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

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