П.17 Найдите значение выражения: a) \( \frac{12}{17} : ( -\frac{1}{7} ) + 5,88 : (-14,7) - 0,1 \) б) \( (8 - 5\frac{3}{4}) \cdot 2\frac{2}{3} + (8 - 6\frac{1}{4}) : 1\frac{3}{4} \) в) \( 5,5 - 3\frac{3}{4} \cdot (1\frac{2}{3} + 1\frac{2}{5}) : 2\frac{9}{25} \) г) \( 2\frac{1}{5} : 1\frac{1}{5} - 4\frac{2}{15} : 1\frac{1}{12} \) д) \( \frac{5}{8} - \frac{5}{14} \cdot ( -\frac{7}{10} + \frac{1}{2}) \) e) \( 2\frac{7}{10} : (3\frac{1}{2} - 1\frac{13}{15}) : 3\frac{3}{5} \)

Решение:

а) \( \frac{12}{17} : ( -\frac{1}{7} ) + 5,88 : (-14,7) - 0,1 \)

1. \( \frac{12}{17} : ( -\frac{1}{7} ) = \frac{12}{17} \cdot (-7) = -\frac{84}{17} \)
2. \( 5,88 : (-14,7) = -0,4 \)
3. \( -\frac{84}{17} - 0,4 - 0,1 = -\frac{84}{17} - 0,5 = -\frac{84}{17} - \frac{8.5}{17} = -\frac{92.5}{17} \approx -5,44 \)

б) \( (8 - 5\frac{3}{4}) \cdot 2\frac{2}{3} + (8 - 6\frac{1}{4}) : 1\frac{3}{4} \)

1. \( 8 - 5\frac{3}{4} = 8 - \frac{23}{4} = \frac{32 - 23}{4} = \frac{9}{4} \)
2. \( \frac{9}{4} \cdot 2\frac{2}{3} = \frac{9}{4} \cdot \frac{8}{3} = \frac{72}{12} = 6 \)
3. \( 8 - 6\frac{1}{4} = 8 - \frac{25}{4} = \frac{32 - 25}{4} = \frac{7}{4} \)
4. \( \frac{7}{4} : 1\frac{3}{4} = \frac{7}{4} : \frac{7}{4} = 1 \)
5. \( 6 + 1 = 7 \)

в) \( 5,5 - 3\frac{3}{4} \cdot (1\frac{2}{3} + 1\frac{2}{5}) : 2\frac{9}{25} \)

1. \( 1\frac{2}{3} + 1\frac{2}{5} = \frac{5}{3} + \frac{7}{5} = \frac{25 + 21}{15} = \frac{46}{15} \)
2. \( 3\frac{3}{4} \cdot \frac{46}{15} = \frac{15}{4} \cdot \frac{46}{15} = \frac{46}{4} = \frac{23}{2} = 11,5 \)
3. \( 11,5 : 2\frac{9}{25} = \frac{23}{2} : \frac{59}{25} = \frac{23}{2} \cdot \frac{25}{59} = \frac{575}{118} \approx 4,87 \)
4. \( 5,5 - 4,87 = 0,63 \)

г) \( 2\frac{1}{5} : 1\frac{1}{5} - 4\frac{2}{15} : 1\frac{1}{12} \)

1. \( 2\frac{1}{5} : 1\frac{1}{5} = \frac{11}{5} : \frac{6}{5} = \frac{11}{5} \cdot \frac{5}{6} = \frac{11}{6} \)
2. \( 4\frac{2}{15} : 1\frac{1}{12} = \frac{62}{15} : \frac{13}{12} = \frac{62}{15} \cdot \frac{12}{13} = \frac{62 \cdot 4}{5 \cdot 13} = \frac{248}{65} \)
3. \( \frac{11}{6} - \frac{248}{65} = \frac{11 \cdot 65 - 248 \cdot 6}{6 \cdot 65} = \frac{715 - 1488}{390} = -\frac{773}{390} \approx -1,98 \)

д) \( \frac{5}{8} - \frac{5}{14} \cdot ( -\frac{7}{10} + \frac{1}{2}) \)

1. \( -\frac{7}{10} + \frac{1}{2} = \frac{-7 + 5}{10} = -\frac{2}{10} = -\frac{1}{5} \)
2. \( \frac{5}{14} \cdot ( -\frac{1}{5} ) = -\frac{1}{14} \)
3. \( \frac{5}{8} - ( -\frac{1}{14} ) = \frac{5}{8} + \frac{1}{14} = \frac{5 \cdot 7 + 1 \cdot 4}{56} = \frac{35 + 4}{56} = \frac{39}{56} \)

е) \( 2\frac{7}{10} : (3\frac{1}{2} - 1\frac{13}{15}) : 3\frac{3}{5} \)

1. \( 3\frac{1}{2} - 1\frac{13}{15} = \frac{7}{2} - \frac{28}{15} = \frac{7 \cdot 15 - 28 \cdot 2}{30} = \frac{105 - 56}{30} = \frac{49}{30} \)
2. \( 2\frac{7}{10} : \frac{49}{30} = \frac{27}{10} : \frac{49}{30} = \frac{27}{10} \cdot \frac{30}{49} = \frac{27 \cdot 3}{49} = \frac{81}{49} \)
3. \( \frac{81}{49} : 3\frac{3}{5} = \frac{81}{49} : \frac{18}{5} = \frac{81}{49} \cdot \frac{5}{18} = \frac{9 \cdot 5}{49 \cdot 2} = \frac{45}{98} \)

Ответ: а) \(-\frac{92.5}{17}\) или ≈ -5,44; б) 7; в) ≈ 0,63; г) \(-\frac{773}{390}\) или ≈ -1,98; д) \(\frac{39}{56}\); е) \(\frac{45}{98}\).