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Решение:

• \( 4 \div \frac{8}{9} = 4 \cdot \frac{9}{8} = \frac{4 \cdot 9}{8} = \frac{36}{8} = \frac{9}{2} = 4 \frac{1}{2} \)
• \( 1 \frac{1}{4} \div \frac{3}{8} = \frac{1 \cdot 4 + 1}{4} \div \frac{3}{8} = \frac{5}{4} \div \frac{3}{8} = \frac{5}{4} \cdot \frac{8}{3} = \frac{5 \cdot 8}{4 \cdot 3} = \frac{5 \cdot (2 \cdot 4)}{4 \cdot 3} = \frac{5 \cdot 2}{3} = \frac{10}{3} = 3 \frac{1}{3} \)
• \( \frac{14}{15} \div \frac{4}{35} = \frac{14}{15} \cdot \frac{35}{4} = \frac{14 \cdot 35}{15 \cdot 4} = \frac{(2 \cdot 7) \cdot (5 \cdot 7)}{(3 \cdot 5) \cdot (2 \cdot 2)} = \frac{7 \cdot 7}{3 \cdot 2} = \frac{49}{6} = 8 \frac{1}{6} \)
• \( 1 \frac{1}{3} \div \frac{7}{10} = \frac{1 \cdot 3 + 1}{3} \div \frac{7}{10} = \frac{4}{3} \div \frac{7}{10} = \frac{4}{3} \cdot \frac{10}{7} = \frac{4 \cdot 10}{3 \cdot 7} = \frac{40}{21} = 1 \frac{19}{21} \)
• \( \frac{12}{21} \div 6 = \frac{12}{21} \cdot \frac{1}{6} = \frac{12}{21 \cdot 6} = \frac{2 \cdot 6}{21 \cdot 6} = \frac{2}{21} \)
• \( \frac{5}{16} \div 3 \frac{1}{8} = \frac{5}{16} \div \frac{3 \cdot 8 + 1}{8} = \frac{5}{16} \div \frac{25}{8} = \frac{5}{16} \cdot \frac{8}{25} = \frac{5 \cdot 8}{16 \cdot 25} = \frac{5 \cdot 8}{(2 \cdot 8) \cdot (5 \cdot 5)} = \frac{1}{2 \cdot 5} = \frac{1}{10} \)

Ответ: 4 \( \frac{1}{2} \); 3 \( \frac{1}{3} \); 8 \( \frac{1}{6} \); 1 \( \frac{19}{21} \); \( \frac{2}{21} \); \( \frac{1}{10} \).