1. **State the problem:** Simplify the expression $\left(4 \frac{1}{15} - 3 \frac{9}{10}\right) \times 6 \frac{6}{7} + 2$. 2. **Convert mixed numbers to improper fractions:** - $4 \frac{1}{15} = \frac{4 \times 15 + 1}{15} = \frac{61}{15}$ - $3 \frac{9}{10} = \frac{3 \times 10 + 9}{10} = \frac{39}{10}$ - $6 \frac{6}{7} = \frac{6 \times 7 + 6}{7} = \frac{48}{7}$ 3. **Subtract the fractions inside the parentheses:** $$\frac{61}{15} - \frac{39}{10}$$ Find common denominator: $\mathrm{lcm}(15,10) = 30$ $$\frac{61}{15} = \frac{61 \times 2}{30} = \frac{122}{30}$$ $$\frac{39}{10} = \frac{39 \times 3}{30} = \frac{117}{30}$$ Subtract: $$\frac{122}{30} - \frac{117}{30} = \frac{5}{30} = \frac{1}{6}$$ 4. **Multiply the result by $6 \frac{6}{7} = \frac{48}{7}$:** $$\frac{1}{6} \times \frac{48}{7} = \frac{48}{42} = \frac{8}{7}$$ 5. **Add 2 to the product:** Convert 2 to fraction with denominator 7: $$2 = \frac{14}{7}$$ Add: $$\frac{8}{7} + \frac{14}{7} = \frac{22}{7}$$ 6. **Convert the improper fraction to a mixed number:** $$\frac{22}{7} = 3 \frac{1}{7}$$ **Final answer:** $3 \frac{1}{7}$