1. The problem is to solve the expression $\frac{3}{8} \times \frac{2}{5}$.\n\n2. The formula for multiplying fractions is:\n$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$\nwhere $a$, $b$, $c$, and $d$ are integers and $b, d \neq 0$.\n\n3. Applying the formula to our problem:\n$$\frac{3}{8} \times \frac{2}{5} = \frac{3 \times 2}{8 \times 5}$$\n\n4. Multiply the numerators and denominators:\n$$\frac{6}{40}$$\n\n5. Simplify the fraction by dividing numerator and denominator by their greatest common divisor (GCD), which is 2:\n$$\frac{6 \div 2}{40 \div 2} = \frac{3}{20}$$\n\n6. Therefore, the final answer is $\frac{3}{20}$.