احسب المجاميع التالية:
\(c = \frac{28}{48} + \frac{15}{36}\)
\(b = \frac{5}{9} + \frac{8}{18}\)
\(a = \frac{5}{16} + \frac{3}{16}\)
\(f = \frac{11}{21} + \frac{9}{8}\)
\(e = \frac{7}{18} + \frac{5}{9}\)
\(d = \frac{8}{15} + \frac{5}{12}\)
\(i = \frac{19}{52} + \frac{9}{39} + \frac{7}{52} + \frac{17}{39}\)
\(h = \frac{9}{17} + \frac{16}{23} + \frac{8}{17} + \frac{7}{23}\)
\(g = \frac{7}{6} + \frac{4}{9} + \frac{8}{15}\)
\(l = \frac{3}{2} + \frac{1}{4} + \frac{5}{8} + \frac{9}{16}\)
\(k = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{6}\)
\(j = \frac{5}{18} + \left(\frac{3}{18} + \frac{15}{27}\right)\)
الحل:
\(c = \frac{28}{48} + \frac{15}{36} = \frac{28 \cdot 3}{48 \cdot 3} + \frac{15 \cdot 4}{36 \cdot 4} = \frac{84}{144} + \frac{60}{144} = \frac{144}{144} = 1\)
\(b = \frac{5}{9} + \frac{8}{18} = \frac{5}{9} + \frac{4}{9} = \frac{9}{9} = 1\)
\(a = \frac{5}{16} + \frac{3}{16} = \frac{8}{16} = \frac{1}{2}\)
\(f = \frac{11}{21} + \frac{9}{8} = \frac{11 \cdot 8}{21 \cdot 8} + \frac{9 \cdot 21}{8 \cdot 21} = \frac{88}{168} + \frac{189}{168} = \frac{277}{168}\)
\(e = \frac{7}{18} + \frac{5}{9} = \frac{7}{18} + \frac{10}{18} = \frac{17}{18}\)
\(d = \frac{8}{15} + \frac{5}{12} = \frac{8 \cdot 4}{15 \cdot 4} + \frac{5 \cdot 5}{12 \cdot 5} = \frac{32}{60} + \frac{25}{60} = \frac{57}{60} = \frac{19}{20}\)
\(i = \frac{19}{52} + \frac{9}{39} + \frac{7}{52} + \frac{17}{39} = \frac{19 + 7}{52} + \frac{9 + 17}{39} = \frac{26}{52} + \frac{26}{39} = \frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{7}{6}\)
\(h = \frac{9}{17} + \frac{16}{23} + \frac{8}{17} + \frac{7}{23} = \frac{9 + 8}{17} + \frac{16 + 7}{23} = \frac{17}{17} + \frac{23}{23} = 1 + 1 = 2\)
\(g = \frac{7}{6} + \frac{4}{9} + \frac{8}{15} = \frac{7 \cdot 45}{6 \cdot 45} + \frac{4 \cdot 30}{9 \cdot 30} + \frac{8 \cdot 18}{15 \cdot 18} = \frac{315}{270} + \frac{120}{270} + \frac{144}{270} = \frac{579}{270} = \frac{193}{90}\)
\(l = \frac{3}{2} + \frac{1}{4} + \frac{5}{8} + \frac{9}{16} = \frac{3 \cdot 8}{2 \cdot 8} + \frac{1 \cdot 4}{4 \cdot 4} + \frac{5 \cdot 2}{8 \cdot 2} + \frac{9}{16} = \frac{24}{16} + \frac{4}{16} + \frac{10}{16} + \frac{9}{16} = \frac{47}{16}\)
\(k = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{6} = \frac{6}{12} + \frac{4}{12} + \frac{3}{12} + \frac{2}{12} = \frac{15}{12} = \frac{5}{4}\)
\(j = \frac{5}{18} + \left(\frac{3}{18} + \frac{15}{27}\right) = \frac{5}{18} + \left(\frac{3}{18} + \frac{10}{18}\right) = \frac{5}{18} + \frac{13}{18} = \frac{18}{18} = 1\)
تمرين 11: احسب ثم اختزل
\(D = \frac{15}{14} - \frac{8}{21}\)
\(C = \frac{35}{45} - \frac{16}{36}\)
\(B = \frac{17}{11} - \frac{14}{22}\)
\(A = \frac{15}{8} - \frac{11}{8}\)
\(G = \frac{97}{63} - \left(\frac{5}{21} + \frac{7}{9}\right)\)
\(F = \frac{19}{24} + \left(\frac{5}{12} - \frac{7}{8}\right)\)
\(E = \frac{11}{16} - \frac{6}{25}\)
\(I = \left(\frac{25}{17} + \frac{101}{102}\right) - \left(\frac{8}{17} + \frac{101}{102}\right)\)
