\[A = \frac{3}{7} + 1\]
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
\[\frac{3}{4} = \frac{9}{12}\]
\[B = \frac{5}{12} + \frac{9}{12} = \frac{14}{12} = \frac{7}{6}\]
\[B = \frac{7}{6}\]
\[I = \left(\frac{17}{3} - \frac{7}{2}\right) + \left(\frac{16}{15} + 3.5\right)\]
التبسيط الجبري: لا توجد كسور متشابهة للإلغاء
نحسب كل قوس على حدة
القوس الأول: \(\frac{17}{3} - \frac{7}{2}\)
PPCM(3, 2) = 6
\[\frac{17}{3} - \frac{7}{2} = \frac{34}{6} - \frac{21}{6} = \frac{13}{6}\]
القوس الثاني: \(\frac{16}{15} + 3.5 = \frac{16}{15} + \frac{7}{2}\)
PPCM(15, 2) = 30
\[\frac{16}{15} + \frac{7}{2} = \frac{32}{30} + \frac{105}{30} = \frac{137}{30}\]
الجمع النهائي: \(\frac{13}{6} + \frac{137}{30}\)
PPCM(6, 30) = 30
\[I = \frac{65}{30} + \frac{137}{30} = \frac{202}{30} = \frac{101}{15}\]
\[I = \frac{101}{15}\]
\[J = \left(\frac{30}{39} - \frac{324}{975}\right) + \left(\frac{4}{13} + \frac{324}{975}\right)\]
التبسيط الجبري:
\[J = \frac{30}{39} - \frac{324}{975} + \frac{4}{13} + \frac{324}{975}\]
نلاحظ أن \(\frac{324}{975}\) تظهر مع + و - فتلغي نفسها:
\[J = \frac{30}{39} + \frac{4}{13}\]
نبسط \(\frac{30}{39} = \frac{10}{13}\)
\[J = \frac{10}{13} + \frac{4}{13} = \frac{14}{13}\]
\[J = \frac{14}{13}\]
\[K = \left(\frac{5}{8} + \frac{3}{4}\right) - \left(\frac{3}{4} - \frac{1}{8}\right)\]
التبسيط الجبري:
\[K = \frac{5}{8} + \frac{3}{4} - \frac{3}{4} + \frac{1}{8}\]
\(\frac{3}{4}\) تلغي نفسها:
\[K = \frac{5}{8} + \frac{1}{8}\]
\[K = \frac{5 + 1}{8} = \frac{6}{8} = \frac{3}{4}\]
\[K = \frac{3}{4}\]
\[L = \left(\frac{7}{12} - \frac{2}{3}\right) + \left(\frac{2}{3} + \frac{5}{12}\right)\]
التبسيط الجبري:
\[L = \frac{7}{12} - \frac{2}{3} + \frac{2}{3} + \frac{5}{12}\]
\(\frac{2}{3}\) تلغي نفسها:
\[L = \frac{7}{12} + \frac{5}{12}\]
\[L = \frac{7 + 5}{12} = \frac{12}{12} = 1\]
\[L = 1\]
\[M = \left(\frac{9}{10} + \frac{1}{5}\right) - \left(\frac{1}{5} - \frac{3}{10}\right)\]
التبسيط الجبري:
\[M = \frac{9}{10} + \frac{1}{5} - \frac{1}{5} + \frac{3}{10}\]
\(\frac{1}{5}\) تلغي نفسها:
\[M = \frac{9}{10} + \frac{3}{10}\]
\[M = \frac{9 + 3}{10} = \frac{12}{10} = \frac{6}{5}\]
\[M = \frac{6}{5}\]
\[N = \left(\frac{4}{7} - \frac{2}{14}\right) + \left(\frac{3}{14} + \frac{2}{14}\right) - \frac{4}{7}\]
التبسيط الجبري:
\[N = \frac{4}{7} - \frac{2}{14} + \frac{3}{14} + \frac{2}{14} - \frac{4}{7}\]
\(\frac{4}{7}\) تلغي نفسها، و \(\frac{2}{14}\) تلغي نفسها:
\[N = \frac{3}{14}\]
\[N = \frac{3}{14}\]
\[O = \left(\frac{5}{6} + \frac{7}{12}\right) - \left(\frac{7}{12} - \frac{1}{6}\right) + \frac{2}{3}\]
التبسيط الجبري:
\[O = \frac{5}{6} + \frac{7}{12} - \frac{7}{12} + \frac{1}{6} + \frac{2}{3}\]
\(\frac{7}{12}\) تلغي نفسها:
\[O = \frac{5}{6} + \frac{1}{6} + \frac{2}{3}\]
\[O = \frac{6}{6} + \frac{2}{3} = 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3}\]
\[O = \frac{5}{3}\]
\[P = \left(\frac{8}{15} - \frac{3}{5}\right) + \left(\frac{3}{5} + \frac{7}{15}\right) - \frac{2}{15}\]
التبسيط الجبري:
\[P = \frac{8}{15} - \frac{3}{5} + \frac{3}{5} + \frac{7}{15} - \frac{2}{15}\]
\(\frac{3}{5}\) تلغي نفسها:
\[P = \frac{8}{15} + \frac{7}{15} - \frac{2}{15}\]
\[P = \frac{8 + 7 - 2}{15} = \frac{13}{15}\]
\[P = \frac{13}{15}\]
\[Q = \left(\frac{11}{8} + \frac{5}{12}\right) - \left(\frac{5}{12} - \frac{3}{8}\right) + \left(\frac{3}{8} - \frac{11}{8}\right)\]
التبسيط الجبري:
\[Q = \frac{11}{8} + \frac{5}{12} - \frac{5}{12} + \frac{3}{8} + \frac{3}{8} - \frac{11}{8}\]
\(\frac{5}{12}\) تلغي نفسها، \(\frac{11}{8}\) تلغي نفسها:
\[Q = \frac{3}{8} + \frac{3}{8}\]
\[Q = \frac{6}{8} = \frac{3}{4}\]
\[Q = \frac{3}{4}\]
\[R = \left(\frac{2}{9} + \frac{4}{3}\right) - \left(\frac{4}{3} - \frac{5}{9}\right) + \left(\frac{5}{9} - \frac{2}{9}\right)\]
التبسيط الجبري:
\[R = \frac{2}{9} + \frac{4}{3} - \frac{4}{3} + \frac{5}{9} + \frac{5}{9} - \frac{2}{9}\]
\(\frac{4}{3}\) تلغي نفسها، \(\frac{2}{9}\) تلغي نفسها:
\[R = \frac{5}{9} + \frac{5}{9}\]
\[R = \frac{10}{9}\]
\[R = \frac{10}{9}\]
\[S = \left(\frac{7}{10} - \frac{3}{20}\right) + \left(\frac{3}{20} + \frac{1}{4}\right) - \left(\frac{1}{4} + \frac{1}{10}\right)\]
التبسيط الجبري:
\[S = \frac{7}{10} - \frac{3}{20} + \frac{3}{20} + \frac{1}{4} - \frac{1}{4} - \frac{1}{10}\]
\(\frac{3}{20}\) تلغي نفسها، \(\frac{1}{4}\) تلغي نفسها:
\[S = \frac{7}{10} - \frac{1}{10}\]
\[S = \frac{7 - 1}{10} = \frac{6}{10} = \frac{3}{5}\]
\[S = \frac{3}{5}\]
\[T = \left(\frac{13}{6} + \frac{5}{18}\right) - \left(\frac{5}{18} - \frac{7}{6}\right) + \left(\frac{7}{6} - \frac{13}{6}\right) + \frac{2}{3}\]
التبسيط الجبري:
\[T = \frac{13}{6} + \frac{5}{18} - \frac{5}{18} + \frac{7}{6} + \frac{7}{6} - \frac{13}{6} + \frac{2}{3}\]
\(\frac{5}{18}\) تلغي نفسها، \(\frac{13}{6}\) تلغي نفسها:
\[T = \frac{7}{6} + \frac{7}{6} + \frac{2}{3}\]
\[T = \frac{14}{6} + \frac{2}{3} = \frac{7}{3} + \frac{2}{3} = \frac{9}{3} = 3\]
\[T = 3\]