🔢 تمرين 1
العمليات الأساسية على الكسور
$$c = -1.2 + \left(-\frac{2}{5}\right)$$
$$b = \frac{1}{2} + \frac{1}{3}$$
$$a = -1 + \left(-\frac{2}{3}\right)$$
الحل:
$$a = -1 + \left(-\frac{2}{7}\right) = -\frac{7}{7} + \left(-\frac{2}{7}\right) = \frac{-7+(-2)}{7} = \frac{-9}{7}$$
$$b = -\frac{1}{2} + \frac{1}{3} = \frac{-3}{6} + \frac{2}{6} = \frac{-3+2}{6} = \frac{1}{6}$$
$$c = -1.2 + \left(-\frac{2}{5}\right) = -\frac{12}{10} + \left(-\frac{2}{5}\right) = -\frac{6}{5} + \left(-\frac{2}{5}\right) = \left[-\frac{8}{5}\right] = -\frac{16}{10} = -1.6$$
🔢 تمرين 2
الجمع والطرح المختلط
$$c = -\frac{2}{3} - 2.5$$
$$b = -\frac{1}{9} + \frac{7}{6}$$
$$a = -2 + \frac{11}{3}$$
الحل:
$$a = -2 + \frac{11}{3} = -\frac{6}{3} + \frac{11}{3} = \frac{-6+11}{3} = \frac{5}{3}$$
$$b = -\frac{1}{9} + \frac{7}{6} = -\frac{2}{18} + \frac{21}{18} = \frac{-2+21}{18} = \frac{19}{18}$$
$$c = -\frac{2}{3} + 2.5 = -\frac{2}{3} + \frac{5}{2} = -\frac{4}{6} + \frac{15}{6} = \frac{-4+15}{6} = \frac{11}{6}$$
🔢 تمرين 3
العبارات المركبة
$$b = \left(-1.2 + \frac{3}{5}\right) + \left[\left(-\frac{1}{2}\right) + \frac{4}{3} + (-2)\right]$$
$$a = \frac{1}{4} + \left(-\frac{7}{3}\right) + \frac{11}{12} + (-0.25)$$
الحل:
$$a = \frac{1}{4} + \left(-\frac{7}{3}\right) + \frac{11}{12} + (-0.25)$$
$$= \frac{3}{12} + \left(-\frac{28}{12}\right) + \frac{11}{12} + \left(-\frac{3}{12}\right)$$
$$= \frac{3-28+11-3}{12} = \frac{-17}{12}$$
$$b = \left(-1.2 + \frac{3}{5}\right) + \left[\left(-\frac{1}{2}\right) + \frac{4}{3} + (-2)\right]$$
$$= \left(-1.2 + 0.6\right) + \left[(-0.5) + \frac{4}{3} + (-2)\right]$$
$$= -0.6 + \left[-2.5 + \frac{4}{3}\right] = -0.6 - 2.5 + \frac{4}{3}$$
$$= -3.1 + \frac{4}{3} = \frac{-93}{30} + \frac{40}{30} = \frac{-53}{30}$$
🔢 تمرين 4
حساب العبارات الجدولية
$$\frac{11}{6} + \left(\frac{7}{6} + 123.1\right) = \ldots$$
$$\left(-\frac{9}{5} + \frac{4}{17}\right) + \left(\frac{11}{5} - \frac{4}{17}\right) = \ldots$$
$$0.52 + \frac{1}{3} + 2.48 + \frac{503}{797} + \left(-\frac{10}{3}\right) = \ldots$$
$$\left(\frac{7}{18} + \left(-\frac{3}{4}\right)\right) + \frac{6}{13} = \ldots$$
الحل:
| $$\frac{11}{6} + \left(\frac{7}{6} + 123.1\right) = \left(\frac{11}{6} + \frac{7}{6}\right) + 123.1 = \frac{18}{6} + 123.1 = 3 + 123.1 = 126.1$$ |