\(H = \left(\frac{37}{39} - \frac{9}{26}\right) - \frac{7}{26}\)
\(K = \left(\frac{56}{7} + \frac{3}{17}\right) + \left(\frac{8}{19} - \frac{3}{17}\right)\)
\(J = \left(\frac{49}{28} - \frac{5}{11}\right) - \left(\frac{21}{28} - \frac{5}{11}\right)\)
الحل:
\(D = \frac{15}{14} - \frac{8}{21} = \frac{15 \cdot 3}{14 \cdot 3} - \frac{8 \cdot 2}{21 \cdot 2} = \frac{45}{42} - \frac{16}{42} = \frac{29}{42}\)
\(C = \frac{35}{45} - \frac{16}{36} = \frac{35 \cdot 4}{45 \cdot 4} - \frac{16 \cdot 5}{36 \cdot 5} = \frac{140}{180} - \frac{80}{180} = \frac{60}{180} = \frac{1}{3}\)
\(B = \frac{17}{11} - \frac{14}{22} = \frac{17 \cdot 2}{11 \cdot 2} - \frac{14}{22} = \frac{34}{22} - \frac{14}{22} = \frac{20}{22} = \frac{10}{11}\)
\(A = \frac{15}{8} - \frac{11}{8} = \frac{15 - 11}{8} = \frac{4}{8} = \frac{1}{2}\)
\(G = \frac{97}{63} - \left(\frac{5}{21} + \frac{7}{9}\right) = \frac{97}{63} - \left(\frac{5 \cdot 3}{21 \cdot 3} + \frac{7 \cdot 7}{9 \cdot 7}\right) = \frac{97}{63} - \left(\frac{15}{63} + \frac{49}{63}\right) = \frac{97}{63} - \frac{64}{63} = \frac{33}{63} = \frac{11}{21}\)
\(F = \frac{19}{24} + \left(\frac{5}{12} - \frac{7}{8}\right) = \frac{19}{24} + \left(\frac{5 \cdot 2}{12 \cdot 2} - \frac{7 \cdot 3}{8 \cdot 3}\right) = \frac{19}{24} + \left(\frac{10}{24} - \frac{21}{24}\right) = \frac{19}{24} + \frac{-11}{24} = \frac{8}{24} = \frac{1}{3}\)
\(E = \frac{11}{16} - \frac{6}{25} = \frac{11 \cdot 25}{16 \cdot 25} - \frac{6 \cdot 16}{25 \cdot 16} = \frac{275}{400} - \frac{96}{400} = \frac{179}{400}\)
\(I = \left(\frac{25}{17} + \frac{101}{102}\right) - \left(\frac{8}{17} + \frac{101}{102}\right) = \frac{25}{17} - \frac{8}{17} = \frac{17}{17} = 1\)
\(H = \left(\frac{37}{39} - \frac{9}{26}\right) - \frac{7}{26} = \frac{37 \cdot 2}{39 \cdot 2} - \frac{9}{26} - \frac{7}{26} = \frac{74}{78} - \frac{9 + 7}{26} = \frac{74}{78} - \frac{16}{26} = \frac{37}{39} - \frac{8}{13} = \frac{37 \cdot 13}{39 \cdot 13} - \frac{8 \cdot 39}{13 \cdot 39} = \frac{481}{507} - \frac{312}{507} = \frac{169}{507}\)
\(K = \left(\frac{56}{7} + \frac{3}{17}\right) + \left(\frac{8}{19} - \frac{3}{17}\right) = \frac{56}{7} + \frac{8}{19} = \frac{56 \cdot 19}{7 \cdot 19} + \frac{8 \cdot 7}{19 \cdot 7} = \frac{1064}{133} + \frac{56}{133} = \frac{1120}{133}\)
\(J = \left(\frac{49}{28} - \frac{5}{11}\right) - \left(\frac{21}{28} - \frac{5}{11}\right) = \frac{49 - 21}{28} = \frac{28}{28} = 1\)
تمرين 12: احسب العبارات التالية
\(c = \frac{34}{21} \times \frac{28}{26} \times \frac{39}{51}\)
\(b = \frac{15}{8} \times \frac{21}{21} \times \frac{14}{5}\)
\(a = \frac{4}{9} \times \frac{3}{7}\)
\(f = \frac{45}{7} \times \frac{11}{18} \times \frac{13}{9} \times \frac{2}{26}\)
\(e = \frac{7}{12} \times \frac{3}{8} \times \frac{2}{9}\)
\(d = \frac{11}{8} \times \frac{7}{9} \times \frac{3}{14}\)
\(i = \frac{9}{5} + \frac{2}{3} \times \frac{9}{8} - \frac{1}{5}\)
