$3 \frac{1}{4} \times \frac{2}{5}$

Step 1 ：Convert the mixed number $3 \frac{1}{4}$ to an improper fraction. The formula for converting a mixed number to an improper fraction is \(\text{whole number} \times \text{denominator} + \text{numerator}\). So, $3 \frac{1}{4}$ becomes $3 \times 4 + 1 = 13$, or $\frac{13}{4}$.

Step 2 ：Next, multiply the two fractions together. The formula for multiplying fractions is \(\frac{\text{numerator1} \times \text{numerator2}}{\text{denominator1} \times \text{denominator2}}\). So, $\frac{13}{4} \times \frac{2}{5}$ becomes $\frac{13 \times 2}{4 \times 5} = \frac{26}{20}$.

Step 3 ：Simplify the fraction $\frac{26}{20}$ by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 26 and 20 is 2. So, $\frac{26}{20}$ simplifies to $\frac{26 \div 2}{20 \div 2} = \frac{13}{10}$.

Step 4 ：Final Answer: The product of $3 \frac{1}{4}$ and $\frac{2}{5}$ is \(\boxed{\frac{13}{10}}\).