1) Calculate the value of the expression: (9 - 5 3/8) * [4 5/12 : 4 2/3 + 3/10 : (1/2 : 4)] / (1/24 + 1/4 : 13 1/3)

Решение:

1. Вычисление выражения в скобках:

• Первая скобка:\[ 9 - 5\frac{3}{8} = 9 - \frac{43}{8} = \frac{72 - 43}{8} = \frac{29}{8} \]
• Вторая скобка:\[ 4\frac{5}{12} : 4\frac{2}{3} + \frac{3}{10} : \left(\frac{1}{2} : 4\right) \]
• \[ 4\frac{5}{12} = \frac{53}{12} \]
• \[ 4\frac{2}{3} = \frac{14}{3} \]
• \[ \frac{53}{12} : \frac{14}{3} = \frac{53}{12} \times \frac{3}{14} = \frac{53}{4 \times 14} = \frac{53}{56} \]
• \[ \frac{1}{2} : 4 = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8} \]
• \[ \frac{3}{10} : \frac{1}{8} = \frac{3}{10} \times 8 = \frac{24}{10} = \frac{12}{5} \]
• \[ \frac{53}{56} + \frac{12}{5} = \frac{53 \times 5 + 12 \times 56}{56 \times 5} = \frac{265 + 672}{280} = \frac{937}{280} \]
• Знаменатель дроби:\[ \frac{1}{24} + \frac{1}{4} : 13\frac{1}{3} \]
• \[ 13\frac{1}{3} = \frac{40}{3} \]
• \[ \frac{1}{4} : \frac{40}{3} = \frac{1}{4} \times \frac{3}{40} = \frac{3}{160} \]
• \[ \frac{1}{24} + \frac{3}{160} = \frac{20 \times 1 + 3 \times 3}{480} = \frac{20 + 9}{480} = \frac{29}{480} \]
• Деление числителя на знаменатель:\[ \frac{29}{8} \times \frac{937}{280} : \frac{29}{480} = \frac{29}{8} \times \frac{937}{280} \times \frac{480}{29} \]
• \[ \frac{1}{8} \times \frac{937}{280} \times 480 = \frac{937 \times 480}{8 \times 280} = \frac{937 \times 60}{280} = \frac{937 \times 6}{28} = \frac{937 \times 3}{14} \]
• \[ \frac{2811}{14} \]

Ответ: \(\frac{2811}{14}\)