Выполните действия: 1) \(1,8 + 1 \frac{2}{7}\) : \(1,8 - 1 \frac{2}{7}\); 2) \(1,5 + 1 \frac{1}{3}\) : \(7,6 - 5 \frac{1}{3}\); 3) \(23 - 15 \frac{1}{3}\) : \(-5 \frac{1}{9}\); 4) \(19 - 13 \frac{2}{3}\) : \(-7 \frac{1}{9}\).

Решение:

1)

• Переведем десятичную дробь в обыкновенную:
• \[ 1,8 = \frac{18}{10} = \frac{9}{5} \]
• \[ \frac{9}{5} + 1 \frac{2}{7} = \frac{9}{5} + \frac{9}{7} = \frac{63 + 45}{35} = \frac{108}{35} \]
• \[ \frac{9}{5} - 1 \frac{2}{7} = \frac{9}{5} - \frac{9}{7} = \frac{63 - 45}{35} = \frac{18}{35} \]
• \[ \frac{108}{35} : \frac{18}{35} = \frac{108}{35} \times \frac{35}{18} = \frac{108}{18} = 6 \]

2)

• \[ 1,5 = \frac{3}{2} \]
• \[ \frac{3}{2} + 1 \frac{1}{3} = \frac{3}{2} + \frac{4}{3} = \frac{9+8}{6} = \frac{17}{6} \]
• \[ 7,6 = \frac{76}{10} = \frac{38}{5} \]
• \[ \frac{38}{5} - 5 \frac{1}{3} = \frac{38}{5} - \frac{16}{3} = \frac{114 - 80}{15} = \frac{34}{15} \]
• \[ \frac{17}{6} : \frac{34}{15} = \frac{17}{6} \times \frac{15}{34} = \frac{17 \times 15}{6 \times 34} = \frac{1 \times 5}{2 \times 2} = \frac{5}{4} = 1,25 \]

3)

• \[ 23 - 15 \frac{1}{3} = 23 - \frac{46}{3} = \frac{69-46}{3} = \frac{23}{3} \]
• \[ -5 \frac{1}{9} = -\frac{46}{9} \]
• \[ \frac{23}{3} : \left(-\frac{46}{9}\right) = \frac{23}{3} \times \left(-\frac{9}{46}\right) = -\frac{23 \times 9}{3 \times 46} = -\frac{1 \times 3}{1 \times 2} = -\frac{3}{2} = -1,5 \]

4)

• \[ 19 - 13 \frac{2}{3} = 19 - \frac{41}{3} = \frac{57-41}{3} = \frac{16}{3} \]
• \[ -7 \frac{1}{9} = -\frac{64}{9} \]
• \[ \frac{16}{3} : \left(-\frac{64}{9}\right) = \frac{16}{3} \times \left(-\frac{9}{64}\right) = -\frac{16 \times 9}{3 \times 64} = -\frac{1 \times 3}{1 \times 4} = -\frac{3}{4} = -0,75 \]

Ответ: 1) 6; 2) 1,25; 3) -1,5; 4) -0,75.