العمليات في مجموعة الأعداد الكسرية النسبية

السنة الثامنة أساسي - ضرب الكسور السالبة والموجبة

تمرين عدد 1: احسب واختزل إلى أقصى حد (مع أعداد سالبة)
$a = -\frac{6}{5} \times \left(\frac{5}{4} + \frac{10}{3}\right)$
$b = \left(-4 + \frac{3}{5}\right) \times \frac{3}{4}$
$c = \frac{1}{5} \times \left(-\frac{7}{5} - \frac{5}{6}\right)$
$d = -\frac{3}{4} \times \left(1.8 - \frac{3}{5}\right)$
$E = -\frac{12}{7} - \left(-\frac{5}{7} + 1\right)$
$F = \frac{19}{5} - \left(-\frac{4}{5} + \frac{7}{4}\right)$
$G = -\frac{19}{5} - \frac{2}{3} \times \left(-\frac{9}{5}\right)$
$H = \left(-\frac{37}{11} - \frac{9}{8}\right) \times \frac{3}{8}$

الحل التفصيلي

$a = -\frac{6}{5} \times \left(\frac{15 + 40}{12}\right) = -\frac{6}{5} \times \frac{55}{12} = -\frac{330}{60} = -\frac{11}{2}$
$b = -\frac{17}{5} \times \frac{3}{4} = -\frac{51}{20}$
$c = \frac{1}{5} \times \left(\frac{-42 - 25}{30}\right) = \frac{1}{5} \times \left(-\frac{67}{30}\right) = -\frac{67}{150}$
$d = -\frac{3}{4} \times \left(\frac{9}{5} - \frac{3}{5}\right) = -\frac{3}{4} \times \frac{6}{5} = -\frac{18}{20} = -\frac{9}{10}$
$E = -\frac{12}{7} - \left(-\frac{5}{7} + \frac{7}{7}\right) = -\frac{12}{7} - \frac{2}{7} = -\frac{14}{7} = -2$
$F = \frac{19}{5} + \frac{4}{5} - \frac{7}{4} = \frac{23}{5} - \frac{7}{4} = \frac{92 - 35}{20} = \frac{57}{20}$
$G = -\frac{19}{5} + \frac{2}{3} \times \frac{9}{5} = -\frac{19}{5} + \frac{6}{5} = -\frac{13}{5}$
$H = \left(\frac{-296 - 99}{88}\right) \times \frac{3}{8} = -\frac{395}{88} \times \frac{3}{8} = -\frac{1185}{704} = -\frac{165}{88}$
تمرين عدد 2: احسب بأيسر طريقة (خاصية التوزيع مع أعداد سالبة)
$e = -\frac{7}{8} \times 6 + 6 \times \frac{3}{4}$
$f = -\frac{5}{3} \times \frac{7}{9} + \left(-\frac{5}{3}\right) \times \frac{4}{9}$
$g = \frac{17}{11} \times \left(-\frac{48}{29}\right) + \frac{17}{11}$
$h = -\frac{9}{11} \times \frac{7}{5} + 1.3 \times \left(-\frac{27}{33}\right)$
$k = -\frac{5}{12} \times 5 - \frac{10}{24} \times 1.8$
$l = -\frac{3}{4} \times 4 - 2 \times 0.5 + 1$
$m = \frac{9}{14} \times \left(-7\right) + 3 \times \frac{5}{6} \times \left(-\frac{1}{5}\right)$

الحل التفصيلي

$e = 6 \times \left(-\frac{7}{8} + \frac{3}{4}\right) = 6 \times \left(-\frac{7}{8} + \frac{6}{8}\right) = 6 \times \left(-\frac{1}{8}\right) = -\frac{3}{4}$
$f = -\frac{5}{3} \times \left(\frac{7}{9} + \frac{4}{9}\right) = -\frac{5}{3} \times \frac{11}{9} = -\frac{55}{27}$
$g = \frac{17}{11} \times \left(-\frac{48}{29} + 1\right) = \frac{17}{11} \times \left(-\frac{48}{29} + \frac{29}{29}\right) = \frac{17}{11} \times \left(-\frac{19}{29}\right) = -\frac{323}{319}$
$h = -\frac{63}{55} - \frac{13 \times 27}{10 \times 33} = -\frac{63}{55} - \frac{39}{110} = -\frac{126}{110} - \frac{39}{110} = -\frac{165}{110} = -\frac{3}{2}$
$k = -\frac{25}{12} - \frac{18}{24} = -\frac{25}{12} - \frac{3}{4} = -\frac{25}{12} - \frac{9}{12} = -\frac{34}{12} = -\frac{17}{6}$
$l = -3 - 1 + 1 = -3$
$m = -\frac{9}{2} - \frac{1}{2} = -5$
تمرين عدد 3: احسب واختزل إلى أقصى حد (عمليات مختلطة)
$p = \left(-\frac{5}{2} + 3\right) \times \left(\frac{9}{4} - 2\right)$
$Q = \left(-1.2 - \frac{2}{5}\right) \times \frac{7}{8} + \frac{1}{3}$
$R = -\frac{4}{3} + \frac{7}{3} \times \left(-\frac{5}{7}\right)$
$S = -\frac{7}{5} \times \frac{4}{3} + \frac{3}{4}$
$T = -\frac{7}{5} + \left(-\frac{3}{5}\right) \times \left(\frac{5}{4} - \frac{3}{8}\right)$

