Solve the following mathematical expressions: a) (3/7 * 33/40 + 5/6) : 1 11/12 b) (6 7/12 - 3 17/36) * 1/2 - 4 1/3 : 13/20 c) (1 11/24 + 13/36) * 1 11/25 - 8/15 * 9/16 d) 2 1/2 * 48 - 3 2/3 : 1/18 + 5 5/12 : 7/36

Решение:

а) \( \left( \frac{3}{7} \cdot \frac{33}{40} + \frac{5}{6} \right) : 1\frac{11}{12} \)

1. \( \frac{3}{7} \cdot \frac{33}{40} = \frac{99}{280} \)
2. \( \frac{99}{280} + \frac{5}{6} = \frac{99 \cdot 3 + 5 \cdot 140}{840} = \frac{297 + 700}{840} = \frac{997}{840} \)
3. \( 1\frac{11}{12} = \frac{23}{12} \)
4. \( \frac{997}{840} : \frac{23}{12} = \frac{997}{840} \cdot \frac{12}{23} = \frac{997 \cdot 12}{840 \cdot 23} = \frac{997}{70 \cdot 23} = \frac{997}{1610} \)

б) \( \left( 6\frac{7}{12} - 3\frac{17}{36} \right) \cdot \frac{1}{2} - 4\frac{1}{3} : \frac{13}{20} \)

1. \( 6\frac{7}{12} - 3\frac{17}{36} = \frac{72+7}{12} - \frac{108+17}{36} = \frac{79}{12} - \frac{125}{36} = \frac{79 \cdot 3 - 125}{36} = \frac{237 - 125}{36} = \frac{112}{36} = \frac{28}{9} \)
2. \( \frac{28}{9} \cdot \frac{1}{2} = \frac{14}{9} \)
3. \( 4\frac{1}{3} = \frac{13}{3} \)
4. \( \frac{13}{3} : \frac{13}{20} = \frac{13}{3} \cdot \frac{20}{13} = \frac{20}{3} \)
5. \( \frac{14}{9} - \frac{20}{3} = \frac{14 - 20 \cdot 3}{9} = \frac{14 - 60}{9} = -\frac{46}{9} \)

в) \( \left( 1\frac{11}{24} + \frac{13}{36} \right) \cdot 1\frac{11}{25} - \frac{8}{15} \cdot \frac{9}{16} \)

1. \( 1\frac{11}{24} + \frac{13}{36} = \frac{24+11}{24} + \frac{13}{36} = \frac{35}{24} + \frac{13}{36} = \frac{35 \cdot 3 + 13 \cdot 2}{72} = \frac{105 + 26}{72} = \frac{131}{72} \)
2. \( 1\frac{11}{25} = \frac{25+11}{25} = \frac{36}{25} \)
3. \( \frac{131}{72} \cdot \frac{36}{25} = \frac{131 \cdot 36}{72 \cdot 25} = \frac{131}{2 \cdot 25} = \frac{131}{50} \)
4. \( \frac{8}{15} \cdot \frac{9}{16} = \frac{8 \cdot 9}{15 \cdot 16} = \frac{1 \cdot 3}{5 \cdot 2} = \frac{3}{10} \)
5. \( \frac{131}{50} - \frac{3}{10} = \frac{131 - 3 \cdot 5}{50} = \frac{131 - 15}{50} = \frac{116}{50} = \frac{58}{25} \)

г) \( 2\frac{1}{2} \cdot 48 - 3\frac{2}{3} : \frac{1}{18} + 5\frac{5}{12} : \frac{7}{36} \)

1. \( 2\frac{1}{2} \cdot 48 = \frac{5}{2} \cdot 48 = 5 \cdot 24 = 120 \)
2. \( 3\frac{2}{3} = \frac{11}{3} \)
3. \( \frac{11}{3} : \frac{1}{18} = \frac{11}{3} \cdot 18 = 11 \cdot 6 = 66 \)
4. \( 5\frac{5}{12} = \frac{65}{12} \)
5. \( \frac{65}{12} : \frac{7}{36} = \frac{65}{12} \cdot \frac{36}{7} = 65 \cdot \frac{3}{7} = \frac{195}{7} \)
6. \( 120 - 66 + \frac{195}{7} = 54 + \frac{195}{7} = \frac{54 \cdot 7 + 195}{7} = \frac{378 + 195}{7} = \frac{573}{7} \)

Ответ: а) \( \frac{997}{1610} \); б) \( -\frac{46}{9} \); в) \( \frac{58}{25} \); г) \( \frac{573}{7} \).