1. **State the problem:** Find the sum of $1 \frac{1}{2}$, $\frac{1}{4}$, and $\frac{2}{5}$.\n\n2. **Convert mixed number to improper fraction:** $1 \frac{1}{2} = \frac{3}{2}$.\n\n3. **Find a common denominator:** The denominators are 2, 4, and 5. The least common denominator (LCD) is 20.\n\n4. **Convert each fraction to have denominator 20:**\n$\frac{3}{2} = \frac{3 \times 10}{2 \times 10} = \frac{30}{20}$\n$\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}$\n$\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}$\n\n5. **Add the fractions:**\n$$\frac{30}{20} + \frac{5}{20} + \frac{8}{20} = \frac{30 + 5 + 8}{20} = \frac{43}{20}$$\n\n6. **Convert the improper fraction to a mixed number:**\n$$\frac{43}{20} = 2 \frac{3}{20}$$\n\n**Final answer:** The sum is $2 \frac{3}{20}$.