5. Все действия с обыкновенными дробями 1).$$\left(\frac{5}{14}+\frac{17}{42}\right) \cdot 1\frac{6}{8}=$$ 2).$$\left(5\frac{8}{9}:1\frac{17}{36}+1\frac{1}{4}\right) \cdot \frac{5}{21}=$$ 3).$$\left(2\frac{1}{4}+4\frac{5}{6}\right):3\frac{2}{5}-\frac{3}{4}:3\frac{3}{5}$$ 4).$$\left(3\frac{1}{4}+3\frac{5}{6}\right):\left(5\frac{3}{4}-3\frac{2}{3}\right)$$ 5).$$\left(6\frac{4}{5}-2\frac{3}{4}\right):\left(2\frac{16}{21}+3\frac{2}{3}\right)$$ 6).$$\left(2\frac{3}{8}+3\frac{7}{24}\right):\left(4\frac{2}{3}-1\frac{1}{8}\right)$$ 7).$$\left(1\frac{3}{4}+2\frac{1}{3}\right):\left(7\frac{1}{2}-1\frac{2}{3}\right)$$ 8).$$\left(\frac{4}{5}+\frac{4}{7}\right)\cdot \left(3\frac{3}{8}+2\frac{11}{4}\right)$$ 9).$$\left(3,6-1\frac{2}{3}\right):\left(4\frac{1}{15}-2\frac{7}{9}\right)$$ 10). $$4\frac{1}{6} \cdot \left(1\frac{1}{2}-\frac{3}{5}\right)+\left(\frac{3}{4}+\frac{5}{6}\right) \cdot 6$$.

Решение:

1. $$\left(\frac{5}{14}+\frac{17}{42}\right) \cdot 1\frac{6}{8}=$$
   1. $$\frac{5}{14}+\frac{17}{42}=\frac{5 \cdot 3}{14 \cdot 3}+\frac{17}{42}=\frac{15}{42}+ \frac{17}{42}=\frac{32}{42}=\frac{16}{21}$$
   2. $$1\frac{6}{8}=\frac{1 \cdot 8+6}{8}=\frac{14}{8}=\frac{7}{4}$$
   3. $$\frac{16}{21} \cdot \frac{7}{4}=\frac{16 \cdot 7}{21 \cdot 4}=\frac{4 \cdot 1}{3 \cdot 1}=\frac{4}{3}=1\frac{1}{3}$$

   Ответ: $$1\frac{1}{3}$$

2. $$\left(5\frac{8}{9}:1\frac{17}{36}+1\frac{1}{4}\right) \cdot \frac{5}{21}=$$
   1. $$5\frac{8}{9}=\frac{5 \cdot 9 +8}{9}=\frac{45+8}{9}=\frac{53}{9}$$
   2. $$1\frac{17}{36}=\frac{1 \cdot 36 +17}{36}=\frac{36+17}{36}=\frac{53}{36}$$
   3. $$1\frac{1}{4}=\frac{1 \cdot 4 +1}{4}=\frac{4+1}{4}=\frac{5}{4}$$
   4. $$\frac{53}{9}:\frac{53}{36}=\frac{53}{9} \cdot \frac{36}{53}=\frac{1}{1} \cdot \frac{4}{1}=4$$
   5. $$4+\frac{5}{4}=\frac{4 \cdot 4}{4}+\frac{5}{4}=\frac{16}{4}+\frac{5}{4}=\frac{21}{4}$$
   6. $$\frac{21}{4} \cdot \frac{5}{21}=\frac{1}{4} \cdot \frac{5}{1}=\frac{5}{4}=1\frac{1}{4}$$

