2.276 Выполните действия: a) \((\frac{4}{9} + \frac{5}{12}) \cdot \frac{18}{31}\); в) \((4 - 3\frac{7}{15}) \cdot \frac{5}{8}\); д) \((1\frac{1}{24} - \frac{5}{12}) \cdot (4\frac{3}{8} - 3\frac{5}{24})\); б) \(\frac{6}{25} \cdot (\frac{11}{15} - \frac{9}{20})\); г) \((5 - 4\frac{4}{7}) \cdot (7\frac{1}{6} - 6\frac{5}{12})\); e) \((1\frac{2}{15} - \frac{11}{15}) \cdot (5\frac{3}{18} - 4\frac{1}{27})\)

a) \((\frac{4}{9} + \frac{5}{12}) \cdot \frac{18}{31}\)

\(\frac{4}{9} + \frac{5}{12} = \frac{4 \cdot 4 + 5 \cdot 3}{36} = \frac{16 + 15}{36} = \frac{31}{36}\)

\(\frac{31}{36} \cdot \frac{18}{31} = \frac{31 \cdot 18}{36 \cdot 31} = \frac{18}{36} = \frac{1}{2}\)

в) \((4 - 3\frac{7}{15}) \cdot \frac{5}{8}\)

\(4 - 3\frac{7}{15} = 4 - \frac{52}{15} = \frac{60 - 52}{15} = \frac{8}{15}\)

\(\frac{8}{15} \cdot \frac{5}{8} = \frac{8 \cdot 5}{15 \cdot 8} = \frac{5}{15} = \frac{1}{3}\)

д) \((1\frac{1}{24} - \frac{5}{12}) \cdot (4\frac{3}{8} - 3\frac{5}{24})\)

\(1\frac{1}{24} - \frac{5}{12} = \frac{25}{24} - \frac{10}{24} = \frac{15}{24} = \frac{5}{8}\)

\(4\frac{3}{8} - 3\frac{5}{24} = \frac{35}{8} - \frac{77}{24} = \frac{105 - 77}{24} = \frac{28}{24} = \frac{7}{6}\)

\(\frac{5}{8} \cdot \frac{7}{6} = \frac{5 \cdot 7}{8 \cdot 6} = \frac{35}{48}\)

б) \(\frac{6}{25} \cdot (\frac{11}{15} - \frac{9}{20})\)

\(\frac{11}{15} - \frac{9}{20} = \frac{11 \cdot 4 - 9 \cdot 3}{60} = \frac{44 - 27}{60} = \frac{17}{60}\)

\(\frac{6}{25} \cdot \frac{17}{60} = \frac{6 \cdot 17}{25 \cdot 60} = \frac{17}{25 \cdot 10} = \frac{17}{250}\)

г) \((5 - 4\frac{4}{7}) \cdot (7\frac{1}{6} - 6\frac{5}{12})\)

\(5 - 4\frac{4}{7} = \frac{35}{7} - \frac{32}{7} = \frac{3}{7}\)

\(7\frac{1}{6} - 6\frac{5}{12} = \frac{43}{6} - \frac{77}{12} = \frac{86 - 77}{12} = \frac{9}{12} = \frac{3}{4}\)

\(\frac{3}{7} \cdot \frac{3}{4} = \frac{3 \cdot 3}{7 \cdot 4} = \frac{9}{28}\)

e) \((1\frac{2}{15} - \frac{11}{15}) \cdot (5\frac{3}{18} - 4\frac{1}{27})\)

\(1\frac{2}{15} - \frac{11}{15} = \frac{17}{15} - \frac{11}{15} = \frac{6}{15} = \frac{2}{5}\)

\(5\frac{3}{18} - 4\frac{1}{27} = 5\frac{1}{6} - 4\frac{1}{27} = \frac{31}{6} - \frac{109}{27} = \frac{31 \cdot 9 - 109 \cdot 2}{54} = \frac{279 - 218}{54} = \frac{61}{54}\)

\(\frac{2}{5} \cdot \frac{61}{54} = \frac{2 \cdot 61}{5 \cdot 54} = \frac{61}{5 \cdot 27} = \frac{61}{135}\)

Ответ: a) \(\frac{1}{2}\); б) \(\frac{17}{250}\); в) \(\frac{1}{3}\); г) \(\frac{9}{28}\); д) \(\frac{35}{48}\); e) \(\frac{61}{135}\)