6. Вычислите, применяя законы сложения для упрощения вычислений:

Решение:

1. \( \left( \frac{13}{25} + \frac{2}{25} \right) + \frac{2}{9} = \frac{13+2}{25} + \frac{2}{9} = \frac{15}{25} + \frac{2}{9} = \frac{3}{5} + \frac{2}{9} = \frac{3 \cdot 9}{5 \cdot 9} + \frac{2 \cdot 5}{9 \cdot 5} = \frac{27}{45} + \frac{10}{45} = \frac{27+10}{45} = \frac{37}{45} \)
2. \( \frac{5}{24} + \left( \frac{3}{7} + \frac{7}{24} \right) = \left( \frac{5}{24} + \frac{7}{24} \right) + \frac{3}{7} = \frac{5+7}{24} + \frac{3}{7} = \frac{12}{24} + \frac{3}{7} = \frac{1}{2} + \frac{3}{7} = \frac{1 \cdot 7}{2 \cdot 7} + \frac{3 \cdot 2}{7 \cdot 2} = \frac{7}{14} + \frac{6}{14} = \frac{7+6}{14} = \frac{13}{14} \)
3. \( \frac{3}{16} + \frac{4}{15} + \frac{11}{16} + \frac{1}{15} = \left( \frac{3}{16} + \frac{11}{16} \right) + \left( \frac{4}{15} + \frac{1}{15} \right) = \frac{3+11}{16} + \frac{4+1}{15} = \frac{14}{16} + \frac{5}{15} = \frac{7}{8} + \frac{1}{3} = \frac{7 \cdot 3}{8 \cdot 3} + \frac{1 \cdot 8}{3 \cdot 8} = \frac{21}{24} + \frac{8}{24} = \frac{21+8}{24} = \frac{29}{24} \)
4. \( \frac{5}{18} + \frac{2}{25} + \frac{11}{18} + \frac{3}{25} = \left( \frac{5}{18} + \frac{11}{18} \right) + \left( \frac{2}{25} + \frac{3}{25} \right) = \frac{5+11}{18} + \frac{2+3}{25} = \frac{16}{18} + \frac{5}{25} = \frac{8}{9} + \frac{1}{5} = \frac{8 \cdot 5}{9 \cdot 5} + \frac{1 \cdot 9}{5 \cdot 9} = \frac{40}{45} + \frac{9}{45} = \frac{40+9}{45} = \frac{49}{45} \)
5. \( \frac{1}{6} + \frac{3}{45} + \frac{7}{36} + \frac{2}{45} + \frac{5}{36} = \frac{1}{6} + \left( \frac{3}{45} + \frac{2}{45} \right) + \left( \frac{7}{36} + \frac{5}{36} \right) = \frac{1}{6} + \frac{5}{45} + \frac{12}{36} = \frac{1}{6} + \frac{1}{9} + \frac{1}{3} = \frac{1 \cdot 3}{6 \cdot 3} + \frac{1 \cdot 2}{9 \cdot 2} + \frac{1 \cdot 6}{3 \cdot 6} = \frac{3}{18} + \frac{2}{18} + \frac{6}{18} = \frac{3+2+6}{18} = \frac{11}{18} \)
6. \( \frac{2}{9} + \frac{7}{60} + \frac{5}{36} + \frac{11}{60} + \frac{1}{36} = \frac{2}{9} + \left( \frac{7}{60} + \frac{11}{60} \right) + \left( \frac{5}{36} + \frac{1}{36} \right) = \frac{2}{9} + \frac{18}{60} + \frac{6}{36} = \frac{2}{9} + \frac{3}{10} + \frac{1}{6} = \frac{2 \cdot 10}{9 \cdot 10} + \frac{3 \cdot 9}{10 \cdot 9} + \frac{1 \cdot 15}{6 \cdot 15} = \frac{20}{90} + \frac{27}{90} + \frac{15}{90} = \frac{20+27+15}{90} = \frac{62}{90} = \frac{31}{45} \)

Ответ: a) \( \frac{37}{45} \); б) \( \frac{13}{14} \); в) \( \frac{29}{24} \); г) \( \frac{49}{45} \); д) \( \frac{11}{18} \); е) \( \frac{31}{45} \).