   Ответ: $$1\frac{1}{4}$$

3. $$\left(2\frac{1}{4}+4\frac{5}{6}\right):3\frac{2}{5}-\frac{3}{4}:3\frac{3}{5}=$$
   1. $$2\frac{1}{4}=\frac{2 \cdot 4 +1}{4}=\frac{8+1}{4}=\frac{9}{4}$$
   2. $$4\frac{5}{6}=\frac{4 \cdot 6 +5}{6}=\frac{24+5}{6}=\frac{29}{6}$$
   3. $$3\frac{2}{5}=\frac{3 \cdot 5 +2}{5}=\frac{15+2}{5}=\frac{17}{5}$$
   4. $$3\frac{3}{5}=\frac{3 \cdot 5 +3}{5}=\frac{15+3}{5}=\frac{18}{5}$$
   5. $$\frac{9}{4}+\frac{29}{6}=\frac{9 \cdot 3}{4 \cdot 3}+\frac{29 \cdot 2}{6 \cdot 2}=\frac{27}{12}+\frac{58}{12}=\frac{85}{12}$$
   6. $$\frac{85}{12}:\frac{17}{5}=\frac{85}{12} \cdot \frac{5}{17}=\frac{5}{12} \cdot \frac{5}{1}=\frac{25}{12}=2\frac{1}{12}$$
   7. $$\frac{3}{4}:\frac{18}{5}=\frac{3}{4} \cdot \frac{5}{18}=\frac{1}{4} \cdot \frac{5}{6}=\frac{5}{24}$$
   8. $$2\frac{1}{12}-\frac{5}{24}=2\frac{2}{24}-\frac{5}{24}=1\frac{26}{24}-\frac{5}{24}=1\frac{21}{24}=1\frac{7}{8}$$

   Ответ: $$1\frac{7}{8}$$

4. $$\left(3\frac{1}{4}+3\frac{5}{6}\right):\left(5\frac{3}{4}-3\frac{2}{3}\right)=$$
   1. $$3\frac{1}{4}=\frac{3 \cdot 4 +1}{4}=\frac{12+1}{4}=\frac{13}{4}$$
   2. $$3\frac{5}{6}=\frac{3 \cdot 6 +5}{6}=\frac{18+5}{6}=\frac{23}{6}$$
   3. $$5\frac{3}{4}=\frac{5 \cdot 4 +3}{4}=\frac{20+3}{4}=\frac{23}{4}$$
   4. $$3\frac{2}{3}=\frac{3 \cdot 3 +2}{3}=\frac{9+2}{3}=\frac{11}{3}$$
   5. $$\frac{13}{4}+\frac{23}{6}=\frac{13 \cdot 3}{4 \cdot 3}+\frac{23 \cdot 2}{6 \cdot 2}=\frac{39}{12}+\frac{46}{12}=\frac{85}{12}$$
   6. $$\frac{23}{4}-\frac{11}{3}=\frac{23 \cdot 3}{4 \cdot 3}-\frac{11 \cdot 4}{3 \cdot 4}=\frac{69}{12}-\frac{44}{12}=\frac{25}{12}$$
   7. $$\frac{85}{12}:\frac{25}{12}=\frac{85}{12} \cdot \frac{12}{25}=\frac{85}{1} \cdot \frac{1}{25}=\frac{17}{1} \cdot \frac{1}{5}=\frac{17}{5}=3\frac{2}{5}$$

   Ответ: $$3\frac{2}{5}$$

5. $$\left(6\frac{4}{5}-2\frac{3}{4}\right):\left(2\frac{16}{21}+3\frac{2}{3}\right)=$$
   1. $$6\frac{4}{5}=\frac{6 \cdot 5 +4}{5}=\frac{30+4}{5}=\frac{34}{5}$$
   2. $$2\frac{3}{4}=\frac{2 \cdot 4 +3}{4}=\frac{8+3}{4}=\frac{11}{4}$$
   3. $$2\frac{16}{21}=\frac{2 \cdot 21 +16}{21}=\frac{42+16}{21}=\frac{58}{21}$$
   4. $$3\frac{2}{3}=\frac{3 \cdot 3 +2}{3}=\frac{9+2}{3}=\frac{11}{3}$$
   5. $$\frac{34}{5}-\frac{11}{4}=\frac{34 \cdot 4}{5 \cdot 4}-\frac{11 \cdot 5}{4 \cdot 5}=\frac{136}{20}-\frac{55}{20}=\frac{81}{20}$$
   6. $$\frac{58}{21}+\frac{11}{3}=\frac{58}{21}+\frac{11 \cdot 7}{3 \cdot 7}=\frac{58}{21}+\frac{77}{21}=\frac{135}{21}=\frac{45}{7}$$
   7. $$\frac{81}{20}:\frac{45}{7}=\frac{81}{20} \cdot \frac{7}{45}=\frac{9}{20} \cdot \frac{7}{5}=\frac{63}{100}$$

