1. Действия с обыкновенными дробями. Найти значение выражений:

Решение:

1. \( \frac{7}{8} : \frac{5}{6} \cdot \frac{9}{20} = \frac{7}{8} \cdot \frac{6}{5} \cdot \frac{9}{20} = \frac{7 \cdot 6 \cdot 9}{8 \cdot 5 \cdot 20} = \frac{378}{800} = \frac{189}{400} \)
2. \( \frac{5}{8} : \frac{9}{14} \cdot \frac{7}{12} = \frac{5}{8} \cdot \frac{14}{9} \cdot \frac{7}{12} = \frac{5 \cdot 14 \cdot 7}{8 \cdot 9 \cdot 12} = \frac{490}{864} = \frac{245}{432} \)
3. \( \frac{8}{9} : \frac{12}{15} \cdot \frac{5}{8} = \frac{8}{9} \cdot \frac{15}{12} \cdot \frac{5}{8} = \frac{8 \cdot 15 \cdot 5}{9 \cdot 12 \cdot 8} = \frac{600}{864} = \frac{25}{36} \)
4. \( \frac{10}{21} : \frac{4}{15} \cdot \frac{5}{14} = \frac{10}{21} \cdot \frac{15}{4} \cdot \frac{5}{14} = \frac{10 \cdot 15 \cdot 5}{21 \cdot 4 \cdot 14} = \frac{750}{1176} = \frac{125}{196} \)
5. \( \frac{2}{9} + \frac{4}{7} \cdot \frac{9}{28} = \frac{2}{9} + \frac{4 \cdot 9}{7 \cdot 28} = \frac{2}{9} + \frac{36}{196} = \frac{2}{9} + \frac{9}{49} = \frac{2 \cdot 49 + 9 \cdot 9}{9 \cdot 49} = \frac{98 + 81}{441} = \frac{179}{441} \)
6. \( \frac{9}{11} + \frac{3}{5} : \frac{11}{40} = \frac{9}{11} + \frac{3}{5} \cdot \frac{40}{11} = \frac{9}{11} + \frac{120}{55} = \frac{9}{11} + \frac{24}{11} = \frac{33}{11} = 3 \)
7. \( \frac{3}{7} + \frac{5}{8} \cdot \frac{7}{40} = \frac{3}{7} + \frac{5 \cdot 7}{8 \cdot 40} = \frac{3}{7} + \frac{35}{320} = \frac{3}{7} + \frac{7}{64} = \frac{3 \cdot 64 + 7 \cdot 7}{7 \cdot 64} = \frac{192 + 49}{448} = \frac{241}{448} \)
8. \( \frac{3}{5} + \frac{2}{3} \cdot \frac{5}{33} = \frac{3}{5} + \frac{2 \cdot 5}{3 \cdot 33} = \frac{3}{5} + \frac{10}{99} = \frac{3 \cdot 99 + 10 \cdot 5}{5 \cdot 99} = \frac{297 + 50}{495} = \frac{347}{495} \)
9. \( 2 \frac{5}{8} + \frac{3}{4} : \frac{1}{2} = \frac{21}{8} + \frac{3}{4} \cdot 2 = \frac{21}{8} + \frac{3}{2} = \frac{21 + 12}{8} = \frac{33}{8} \)
10. \( \frac{1}{4} : \frac{6}{2} + \frac{1}{3} = \frac{1}{4} \cdot \frac{2}{6} + \frac{1}{3} = \frac{2}{24} + \frac{1}{3} = \frac{1}{12} + \frac{1}{3} = \frac{1 + 4}{12} = \frac{5}{12} \)
11. \( 2 \frac{1}{16} + \frac{3}{4} : \frac{1}{4} = \frac{33}{16} + \frac{3}{4} \cdot 4 = \frac{33}{16} + 3 = \frac{33 + 48}{16} = \frac{81}{16} \)
12. \( 3 \frac{11}{15} + \frac{4}{5} : \frac{1}{3} = \frac{56}{15} + \frac{4}{5} \cdot 3 = \frac{56}{15} + \frac{12}{5} = \frac{56 + 36}{15} = \frac{92}{15} \)
13. \( 1 - \frac{3}{8} \cdot \frac{3}{5} = 1 - \frac{9}{40} = \frac{40 - 9}{40} = \frac{31}{40} \)
