5.523 Найдите частное: a) \(\frac{4}{5} : \frac{9}{11}\); в) \(\frac{2}{7} : \frac{2}{9}\); д) \(\frac{4}{7} : \frac{16}{49}\); б) \(\frac{1}{7} : \frac{5}{8}\); г) \(\frac{5}{8} : \frac{7}{12}\); е) \(\frac{5}{9} : \frac{1}{2}\); ж) \(\frac{12}{25} : \frac{8}{15}\); з) \(\frac{9}{14} : \frac{18}{35}\)

Решение:

a) \(\frac{4}{5} : \frac{9}{11} = \frac{4}{5} \cdot \frac{11}{9} = \frac{4 \cdot 11}{5 \cdot 9} = \frac{44}{45}\)

б) \(\frac{1}{7} : \frac{5}{8} = \frac{1}{7} \cdot \frac{8}{5} = \frac{1 \cdot 8}{7 \cdot 5} = \frac{8}{35}\)

в) \(\frac{2}{7} : \frac{2}{9} = \frac{2}{7} \cdot \frac{9}{2} = \frac{2 \cdot 9}{7 \cdot 2} = \frac{9}{7} = 1\frac{2}{7}\)

г) \(\frac{5}{8} : \frac{7}{12} = \frac{5}{8} \cdot \frac{12}{7} = \frac{5 \cdot 12}{8 \cdot 7} = \frac{5 \cdot 3 \cdot 4}{2 \cdot 4 \cdot 7} = \frac{15}{14} = 1\frac{1}{14}\)

д) \(\frac{4}{7} : \frac{16}{49} = \frac{4}{7} \cdot \frac{49}{16} = \frac{4 \cdot 49}{7 \cdot 16} = \frac{4 \cdot 7 \cdot 7}{7 \cdot 4 \cdot 4} = \frac{7}{4} = 1\frac{3}{4}\)

е) \(\frac{5}{9} : \frac{1}{2} = \frac{5}{9} \cdot \frac{2}{1} = \frac{5 \cdot 2}{9 \cdot 1} = \frac{10}{9} = 1\frac{1}{9}\)

ж) \(\frac{12}{25} : \frac{8}{15} = \frac{12}{25} \cdot \frac{15}{8} = \frac{12 \cdot 15}{25 \cdot 8} = \frac{3 \cdot 4 \cdot 3 \cdot 5}{5 \cdot 5 \cdot 2 \cdot 4} = \frac{9}{10}\)

з) \(\frac{9}{14} : \frac{18}{35} = \frac{9}{14} \cdot \frac{35}{18} = \frac{9 \cdot 35}{14 \cdot 18} = \frac{9 \cdot 5 \cdot 7}{2 \cdot 7 \cdot 2 \cdot 9} = \frac{5}{4} = 1\frac{1}{4}\)

Ответ: a) \(\frac{44}{45}\); б) \(\frac{8}{35}\); в) \(1\frac{2}{7}\); г) \(1\frac{1}{14}\); д) \(1\frac{3}{4}\); е) \(1\frac{1}{9}\); ж) \(\frac{9}{10}\); з) \(1\frac{1}{4}\)