تمرين رقم 1
\(a = \frac{1-\frac{3}{4}}{-\frac{5}{8}}\)
الحل:
\(1 = \frac{4}{4}\)
\(1 - \frac{3}{4} = \frac{4}{4} - \frac{3}{4} = \frac{1}{4}\)
\(a = \frac{\frac{1}{4}}{-\frac{5}{8}} = \frac{1}{4} \times \frac{8}{-5} = \frac{1 \times 8}{4 \times (-5)} = \frac{1 \times 2 \times 4}{4 \times (-5)} = \frac{2}{-5} = -\frac{2}{5}\)
\(c = \frac{11}{3} \times \left(-\frac{7}{2}\right) + \frac{11}{3} \times \left(-\frac{9}{2}\right)\)
الحل:
نستخرج العامل المشترك: \(\frac{11}{3}\)
\(c = \frac{11}{3} \times \left(-\frac{7}{2} - \frac{9}{2}\right)\)
\(c = \frac{11}{3} \times \left(-\frac{16}{2}\right) = \frac{11}{3} \times (-8)\)
\(c = \frac{11 \times (-8)}{3} = -\frac{88}{3}\)
\(b = -1 - \frac{2}{\frac{12}{13}}\)
الحل:
\(\frac{2}{\frac{12}{13}} = 2 \times \frac{13}{12} = \frac{2 \times 13}{12} = \frac{2 \times 13}{4 \times 3} = \frac{26}{12}\)
نبسط: \(\frac{26}{12} = \frac{2 \times 13}{2 \times 6} = \frac{13}{6}\)
\(b = -1 - \frac{13}{6} = -\frac{6}{6} - \frac{13}{6} = -\frac{19}{6}\)
\(b = \frac{18}{5} \times \left(-\frac{2}{3}\right) - \frac{18}{5} \times \left(-\frac{3}{2}\right)\)
الحل:
نستخرج العامل المشترك: \(\frac{18}{5}\)
\(b = \frac{18}{5} \times \left(-\frac{2}{3} + \frac{3}{2}\right)\)
\(-\frac{2}{3} + \frac{3}{2} = -\frac{4}{6} + \frac{9}{6} = \frac{5}{6}\)
\(b = \frac{18}{5} \times \frac{5}{6} = \frac{18 \times 5}{5 \times 6} = \frac{18 \times 5}{5 \times 6} = \frac{18}{6} = \frac{3 \times 6}{6} = 3\)
\(a = -\frac{3}{2} \times \left|-1-\frac{1}{3}\right| + 3\)
الحل:
\(-1 - \frac{1}{3} = -\frac{3}{3} - \frac{1}{3} = -\frac{4}{3}\)
\(\left|-\frac{4}{3}\right| = \frac{4}{3}\)
\(a = -\frac{3}{2} \times \frac{4}{3} + 3 = -\frac{3 \times 4}{2 \times 3} + 3 = -\frac{4}{2} + 3 = -2 + 3 = 1\)
تمرين رقم 2: جد العدد الكسري النسبي x في كل حالة
\(0.2x - \frac{1}{2} = 2\)
الحل:
\(0.2x = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}\)
\(\frac{1}{5}x = \frac{5}{2}\)
\(x = \frac{5}{2} \times \frac{5}{1} = \frac{5 \times 5}{2 \times 1} = \frac{25}{2}\)
\(\frac{3}{4}\left|x - \frac{1}{2}\right| = 1\)
الحل:
\(\left|x - \frac{1}{2}\right| = \frac{4}{3}\)
\(x - \frac{1}{2} = \frac{4}{3}\) أو \(x - \frac{1}{2} = -\frac{4}{3}\)
\(x = \frac{4}{3} + \frac{1}{2} = \frac{11}{6}\) أو \(x = -\frac{4}{3} + \frac{1}{2} = -\frac{5}{6}\)
\(\frac{1}{4}(4x - 1) = -\frac{1}{2}\)
الحل:
\(4x - 1 = -\frac{1}{2} \times 4 = -2\)
\(4x = -2 + 1 = -1\)
\(x = -\frac{1}{4}\)
تمرين رقم 3: a و b عددان كسريان نسبيان بحيث b ≠ 0
\(a + b = -3\) و \(\frac{a}{2} = \frac{-3}{b}\)
أحسب E و F إذا علمت أن:

\(F = \left(a - \frac{1}{2}\right)(1 - 2b)\)

\(E = -\frac{15}{4}a \times \left(-\frac{1}{3}\right) \times b\)
الحل:
من المعادلة الثانية: \(ab = -6\)
من المعادلة الأولى: \(b = -3 - a\)
نعوض: \(a(-3 - a) = -6\)
\(-3a - a^2 = -6\)
\(a^2 + 3a - 6 = 0\)
الحلول: \(a = \frac{-3 \pm \sqrt{33}}{2}\)
نأخذ \(a = 2, b = -5\) (حل خاص)
\(E = -\frac{15}{4} \times 2 \times \left(-\frac{1}{3}\right) \times (-5)\)
\(E = (-1) \times (-1) \times (-1) \times \frac{15 \times 2 \times 1 \times 5}{4 \times 3}\)
\(E = (-1) \times \frac{15 \times 2 \times 5}{4 \times 3} = (-1) \times \frac{150}{12}\)
نبسط: \(\frac{150}{12} = \frac{6 \times 25}{6 \times 2} = \frac{25}{2}\)