| $$\left(-\frac{9}{5} + \frac{4}{17}\right) + \left(\frac{11}{5} - \frac{4}{17}\right) = \frac{-9+11}{5} + \frac{4-4}{17} = \frac{2}{5}$$ |
| $$0.52 + \frac{1}{3} + 2.48 + \frac{503}{797} + \left(-\frac{10}{3}\right) = 3 + \left(-\frac{9}{3}\right) + \frac{503}{797} = \frac{503}{797}$$ |
| $$\left(\frac{7}{18} + \left(-\frac{3}{4}\right)\right) + \frac{6}{13} = \frac{14}{36} + \left(-\frac{27}{36}\right) + \frac{6}{13} = -\frac{13}{36} + \frac{6}{13}$$ |
🔢 تمرين 5
شبكة العمليات
$$-1.4 - \frac{5}{4}$$
$$\frac{6}{7} + \left(-\frac{2}{21}\right)$$
$$0 - \left(-\frac{2}{3}\right)$$
$$\frac{13}{6} - \frac{5}{12}$$
$$\frac{2}{7} - \frac{8}{3}$$
$$\frac{8}{15} - \frac{1}{5}$$
الحل:
| $$-1.4 - \frac{5}{4}$$ | $$\frac{6}{7} + \left(-\frac{2}{21}\right)$$ | $$0 - \left(-\frac{2}{3}\right)$$ |
|---|---|---|
| $$= -\frac{14}{10} - \frac{5}{4} = -\frac{7}{5} - \frac{5}{4} = \frac{-28-25}{20} = -\frac{53}{20}$$ | $$= \frac{18}{21} - \frac{2}{21} = \frac{16}{21}$$ | $$= 0 + \frac{2}{3} = \frac{2}{3}$$ |
| $$\frac{13}{6} - \frac{5}{12}$$ | $$\frac{2}{7} - \frac{8}{3}$$ | $$\frac{8}{15} - \frac{1}{5}$$ |
| $$= \frac{26-5}{12} = \frac{21}{12} = \frac{7}{4}$$ | $$= \frac{6-56}{21} = \frac{-50}{21}$$ | $$= \frac{8-3}{15} = \frac{5}{15} = \frac{1}{3}$$ |
🔢 تمرين 6
الحساب بأيسر طريقة
احسب العبارات التالية بأيسر طريقة:
$$a = \frac{25}{13} + \left(0.95 + \frac{12}{13}\right)$$
$$b = \left(\frac{23}{7} + \frac{819}{111}\right) - \left(\frac{4}{14} + \frac{819}{111}\right) + \left(\frac{9}{15} - 9.831\right) - \left(\frac{3}{5} - 9.831\right)$$
الحل:
$$a = \frac{25}{13} + \left(0.95 + \frac{12}{13}\right) = \frac{25}{13} + 0.95 + \frac{12}{13}$$
$$= \frac{25 + 12}{13} + 0.95 = \frac{37}{13} + 0.95$$
$$= \frac{37}{13} + \frac{95}{100} = \frac{37}{13} + \frac{19}{20}$$
$$b = \left(\frac{23}{7} + \frac{819}{111}\right) - \left(\frac{4}{14} + \frac{819}{111}\right) + \left(\frac{9}{15} - 9.831\right) - \left(\frac{3}{5} - 9.831\right)$$
بتجميع الحدود المتشابهة:
$$= \left(\frac{23}{7} - \frac{4}{14}\right) + \left(\frac{819}{111} - \frac{819}{111}\right) + \left(\frac{9}{15} - \frac{3}{5}\right) + \left(-9.831 + 9.831\right)$$
$$= \left(\frac{23}{7} - \frac{2}{7}\right) + 0 + \left(\frac{9}{15} - \frac{9}{15}\right) + 0$$
$$= \frac{21}{7} + 0 = 3$$
📏 مسألة هندسية
المستقيم المدرج والإحداثيات
