1. **State the problem:** Add the mixed fractions $3 \frac{4}{9}$ and $2 \frac{2}{4}$.\n\n2. **Convert mixed fractions to improper fractions:**\n- $3 \frac{4}{9} = \frac{3 \times 9 + 4}{9} = \frac{27 + 4}{9} = \frac{31}{9}$\n- $2 \frac{2}{4} = \frac{2 \times 4 + 2}{4} = \frac{8 + 2}{4} = \frac{10}{4}$\n\n3. **Find a common denominator:**\n- The denominators are 9 and 4. The least common denominator (LCD) is $\mathrm{lcm}(9,4) = 36$.\n\n4. **Convert fractions to have the common denominator:**\n- $\frac{31}{9} = \frac{31 \times 4}{9 \times 4} = \frac{124}{36}$\n- $\frac{10}{4} = \frac{10 \times 9}{4 \times 9} = \frac{90}{36}$\n\n5. **Add the fractions:**\n$$\frac{124}{36} + \frac{90}{36} = \frac{124 + 90}{36} = \frac{214}{36}$$\n\n6. **Simplify the fraction:**\n- Find the greatest common divisor (GCD) of 214 and 36, which is 2.\n- Divide numerator and denominator by 2:\n$$\frac{214 \div 2}{36 \div 2} = \frac{107}{18}$$\n\n7. **Convert back to a mixed number:**\n- Divide 107 by 18: $107 \div 18 = 5$ remainder $17$.\n- So, $\frac{107}{18} = 5 \frac{17}{18}$.\n\n**Final answer:** $3 \frac{4}{9} + 2 \frac{2}{4} = 5 \frac{17}{18}$