🧮 العمليات في الأقواس

أحسب بأيسر طريقة - مجموعة تمارين شاملة

1 التمرين الأول - أحسب بأيسر طريقة
$$A = \left(-\frac{7}{5} - \frac{1}{11}\right) - \left(\frac{3}{10} - \frac{1}{11}\right)$$
$$B = \left(\frac{5}{12} + \frac{1}{100}\right) - \left(-\frac{5}{18} + \frac{1}{100}\right)$$
$$C = \left(\frac{17}{5} + \frac{4}{7} - \frac{2}{3}\right) - \left(\frac{4}{7} + \frac{3}{2} - \frac{2}{3}\right)$$
$$D = -\frac{13}{15} - \left(\frac{2}{5} + \frac{1}{7}\right)$$

✨ الحل التفصيلي

$$\begin{align} A &= \left(-\frac{7}{5} - \frac{1}{11}\right) - \left(\frac{3}{10} - \frac{1}{11}\right)\\ &= -\frac{7}{5} - \frac{1}{11} - \frac{3}{10} + \frac{1}{11}\\ &= -\frac{7}{5} - \frac{3}{10}\\ &= -\frac{14}{10} - \frac{3}{10} = -\frac{17}{10} \end{align}$$
$$\begin{align} B &= \left(\frac{5}{12} + \frac{1}{100}\right) - \left(-\frac{5}{18} + \frac{1}{100}\right)\\ &= \frac{5}{12} + \frac{1}{100} + \frac{5}{18} - \frac{1}{100}\\ &= \frac{5}{12} + \frac{5}{18}\\ &= \frac{15}{36} + \frac{10}{36} = \frac{25}{36} \end{align}$$
$$\begin{align} C &= \left(\frac{17}{5} + \frac{4}{7} - \frac{2}{3}\right) - \left(\frac{4}{7} + \frac{3}{2} - \frac{2}{3}\right)\\ &= \frac{17}{5} + \frac{4}{7} - \frac{2}{3} - \frac{4}{7} - \frac{3}{2} + \frac{2}{3}\\ &= \frac{17}{5} - \frac{3}{2}\\ &= \frac{34}{10} - \frac{15}{10} = \frac{19}{10} \end{align}$$
$$\begin{align} D &= -\frac{13}{15} - \left(\frac{2}{5} + \frac{1}{7}\right)\\ &= -\frac{13}{15} - \frac{2}{5} - \frac{1}{7}\\ &= -\frac{91}{105} - \frac{42}{105} - \frac{15}{105}\\ &= -\frac{148}{105} \end{align}$$
2 التمرين الثاني - أحسب بأيسر طريقة
$$A = \left(\frac{5}{7} - \frac{3}{11}\right) + \left(\frac{2}{7} + \frac{3}{11}\right)$$
$$B = \left(-\frac{4}{9} + \frac{5}{13}\right) - \left(\frac{5}{9} - \frac{5}{13}\right)$$
$$C = \left(\frac{7}{3} - \frac{5}{4} + \frac{1}{6}\right) + \left(-\frac{1}{3} + \frac{9}{4} - \frac{1}{6}\right)$$
$$D = -\frac{9}{14} + \left(\frac{3}{7} - \frac{2}{5}\right)$$

✨ الحل التفصيلي

$$\begin{align} A &= \left(\frac{5}{7} - \frac{3}{11}\right) + \left(\frac{2}{7} + \frac{3}{11}\right)\\ &= \frac{5}{7} - \frac{3}{11} + \frac{2}{7} + \frac{3}{11}\\ &= \frac{5}{7} + \frac{2}{7} = \frac{7}{7} = 1 \end{align}$$
$$\begin{align} B &= \left(-\frac{4}{9} + \frac{5}{13}\right) - \left(\frac{5}{9} - \frac{5}{13}\right)\\ &= -\frac{4}{9} + \frac{5}{13} - \frac{5}{9} + \frac{5}{13}\\ &= -\frac{9}{9} + \frac{10}{13} = -1 + \frac{10}{13} = -\frac{3}{13} \end{align}$$
$$\begin{align} C &= \left(\frac{7}{3} - \frac{5}{4} + \frac{1}{6}\right) + \left(-\frac{1}{3} + \frac{9}{4} - \frac{1}{6}\right)\\ &= \frac{7}{3} - \frac{5}{4} + \frac{1}{6} - \frac{1}{3} + \frac{9}{4} - \frac{1}{6}\\ &= \frac{6}{3} + \frac{4}{4} = 2 + 1 = 3 \end{align}$$
$$\begin{align} D &= -\frac{9}{14} + \left(\frac{3}{7} - \frac{2}{5}\right)\\ &= -\frac{9}{14} + \frac{3}{7} - \frac{2}{5}\\ &= -\frac{9}{14} + \frac{6}{14} - \frac{2}{5}\\ &= -\frac{3}{14} - \frac{2}{5} = -\frac{15}{70} - \frac{28}{70} = -\frac{43}{70} \end{align}$$
3 التمرين الثالث - اختصر العبارات التالية
$$A = \frac{7}{5} - \left(\frac{3}{10} + \frac{7}{5}\right)$$
$$B = \frac{9}{8} + \left(\frac{5}{12} - \frac{9}{8}\right)$$
$$C = -\frac{11}{6} - \left(\frac{2}{3} - \frac{11}{6}\right)$$
$$D = \frac{15}{14} + \left(-\frac{3}{7} + \frac{15}{14}\right)$$

