1. The problem is to divide the mixed number $1 \frac{5}{6}$ by the fraction $\frac{5}{6}$.\n2. First, convert the mixed number to an improper fraction.\n\n$$1 \frac{5}{6} = \frac{(1 \times 6) + 5}{6} = \frac{6 + 5}{6} = \frac{11}{6}$$\n\n3. Division of fractions is done by multiplying the first fraction by the reciprocal of the second fraction.\n\nFormula: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$\n\n4. Apply the formula: $$\frac{11}{6} \div \frac{5}{6} = \frac{11}{6} \times \frac{6}{5}$$\n\n5. Multiply the numerators and denominators: $$\frac{11 \times 6}{6 \times 5} = \frac{66}{30}$$\n\n6. Simplify the fraction by canceling common factors: $$\frac{\cancel{66}^{11 \times \cancel{6}}}{\cancel{30}^{5 \times \cancel{6}}} = \frac{11}{5}$$\n\n7. The simplified answer is $$\frac{11}{5}$$ or as a mixed number $2 \frac{1}{5}$.