1. **State the problem:** Multiply the mixed numbers $1\frac{1}{5}$, $\frac{3}{4}$, and $2\frac{1}{2}$.\n\n2. **Convert mixed numbers to improper fractions:**\n- $1\frac{1}{5} = \frac{6}{5}$ because $1 \times 5 + 1 = 6$.\n- $2\frac{1}{2} = \frac{5}{2}$ because $2 \times 2 + 1 = 5$.\n\n3. **Write the multiplication expression:**\n$$\frac{6}{5} \times \frac{3}{4} \times \frac{5}{2}$$\n\n4. **Multiply numerators and denominators:**\nNumerator: $6 \times 3 \times 5 = 90$\nDenominator: $5 \times 4 \times 2 = 40$\n\n5. **Simplify the fraction:**\n$$\frac{90}{40} = \frac{9}{4}$$ after dividing numerator and denominator by 10.\n\n6. **Convert improper fraction to mixed number:**\n$\frac{9}{4} = 2\frac{1}{4}$ because $9 \div 4 = 2$ remainder $1$.\n\n**Final answer:** $2\frac{1}{4}$