14. \( 1 - \frac{4}{5} \cdot \frac{2}{9} = 1 - \frac{8}{45} = \frac{45 - 8}{45} = \frac{37}{45} \)
15. \( 1 - \frac{3}{7} \cdot \frac{4}{5} = 1 - \frac{12}{35} = \frac{35 - 12}{35} = \frac{23}{35} \)
16. \( 1 - \frac{2}{4} \cdot \frac{23}{7} = 1 - \frac{1}{2} \cdot \frac{23}{7} = 1 - \frac{23}{14} = \frac{14 - 23}{14} = -\frac{9}{14} \)
17. \( 1 \frac{1}{4} - \frac{2}{7} \cdot \frac{23}{28} = \frac{5}{4} - \frac{46}{196} = \frac{5}{4} - \frac{23}{98} = \frac{5 \cdot 49 - 23 \cdot 2}{196} = \frac{245 - 46}{196} = \frac{199}{196} \)
18. \( 1 \frac{2}{5} - \frac{3}{8} \cdot \frac{39}{40} = \frac{7}{5} - \frac{117}{320} = \frac{7 \cdot 64 - 117}{320} = \frac{448 - 117}{320} = \frac{331}{320} \)
19. \( \frac{2}{11} + \frac{37}{55} = \frac{10 + 37}{55} = \frac{47}{55} \)
20. \( \frac{1}{7} + \frac{4}{11} \cdot \frac{60}{77} = \frac{1}{7} + \frac{240}{847} = \frac{121 + 240}{847} = \frac{361}{847} \)
21. \( \frac{25}{4} - \frac{6}{5} \cdot \frac{15}{8} = \frac{25}{4} - \frac{90}{40} = \frac{25}{4} - \frac{9}{4} = \frac{16}{4} = 4 \)
22. \( \frac{24}{5} : \frac{17}{15} - \frac{1}{12} \cdot \frac{20}{3} = \frac{24}{5} \cdot \frac{15}{17} - \frac{20}{36} = \frac{360}{85} - \frac{5}{9} = \frac{72}{17} - \frac{5}{9} = \frac{648 - 85}{153} = \frac{563}{153} \)
23. \( \frac{7}{2} : \frac{7}{8} \cdot \left( \frac{7}{8} - \frac{3}{4} \right) = \frac{7}{2} \cdot \frac{8}{7} \cdot \left( \frac{7-6}{8} \right) = 4 \cdot \frac{1}{8} = \frac{1}{2} \)
24. \( \frac{2}{9} + \frac{4}{21} \cdot \frac{3}{7} = \frac{2}{9} + \frac{12}{147} = \frac{2}{9} + \frac{4}{49} = \frac{98 + 36}{441} = \frac{134}{441} \)
25. \( \frac{15}{7} + \frac{9}{7} \cdot \frac{2}{3} = \frac{15}{7} + \frac{18}{21} = \frac{15}{7} + \frac{6}{7} = \frac{21}{7} = 3 \)
26. \( \frac{45}{8} : \frac{6}{5} \cdot \frac{9}{4} = \frac{45}{8} \cdot \frac{5}{6} \cdot \frac{9}{4} = \frac{2025}{192} = \frac{675}{64} \)
27. \( \frac{6}{5} \left( \frac{9}{10} - \frac{1}{5} \right) = \frac{6}{5} \left( \frac{9-2}{10} \right) = \frac{6}{5} \cdot \frac{7}{10} = \frac{42}{50} = \frac{21}{25} \)
28. \( \frac{22}{3} : \frac{11}{48} \cdot \frac{5}{8} = \frac{22}{3} \cdot \frac{48}{11} \cdot \frac{5}{8} = 2 \cdot 16 \cdot \frac{5}{8} = 32 \cdot \frac{5}{8} = 4 \cdot 5 = 20 \)
29. \( \frac{49}{6} : \frac{12}{7} \cdot \frac{7}{4} = \frac{49}{6} \cdot \frac{7}{12} \cdot \frac{7}{4} = \frac{343}{288} \)
30. \( \frac{1}{3} + \frac{7}{4} \cdot \frac{3}{2} = \frac{1}{3} + \frac{21}{8} = \frac{8 + 63}{24} = \frac{71}{24} \)