الحل التفصيلي

$p = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$
$Q = \left(-\frac{6}{5} - \frac{2}{5}\right) \times \frac{7}{8} + \frac{1}{3} = -\frac{8}{5} \times \frac{7}{8} + \frac{1}{3} = -\frac{7}{5} + \frac{1}{3} = \frac{-21 + 5}{15} = -\frac{16}{15}$
$R = -\frac{4}{3} - \frac{5}{3} = -\frac{9}{3} = -3$
$S = -\frac{28}{15} + \frac{3}{4} = \frac{-112 + 45}{60} = -\frac{67}{60}$
$T = -\frac{7}{5} - \frac{3}{5} \times \frac{7}{8} = -\frac{7}{5} - \frac{21}{40} = \frac{-56 - 21}{40} = -\frac{77}{40}$
تمرين عدد 4: ضرب عدة كسور (مع أعداد سالبة)
$A = -\frac{75}{4} \times \frac{16}{8} \times \left(-\frac{2}{25}\right)$
$B = \frac{25}{12} \times \left(-\frac{39}{44}\right) \times \frac{121}{65}$
$C = -2.6 \times \frac{75}{260} \times 3.6 \times \left(-\frac{5}{18}\right)$
$D = -\frac{42}{17} \times \frac{1}{5} \times \left(-\frac{34}{21}\right) \times \frac{3}{4}$
$E = \frac{57}{27} \times \left(-\frac{43}{18}\right) \times 0 \times \frac{45}{26}$
$F = -\frac{25}{28} \times 12 \times \frac{1}{30} \times \left(-\frac{24}{51}\right)$

الحل التفصيلي

$A = -\frac{75}{4} \times 2 \times \left(-\frac{2}{25}\right) = (-) \times (-) = (+)$ $A = \frac{75 \times 2 \times 2}{4 \times 25} = \frac{300}{100} = 3$
$B = \frac{25}{12} \times \left(-\frac{39}{44}\right) \times \frac{121}{65}$ (عدد سالب) $B = -\frac{25 \times 39 \times 121}{12 \times 44 \times 65} = -\frac{55}{16}$
$C = -\frac{13}{5} \times \frac{75}{260} \times \frac{18}{5} \times \left(-\frac{5}{18}\right) = (+)$ $C = \frac{13 \times 75}{5 \times 260} = \frac{3}{4}$
$D = -\frac{42}{17} \times \frac{1}{5} \times \left(-\frac{34}{21}\right) \times \frac{3}{4} = (+)$ $D = \frac{42 \times 34 \times 3}{17 \times 5 \times 21 \times 4} = \frac{12}{55}$
$E = 0$ (أي عدد مضروب في صفر يساوي صفر)
$F = -\frac{25}{28} \times 12 \times \frac{1}{30} \times \left(-\frac{24}{51}\right) = (+)$ $F = \frac{25 \times 12 \times 24}{28 \times 30 \times 51} = \frac{2}{17}$
تمرين عدد 5: قواعد الإشارات في الضرب
$A = \left(-\frac{3}{5}\right) \times \left(-\frac{2}{7}\right)$
$B = \frac{4}{9} \times \left(-\frac{5}{6}\right)$
$C = \left(-\frac{7}{8}\right) \times \frac{3}{4} \times \left(-\frac{2}{3}\right)$
$D = \left(-\frac{5}{6}\right) \times \left(-\frac{3}{10}\right) \times \left(-\frac{4}{5}\right)$
$E = \frac{11}{12} \times \left(-\frac{6}{11}\right) \times \left(-\frac{4}{3}\right)$

الحل التفصيلي

$A = \left(-\frac{3}{5}\right) \times \left(-\frac{2}{7}\right) = (+) \times (+) = \frac{6}{35}$ (سالب × سالب = موجب)
$B = \frac{4}{9} \times \left(-\frac{5}{6}\right) = -\frac{20}{54} = -\frac{10}{27}$ (موجب × سالب = سالب)
$C = \left(-\frac{7}{8}\right) \times \frac{3}{4} \times \left(-\frac{2}{3}\right)$ $(-) \times (+) \times (-) = (+)$ $C = \frac{7 \times 3 \times 2}{8 \times 4 \times 3} = \frac{7}{16}$
$D = \left(-\frac{5}{6}\right) \times \left(-\frac{3}{10}\right) \times \left(-\frac{4}{5}\right)$ $(-) \times (-) \times (-) = (-)$ $D = -\frac{5 \times 3 \times 4}{6 \times 10 \times 5} = -\frac{1}{5}$
$E = \frac{11}{12} \times \left(-\frac{6}{11}\right) \times \left(-\frac{4}{3}\right)$ $(+) \times (-) \times (-) = (+)$ $E = \frac{11 \times 6 \times 4}{12 \times 11 \times 3} = \frac{2}{3}$