\(h = \left(\frac{7}{4} + \frac{11}{6}\right) \times \left(\frac{7}{4} - \frac{3}{4}\right)\)
\(g = \frac{7}{8} + \frac{13}{9} \times \left(\frac{8}{3} - \frac{5}{6}\right)\)
الحل:
\(c = \frac{34}{21} \times \frac{28}{26} \times \frac{39}{51} = \frac{34 \times 28 \times 39}{21 \times 26 \times 51} = \frac{37128}{27846} = \frac{4}{3}\)
\(b = \frac{15}{8} \times \frac{21}{21} \times \frac{14}{5} = \frac{15 \times 1 \times 14}{8 \times 1 \times 5} = \frac{210}{40} = \frac{21}{4}\)
\(a = \frac{4}{9} \times \frac{3}{7} = \frac{4 \times 3}{9 \times 7} = \frac{12}{63} = \frac{4}{21}\)
\(f = \frac{45}{7} \times \frac{11}{18} \times \frac{13}{9} \times \frac{2}{26} = \frac{45 \times 11 \times 13 \times 2}{7 \times 18 \times 9 \times 26} = \frac{13585}{29484} = \frac{13585}{29484}\)
\(e = \frac{7}{12} \times \frac{3}{8} \times \frac{2}{9} = \frac{7 \times 3 \times 2}{12 \times 8 \times 9} = \frac{42}{864} = \frac{1}{20.57} \approx \frac{1}{21}\)
\(d = \frac{11}{8} \times \frac{7}{9} \times \frac{3}{14} = \frac{11 \times 7 \times 3}{8 \times 9 \times 14} = \frac{231}{1008} = \frac{231}{1008} = \frac{11}{48}\)
\(i = \frac{9}{5} + \frac{2}{3} \times \frac{9}{8} - \frac{1}{5} = \frac{9}{5} + \frac{2 \times 9}{3 \times 8} - \frac{1}{5} = \frac{9}{5} + \frac{18}{24} - \frac{1}{5} = \frac{9}{5} + \frac{3}{4} - \frac{1}{5} = \frac{36}{20} + \frac{15}{20} - \frac{4}{20} = \frac{47}{20}\)
\(h = \left(\frac{7}{4} + \frac{11}{6}\right) \times \left(\frac{7}{4} - \frac{3}{4}\right) = \left(\frac{21}{12} + \frac{22}{12}\right) \times \left(\frac{7 - 3}{4}\right) = \frac{43}{12} \times \frac{4}{4} = \frac{43}{12} \times 1 = \frac{43}{12}\)
\(g = \frac{7}{8} + \frac{13}{9} \times \left(\frac{8}{3} - \frac{5}{6}\right) = \frac{7}{8} + \frac{13}{9} \times \left(\frac{16}{6} - \frac{5}{6}\right) = \frac{7}{8} + \frac{13}{9} \times \frac{11}{6} = \frac{7}{8} + \frac{13 \times 11}{9 \times 6} = \frac{7}{8} + \frac{143}{54} = \frac{21}{24} + \frac{143}{54} = \frac{63}{72} + \frac{191}{72} = \frac{254}{72} = \frac{127}{36}\)
تمرين 13: احسب العبارات التالية
\(Z = \frac{9}{14} \times \left(14 - \frac{14}{3}\right)\)
\(Y = \frac{5}{7} \times \left(\frac{42}{25} - \frac{14}{15}\right)\)
\(X = \frac{15}{4} \times \left(\frac{6}{5} + \frac{2}{5}\right)\)
\(V = \frac{15}{28} \times \frac{13}{5} \times \frac{20}{36} \times \frac{2}{3}\)
\(U = \frac{3}{8} \times \frac{17}{21} - \frac{3}{8} \times \frac{3}{7}\)
\(T = \frac{2}{9} \times \frac{3}{11} + \frac{2}{9} \times \frac{8}{11}\)
الحل:
\(Z = \frac{9}{14} \times \left(14 - \frac{14}{3}\right) = \frac{9}{14} \times \left(\frac{42}{3} - \frac{14}{3}\right) = \frac{9}{14} \times \frac{28}{3} = \frac{9 \times 28}{14 \times 3} = \frac{252}{42} = 6\)
\(Y = \frac{5}{7} \times \left(\frac{42}{25} - \frac{14}{15}\right) = \frac{5}{7} \times \left(\frac{42 \times 3}{25 \times 3} - \frac{14 \times 5}{15 \times 5}\right) = \frac{5}{7} \times \left(\frac{126}{75} - \frac{70}{75}\right) = \frac{5}{7} \times \frac{56}{75} = \frac{5 \times 56}{7 \times 75} = \frac{280}{525} = \frac{8}{15}\)
