العمليات في مجموعة الأعداد الكسرية - صفحة 2

تمارين متقدمة في ضرب وتبسيط الكسور

تمرين عدد 1: احسب واختزل إلى أقصى حد
\(a = \frac{6}{5} \times \left(\frac{5}{4} + \frac{10}{3}\right)\)
\(b = \left(4 + \frac{3}{5}\right) \times \frac{3}{4}\)
\(c = \frac{1}{5} \times \left(\frac{7}{5} - \frac{5}{6}\right)\)
\(d = \frac{3}{4} \times \left(1.8 - \frac{3}{5}\right)\)
\(E = \frac{12}{7} - \left(\frac{5}{7} + 1\right)\)
\(F = \frac{19}{5} - \left(\frac{4}{5} + \frac{7}{4}\right)\)
\(G = \frac{19}{5} - \frac{2}{3} \times \frac{9}{5}\)
\(H = \left(\frac{37}{11} - \frac{9}{8}\right) \times \frac{3}{8}\)
\(I = \frac{13}{8} + \left(1 - \frac{3}{8}\right)\)
\(J = \left(\frac{17}{9} - \frac{3}{4}\right) + \frac{2}{9}\)
\(K = \left(\frac{19}{11} + 4\right) - \frac{8}{11}\)

الحل التفصيلي

\(a = \frac{6}{5} \times \left(\frac{15 + 40}{12}\right) = \frac{6}{5} \times \frac{55}{12} = \frac{330}{60} = \frac{11}{2}\)
\(b = \frac{23}{5} \times \frac{3}{4} = \frac{69}{20}\)
\(c = \frac{1}{5} \times \left(\frac{42 - 25}{30}\right) = \frac{1}{5} \times \frac{17}{30} = \frac{17}{150}\)
\(d = \frac{3}{4} \times \left(\frac{9}{5} - \frac{3}{5}\right) = \frac{3}{4} \times \frac{6}{5} = \frac{18}{20} = \frac{9}{10}\)
\(E = \frac{12}{7} - \frac{12}{7} = 0\)
\(F = \frac{19}{5} - \frac{4}{5} - \frac{7}{4} = 3 - \frac{7}{4} = \frac{5}{4}\)
تمرين عدد 2: احسب بأيسر طريقة
\(e = \frac{7}{8} \times 6 + 6 \times \frac{3}{4}\)
\(f = \frac{5}{3} \times \frac{7}{9} + \frac{5}{3} \times \frac{4}{9}\)
\(g = \frac{17}{11} \times \frac{48}{29} + \frac{17}{11}\)
\(h = \frac{9}{11} \times \frac{7}{5} + 1.3 \times \frac{27}{33}\)
\(k = \frac{5}{12} \times 5 - \frac{10}{24} \times 1.8\)
\(l = \frac{3}{4} \times 4 - 2 \times 0.5 + 1\)
\(m = \frac{9}{14} \times 7 + 3 \times \frac{5}{6} \times \frac{1}{5}\)

الحل التفصيلي

\(e = 6 \times \left(\frac{7}{8} + \frac{3}{4}\right) = 6 \times \frac{13}{8} = \frac{39}{4}\)
\(f = \frac{5}{3} \times \left(\frac{7}{9} + \frac{4}{9}\right) = \frac{5}{3} \times \frac{11}{9} = \frac{55}{27}\)
\(g = \frac{17}{11} \times \left(\frac{48}{29} + 1\right) = \frac{17}{11} \times \frac{77}{29} = \frac{17 \times 7}{11} = \frac{119}{11}\)
\(h = \frac{9 \times 7}{11 \times 5} + \frac{13 \times 27}{10 \times 33} = \frac{63}{55} + \frac{351}{330} = \frac{63}{55} + \frac{39}{110}\)
\(k = \frac{25}{12} - \frac{18}{24} = \frac{25}{12} - \frac{3}{4} = \frac{25 - 9}{12} = \frac{16}{12} = \frac{4}{3}\)
\(l = 3 - 1 + 1 = 3\)
\(m = \frac{9}{2} + \frac{1}{2} = 5\)
تمرين عدد 3: احسب واختزل إلى أقصى حد
\(p = \left(\frac{5}{2} + 3\right) \times \left(\frac{9}{4} - 2\right)\)
\(Q = \left(1.2 - \frac{2}{5}\right) \times \frac{7}{8} + \frac{1}{3}\)
\(R = \frac{4}{3} + \frac{7}{3} \times \frac{5}{7}\)
\(S = \frac{7}{5} \times \frac{4}{3} + \frac{3}{4}\)
\(T = \frac{7}{5} + \frac{3}{5} \times \left(\frac{5}{4} - \frac{3}{8}\right)\)

