $$\frac{8}{5} - \frac{7}{3} + \frac{2}{5} \div 3$$ 4. **Division by 3 is multiplication by its reciprocal:** $$\frac{2}{5} \div 3 = \frac{2}{5} \times \frac{1}{3} = \frac{2}{15}$$ 5. **Substitute back:** $$\frac{8}{5} - \frac{7}{3} + \frac{2}{15}$$ 6. **Find common denominator for all fractions:** The denominators are 5, 3, and 15. The least common denominator (LCD) is 15. 7. **Convert each fraction to denominator 15:** $$\frac{8}{5} = \frac{8 \times 3}{5 \times 3} = \frac{24}{15}$$ $$\frac{7}{3} = \frac{7 \times 5}{3 \times 5} = \frac{35}{15}$$ $$\frac{2}{15} = \frac{2}{15}$$ 8. **Rewrite expression with common denominator:** $$\frac{24}{15} - \frac{35}{15} + \frac{2}{15} = \frac{24 - 35 + 2}{15} = \frac{-9}{15}$$ 9. **Simplify the fraction:** $$\frac{-9}{15} = \frac{\cancel{-9}}{\cancel{15}} \times \frac{-1}{-1} = \frac{-3}{5}$$ 10. **Final answer:** $$-\frac{3}{5}$$