1. **State the problem:** Add the mixed numbers $1 \frac{8}{11}$ and $7 \frac{2}{11}$.\n\n2. **Recall the rule for adding mixed numbers with the same denominator:** Add the whole numbers and add the fractions separately. If the fraction sum is improper, convert it to a mixed number.\n\n3. **Add the whole numbers:** $1 + 7 = 8$.\n\n4. **Add the fractions:** $\frac{8}{11} + \frac{2}{11} = \frac{8+2}{11} = \frac{10}{11}$.\n\n5. **Combine the sums:** $8 + \frac{10}{11} = 8 \frac{10}{11}$.\n\n6. **Check if the fraction can be simplified:** $\frac{10}{11}$ is already in lowest terms since 10 and 11 have no common factors other than 1.\n\n**Final answer:** $8 \frac{10}{11}$