Решение:

a) $$4 \frac{7}{9} + 5 \frac{6}{39} + 4 \frac{4}{78} = 4 \frac{7}{9} + 5 \frac{2}{13} + 4 \frac{2}{39} = 4 \frac{7}{9} + 5 \frac{6}{39} + 4 \frac{2}{39} = 4 \frac{7}{9} + 5 \frac{8}{39} = \frac{4 \cdot 9 + 7}{9} + \frac{5 \cdot 39 + 8}{39} = \frac{43}{9} + \frac{203}{39} = \frac{43 \cdot 13}{9 \cdot 13} + \frac{203 \cdot 3}{39 \cdot 3} = \frac{559}{117} + \frac{609}{117} = \frac{1168}{117} = 9 \frac{115}{117}$$

б) $$7 \frac{13}{15} - 3 \frac{3}{30} = 7 \frac{13}{15} - 3 \frac{1}{10} = \frac{7 \cdot 15 + 13}{15} - \frac{3 \cdot 10 + 1}{10} = \frac{118}{15} - \frac{31}{10} = \frac{118 \cdot 2}{15 \cdot 2} - \frac{31 \cdot 3}{10 \cdot 3} = \frac{236}{30} - \frac{93}{30} = \frac{143}{30} = 4 \frac{23}{30}$$

в) $$\frac{6}{7} : (\frac{9}{8} - \frac{7}{6}) = \frac{6}{7} : (\frac{9 \cdot 3}{8 \cdot 3} - \frac{7 \cdot 4}{6 \cdot 4}) = \frac{6}{7} : (\frac{27}{24} - \frac{28}{24}) = \frac{6}{7} : (\frac{-1}{24}) = \frac{6}{7} \cdot (-24) = -\frac{144}{7} = -20 \frac{4}{7}$$

г) $$7 \frac{4}{8} : \frac{35}{9} = 7 \frac{1}{2} : \frac{35}{9} = \frac{7 \cdot 2 + 1}{2} : \frac{35}{9} = \frac{15}{2} : \frac{35}{9} = \frac{15}{2} \cdot \frac{9}{35} = \frac{3}{2} \cdot \frac{9}{7} = \frac{27}{14} = 1 \frac{13}{14}$$

е) $$(1 \frac{3}{4} - \frac{4}{9}) : \frac{5}{3} = (\frac{1 \cdot 4 + 3}{4} - \frac{4}{9}) : \frac{5}{3} = (\frac{7}{4} - \frac{4}{9}) : \frac{5}{3} = (\frac{7 \cdot 9}{4 \cdot 9} - \frac{4 \cdot 4}{9 \cdot 4}) : \frac{5}{3} = (\frac{63}{36} - \frac{16}{36}) : \frac{5}{3} = \frac{47}{36} : \frac{5}{3} = \frac{47}{36} \cdot \frac{3}{5} = \frac{47}{12} \cdot \frac{1}{5} = \frac{47}{60}$$

Ответ: а) $$9 \frac{115}{117}$$; б) $$4 \frac{23}{30}$$; в) $$-20 \frac{4}{7}$$; г) $$1 \frac{13}{14}$$; е) $$\frac{47}{60}$$