$$A = \frac{3}{7} + 1$$
نحول العدد الصحيح إلى كسر:
$$1 = \frac{7}{7}$$
نجمع الكسرين:
$$A = \frac{3}{7} + \frac{7}{7} = \frac{10}{7}$$
$$A = \frac{10}{7}$$
$$B = \frac{5}{12} + \frac{3}{4}$$
نجد المضاعف المشترك الأصغر: PPCM(12, 4) = 12
$$\frac{3}{4} = \frac{9}{12}$$
$$B = \frac{5}{12} + \frac{9}{12} = \frac{14}{12} = \frac{7}{6}$$
$$B = \frac{7}{6}$$
$$J = \left(\frac{30}{39} - \frac{324}{975}\right) + \left(\frac{4}{13} + \frac{324}{975}\right)$$
نطبق القاعدة: $(a-c)+(b+c) = a+b$
حيث $a = \frac{30}{39}$، $b = \frac{4}{13}$، $c = \frac{324}{975}$
$$J = \frac{30}{39} + \frac{4}{13}$$
نبسط $\frac{30}{39} = \frac{10}{13}$
$$J = \frac{10}{13} + \frac{4}{13} = \frac{14}{13}$$
$$J = \frac{14}{13}$$
$$K = \left(\frac{5}{8} + \frac{3}{4}\right) + \left(\frac{1}{8} - \frac{3}{4}\right)$$
نطبق القاعدة: $(a+c)+(b-c) = a+b$
حيث $a = \frac{5}{8}$، $b = \frac{1}{8}$، $c = \frac{3}{4}$
$$K = \frac{5}{8} + \frac{1}{8}$$
$$K = \frac{5 + 1}{8} = \frac{6}{8} = \frac{3}{4}$$
$$K = \frac{3}{4}$$
$$L = \left(\frac{7}{12} + \frac{2}{3}\right) + \left(\frac{5}{12} - \frac{2}{3}\right)$$
نطبق القاعدة: $(a+c)+(b-c) = a+b$
حيث $a = \frac{7}{12}$، $b = \frac{5}{12}$، $c = \frac{2}{3}$
$$L = \frac{7}{12} + \frac{5}{12}$$
$$L = \frac{7 + 5}{12} = \frac{12}{12} = 1$$
$$L = 1$$
$$M = \left(\frac{9}{10} + \frac{1}{5}\right) - \left(\frac{3}{10} + \frac{1}{5}\right)$$
نطبق القاعدة: $(a+c)-(b+c) = a-b$
حيث $a = \frac{9}{10}$، $b = \frac{3}{10}$، $c = \frac{1}{5}$
$$M = \frac{9}{10} - \frac{3}{10}$$
$$M = \frac{9 - 3}{10} = \frac{6}{10} = \frac{3}{5}$$
$$M = \frac{3}{5}$$
$$N = \left(\frac{4}{7} + \frac{3}{14}\right) - \left(\frac{2}{14} + \frac{3}{14}\right)$$
نطبق القاعدة: $(a+c)-(b+c) = a-b$
حيث $a = \frac{4}{7}$، $b = \frac{2}{14} = \frac{1}{7}$، $c = \frac{3}{14}$
$$N = \frac{4}{7} - \frac{1}{7}$$
$$N = \frac{4 - 1}{7} = \frac{3}{7}$$
$$N = \frac{3}{7}$$
$$O = \left(\frac{5}{6} - \frac{7}{12}\right) - \left(\frac{1}{6} - \frac{7}{12}\right)$$
نطبق القاعدة: $(a-c)-(b-c) = a-b$
حيث $a = \frac{5}{6}$، $b = \frac{1}{6}$، $c = \frac{7}{12}$
$$O = \frac{5}{6} - \frac{1}{6}$$
$$O = \frac{5 - 1}{6} = \frac{4}{6} = \frac{2}{3}$$
$$O = \frac{2}{3}$$
$$P = \left(\frac{8}{15} - \frac{3}{5}\right) - \left(\frac{2}{15} - \frac{3}{5}\right)$$
نطبق القاعدة: $(a-c)-(b-c) = a-b$
حيث $a = \frac{8}{15}$، $b = \frac{2}{15}$، $c = \frac{3}{5}$
$$P = \frac{8}{15} - \frac{2}{15}$$
$$P = \frac{8 - 2}{15} = \frac{6}{15} = \frac{2}{5}$$
$$P = \frac{2}{5}$$
$$Q = \left(\frac{11}{8} + \frac{5}{12}\right) + \left(\frac{3}{8} - \frac{5}{12}\right)$$
نطبق القاعدة: $(a+c)+(b-c) = a+b$
حيث $a = \frac{11}{8}$، $b = \frac{3}{8}$، $c = \frac{5}{12}$
$$Q = \frac{11}{8} + \frac{3}{8}$$
$$Q = \frac{11 + 3}{8} = \frac{14}{8} = \frac{7}{4}$$
$$Q = \frac{7}{4}$$
$$R = \left(\frac{2}{9} + \frac{4}{3}\right) - \left(\frac{5}{9} + \frac{4}{3}\right)$$
نطبق القاعدة: $(a+c)-(b+c) = a-b$
حيث $a = \frac{2}{9}$، $b = \frac{5}{9}$، $c = \frac{4}{3}$
$$R = \frac{2}{9} - \frac{5}{9}$$
$$R = \frac{2 - 5}{9} = \frac{-3}{9} = -\frac{1}{3}$$
$$R = -\frac{1}{3}$$
$$S = \left(\frac{7}{10} - \frac{3}{20}\right) - \left(\frac{1}{10} - \frac{3}{20}\right)$$
نطبق القاعدة: $(a-c)-(b-c) = a-b$
حيث $a = \frac{7}{10}$، $b = \frac{1}{10}$، $c = \frac{3}{20}$
$$S = \frac{7}{10} - \frac{1}{10}$$
$$S = \frac{7 - 1}{10} = \frac{6}{10} = \frac{3}{5}$$
$$S = \frac{3}{5}$$
$$T = \left(\frac{13}{6} + \frac{5}{18}\right) + \left(\frac{7}{6} - \frac{5}{18}\right)$$
نطبق القاعدة: $(a+c)+(b-c) = a+b$
حيث $a = \frac{13}{6}$، $b = \frac{7}{6}$، $c = \frac{5}{18}$
$$T = \frac{13}{6} + \frac{7}{6}$$
$$T = \frac{13 + 7}{6} = \frac{20}{6} = \frac{10}{3}$$
$$T = \frac{10}{3}$$