   Ответ: $$\frac{63}{100}$$

6. $$\left(2\frac{3}{8}+3\frac{7}{24}\right):\left(4\frac{2}{3}-1\frac{1}{8}\right)=$$
   1. $$2\frac{3}{8}=\frac{2 \cdot 8 +3}{8}=\frac{16+3}{8}=\frac{19}{8}$$
   2. $$3\frac{7}{24}=\frac{3 \cdot 24 +7}{24}=\frac{72+7}{24}=\frac{79}{24}$$
   3. $$4\frac{2}{3}=\frac{4 \cdot 3 +2}{3}=\frac{12+2}{3}=\frac{14}{3}$$
   4. $$1\frac{1}{8}=\frac{1 \cdot 8 +1}{8}=\frac{8+1}{8}=\frac{9}{8}$$
   5. $$\frac{19}{8}+\frac{79}{24}=\frac{19 \cdot 3}{8 \cdot 3}+\frac{79}{24}=\frac{57}{24}+\frac{79}{24}=\frac{136}{24}=\frac{17}{3}$$
   6. $$\frac{14}{3}-\frac{9}{8}=\frac{14 \cdot 8}{3 \cdot 8}-\frac{9 \cdot 3}{8 \cdot 3}=\frac{112}{24}-\frac{27}{24}=\frac{85}{24}$$
   7. $$\frac{17}{3}:\frac{85}{24}=\frac{17}{3} \cdot \frac{24}{85}=\frac{1}{1} \cdot \frac{8}{5}=\frac{8}{5}=1\frac{3}{5}$$

   Ответ: $$1\frac{3}{5}$$

7. $$\left(1\frac{3}{4}+2\frac{1}{3}\right):\left(7\frac{1}{2}-1\frac{2}{3}\right)=$$
   1. $$1\frac{3}{4}=\frac{1 \cdot 4 +3}{4}=\frac{4+3}{4}=\frac{7}{4}$$
   2. $$2\frac{1}{3}=\frac{2 \cdot 3 +1}{3}=\frac{6+1}{3}=\frac{7}{3}$$
   3. $$7\frac{1}{2}=\frac{7 \cdot 2 +1}{2}=\frac{14+1}{2}=\frac{15}{2}$$
   4. $$1\frac{2}{3}=\frac{1 \cdot 3 +2}{3}=\frac{3+2}{3}=\frac{5}{3}$$
   5. $$\frac{7}{4}+\frac{7}{3}=\frac{7 \cdot 3}{4 \cdot 3}+\frac{7 \cdot 4}{3 \cdot 4}=\frac{21}{12}+\frac{28}{12}=\frac{49}{12}$$
   6. $$\frac{15}{2}-\frac{5}{3}=\frac{15 \cdot 3}{2 \cdot 3}-\frac{5 \cdot 2}{3 \cdot 2}=\frac{45}{6}-\frac{10}{6}=\frac{35}{6}$$
   7. $$\frac{49}{12}:\frac{35}{6}=\frac{49}{12} \cdot \frac{6}{35}=\frac{7}{2} \cdot \frac{1}{5}=\frac{7}{10}$$

