461. Найдите значение выражения: 1) (2 13/48 + 2 5/12) : 3 3/4 - 9 3/4 : 12; 2) (8 2 10/19 - 1 13/15 * 1 6/49) : (3 1/12 - 1 25/36)

Решение:

461.

1. \( \left( 2\frac{13}{48} + 2\frac{5}{12} \right) : 3\frac{3}{4} - 9\frac{3}{4} : 12 \)


2. \( 2\frac{13}{48} = \frac{2 \cdot 48 + 13}{48} = \frac{96 + 13}{48} = \frac{109}{48} \)


3. \( 2\frac{5}{12} = \frac{2 \cdot 12 + 5}{12} = \frac{24 + 5}{12} = \frac{29}{12} = \frac{29 \cdot 4}{12 \cdot 4} = \frac{116}{48} \)


4. \( \frac{109}{48} + \frac{116}{48} = \frac{109 + 116}{48} = \frac{225}{48} \)


5. \( 3\frac{3}{4} = \frac{3 \cdot 4 + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4} \)


6. \( \frac{225}{48} : \frac{15}{4} = \frac{225}{48} \cdot \frac{4}{15} = \frac{225 \cdot 4}{48 \cdot 15} = \frac{(15 \cdot 15) \cdot 4}{(12 \cdot 4) \cdot 15} = \frac{15}{12} = \frac{5}{4} \)


7. \( 9\frac{3}{4} = \frac{9 \cdot 4 + 3}{4} = \frac{36 + 3}{4} = \frac{39}{4} \)


8. \( \frac{39}{4} : 12 = \frac{39}{4 \cdot 12} = \frac{39}{48} = \frac{13}{16} \)


9. \( \frac{5}{4} - \frac{13}{16} = \frac{5 \cdot 4}{4 \cdot 4} - \frac{13}{16} = \frac{20}{16} - \frac{13}{16} = \frac{20 - 13}{16} = \frac{7}{16} \)

Ответ: \( \frac{7}{16} \)

2) \( \left( 8:2\frac{10}{19} - 1\frac{13}{15} \cdot 1\frac{6}{49} \right) : \left( 3\frac{1}{12} - 1\frac{25}{36} \right) \)

1. \( 8:2\frac{10}{19} = 8 : \frac{2 \cdot 19 + 10}{19} = 8 : \frac{38 + 10}{19} = 8 : \frac{48}{19} = 8 \cdot \frac{19}{48} = \frac{8 \cdot 19}{48} = \frac{19}{6} \)


2. \( 1\frac{13}{15} = \frac{1 \cdot 15 + 13}{15} = \frac{15 + 13}{15} = \frac{28}{15} \)


3. \( 1\frac{6}{49} = \frac{1 \cdot 49 + 6}{49} = \frac{49 + 6}{49} = \frac{55}{49} \)


4. \( \frac{28}{15} \cdot \frac{55}{49} = \frac{28 \cdot 55}{15 \cdot 49} = \frac{(4 \cdot 7) \cdot (5 \cdot 11)}{(3 \cdot 5) \cdot (7 \cdot 7)} = \frac{4 \cdot 11}{3 \cdot 7} = \frac{44}{21} \)


5. \( \frac{19}{6} - \frac{44}{21} = \frac{19 \cdot 7}{6 \cdot 7} - \frac{44 \cdot 2}{21 \cdot 2} = \frac{133}{42} - \frac{88}{42} = \frac{133 - 88}{42} = \frac{45}{42} = \frac{15}{14} \)


6. \( 3\frac{1}{12} = \frac{3 \cdot 12 + 1}{12} = \frac{36 + 1}{12} = \frac{37}{12} \)


7. \( 1\frac{25}{36} = \frac{1 \cdot 36 + 25}{36} = \frac{36 + 25}{36} = \frac{61}{36} \)


8. \( \frac{37}{12} - \frac{61}{36} = \frac{37 \cdot 3}{12 \cdot 3} - \frac{61}{36} = \frac{111}{36} - \frac{61}{36} = \frac{111 - 61}{36} = \frac{50}{36} = \frac{25}{18} \)


9. \( \frac{15}{14} : \frac{25}{18} = \frac{15}{14} \cdot \frac{18}{25} = \frac{15 \cdot 18}{14 \cdot 25} = \frac{(3 \cdot 5) \cdot (2 \cdot 9)}{(2 \cdot 7) \cdot (5 \cdot 5)} = \frac{3 \cdot 9}{7 \cdot 5} = \frac{27}{35} \)

Ответ: \( \frac{27}{35} \)