31. \( \frac{33}{4} : \frac{8}{3} \cdot \frac{11}{6} = \frac{33}{4} \cdot \frac{3}{8} \cdot \frac{11}{6} = \frac{1089}{192} = \frac{363}{64} \)
32. \( \frac{45}{7} : \frac{15}{56} \cdot \frac{3}{8} = \frac{45}{7} \cdot \frac{56}{15} \cdot \frac{3}{8} = 3 \cdot 8 \cdot \frac{3}{8} = 24 \cdot \frac{3}{8} = 3 \cdot 3 = 9 \)
33. \( 1 \frac{2}{7} - \frac{1}{2} \cdot \frac{28}{1} = \frac{9}{7} - 14 = \frac{9 - 98}{7} = -\frac{89}{7} \)
34. \( \frac{22}{15} + \frac{4}{5} : \frac{3}{2} = \frac{22}{15} + \frac{4}{5} \cdot \frac{2}{3} = \frac{22}{15} + \frac{8}{15} = \frac{30}{15} = 2 \)
35. \( \frac{7}{8} \left( \frac{3}{4} - \frac{1}{2} \right) = \frac{7}{8} \left( \frac{3-2}{4} \right) = \frac{7}{8} \cdot \frac{1}{4} = \frac{7}{32} \)
36. \( \frac{11}{9} - \frac{7}{2} \cdot \frac{5}{18} = \frac{11}{9} - \frac{35}{36} = \frac{44 - 35}{36} = \frac{9}{36} = \frac{1}{4} \)
37. \( \frac{9}{14} : \frac{5}{2} - \frac{4}{35} = \frac{9}{14} \cdot \frac{2}{5} - \frac{4}{35} = \frac{18}{70} - \frac{4}{35} = \frac{9}{35} - \frac{4}{35} = \frac{5}{35} = \frac{1}{7} \)
38. \( \frac{13}{6} : \frac{3}{2} - \frac{1}{9} = \frac{13}{6} \cdot \frac{2}{3} - \frac{1}{9} = \frac{26}{18} - \frac{1}{9} = \frac{13}{9} - \frac{1}{9} = \frac{12}{9} = \frac{4}{3} \)
39. \( \frac{32}{7} : \frac{16}{35} \cdot \frac{5}{5} = \frac{32}{7} \cdot \frac{35}{16} \cdot 1 = 2 \cdot 5 = 10 \)
40. \( \frac{9}{8} : \frac{5}{2} - \frac{3}{20} = \frac{9}{8} \cdot \frac{2}{5} - \frac{3}{20} = \frac{18}{40} - \frac{3}{20} = \frac{9}{20} - \frac{3}{20} = \frac{6}{20} = \frac{3}{10} \)
41. \( \frac{26}{9} + \frac{8}{9} \cdot \frac{5}{4} = \frac{26}{9} + \frac{40}{36} = \frac{26}{9} + \frac{10}{9} = \frac{36}{9} = 4 \)
42. \( \frac{5}{3} - \frac{5}{4} \cdot \frac{1}{12} = \frac{5}{3} - \frac{5}{48} = \frac{80 - 5}{48} = \frac{75}{48} = \frac{25}{16} \)
43. \( \frac{21}{8} : \frac{4}{3} \cdot \frac{7}{6} = \frac{21}{8} \cdot \frac{3}{4} \cdot \frac{7}{6} = \frac{441}{192} = \frac{147}{64} \)
44. \( \frac{19}{7} : \frac{38}{63} \cdot \frac{8}{9} = \frac{19}{7} \cdot \frac{63}{38} \cdot \frac{8}{9} = \frac{1}{1} \cdot \frac{9}{2} \cdot \frac{8}{9} = \frac{8}{2} = 4 \)
45. \( \left( \frac{13}{4} - \frac{11}{10} \right) : \frac{43}{80} = \left( \frac{65 - 22}{20} \right) : \frac{43}{80} = \frac{43}{20} \cdot \frac{80}{43} = \frac{80}{20} = 4 \)
46. \( \frac{5}{6} - \frac{7}{12} : \frac{7}{2} = \frac{5}{6} - \frac{7}{12} \cdot \frac{2}{7} = \frac{5}{6} - \frac{2}{12} = \frac{5}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3} \)