1. Problem 1: Simplify $ \frac{2+4}{3+\frac{5}{3-\frac{1}{2}}} $. - Evaluate inner denominator: $3-\frac{1}{2} = \frac{6}{2}-\frac{1}{2} = \frac{5}{2}$. - Then $3 + \frac{5}{3-\frac{1}{2}} = 3 + \frac{5}{\frac{5}{2}} = 3 + 2 = 5$. - Numerator $2+4 = 6$. - So expression is $\frac{6}{5}$. 2. Problem 2: Simplify $\frac{4}{7} \div \frac{6}{5}$. - Division of fractions: $\frac{4}{7} \times \frac{5}{6} = \frac{20}{42} = \frac{10}{21}$ after simplifying. 3. Problem 3: Simplify $\frac{2+4+2-\frac{1}{4}}{3\div 7}$. - Calculate numerator: $2+4+2 = 8$, then $8 - \frac{1}{4} = \frac{32}{4} - \frac{1}{4} = \frac{31}{4}$. - Denominator: $3 \div 7 = \frac{3}{7}$. - So expression becomes $\frac{31}{4} \div \frac{3}{7} = \frac{31}{4} \times \frac{7}{3} = \frac{217}{12}$. 4. Problem 4: Simplify $(\frac{2}{7} + \frac{5}{3}) \div (\frac{2}{7} - \frac{1}{4})$. - Compute numerator: $\frac{2}{7} + \frac{5}{3} = \frac{6}{21} + \frac{35}{21} = \frac{41}{21}$. - Compute denominator: $\frac{2}{7} - \frac{1}{4} = \frac{8}{28} - \frac{7}{28} = \frac{1}{28}$. - Division: $\frac{41}{21} \div \frac{1}{28} = \frac{41}{21} \times 28 = \frac{41 \times 28}{21} = \frac{1148}{21}$. 5. Problem 5: Simplify $\frac{5}{3} \div \left(\frac{5}{3}, \frac{2}{7}\right) + \frac{1}{3} \times \frac{15}{20} - \frac{1}{12}$. - Here, colon between $\frac{5}{3}$ and $\frac{2}{7}$ suggests division $\frac{5}{3} \div \frac{2}{7} = \frac{5}{3} \times \frac{7}{2} = \frac{35}{6}$. - So first term: $\frac{5}{3} \div \frac{35}{6} = \frac{5}{3} \times \frac{6}{35} = \frac{30}{105} = \frac{2}{7}$. - Second term: $\frac{1}{3} \times \frac{15}{20} = \frac{1}{3} \times \frac{3}{4} = \frac{1}{4}$. - Sum: $\frac{2}{7} + \frac{1}{4} - \frac{1}{12} = \frac{24}{84} + \frac{21}{84} - \frac{7}{84} = \frac{38}{84} = \frac{19}{42}$. Final answers: 1) $\frac{6}{5}$ 2) $\frac{10}{21}$ 3) $\frac{217}{12}$ 4) $\frac{1148}{21}$ 5) $\frac{19}{42}$