1. **State the problem:** Multiply the fraction $\frac{4}{6}$ by the mixed number $3 \frac{3}{5}$. 2. **Convert the mixed number to an improper fraction:** $$3 \frac{3}{5} = \frac{3 \times 5 + 3}{5} = \frac{15 + 3}{5} = \frac{18}{5}$$ 3. **Write the multiplication of fractions:** $$\frac{4}{6} \times \frac{18}{5}$$ 4. **Simplify before multiplying:** - Factor numerator and denominator to find common factors. - $4 = 2 \times 2$ - $6 = 2 \times 3$ - $18 = 2 \times 3 \times 3$ 5. **Cancel common factors:** $$\frac{\cancel{2} \times 2}{\cancel{2} \times 3} \times \frac{2 \times 3 \times 3}{5} = \frac{2}{3} \times \frac{18}{5}$$ 6. **Multiply numerators and denominators:** $$\frac{2 \times 18}{3 \times 5} = \frac{36}{15}$$ 7. **Simplify the fraction:** - Both numerator and denominator are divisible by 3. $$\frac{\cancel{3} \times 12}{\cancel{3} \times 5} = \frac{12}{5}$$ 8. **Convert back to a mixed number:** $$\frac{12}{5} = 2 \frac{2}{5}$$ **Final answer:** $2 \frac{2}{5}$