   Ответ: $$\frac{7}{10}$$

8. $$\left(\frac{4}{5}+\frac{4}{7}\right)\cdot \left(3\frac{3}{8}+2\frac{11}{4}\right)=$$
   1. $$3\frac{3}{8}=\frac{3 \cdot 8 +3}{8}=\frac{24+3}{8}=\frac{27}{8}$$
   2. $$2\frac{11}{4}=\frac{2 \cdot 4 +11}{4}=\frac{8+11}{4}=\frac{19}{4}$$
   3. $$\frac{4}{5}+\frac{4}{7}=\frac{4 \cdot 7}{5 \cdot 7}+\frac{4 \cdot 5}{7 \cdot 5}=\frac{28}{35}+\frac{20}{35}=\frac{48}{35}$$
   4. $$\frac{27}{8}+\frac{19}{4}=\frac{27}{8}+\frac{19 \cdot 2}{4 \cdot 2}=\frac{27}{8}+\frac{38}{8}=\frac{65}{8}$$
   5. $$\frac{48}{35} \cdot \frac{65}{8}=\frac{6}{7} \cdot \frac{13}{1}=\frac{78}{7}=11\frac{1}{7}$$

   Ответ: $$11\frac{1}{7}$$

9. $$\left(3,6-1\frac{2}{3}\right):\left(4\frac{1}{15}-2\frac{7}{9}\right)=$$
   1. $$1\frac{2}{3}=\frac{1 \cdot 3 +2}{3}=\frac{3+2}{3}=\frac{5}{3}$$
   2. $$4\frac{1}{15}=\frac{4 \cdot 15 +1}{15}=\frac{60+1}{15}=\frac{61}{15}$$
   3. $$2\frac{7}{9}=\frac{2 \cdot 9 +7}{9}=\frac{18+7}{9}=\frac{25}{9}$$
   4. $$3,6-1\frac{2}{3}=3\frac{6}{10}-1\frac{2}{3}=3\frac{3}{5}-1\frac{2}{3}=3\frac{9}{15}-1\frac{10}{15}=2\frac{24}{15}-1\frac{10}{15}=1\frac{14}{15}=\frac{29}{15}$$
   5. $$4\frac{1}{15}-2\frac{7}{9}=\frac{61}{15}-\frac{25}{9}=\frac{61 \cdot 3}{15 \cdot 3}-\frac{25 \cdot 5}{9 \cdot 5}=\frac{183}{45}-\frac{125}{45}=\frac{58}{45}$$
   6. $$\frac{29}{15}:\frac{58}{45}=\frac{29}{15} \cdot \frac{45}{58}=\frac{1}{1} \cdot \frac{3}{2}=\frac{3}{2}=1\frac{1}{2}$$

   Ответ: $$1\frac{1}{2}$$

10. $$4\frac{1}{6} \cdot \left(1\frac{1}{2}-\frac{3}{5}\right)+\left(\frac{3}{4}+\frac{5}{6}\right) \cdot 6.$$
    1. $$4\frac{1}{6}=\frac{4 \cdot 6 +1}{6}=\frac{24+1}{6}=\frac{25}{6}$$
    2. $$1\frac{1}{2}=\frac{1 \cdot 2 +1}{2}=\frac{2+1}{2}=\frac{3}{2}$$
    3. $$\frac{3}{2}-\frac{3}{5}=\frac{3 \cdot 5}{2 \cdot 5}-\frac{3 \cdot 2}{5 \cdot 2}=\frac{15}{10}-\frac{6}{10}=\frac{9}{10}$$
    4. $$\frac{3}{4}+\frac{5}{6}=\frac{3 \cdot 3}{4 \cdot 3}+\frac{5 \cdot 2}{6 \cdot 2}=\frac{9}{12}+\frac{10}{12}=\frac{19}{12}$$
    5. $$\frac{25}{6} \cdot \frac{9}{10}=\frac{5}{2} \cdot \frac{3}{2}=\frac{15}{4}$$
    6. $$\frac{19}{12} \cdot 6=\frac{19}{2} \cdot 1=\frac{19}{2}$$
    7. $$\frac{15}{4}+\frac{19}{2}=\frac{15}{4}+\frac{19 \cdot 2}{2 \cdot 2}=\frac{15}{4}+\frac{38}{4}=\frac{53}{4}=13\frac{1}{4}$$

    Ответ: $$13\frac{1}{4}$$