445. Найдите значение выражения: a) (\frac{5}{12} + \frac{3}{8}) \cdot \frac{12}{19}; б) \frac{6}{7} \cdot (\frac{11}{18} - \frac{5}{12}); в) (3 \frac{1}{14} - 2 \frac{5}{7}) \cdot (7 - 6 \frac{3}{5}); г) (3 \frac{1}{12} - 2 \frac{3}{4}) \cdot (1 \frac{1}{6} - \frac{5}{12}); д) (6 \frac{7}{12} - 5 \frac{11}{15}) \cdot (1 \frac{3}{17} - \frac{10}{17}); е) \frac{2}{3} \cdot \frac{9}{16} - \frac{5}{24} \cdot \frac{2}{5} - \frac{1}{6}.

Ответ:

a)

\[(\frac{5}{12} + \frac{3}{8}) \cdot \frac{12}{19} = (\frac{5 \cdot 2}{12 \cdot 2} + \frac{3 \cdot 3}{8 \cdot 3}) \cdot \frac{12}{19} = (\frac{10}{24} + \frac{9}{24}) \cdot \frac{12}{19} = \frac{19}{24} \cdot \frac{12}{19} = \frac{1}{2} \cdot \frac{1}{1} = \frac{1}{2}\]

б)

\[\frac{6}{7} \cdot (\frac{11}{18} - \frac{5}{12}) = \frac{6}{7} \cdot (\frac{11 \cdot 2}{18 \cdot 2} - \frac{5 \cdot 3}{12 \cdot 3}) = \frac{6}{7} \cdot (\frac{22}{36} - \frac{15}{36}) = \frac{6}{7} \cdot \frac{7}{36} = \frac{1}{1} \cdot \frac{1}{6} = \frac{1}{6}\]

в)

\[(3 \frac{1}{14} - 2 \frac{5}{7}) \cdot (7 - 6 \frac{3}{5}) = (\frac{3 \cdot 14 + 1}{14} - \frac{2 \cdot 7 + 5}{7}) \cdot (\frac{7 \cdot 5}{5} - \frac{6 \cdot 5 + 3}{5}) = (\frac{43}{14} - \frac{19}{7}) \cdot (\frac{35}{5} - \frac{33}{5}) = (\frac{43}{14} - \frac{19 \cdot 2}{7 \cdot 2}) \cdot \frac{2}{5} = (\frac{43}{14} - \frac{38}{14}) \cdot \frac{2}{5} = \frac{5}{14} \cdot \frac{2}{5} = \frac{1}{7} \cdot \frac{1}{1} = \frac{1}{7}\]

г)

\[(3 \frac{1}{12} - 2 \frac{3}{4}) \cdot (1 \frac{1}{6} - \frac{5}{12}) = (\frac{3 \cdot 12 + 1}{12} - \frac{2 \cdot 4 + 3}{4}) \cdot (\frac{1 \cdot 6 + 1}{6} - \frac{5}{12}) = (\frac{37}{12} - \frac{11}{4}) \cdot (\frac{7}{6} - \frac{5}{12}) = (\frac{37}{12} - \frac{11 \cdot 3}{4 \cdot 3}) \cdot (\frac{7 \cdot 2}{6 \cdot 2} - \frac{5}{12}) = (\frac{37}{12} - \frac{33}{12}) \cdot (\frac{14}{12} - \frac{5}{12}) = \frac{4}{12} \cdot \frac{9}{12} = \frac{1}{3} \cdot \frac{3}{4} = \frac{1}{4}\]

д)

\[(6 \frac{7}{12} - 5 \frac{11}{15}) \cdot (1 \frac{3}{17} - \frac{10}{17}) = (\frac{6 \cdot 12 + 7}{12} - \frac{5 \cdot 15 + 11}{15}) \cdot (\frac{1 \cdot 17 + 3}{17} - \frac{10}{17}) = (\frac{79}{12} - \frac{86}{15}) \cdot (\frac{20}{17} - \frac{10}{17}) = (\frac{79 \cdot 5}{12 \cdot 5} - \frac{86 \cdot 4}{15 \cdot 4}) \cdot \frac{10}{17} = (\frac{395}{60} - \frac{344}{60}) \cdot \frac{10}{17} = \frac{51}{60} \cdot \frac{10}{17} = \frac{3}{6} \cdot \frac{1}{1} = \frac{1}{2}\]

е)

\[\frac{2}{3} \cdot \frac{9}{16} - \frac{5}{24} \cdot \frac{2}{5} - \frac{1}{6} = \frac{1}{1} \cdot \frac{3}{8} - \frac{1}{12} \cdot \frac{1}{1} - \frac{1}{6} = \frac{3}{8} - \frac{1}{12} - \frac{1}{6} = \frac{3 \cdot 3}{8 \cdot 3} - \frac{1 \cdot 2}{12 \cdot 2} - \frac{1 \cdot 4}{6 \cdot 4} = \frac{9}{24} - \frac{2}{24} - \frac{4}{24} = \frac{3}{24} = \frac{1}{8}\]

Ответ: