العمليات في مجموعة الأعداد الكسرية

تمارين شاملة في جمع وطرح وتبسيط الكسور

تمرين عدد 1: احسب واختزل إلى أقصى حد
\(\frac{5}{4} - \frac{7}{9}\)
\(\frac{2}{3} - \frac{3}{5}\)
\(\frac{7}{4} - \frac{4}{3}\)
\(\frac{12}{7} - 1\)
\(1.5 - \frac{6}{9}\)
\(\frac{1}{2} - 0.6\)

الحل التفصيلي

\(\frac{5}{4} - \frac{7}{9} = \frac{5 \times 9}{4 \times 9} - \frac{7 \times 4}{9 \times 4} = \frac{45}{36} - \frac{28}{36} = \frac{17}{36}\)
\(\frac{2}{3} - \frac{3}{5} = \frac{2 \times 5}{3 \times 5} - \frac{3 \times 3}{5 \times 3} = \frac{10}{15} - \frac{9}{15} = \frac{1}{15}\)
\(\frac{7}{4} - \frac{4}{3} = \frac{7 \times 3}{4 \times 3} - \frac{4 \times 4}{3 \times 4} = \frac{21}{12} - \frac{16}{12} = \frac{5}{12}\)
\(\frac{12}{7} - 1 = \frac{12}{7} - \frac{7}{7} = \frac{5}{7}\)
\(1.5 - \frac{6}{9} = \frac{3}{2} - \frac{2}{3} = \frac{9}{6} - \frac{4}{6} = \frac{5}{6}\)
\(\frac{1}{2} - 0.6 = \frac{1}{2} - \frac{3}{5} = \frac{5}{10} - \frac{6}{10} = -\frac{1}{10}\)
تمرين عدد 2: احسب واختزل إلى أقصى حد
\(\frac{1}{4} + \frac{1}{8}\)
\(\frac{3}{5} + \frac{3}{7}\)
\(3 + \frac{6}{7}\)
\(5 + \frac{3}{4}\)
\(\frac{4}{5} + \frac{16}{15}\)
\(\frac{1}{2} + \frac{4}{3}\)
\(\frac{20}{45} + \frac{7}{3}\)
\(\frac{7}{26} + \frac{11}{39}\)

الحل التفصيلي

\(\frac{1}{4} + \frac{1}{8} = \frac{2}{8} + \frac{1}{8} = \frac{3}{8}\)
\(\frac{3}{5} + \frac{3}{7} = \frac{21}{35} + \frac{15}{35} = \frac{36}{35}\)
\(3 + \frac{6}{7} = \frac{21}{7} + \frac{6}{7} = \frac{27}{7}\)
\(5 + \frac{3}{4} = \frac{20}{4} + \frac{3}{4} = \frac{23}{4}\)
\(\frac{4}{5} + \frac{16}{15} = \frac{12}{15} + \frac{16}{15} = \frac{28}{15}\)
\(\frac{1}{2} + \frac{4}{3} = \frac{3}{6} + \frac{8}{6} = \frac{11}{6}\)
\(\frac{20}{45} + \frac{7}{3} = \frac{4}{9} + \frac{7}{3} = \frac{4}{9} + \frac{21}{9} = \frac{25}{9}\)
\(\frac{7}{26} + \frac{11}{39} = \frac{21}{78} + \frac{22}{78} = \frac{43}{78}\)
تمرين عدد 3: ابحث عن العدد الكسري x في كل حالة
\(x - 3 = \frac{5}{3}\)
\(2.8 - x = \frac{7}{8}\)
\(\frac{126}{54} - x = \frac{3}{4}\)
\(\frac{5}{4} - x = \frac{7}{12}\)
\(x - \frac{2}{9} = 0.75\)
\(\frac{13}{6} - x = 2\)

الحل التفصيلي

\(x - 3 = \frac{5}{3} \Rightarrow x = \frac{5}{3} + 3 = \frac{5}{3} + \frac{9}{3} = \frac{14}{3}\)
\(2.8 - x = \frac{7}{8} \Rightarrow x = 2.8 - \frac{7}{8} = \frac{14}{5} - \frac{7}{8} = \frac{112 - 35}{40} = \frac{77}{40}\)
\(\frac{126}{54} - x = \frac{3}{4} \Rightarrow x = \frac{126}{54} - \frac{3}{4} = \frac{7}{3} - \frac{3}{4} = \frac{28 - 9}{12} = \frac{19}{12}\)
\(\frac{5}{4} - x = \frac{7}{12} \Rightarrow x = \frac{5}{4} - \frac{7}{12} = \frac{15 - 7}{12} = \frac{8}{12} = \frac{2}{3}\)
\(x - \frac{2}{9} = 0.75 \Rightarrow x = \frac{3}{4} + \frac{2}{9} = \frac{27 + 8}{36} = \frac{35}{36}\)
\(\frac{13}{6} - x = 2 \Rightarrow x = \frac{13}{6} - 2 = \frac{13 - 12}{6} = \frac{1}{6}\)
تمرين عدد 4: احسب بأيسر طريقة واختزل إلى أقصى حد
\(\frac{5}{3} + \frac{11}{9} + \frac{2}{3}\)
\(1 + \frac{1}{2} + \frac{1}{4}\)
\(4 + \frac{2}{3} + \frac{28}{12}\)
\(\frac{17}{25} + 2 + \frac{13}{25}\)
\(1 + \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5}\)
\(0.75 + \frac{2}{3} + \frac{7}{4} + 0.25 + \frac{4}{3}\)

