1. **State the problem:** Simplify the expression $\frac{1}{2} \div \frac{1}{4} + \frac{3}{4} + \frac{4}{5}$.\n\n2. **Recall the division rule for fractions:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $a \div b = a \times \frac{1}{b}$.\n\n3. **Apply the division rule:** $$\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1}$$ \n4. **Multiply the fractions:** $$\frac{1}{2} \times \frac{4}{1} = \frac{1 \times 4}{2 \times 1} = \frac{4}{2}$$ \n5. **Simplify the fraction by canceling common factors:** $$\frac{\cancel{4}}{\cancel{2}} = 2$$ \n6. **Rewrite the expression with the simplified division result:** $$2 + \frac{3}{4} + \frac{4}{5}$$ \n7. **Find a common denominator to add the fractions:** The denominators are 1 (for 2), 4, and 5. The least common denominator (LCD) is 20. \n8. **Convert each term to have denominator 20:** $$2 = \frac{2 \times 20}{20} = \frac{40}{20}$$ $$\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}$$ $$\frac{4}{5} = \frac{4 \times 4}{5 \times 4} = \frac{16}{20}$$ \n9. **Add the fractions:** $$\frac{40}{20} + \frac{15}{20} + \frac{16}{20} = \frac{40 + 15 + 16}{20} = \frac{71}{20}$$ \n10. **Final answer:** $$\boxed{\frac{71}{20}}$$ or as a mixed number $3 \frac{11}{20}$.