Вычислите 6 2\(\frac{2}{7}\) \(\times\) 2.4 \(\times\) 1\(\frac{5}{9}\) \(\times\) 1\(\frac{9}{16}\) / 3\(\frac{1}{3}\) \(\times\) 1.125 \(\times\) 1\(\frac{5}{7}\) \(\times\) 1\(\frac{7}{9}\)

Решение:

Запишем смешанные числа в виде неправильных дробей:

• \( 2\frac{2}{7} = \frac{2 \times 7 + 2}{7} = \frac{16}{7} \)
• \( 1\frac{5}{9} = \frac{1 \times 9 + 5}{9} = \frac{14}{9} \)
• \( 1\frac{9}{16} = \frac{1 \times 16 + 9}{16} = \frac{25}{16} \)
• \( 3\frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3} \)
• \( 1\frac{5}{7} = \frac{1 \times 7 + 5}{7} = \frac{12}{7} \)
• \( 1\frac{7}{9} = \frac{1 \times 9 + 7}{9} = \frac{16}{9} \)

Запишем десятичные дроби в виде обыкновенных:

• \( 2.4 = \frac{24}{10} = \frac{12}{5} \)
• \( 1.125 = \frac{1125}{1000} = \frac{9}{8} \)

Подставим полученные дроби в исходное выражение:

\[ \frac{\frac{16}{7} \times \frac{12}{5} \times \frac{14}{9} \times \frac{25}{16}}{\frac{10}{3} \times \frac{9}{8} \times \frac{12}{7} \times \frac{16}{9}} \]\[ = \frac{\frac{16 \times 12 \times 14 \times 25}{7 \times 5 \times 9 \times 16}}{\frac{10 \times 9 \times 12 \times 16}{3 \times 8 \times 7 \times 9}} \]\[ = \frac{16 \times 12 \times 14 \times 25}{7 \times 5 \times 9 \times 16} \times \frac{3 \times 8 \times 7 \times 9}{10 \times 9 \times 12 \times 16} \]\[ = \frac{16 \times 12 \times 14 \times 25 \times 3 \times 8 \times 7 \times 9}{7 \times 5 \times 9 \times 16 \times 10 \times 9 \times 12 \times 16} \]\[ = \frac{25 \times 3 \times 8 \times 7}{5 \times 10 \times 7 \times 16} \]\[ = \frac{25 \times 3 \times 8}{5 \times 10 \times 16} \]\[ = \frac{5 \times 3 \times 8}{10 \times 16} \]\[ = \frac{5 \times 3 \times 1}{10 \times 2} \]\[ = \frac{15}{20} \]\[ = \frac{3}{4} \]

Ответ: 6 \(\frac{3}{4}\).