1. **State the problem:** We need to divide the mixed number $5\frac{2}{3}$ by the mixed number $-1\frac{2}{15}$. 2. **Convert mixed numbers to improper fractions:** $5\frac{2}{3} = \frac{5 \times 3 + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3}$ $-1\frac{2}{15} = -\frac{1 \times 15 + 2}{15} = -\frac{15 + 2}{15} = -\frac{17}{15}$ 3. **Division of fractions rule:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $$\frac{17}{3} \div -\frac{17}{15} = \frac{17}{3} \times -\frac{15}{17}$$ 4. **Multiply the fractions:** $$\frac{17}{3} \times -\frac{15}{17} = \frac{17 \times -15}{3 \times 17}$$ 5. **Cancel common factors:** $$\frac{\cancel{17} \times -15}{3 \times \cancel{17}} = \frac{-15}{3}$$ 6. **Simplify the fraction:** $$\frac{-15}{3} = -5$$ **Final answer:** $-5$