\(X = \frac{15}{4} \times \left(\frac{6}{5} + \frac{2}{5}\right) = \frac{15}{4} \times \frac{8}{5} = \frac{15 \times 8}{4 \times 5} = \frac{120}{20} = 6\)
\(V = \frac{15}{28} \times \frac{13}{5} \times \frac{20}{36} \times \frac{2}{3} = \frac{15 \times 13 \times 20 \times 2}{28 \times 5 \times 36 \times 3} = \frac{7800}{15120} = \frac{65}{126} = \frac{5}{9.69} \approx \frac{1}{2}\)
\(U = \frac{3}{8} \times \frac{17}{21} - \frac{3}{8} \times \frac{3}{7} = \frac{3}{8} \times \left(\frac{17}{21} - \frac{3}{7}\right) = \frac{3}{8} \times \left(\frac{17 \times 1}{21 \times 1} - \frac{3 \times 3}{7 \times 3}\right) = \frac{3}{8} \times \left(\frac{17}{21} - \frac{9}{21}\right) = \frac{3}{8} \times \frac{8}{21} = \frac{3 \times 8}{8 \times 21} = \frac{24}{168} = \frac{1}{7}\)
\(T = \frac{2}{9} \times \frac{3}{11} + \frac{2}{9} \times \frac{8}{11} = \frac{2}{9} \times \left(\frac{3}{11} + \frac{8}{11}\right) = \frac{2}{9} \times \frac{11}{11} = \frac{2}{9} \times 1 = \frac{2}{9}\)
تمرين 14: احسب
\(\frac{2}{3} \div 5\)
\(\frac{2}{3} \div \frac{5}{4}\)
\(\frac{4}{15} \div \frac{2}{3}\)
\(\frac{1}{0.8}\)
\(\frac{1}{7} \div \frac{6}{5}\)
\(\frac{1}{5} \div \frac{8}{9}\)
الحل:
\(\frac{2}{3} \div 5 = \frac{2}{3} \div \frac{5}{1} = \frac{2}{3} \times \frac{1}{5} = \frac{2 \times 1}{3 \times 5} = \frac{2}{15}\)
\(\frac{2}{3} \div \frac{5}{4} = \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}\)
\(\frac{4}{15} \div \frac{2}{3} = \frac{4}{15} \times \frac{3}{2} = \frac{4 \times 3}{15 \times 2} = \frac{12}{30} = \frac{2}{5}\)
\(\frac{1}{0.8} = \frac{1}{\frac{8}{10}} = \frac{10}{8} = \frac{5}{4}\)
\(\frac{1}{7} \div \frac{6}{5} = \frac{1}{7} \times \frac{5}{6} = \frac{1 \times 5}{7 \times 6} = \frac{5}{42}\)
\(\frac{1}{5} \div \frac{8}{9} = \frac{1}{5} \times \frac{9}{8} = \frac{1 \times 9}{5 \times 8} = \frac{9}{40}\)
تمرين 15: احسب ثم اختزل إلى أقصى حد
\(\frac{4}{9} \div \frac{5}{3} \div \frac{7}{2}\)
\(\frac{11}{15} \div \frac{9}{37} \div \frac{5}{6}\)
\(\frac{2}{5} \div \frac{3}{4}\)
\(\frac{7}{9} - \frac{8}{3}\)
\(\frac{18}{3} + \frac{9}{24}\)
الحل:
\(\frac{4}{9} \div \frac{5}{3} \div \frac{7}{2} = \frac{4}{9} \times \frac{3}{5} \times \frac{2}{7} = \frac{4 \times 3 \times 2}{9 \times 5 \times 7} = \frac{24}{315} = \frac{8}{105}\)
\(\frac{11}{15} \div \frac{9}{37} \div \frac{5}{6} = \frac{11}{15} \times \frac{37}{9} \times \frac{6}{5} = \frac{11 \times 37 \times 6}{15 \times 9 \times 5} = \frac{2442}{675} = \frac{814}{225}\)
\(\frac{2}{5} \div \frac{3}{4} = \frac{2}{5} \times \frac{4}{3} = \frac{2 \times 4}{5 \times 3} = \frac{8}{15}\)
\(\frac{7}{9} - \frac{8}{3} = \frac{7}{9} - \frac{24}{9} = \frac{7 - 24}{9} = \frac{-17}{9}\)
\(\frac{18}{3} + \frac{9}{24} = 6 + \frac{9}{24} = 6 + \frac{3}{8} = \frac{48}{8} + \frac{3}{8} = \frac{51}{8}\)
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