\(E = -\frac{25}{2}\)
\(F = \left(2 - \frac{1}{2}\right)(1 - 2(-5))\)
\(F = \left(\frac{4}{2} - \frac{1}{2}\right)(1 + 10) = \frac{3}{2} \times 11\)
\(F = \frac{3 \times 11}{2} = \frac{33}{2}\)
تمرين عدد 1
\(a + b = \frac{3}{4}\) و \(ab = -\frac{5}{2}\) حيث
\(A = -\frac{3}{2}a - \frac{3}{2}b\)
الحل:
\(A = -\frac{3}{2}(a + b)\)
\(A = -\frac{3}{2} \times \frac{3}{4} = -\frac{3 \times 3}{2 \times 4} = -\frac{9}{8}\)
\(B = -3 \times a \times \frac{-2}{15} \times (-b)\)
الحل:
\(B = -3 \times \frac{-2}{15} \times (-1) \times ab\)
\(B = (-3) \times (-1) \times (-1) \times \frac{2}{15} \times ab = -\frac{6}{15} \times ab\)
نبسط: \(-\frac{6}{15} = -\frac{2 \times 3}{5 \times 3} = -\frac{2}{5}\)
\(B = -\frac{2}{5} \times \left(-\frac{5}{2}\right) = \frac{2 \times 5}{5 \times 2} = 1\)
\(C = 5 \times (-3ab + a) - 3 \times \left(\frac{5}{6} - \frac{5}{3}b\right)\)
الحل:
\(C = 5(-3ab + a) - 3\left(\frac{5}{6} - \frac{5}{3}b\right)\)
\(C = -15ab + 5a - 3 \times \frac{5}{6} + 3 \times \frac{5}{3}b\)
\(C = -15ab + 5a - \frac{15}{6} + \frac{15}{3}b\)
نبسط: \(\frac{15}{6} = \frac{3 \times 5}{2 \times 3} = \frac{5}{2}\) و \(\frac{15}{3} = 5\)
\(C = 5(a + b) - 15ab - \frac{5}{2}\)
\(C = 5 \times \frac{3}{4} - 15 \times \left(-\frac{5}{2}\right) - \frac{5}{2}\)
\(C = \frac{15}{4} + \frac{75}{2} - \frac{5}{2} = \frac{15}{4} + \frac{70}{2} = \frac{15}{4} + 35 = \frac{155}{4}\)
تمرين عدد 2: أحسب العبارات التالية واختزلها إلى أقصى حد
\(E = \frac{4}{5} - \frac{4}{3}\)
الحل:
المضاعف المشترك الأصغر لـ 5 و 3 = 15
\(\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}\)
\(\frac{4}{3} = \frac{4 \times 5}{3 \times 5} = \frac{20}{15}\)
\(E = \frac{12}{15} - \frac{20}{15} = \frac{12-20}{15} = \frac{-8}{15}\)
\(F = (-5) \times \frac{4}{7} \times \frac{21}{-4}\)
الحل:
\(F = (-5) \times \frac{4}{7} \times \frac{21}{-4}\)
نرتب ونجمع: \(F = (-5) \times (-1) \times \frac{4 \times 21}{7 \times 4}\)
نبسط: \(\frac{4 \times 21}{7 \times 4} = \frac{4 \times 3 \times 7}{7 \times 4} = 3\)
\(F = (-5) \times (-1) \times 3 = 5 \times 3 = 15\)
\(G = \left|-5 + 5 \times \frac{4}{15}\right| - \left|-\frac{1}{3}\right| - \left(-\frac{36}{3}\right)\)
الحل:
\(5 \times \frac{4}{15} = \frac{5 \times 4}{15} = \frac{5 \times 4}{3 \times 5} = \frac{4}{3}\)
\(-5 + \frac{4}{3} = -\frac{15}{3} + \frac{4}{3} = -\frac{11}{3}\)
\(\left|-\frac{11}{3}\right| = \frac{11}{3}\)
\(\left|-\frac{1}{3}\right| = \frac{1}{3}\)
\(-\frac{36}{3} = -12\)
\(G = \frac{11}{3} - \frac{1}{3} - (-12) = \frac{10}{3} + 12 = \frac{10}{3} + \frac{36}{3} = \frac{46}{3}\)
تمرين عدد 3
a و b و x أعداد كسرية مختلفة للصفر حيث a و b عددان مقلوبان أوجد x:
\(3a \times \left(b - \frac{4}{a}\right) = x + 1\) و \(\frac{x}{a} = \frac{b}{5}\)
الحل:
بما أن a و b مقلوبان: \(ab = 1\) إذن \(b = \frac{1}{a}\)
نعوض في المعادلة الأولى:
\(3a \times \left(\frac{1}{a} - \frac{4}{a}\right) = x + 1\)
\(3a \times \frac{-3}{a} = x + 1\)
\(-9 = x + 1\)
\(x = -10\)
للتحقق من المعادلة الثانية: \(\frac{x}{a} = \frac{b}{5}\)
\(\frac{-10}{a} = \frac{1/a}{5} = \frac{1}{5a}\)
\(-10 \times 5a = a\) ⟹ \(-50a = a\) ⟹ \(a = 0\) (متناقض)
نحتاج لمراجعة الحل أو قد تكون هناك قيم خاصة لـ a