1. **State the problem:** Subtract the mixed numbers $4 \frac{1}{6}$ and $1 \frac{6}{9}$. 2. **Convert mixed numbers to improper fractions:** $4 \frac{1}{6} = \frac{4 \times 6 + 1}{6} = \frac{25}{6}$ $1 \frac{6}{9} = \frac{1 \times 9 + 6}{9} = \frac{15}{9}$ 3. **Simplify fractions if possible:** $\frac{15}{9}$ can be simplified by dividing numerator and denominator by 3: $\frac{15}{9} = \frac{5}{3}$ 4. **Find common denominator:** Denominators are 6 and 3. The least common denominator (LCD) is 6. 5. **Convert fractions to have the same denominator:** $\frac{25}{6}$ stays the same. $\frac{5}{3} = \frac{5 \times 2}{3 \times 2} = \frac{10}{6}$ 6. **Subtract the fractions:** $\frac{25}{6} - \frac{10}{6} = \frac{25 - 10}{6} = \frac{15}{6}$ 7. **Simplify the result:** $\frac{15}{6}$ can be simplified by dividing numerator and denominator by 3: $\frac{15}{6} = \frac{5}{2}$ 8. **Convert back to mixed number:** $\frac{5}{2} = 2 \frac{1}{2}$ **Final answer:** $2 \frac{1}{2}$