الحل التفصيلي

\(\frac{5}{3} + \frac{11}{9} + \frac{2}{3} = \frac{5+2}{3} + \frac{11}{9} = \frac{7}{3} + \frac{11}{9} = \frac{21+11}{9} = \frac{32}{9}\)
\(1 + \frac{1}{2} + \frac{1}{4} = \frac{4}{4} + \frac{2}{4} + \frac{1}{4} = \frac{7}{4}\)
\(4 + \frac{2}{3} + \frac{28}{12} = 4 + \frac{2}{3} + \frac{7}{3} = 4 + \frac{9}{3} = 4 + 3 = 7\)
\(\frac{17}{25} + 2 + \frac{13}{25} = \frac{17+13}{25} + 2 = \frac{30}{25} + 2 = \frac{6}{5} + 2 = \frac{16}{5}\)
\(1 + \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} = \frac{60 + 30 + 40 + 45 + 48}{60} = \frac{223}{60}\)
\(0.75 + \frac{2}{3} + \frac{7}{4} + 0.25 + \frac{4}{3} = 1 + \frac{6}{3} + \frac{7}{4} = 1 + 2 + \frac{7}{4} = \frac{19}{4}\)
تمرين عدد 5: احسب بأيسر طريقة واختزل إلى أقصى حد
\(A = \left(\frac{7}{3} + \frac{2125}{2124}\right) - \left(2 + \frac{2125}{2124}\right)\)
\(B = \left(\frac{19}{45} + \frac{4}{23}\right) - \left(\frac{4}{23} + \frac{1}{3}\right)\)
\(C = \left(3 + \frac{7}{3}\right) + \left(5 - \frac{21}{9}\right)\)
\(D = \left(\frac{17}{2} - \frac{5}{4}\right) + \left(\frac{10}{8} + \frac{19}{2}\right)\)

الحل التفصيلي

\(A = \frac{7}{3} + \frac{2125}{2124} - 2 - \frac{2125}{2124} = \frac{7}{3} - 2 = \frac{7-6}{3} = \frac{1}{3}\)
\(B = \frac{19}{45} + \frac{4}{23} - \frac{4}{23} - \frac{1}{3} = \frac{19}{45} - \frac{1}{3} = \frac{19}{45} - \frac{15}{45} = \frac{4}{45}\)
\(C = 3 + \frac{7}{3} + 5 - \frac{21}{9} = 8 + \frac{7}{3} - \frac{7}{3} = 8\)
\(D = \frac{17}{2} - \frac{5}{4} + \frac{10}{8} + \frac{19}{2} = \frac{36}{2} + \frac{10-5}{8} = 18 + \frac{5}{8} = \frac{149}{8}\)
تمرين عدد 6: ابحث عن العدد الكسري x في كل حالة من الحالات التالية
\(\frac{9}{4} + x = 3\)
\(\frac{11}{2} + x = 10\)
\(x + \frac{3}{13} = 1\)
\(x + \frac{5}{7} = \frac{19}{14}\)
\(\frac{11}{30} + x = \frac{37}{40}\)
\(x + \frac{15}{28} = \frac{24}{21}\)

الحل التفصيلي

\(\frac{9}{4} + x = 3 \Rightarrow x = 3 - \frac{9}{4} = \frac{12-9}{4} = \frac{3}{4}\)
\(\frac{11}{2} + x = 10 \Rightarrow x = 10 - \frac{11}{2} = \frac{20-11}{2} = \frac{9}{2}\)
\(x + \frac{3}{13} = 1 \Rightarrow x = 1 - \frac{3}{13} = \frac{13-3}{13} = \frac{10}{13}\)
\(x + \frac{5}{7} = \frac{19}{14} \Rightarrow x = \frac{19}{14} - \frac{5}{7} = \frac{19-10}{14} = \frac{9}{14}\)
\(\frac{11}{30} + x = \frac{37}{40} \Rightarrow x = \frac{37}{40} - \frac{11}{30} = \frac{111-44}{120} = \frac{67}{120}\)
\(x + \frac{15}{28} = \frac{24}{21} \Rightarrow x = \frac{8}{7} - \frac{15}{28} = \frac{32-15}{28} = \frac{17}{28}\)