Вычислите $\frac{5}{6} - \frac{3}{22}$

1. $$ \frac{5}{6} - \frac{3}{22} = \frac{5 \cdot 11}{6 \cdot 11} - \frac{3 \cdot 3}{22 \cdot 3} = \frac{55}{66} - \frac{9}{66} = \frac{55 - 9}{66} = \frac{46}{66} = \frac{23}{33} $$ **Ответ:** $\frac{23}{33}$ 2. $$ 0.75 - \left( \frac{3}{16} + \frac{1}{24} \right) $$ $$ = \frac{3}{4} - \left( \frac{3 \cdot 3}{16 \cdot 3} + \frac{1 \cdot 2}{24 \cdot 2} \right) $$ $$ = \frac{3}{4} - \left( \frac{9}{48} + \frac{2}{48} \right) $$ $$ = \frac{3}{4} - \frac{11}{48} $$ $$ = \frac{3 \cdot 12}{4 \cdot 12} - \frac{11}{48} $$ $$ = \frac{36}{48} - \frac{11}{48} = \frac{25}{48} $$ **Ответ:** $\frac{25}{48}$ 3. $$ \frac{9}{16} + \left( 0.25 - \frac{5}{24} \right) $$ $$ = \frac{9}{16} + \left( \frac{1}{4} - \frac{5}{24} \right) $$ $$ = \frac{9}{16} + \left( \frac{1 \cdot 6}{4 \cdot 6} - \frac{5}{24} \right) $$ $$ = \frac{9}{16} + \left( \frac{6}{24} - \frac{5}{24} \right) $$ $$ = \frac{9}{16} + \frac{1}{24} $$ $$ = \frac{9 \cdot 3}{16 \cdot 3} + \frac{1 \cdot 2}{24 \cdot 2} $$ $$ = \frac{27}{48} + \frac{2}{48} = \frac{29}{48} $$ **Ответ:** $\frac{29}{48}$ 4. $$ 0.88 - \frac{2}{15} $$ $$ = \frac{88}{100} - \frac{2}{15} $$ $$ = \frac{22}{25} - \frac{2}{15} $$ $$ = \frac{22 \cdot 3}{25 \cdot 3} - \frac{2 \cdot 5}{15 \cdot 5} $$ $$ = \frac{66}{75} - \frac{10}{75} = \frac{56}{75} $$ **Ответ:** $\frac{56}{75}$ 5. $$ \left( \frac{1}{6} + \frac{5}{18} \right) - \left( \frac{1}{18} + \frac{1}{24} \right) $$ $$ = \left( \frac{1 \cdot 3}{6 \cdot 3} + \frac{5}{18} \right) - \left( \frac{1 \cdot 4}{18 \cdot 4} + \frac{1 \cdot 3}{24 \cdot 3} \right) $$ $$ = \left( \frac{3}{18} + \frac{5}{18} \right) - \left( \frac{4}{72} + \frac{3}{72} \right) $$ $$ = \frac{8}{18} - \frac{7}{72} $$ $$ = \frac{4}{9} - \frac{7}{72} $$ $$ = \frac{4 \cdot 8}{9 \cdot 8} - \frac{7}{72} $$ $$ = \frac{32}{72} - \frac{7}{72} = \frac{25}{72} $$ **Ответ:** $\frac{25}{72}$ 6. $$ 4 - 2\frac{4}{7} $$ $$ = 4 - \frac{2 \cdot 7 + 4}{7} $$ $$ = 4 - \frac{18}{7} $$ $$ = \frac{4 \cdot 7}{7} - \frac{18}{7} $$ $$ = \frac{28 - 18}{7} = \frac{10}{7} = 1\frac{3}{7} $$ **Ответ:** $1\frac{3}{7}$ 7. $$ 2.5 + 1\frac{2}{3} - 3\frac{1}{6} $$ $$ = \frac{5}{2} + \frac{5}{3} - \frac{19}{6} $$ $$ = \frac{5 \cdot 3}{2 \cdot 3} + \frac{5 \cdot 2}{3 \cdot 2} - \frac{19}{6} $$ $$ = \frac{15}{6} + \frac{10}{6} - \frac{19}{6} $$ $$ = \frac{15 + 10 - 19}{6} = \frac{25 - 19}{6} = \frac{6}{6} = 1 $$ **Ответ:** 1 8. $$ 0.36 - \frac{9}{75} + \frac{2}{15} $$ $$ = \frac{36}{100} - \frac{9}{75} + \frac{2}{15} $$ $$ = \frac{9}{25} - \frac{3 \cdot 3}{25 \cdot 3} + \frac{2 \cdot 5}{15 \cdot 5} $$ $$ = \frac{9}{25} - \frac{3}{25} + \frac{10}{75} $$ $$ = \frac{6}{25} + \frac{10}{75} $$ $$ = \frac{6 \cdot 3}{25 \cdot 3} + \frac{10}{75} $$ $$ = \frac{18}{75} + \frac{10}{75} = \frac{28}{75} $$ **Ответ:** $\frac{28}{75}$ 9. $$ \left( \frac{1}{2} - \frac{1}{3} \right) - \left( \frac{1}{4} - \frac{1}{5} \right) $$ $$ = \left( \frac{3}{6} - \frac{2}{6} \right) - \left( \frac{5}{20} - \frac{4}{20} \right) $$ $$ = \frac{1}{6} - \frac{1}{20} $$ $$ = \frac{1 \cdot 10}{6 \cdot 10} - \frac{1 \cdot 3}{20 \cdot 3} $$ $$ = \frac{10}{60} - \frac{3}{60} = \frac{7}{60} $$ **Ответ:** $\frac{7}{60}$ 10. $$ 3\frac{2}{9} + 4\frac{1}{6} - 1\frac{8}{27} $$ $$ = \frac{29}{9} + \frac{25}{6} - \frac{35}{27} $$ $$ = \frac{29 \cdot 6}{9 \cdot 6} + \frac{25 \cdot 9}{6 \cdot 9} - \frac{35 \cdot 2}{27 \cdot 2} \quad \text{ (общий знаменатель 54)} $$ $$ = \frac{174}{54} + \frac{225}{54} - \frac{70}{54} $$ $$ = \frac{174 + 225 - 70}{54} = \frac{399 - 70}{54} = \frac{329}{54} = 6\frac{5}{54} $$ **Ответ:** $6\frac{5}{54}$ 11. $$ 11 - 6\frac{2}{3} - 3\frac{5}{7} $$ $$ = 11 - \frac{20}{3} - \frac{26}{7} $$ $$ = \frac{11 \cdot 21}{21} - \frac{20 \cdot 7}{3 \cdot 7} - \frac{26 \cdot 3}{7 \cdot 3} $$ $$ = \frac{231}{21} - \frac{140}{21} - \frac{78}{21} $$ $$ = \frac{231 - 140 - 78}{21} = \frac{91 - 78}{21} = \frac{13}{21} $$ **Ответ:** $\frac{13}{21}$ 12. $$ \frac{27}{75} + 0.32 - \frac{7}{15} $$ $$ = \frac{9}{25} + \frac{32}{100} - \frac{7}{15} $$ $$ = \frac{9}{25} + \frac{8}{25} - \frac{7}{15} $$ $$ = \frac{17}{25} - \frac{7}{15} $$ $$ = \frac{17 \cdot 3}{25 \cdot 3} - \frac{7 \cdot 5}{15 \cdot 5} $$ $$ = \frac{51}{75} - \frac{35}{75} = \frac{16}{75} $$ **Ответ:** $\frac{16}{75}$ 13. $$ \left( \frac{1}{3} - \frac{1}{4} \right) - \left( \frac{1}{5} - \frac{1}{6} \right) $$ $$ = \left( \frac{4}{12} - \frac{3}{12} \right) - \left( \frac{6}{30} - \frac{5}{30} \right) $$ $$ = \frac{1}{12} - \frac{1}{30} $$ $$ = \frac{1 \cdot 5}{12 \cdot 5} - \frac{1 \cdot 2}{30 \cdot 2} $$ $$ = \frac{5}{60} - \frac{2}{60} = \frac{3}{60} = \frac{1}{20} $$ **Ответ:** $\frac{1}{20}$ 14. $$ 5\frac{7}{12} - 2\frac{4}{9} + 1\frac{5}{18} $$ $$ = \frac{67}{12} - \frac{22}{9} + \frac{23}{18} $$ $$ = \frac{67 \cdot 3}{12 \cdot 3} - \frac{22 \cdot 4}{9 \cdot 4} + \frac{23 \cdot 2}{18 \cdot 2} \quad \text{ (общий знаменатель 36)} $$ $$ = \frac{201}{36} - \frac{88}{36} + \frac{46}{36} $$ $$ = \frac{201 - 88 + 46}{36} = \frac{113 + 46}{36} = \frac{159}{36} = 4\frac{15}{36} = 4\frac{5}{12} $$ **Ответ:** $4\frac{5}{12}$ 15. $$ 10 - 3\frac{4}{5} - 5\frac{2}{3} $$ $$ = 10 - \frac{19}{5} - \frac{17}{3} $$ $$ = \frac{10 \cdot 15}{15} - \frac{19 \cdot 3}{5 \cdot 3} - \frac{17 \cdot 5}{3 \cdot 5} $$ $$ = \frac{150}{15} - \frac{57}{15} - \frac{85}{15} $$ $$ = \frac{150 - 57 - 85}{15} = \frac{93 - 85}{15} = \frac{8}{15} $$ **Ответ:** $\frac{8}{15}$ 16. $$ \frac{3}{4} \cdot \frac{8}{9} = \frac{3 \cdot 8}{4 \cdot 9} = \frac{24}{36} = \frac{2}{3} $$ **Ответ:** $\frac{2}{3}$ 17. $$ 1\frac{7}{8} \cdot 0.4 $$ $$ = \frac{15}{8} \cdot \frac{4}{10} $$ $$ = \frac{15}{8} \cdot \frac{2}{5} $$ $$ = \frac{15 \cdot 2}{8 \cdot 5} = \frac{30}{40} = \frac{3}{4} $$ **Ответ:** $\frac{3}{4}$ 18. $$ 1\frac{1}{3} \cdot \frac{3}{7} + \frac{4}{7} \cdot 1\frac{1}{3} $$ $$ = 1\frac{1}{3} \cdot \left( \frac{3}{7} + \frac{4}{7} \right) $$ $$ = 1\frac{1}{3} \cdot \frac{7}{7} $$ $$ = 1\frac{1}{3} \cdot 1 = 1\frac{1}{3} = \frac{4}{3} $$ **Ответ:** $1\frac{1}{3}$ или $\frac{4}{3}$ 19. $$ \frac{5}{7} \cdot \frac{21}{25} = \frac{5 \cdot 21}{7 \cdot 25} = \frac{1 \cdot 3}{1 \cdot 5} = \frac{3}{5} $$ **Ответ:** $\frac{3}{5}$ 20. $$ 2\frac{1}{12} \cdot 0.8 $$ $$ = \frac{25}{12} \cdot \frac{8}{10} $$ $$ = \frac{25}{12} \cdot \frac{4}{5} $$ $$ = \frac{25 \cdot 4}{12 \cdot 5} = \frac{5 \cdot 1}{3 \cdot 1} = \frac{5}{3} = 1\frac{2}{3} $$ **Ответ:** $1\frac{2}{3}$ или $\frac{5}{3}$ 21. $$ 2\frac{1}{4} \cdot \frac{7}{9} + \frac{2}{9} \cdot 2\frac{1}{4} $$ $$ = 2\frac{1}{4} \cdot \left( \frac{7}{9} + \frac{2}{9} \right) $$ $$ = 2\frac{1}{4} \cdot \frac{9}{9} $$ $$ = 2\frac{1}{4} \cdot 1 = 2\frac{1}{4} = \frac{9}{4} $$ **Ответ:** $2\frac{1}{4}$ или $\frac{9}{4}$