1. **State the problem:** Simplify the expression $$\frac{1}{7} + \frac{4}{7} \div \frac{10}{11}$$ and write the answer in its simplest form. 2. **Recall the division rule for fractions:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $$\frac{4}{7} \div \frac{10}{11} = \frac{4}{7} \times \frac{11}{10}$$ 3. **Multiply the fractions:** $$\frac{4}{7} \times \frac{11}{10} = \frac{4 \times 11}{7 \times 10} = \frac{44}{70}$$ 4. **Simplify the fraction $$\frac{44}{70}$$:** Both numerator and denominator are divisible by 2. $$\frac{44}{70} = \frac{\cancel{44}^{22}}{\cancel{70}^{35}} = \frac{22}{35}$$ 5. **Rewrite the original expression with this result:** $$\frac{1}{7} + \frac{22}{35}$$ 6. **Find a common denominator:** The least common denominator of 7 and 35 is 35. Convert $$\frac{1}{7}$$ to $$\frac{5}{35}$$ because $$1 \times 5 = 5$$ and $$7 \times 5 = 35$$. 7. **Add the fractions:** $$\frac{5}{35} + \frac{22}{35} = \frac{5 + 22}{35} = \frac{27}{35}$$ 8. **Check if $$\frac{27}{35}$$ can be simplified:** 27 and 35 have no common factors other than 1, so this is the simplest form. **Final answer:** $$\frac{27}{35}$$