1) \(\frac{1}{2}+(5 \frac{1}{6}-3 \frac{3}{4}+\frac{1}{2}) \cdot \frac{10}{23}\) 2) \(2 \frac{1}{2} \cdot 48 - 3 \frac{2}{3} : \frac{1}{18} + 5 \frac{5}{12} : \frac{4}{36}\) 3) \((1 \frac{8}{13} \cdot \frac{13}{42} + 5 \frac{5}{7} : \frac{8}{21}) : (8 \frac{1}{8} + 3 \frac{1}{2})\)

\[5 \frac{1}{6}-3 \frac{3}{4}+\frac{1}{2} = \frac{31}{6} - \frac{15}{4} + \frac{1}{2} = \frac{31 \cdot 2}{6 \cdot 2} - \frac{15 \cdot 3}{4 \cdot 3} + \frac{1 \cdot 6}{2 \cdot 6} = \frac{62}{12} - \frac{45}{12} + \frac{6}{12} = \frac{62 - 45 + 6}{12} = \frac{23}{12}\]

\[\frac{23}{12} \cdot \frac{10}{23} = \frac{23 \cdot 10}{12 \cdot 23} = \frac{10}{12} = \frac{5}{6}\]

\[\frac{1}{2} + \frac{5}{6} = \frac{1 \cdot 3}{2 \cdot 3} + \frac{5}{6} = \frac{3}{6} + \frac{5}{6} = \frac{8}{6} = \frac{4}{3} = 1 \frac{1}{3}\]

\[2 \frac{1}{2} \cdot 48 = \frac{5}{2} \cdot 48 = \frac{5 \cdot 48}{2} = 5 \cdot 24 = 120\]

\[3 \frac{2}{3} : \frac{1}{18} = \frac{11}{3} : \frac{1}{18} = \frac{11}{3} \cdot 18 = \frac{11 \cdot 18}{3} = 11 \cdot 6 = 66\]

\[5 \frac{5}{12} : \frac{4}{36} = \frac{65}{12} : \frac{4}{36} = \frac{65}{12} \cdot \frac{36}{4} = \frac{65 \cdot 36}{12 \cdot 4} = \frac{65 \cdot 3}{4} = \frac{195}{4} = 48 \frac{3}{4}\]

\[120 - 66 + 48 \frac{3}{4} = 54 + 48 \frac{3}{4} = 102 \frac{3}{4}\]

\[1 \frac{8}{13} \cdot \frac{13}{42} = \frac{21}{13} \cdot \frac{13}{42} = \frac{21 \cdot 13}{13 \cdot 42} = \frac{21}{42} = \frac{1}{2}\]

\[5 \frac{5}{7} : \frac{8}{21} = \frac{40}{7} : \frac{8}{21} = \frac{40}{7} \cdot \frac{21}{8} = \frac{40 \cdot 21}{7 \cdot 8} = \frac{5 \cdot 3}{1} = 15\]

\[\frac{1}{2} + 15 = 15 \frac{1}{2} = \frac{31}{2}\]

\[8 \frac{1}{8} + 3 \frac{1}{2} = \frac{65}{8} + \frac{7}{2} = \frac{65}{8} + \frac{7 \cdot 4}{2 \cdot 4} = \frac{65}{8} + \frac{28}{8} = \frac{93}{8}\]

\[\frac{31}{2} : \frac{93}{8} = \frac{31}{2} \cdot \frac{8}{93} = \frac{31 \cdot 8}{2 \cdot 93} = \frac{31 \cdot 4}{93} = \frac{4}{3} = 1 \frac{1}{3}\]