يعتبر المستقيم $$\Delta$$ المقل المدرج بواسطة المعين $$(O,I)$$
نرقم فاصلة كل من $$O$$ و $$I$$ و $$A$$ و $$B$$ حسب المعين $$(O,I)$$
نرقم فاصلة النقاط $$M$$ و $$N$$ و $$P$$ التي فاصلاتها على التوالي $$2.5$$ و $$\frac{1}{2}$$ و $$-3.5$$
ماذا نستنتج بالنسبة إلى النقطة $$N$$؟
نعتبر البعدين $$MN$$ و $$PN$$
أحسب البعد $$MN$$ و $$PN$$
P O N I M
| | | | |
-3.5 0 0.5 1 2.5
| | | | |
-3.5 0 0.5 1 2.5
تحليل موقع النقاط:
النقطة $$N$$ فاصلتها $$\frac{1}{2} = 0.5$$ تقع بين $$O$$ (فاصلتها 0) و $$I$$ (فاصلتها 1)
استنتاج: النقطة $$N$$ تقع على القطعة $$[OI]$$ وتقسمها بنسبة $$1:1$$
حساب المسافات:
$$MN = |x_M - x_N| = |2.5 - 0.5| = |2| = 2$$
$$PN = |x_P - x_N| = |-3.5 - 0.5| = |-4| = 4$$
🔢 تمرين 7
المعادلات والمتغيرات
$$a - b = -\frac{4}{3}$$ بحيث $$b \in \mathbb{Q}$$ و $$a \in \mathbb{Q}$$
$$Y = b + \left(\frac{7}{3} - a\right)$$ و $$X = a - (b - 1)$$
الحل:
$$X = a - (b - 1) = a - b + 1 = -\frac{4}{3} + 1 = -\frac{4}{3} + \frac{3}{3} = -\frac{1}{3}$$
$$Y = b + \left(\frac{7}{3} - a\right) = b - a + \frac{7}{3} = -(a-b) + \frac{7}{3}$$
$$= -\left(-\frac{4}{3}\right) + \frac{7}{3} = \frac{4}{3} + \frac{7}{3} = \frac{11}{3}$$
🔢 تمرين 8
حل المعادلات الجبرية
$$x \in \mathbb{Q}$$ حيث $$A = -\frac{3}{2} + \left(x - \frac{1}{6}\right)$$
إذا $$A = -\frac{4}{3} + x$$
احسب $$A$$ في الحالتين $$6x - 1 = -9$$ و $$x = -2$$
أوجد قيمة $$x$$ إذا $$A = \frac{19}{9}$$
الحل:
$$A = -\frac{3}{2} + \left(x - \frac{1}{6}\right) = -\frac{3}{2} + x - \frac{1}{6}$$
$$= -\frac{9}{6} + x - \frac{1}{6} = x - \frac{10}{6} = x - \frac{5}{3}$$
في الحالة الأولى: $$6x - 1 = -9$$
$$6x = -8 \Rightarrow x = -\frac{4}{3}$$
$$A = -\frac{4}{3} + \left(-\frac{4}{3}\right) = -\frac{8}{3}$$
$$6x = -8 \Rightarrow x = -\frac{4}{3}$$
$$A = -\frac{4}{3} + \left(-\frac{4}{3}\right) = -\frac{8}{3}$$
في الحالة الثانية: $$x = -2$$
$$A = -\frac{4}{3} + (-2) = -\frac{4}{3} - \frac{6}{3} = -\frac{10}{3}$$
$$A = -\frac{4}{3} + (-2) = -\frac{4}{3} - \frac{6}{3} = -\frac{10}{3}$$
إيجاد قيمة x عندما A = 19/9:
$$A = \frac{19}{9} \Leftrightarrow -\frac{4}{3} + x = \frac{19}{9}$$
$$x = \frac{19}{9} + \frac{4}{3} = \frac{19}{9} + \frac{12}{9} = \frac{31}{9}$$
$$A = \frac{19}{9} \Leftrightarrow -\frac{4}{3} + x = \frac{19}{9}$$
$$x = \frac{19}{9} + \frac{4}{3} = \frac{19}{9} + \frac{12}{9} = \frac{31}{9}$$