1. Stating the problem: Calculate $3 \frac{5}{6} - 1 \frac{7}{12}$. 2. Convert mixed numbers to improper fractions: $$3 \frac{5}{6} = \frac{3 \times 6 + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6}$$ $$1 \frac{7}{12} = \frac{1 \times 12 + 7}{12} = \frac{12 + 7}{12} = \frac{19}{12}$$ 3. Find a common denominator to subtract the fractions. The denominators are 6 and 12, so the least common denominator (LCD) is 12. 4. Convert $\frac{23}{6}$ to twelfths: $$\frac{23}{6} = \frac{23 \times 2}{6 \times 2} = \frac{46}{12}$$ 5. Now subtract the fractions: $$\frac{46}{12} - \frac{19}{12} = \frac{46 - 19}{12} = \frac{27}{12}$$ 6. Simplify the fraction $\frac{27}{12}$ by dividing numerator and denominator by their greatest common divisor (GCD), which is 3: $$\frac{\cancel{27}^{9}}{\cancel{12}^{4}} = \frac{9}{4}$$ 7. Convert the improper fraction back to a mixed number: $$\frac{9}{4} = 2 \frac{1}{4}$$ Final answer: $2 \frac{1}{4}$.