العمليات في مجموعة الأعداد الكسرية - صفحة 3

$A = \frac{\frac{2}{3} + \frac{12}{7}}{\frac{19}{3} - \frac{7}{4}} = \frac{\frac{14 + 36}{21}}{\frac{76 - 21}{12}} = \frac{\frac{50}{21}}{\frac{55}{12}} = \frac{50}{21} \times \frac{12}{55} = \frac{40}{77}$

$B = \frac{\frac{9}{4}}{\frac{4}{3}} \div \frac{\frac{4}{5}}{1} = \frac{9}{4} \times \frac{3}{4} \times \frac{5}{4} = \frac{135}{64}$

$C = \frac{\frac{5}{4} + \frac{23}{30}}{\frac{3}{5} + \frac{7}{6}} = \frac{\frac{75 + 46}{60}}{\frac{18 + 35}{30}} = \frac{\frac{121}{60}}{\frac{53}{30}} = \frac{121}{106}$

$D = \frac{\frac{3}{4} + \frac{3}{5}}{\frac{1}{3} - \frac{4}{3}} = \frac{\frac{27}{20}}{-\frac{1}{3}} = -\frac{81}{20}$

$E = \frac{2}{2 + \frac{2}{\frac{9}{2}}} = \frac{2}{2 + \frac{4}{9}} = \frac{2}{\frac{22}{9}} = \frac{18}{22} = \frac{9}{11}$

الخطوة 1: تحويل جميع الأعداد إلى كسور:
$\frac{1}{4}$، $3.6 = \frac{18}{5}$، $0.2 = \frac{1}{5}$، $\frac{1}{29}$، $35 = \frac{35}{1}$
$0.55 = \frac{11}{20}$، $\frac{22}{74} = \frac{11}{37}$، $0.8 = \frac{4}{5}$، $\frac{22}{7}$

الخطوة 2: فصل الأعداد الأصغر من 1 والأكبر من 1:
أصغر من 1: $\frac{1}{4}$، $\frac{1}{5}$، $\frac{1}{29}$، $\frac{11}{20}$، $\frac{11}{37}$، $\frac{4}{5}$
أكبر من 1: $\frac{18}{5}$، $\frac{35}{1}$، $\frac{22}{7}$

الخطوة 3: ترتيب الأعداد الأصغر من 1 (المقام المشترك = 580):
$\frac{1}{29} = \frac{20}{580}$، $\frac{1}{5} = \frac{116}{580}$، $\frac{1}{4} = \frac{145}{580}$
$\frac{11}{37} = \frac{172}{580}$، $\frac{11}{20} = \frac{319}{580}$، $\frac{4}{5} = \frac{464}{580}$

الخطوة 4: ترتيب الأعداد الأكبر من 1 (المقام المشترك = 35):
$\frac{22}{7} = \frac{110}{35}$، $\frac{18}{5} = \frac{126}{35}$، $\frac{35}{1} = \frac{1225}{35}$

الترتيب النهائي التصاعدي:
$\frac{1}{29} < 0.2 < \frac{1}{4} < \frac{11}{37} < 0.55 < 0.8 < \frac{22}{7} < 3.6 < 35$

$\frac{3}{8} \times x = 1 \Rightarrow x = \frac{8}{3}$

$\frac{5}{4} \times x = 1 \Rightarrow x = \frac{4}{5}$

$x \times 8 = 1 \Rightarrow x = \frac{1}{8}$

$\frac{1}{15} \times x = 1 \Rightarrow x = 15$

$\frac{1}{x} = \frac{11}{6} \Rightarrow x = \frac{6}{11}$

$0.1x = 1 \Rightarrow x = 10$

$8.2x = 1 \Rightarrow x = \frac{1}{8.2} = \frac{5}{41}$

$\frac{11}{3} \div 15 = \frac{11}{3} \times \frac{1}{15} = \frac{11}{45}$

$6 \div \frac{6}{5} = 6 \times \frac{5}{6} = 5$

$\frac{4}{3} \div \frac{49}{36} = \frac{4}{3} \times \frac{36}{49} = \frac{48}{49}$

$\frac{7}{5} \div 0.8 = \frac{7}{5} \div \frac{4}{5} = \frac{7}{5} \times \frac{5}{4} = \frac{7}{4}$

$\frac{7.5}{2} \div \frac{11}{4} = \frac{15}{4} \times \frac{4}{11} = \frac{15}{11}$

$\frac{8}{27} \div \frac{75}{34} = \frac{8}{27} \times \frac{34}{75} = \frac{272}{2025}$

$\frac{45}{51} \div 50 = \frac{45}{51} \times \frac{1}{50} = \frac{9}{510} = \frac{3}{170}$

$\frac{5}{3} \div \frac{6}{5} \times \frac{7}{2} = \frac{5}{3} \times \frac{5}{6} \times \frac{7}{2} = \frac{175}{36}$