1. **State the problem:** We need to add the mixed numbers $7 \frac{7}{10}$ and $3 \frac{5}{6}$. 2. **Convert mixed numbers to improper fractions:** $7 \frac{7}{10} = \frac{7 \times 10 + 7}{10} = \frac{70 + 7}{10} = \frac{77}{10}$ $3 \frac{5}{6} = \frac{3 \times 6 + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6}$ 3. **Find a common denominator:** The denominators are 10 and 6. The least common denominator (LCD) is 30. 4. **Convert fractions to have the LCD:** $\frac{77}{10} = \frac{77 \times 3}{10 \times 3} = \frac{231}{30}$ $\frac{23}{6} = \frac{23 \times 5}{6 \times 5} = \frac{115}{30}$ 5. **Add the fractions:** $$\frac{231}{30} + \frac{115}{30} = \frac{231 + 115}{30} = \frac{346}{30}$$ 6. **Simplify the fraction:** Divide numerator and denominator by 2: $$\frac{\cancel{346}^{173}}{\cancel{30}^{15}} = \frac{173}{15}$$ 7. **Convert back to a mixed number:** $173 \div 15 = 11$ remainder $8$, so $$\frac{173}{15} = 11 \frac{8}{15}$$ **Final answer:** $7 \frac{7}{10} + 3 \frac{5}{6} = 11 \frac{8}{15}$