1. **State the problem:** Multiply the mixed number $3 \frac{3}{8}$ by the fraction $\frac{2}{5}$. 2. **Convert the mixed number to an improper fraction:** $$3 \frac{3}{8} = \frac{3 \times 8 + 3}{8} = \frac{24 + 3}{8} = \frac{27}{8}$$ 3. **Write the multiplication of fractions:** $$\frac{27}{8} \times \frac{2}{5}$$ 4. **Multiply the numerators and denominators:** $$\frac{27 \times 2}{8 \times 5} = \frac{54}{40}$$ 5. **Simplify the fraction by dividing numerator and denominator by their greatest common divisor (GCD):** The GCD of 54 and 40 is 2, so $$\frac{\cancel{54}^{27 \times 2}}{\cancel{40}^{8 \times 5}} = \frac{27}{20}$$ 6. **Convert the improper fraction back to a mixed number:** $$\frac{27}{20} = 1 \frac{7}{20}$$ **Final answer:** $$1 \frac{7}{20}$$