✨ الحل التفصيلي

$$\begin{align} A &= \frac{7}{5} - \left(\frac{3}{10} + \frac{7}{5}\right)\\ &= \frac{7}{5} - \frac{3}{10} - \frac{7}{5}\\ &= -\frac{3}{10} \end{align}$$
$$\begin{align} B &= \frac{9}{8} + \left(\frac{5}{12} - \frac{9}{8}\right)\\ &= \frac{9}{8} + \frac{5}{12} - \frac{9}{8}\\ &= \frac{5}{12} \end{align}$$
$$\begin{align} C &= -\frac{11}{6} - \left(\frac{2}{3} - \frac{11}{6}\right)\\ &= -\frac{11}{6} - \frac{2}{3} + \frac{11}{6}\\ &= -\frac{2}{3} \end{align}$$
$$\begin{align} D &= \frac{15}{14} + \left(-\frac{3}{7} + \frac{15}{14}\right)\\ &= \frac{15}{14} - \frac{3}{7} + \frac{15}{14}\\ &= \frac{30}{14} - \frac{6}{14} = \frac{24}{14} = \frac{12}{7} \end{align}$$
4 التمرين الرابع - اختصر العبارات التالية
$$A = \left(4 - \frac{3}{5}\right) - \left(2 + \frac{3}{5}\right)$$
$$B = \left(-3 + \frac{7}{9}\right) + \left(5 - \frac{7}{9}\right)$$
$$C = \left(\frac{5}{6} - 2\right) - \left(\frac{5}{6} + 3\right)$$
$$D = \left(\frac{11}{8} + 1\right) + \left(\frac{11}{8} - 4\right)$$

✨ الحل التفصيلي

$$\begin{align} A &= \left(4 - \frac{3}{5}\right) - \left(2 + \frac{3}{5}\right)\\ &= 4 - \frac{3}{5} - 2 - \frac{3}{5}\\ &= 2 - \frac{6}{5} = \frac{10}{5} - \frac{6}{5} = \frac{4}{5} \end{align}$$
$$\begin{align} B &= \left(-3 + \frac{7}{9}\right) + \left(5 - \frac{7}{9}\right)\\ &= -3 + \frac{7}{9} + 5 - \frac{7}{9}\\ &= 2 \end{align}$$
$$\begin{align} C &= \left(\frac{5}{6} - 2\right) - \left(\frac{5}{6} + 3\right)\\ &= \frac{5}{6} - 2 - \frac{5}{6} - 3\\ &= -5 \end{align}$$
$$\begin{align} D &= \left(\frac{11}{8} + 1\right) + \left(\frac{11}{8} - 4\right)\\ &= \frac{11}{8} + 1 + \frac{11}{8} - 4\\ &= \frac{22}{8} - 3 = \frac{11}{4} - 3 = -\frac{1}{4} \end{align}$$
5 التمرين الخامس - اختصر العبارات التالية
$$A = 3 - \left(7 - \frac{2}{3}\right) + \left(\frac{2}{3} + 5\right)$$
$$B = -2 + \left(6 + \frac{5}{8}\right) - \left(\frac{5}{8} - 1\right)$$
$$C = \left(\frac{9}{10} - 4\right) + \left(2 - \frac{9}{10}\right) + 5$$
$$D = \left(-\frac{7}{12} + 3\right) - \left(1 + \frac{7}{12}\right) - 8$$

✨ الحل التفصيلي

$$\begin{align} A &= 3 - \left(7 - \frac{2}{3}\right) + \left(\frac{2}{3} + 5\right)\\ &= 3 - 7 + \frac{2}{3} + \frac{2}{3} + 5\\ &= 1 + \frac{4}{3} = \frac{7}{3} \end{align}$$
$$\begin{align} B &= -2 + \left(6 + \frac{5}{8}\right) - \left(\frac{5}{8} - 1\right)\\ &= -2 + 6 + \frac{5}{8} - \frac{5}{8} + 1\\ &= 5 \end{align}$$
$$\begin{align} C &= \left(\frac{9}{10} - 4\right) + \left(2 - \frac{9}{10}\right) + 5\\ &= \frac{9}{10} - 4 + 2 - \frac{9}{10} + 5\\ &= 3 \end{align}$$
$$\begin{align} D &= \left(-\frac{7}{12} + 3\right) - \left(1 + \frac{7}{12}\right) - 8\\ &= -\frac{7}{12} + 3 - 1 - \frac{7}{12} - 8\\ &= -6 - \frac{14}{12} = -6 - \frac{7}{6} = -\frac{43}{6} \end{align}$$