الحل التفصيلي

\(p = \frac{11}{2} \times \frac{1}{4} = \frac{11}{8}\)
\(Q = \frac{4}{5} \times \frac{7}{8} + \frac{1}{3} = \frac{7}{10} + \frac{1}{3} = \frac{21 + 10}{30} = \frac{31}{30}\)
\(R = \frac{4}{3} + \frac{5}{3} = \frac{9}{3} = 3\)
\(S = \frac{28}{15} + \frac{3}{4} = \frac{112 + 45}{60} = \frac{157}{60}\)
\(T = \frac{7}{5} + \frac{3}{5} \times \frac{7}{8} = \frac{7}{5} + \frac{21}{40} = \frac{56 + 21}{40} = \frac{77}{40}\)
تمرين عدد 7 (الجزء الأول): احسب واختزل إلى أقصى حد الجداءات التالية
\(\frac{1}{3} \times \frac{5}{8} \div \frac{4}{3} \times \frac{2}{9}\)
\(\frac{7}{3} \times \frac{2}{5} \div \frac{13}{7} \times \frac{28}{39}\)
\(3.6 \times \frac{35}{9} \div \frac{32}{45} \times \frac{27}{24}\)
\(\frac{65}{18} \times \frac{27}{52}\)
\(\frac{15}{46} \times 2.3 \div \frac{75}{36} \times \frac{54}{55}\)

الحل التفصيلي

\(\frac{1}{3} \times \frac{5}{8} \times \frac{3}{4} \times \frac{2}{9} = \frac{1 \times 5 \times 3 \times 2}{3 \times 8 \times 4 \times 9} = \frac{30}{864} = \frac{5}{144}\)
\(\frac{7}{3} \times \frac{2}{5} \times \frac{7}{13} \times \frac{28}{39} = \frac{7 \times 2 \times 7 \times 28}{3 \times 5 \times 13 \times 39} = \frac{8}{15}\)
\(\frac{18}{5} \times \frac{35}{9} \times \frac{45}{32} \times \frac{27}{24} = \frac{18 \times 35 \times 45 \times 27}{5 \times 9 \times 32 \times 24} = \frac{189}{8}\)
\(\frac{65}{18} \times \frac{27}{52} = \frac{65 \times 27}{18 \times 52} = \frac{1755}{936} = \frac{65}{52} = \frac{5}{4}\)
\(\frac{15}{46} \times \frac{23}{10} \times \frac{36}{75} \times \frac{54}{55} = \frac{27}{25}\)
تمرين عدد 7 (الجزء الثاني): احسب واختزل إلى أقصى حد
\(A = \frac{75}{4} \times \frac{16}{8} \times \frac{2}{25}\)
\(B = \frac{25}{12} \times \frac{39}{44} \times \frac{121}{65}\)
\(C = 2.6 \times \frac{75}{260} \times 3.6 \times \frac{5}{18}\)
\(D = \frac{42}{17} \times \frac{1}{5} \times \frac{34}{21} \times \frac{3}{4} \times \frac{12}{33}\)
\(E = \frac{57}{27} \times \frac{43}{18} \times 0 \times \frac{45}{26} \times \frac{12}{35}\)
\(F = \frac{25}{28} \times 12 \times \frac{1}{30} \times \frac{24}{51} \times \frac{15}{48} \times \frac{1}{5}\)

الحل التفصيلي

\(A = \frac{75 \times 16 \times 2}{4 \times 8 \times 25} = \frac{2400}{800} = 3\)
\(B = \frac{25 \times 39 \times 121}{12 \times 44 \times 65} = \frac{25 \times 3 \times 11}{12 \times 4 \times 5} = \frac{275}{80} = \frac{55}{16}\)
\(C = \frac{13}{5} \times \frac{75}{260} \times \frac{18}{5} \times \frac{5}{18} = \frac{13 \times 75}{5 \times 260} = \frac{3}{4}\)
\(D = \frac{42 \times 1 \times 34 \times 3 \times 12}{17 \times 5 \times 21 \times 4 \times 33} = \frac{2 \times 2 \times 3 \times 12}{5 \times 4 \times 33} = \frac{12}{55}\)
\(E = 0\) (أي عدد مضروب في صفر يساوي صفر)
\(F = \frac{25 \times 12 \times 1 \times 24 \times 15 \times 1}{28 \times 30 \times 51 \times 48 \times 5} = \frac{1}{17}\)