\[A = \frac{3}{7} + 1\]
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 \(a = \frac{13}{6}\)، \(b = \frac{7}{6}\)، \(c = \frac{5}{18}\)
\[T = \frac{13}{6} + \frac{7}{6}\]
\[T = \frac{13 + 7}{6} = \frac{20}{6} = \frac{10}{3}\]
\[T = \frac{10}{3}\]