462. Найдите значение выражения: 1) (2 5/9 - 1 20/21) : 1 8/49 + 1 8/9 : 6; 2) (1 17/18 * 1 13/14 - 2 5/8 : 1 19/20) : (2 25/78 - 1 1/26)

Решение:

462.

1. \( \left( 2\frac{5}{9} - 1\frac{20}{21} \right) : 1\frac{8}{49} + 1\frac{8}{9} : 6 \)


2. \( 2\frac{5}{9} = \frac{2 \cdot 9 + 5}{9} = \frac{18 + 5}{9} = \frac{23}{9} \)


3. \( 1\frac{20}{21} = \frac{1 \cdot 21 + 20}{21} = \frac{21 + 20}{21} = \frac{41}{21} \)


4. \( \frac{23}{9} - \frac{41}{21} = \frac{23 \cdot 7}{9 \cdot 7} - \frac{41 \cdot 3}{21 \cdot 3} = \frac{161}{63} - \frac{123}{63} = \frac{161 - 123}{63} = \frac{38}{63} \)


5. \( 1\frac{8}{49} = \frac{1 \cdot 49 + 8}{49} = \frac{49 + 8}{49} = \frac{57}{49} \)


6. \( \frac{38}{63} : \frac{57}{49} = \frac{38}{63} \cdot \frac{49}{57} = \frac{38 \cdot 49}{63 \cdot 57} = \frac{(2 \cdot 19) \cdot (7 \cdot 7)}{(7 \cdot 9) \cdot (3 \cdot 19)} = \frac{2 \cdot 7}{9 \cdot 3} = \frac{14}{27} \)


7. \( 1\frac{8}{9} = \frac{1 \cdot 9 + 8}{9} = \frac{9 + 8}{9} = \frac{17}{9} \)


8. \( \frac{17}{9} : 6 = \frac{17}{9 \cdot 6} = \frac{17}{54} \)


9. \( \frac{14}{27} + \frac{17}{54} = \frac{14 \cdot 2}{27 \cdot 2} + \frac{17}{54} = \frac{28}{54} + \frac{17}{54} = \frac{28 + 17}{54} = \frac{45}{54} = \frac{5}{6} \)

Ответ: \( \frac{5}{6} \)

2) \( \left( 1\frac{17}{18} \cdot 1\frac{13}{14} - 2\frac{5}{8} : 1\frac{19}{20} \right) : \left( 2\frac{25}{78} - 1\frac{1}{26} \right) \)

1. \( 1\frac{17}{18} = \frac{1 \cdot 18 + 17}{18} = \frac{18 + 17}{18} = \frac{35}{18} \)


2. \( 1\frac{13}{14} = \frac{1 \cdot 14 + 13}{14} = \frac{14 + 13}{14} = \frac{27}{14} \)


3. \( \frac{35}{18} \cdot \frac{27}{14} = \frac{35 \cdot 27}{18 \cdot 14} = \frac{(5 \cdot 7) \cdot (3 \cdot 9)}{(2 \cdot 9) \cdot (2 \cdot 7)} = \frac{5 \cdot 3}{2 \cdot 2} = \frac{15}{4} \)


4. \( 2\frac{5}{8} = \frac{2 \cdot 8 + 5}{8} = \frac{16 + 5}{8} = \frac{21}{8} \)


5. \( 1\frac{19}{20} = \frac{1 \cdot 20 + 19}{20} = \frac{20 + 19}{20} = \frac{39}{20} \)


6. \( \frac{21}{8} : \frac{39}{20} = \frac{21}{8} \cdot \frac{20}{39} = \frac{21 \cdot 20}{8 \cdot 39} = \frac{(3 \cdot 7) \cdot (4 \cdot 5)}{(2 \cdot 4) \cdot (3 \cdot 13)} = \frac{7 \cdot 5}{2 \cdot 13} = \frac{35}{26} \)


7. \( \frac{15}{4} - \frac{35}{26} = \frac{15 \cdot 13}{4 \cdot 13} - \frac{35 \cdot 2}{26 \cdot 2} = \frac{195}{52} - \frac{70}{52} = \frac{195 - 70}{52} = \frac{125}{52} \)


8. \( 2\frac{25}{78} = \frac{2 \cdot 78 + 25}{78} = \frac{156 + 25}{78} = \frac{181}{78} \)


9. \( 1\frac{1}{26} = \frac{1 \cdot 26 + 1}{26} = \frac{26 + 1}{26} = \frac{27}{26} \)


10. \( \frac{181}{78} - \frac{27}{26} = \frac{181}{78} - \frac{27 \cdot 3}{26 \cdot 3} = \frac{181}{78} - \frac{81}{78} = \frac{181 - 81}{78} = \frac{100}{78} = \frac{50}{39} \)


11. \( \frac{125}{52} : \frac{50}{39} = \frac{125}{52} \cdot \frac{39}{50} = \frac{125 \cdot 39}{52 \cdot 50} = \frac{(25 \cdot 5) \cdot (3 \cdot 13)}{(4 \cdot 13) \cdot (25 \cdot 2)} = \frac{5 \cdot 3}{4 \cdot 2} = \frac{15}{8} \)

Ответ: \( \frac{15}{8} \)