تمارين في الرياضيات

تصحيح التمرين رقم 1:

a) $$ \frac{4}{9} \times \frac{3}{8} = \frac{4 \times 3}{9 \times 8} = \frac{12}{72} = \frac{1}{6} $$

b) $$ -\frac{12}{17} \times \frac{4}{5} = -\frac{12 \times 4}{17 \times 5} = -\frac{48}{85} $$

c) $$ \frac{-9}{4} \times \frac{-5}{3} = \frac{(-9) \times (-5)}{4 \times 3} = \frac{45}{12} = \frac{15}{4} $$

d) $$ \frac{8}{-11} \times \frac{-33}{20} = \frac{8 \times (-33)}{(-11) \times 20} = \frac{264}{220} = \frac{66}{55} = \frac{6}{5} $$

e) $$ \frac{-2}{-3} \times \frac{-15}{10} = \frac{2}{3} \times (-\frac{15}{10}) = \frac{2}{3} \times (-\frac{3}{2}) = -1 $$

f) $$ \frac{-12}{-7} \times \frac{-14}{-18} = \frac{12}{7} \times \frac{14}{18} = \frac{12}{7} \times \frac{7}{9} = \frac{12}{9} = \frac{4}{3} $$

g) $$ \frac{36}{45} \times \frac{15}{18} = \frac{4}{5} \times \frac{5}{6} = \frac{4}{6} = \frac{2}{3} $$

h) $$ \frac{3}{5} \times \frac{5}{4} \times \frac{(-12)}{18} = \frac{3}{4} \times (-\frac{2}{3}) = -\frac{2}{4} = -\frac{1}{2} $$

تصحيح التمرين رقم 2:

نشر ثم اختصار العبارات:

X:

$$ X = -5(a-\frac{3}{10}b)+\frac{1}{2}(10a-b) $$

$$ X = -5a + \frac{15}{10}b + \frac{10}{2}a - \frac{1}{2}b $$

$$ X = -5a + \frac{3}{2}b + 5a - \frac{1}{2}b $$

$$ X = (-5a + 5a) + (\frac{3}{2}b - \frac{1}{2}b) $$

$$ X = 0 + \frac{2}{2}b = b $$

Y:

$$ Y = \frac{-4}{3}(\frac{3}{4}a-\frac{1}{2}b+3)-\frac{2}{5}(\frac{5}{3}b-5a-10) $$

$$ Y = (-\frac{4}{3} \times \frac{3}{4}a) + (-\frac{4}{3} \times -\frac{1}{2}b) + (-\frac{4}{3} \times 3) + (-\frac{2}{5} \times \frac{5}{3}b) + (-\frac{2}{5} \times -5a) + (-\frac{2}{5} \times -10) $$

$$ Y = -a + \frac{2}{3}b - 4 - \frac{2}{3}b + 2a + 4 $$

$$ Y = (-a + 2a) + (\frac{2}{3}b - \frac{2}{3}b) + (-4 + 4) $$

$$ Y = a $$

Z:

$$ Z = \frac{1}{2}a(\frac{6}{5}-b)-5b(\frac{1}{5}-\frac{3}{10}a)+\frac{3}{5}(-a+\frac{5}{3}) $$

$$ Z = \frac{3}{5}a - \frac{1}{2}ab - b + \frac{3}{2}ab - \frac{3}{5}a + 1 $$

$$ Z = (\frac{3}{5}a - \frac{3}{5}a) + (-\frac{1}{2}ab + \frac{3}{2}ab) - b + 1 $$

$$ Z = ab - b + 1 $$

T:

$$ T = (\frac{3}{2}a-5)(1-b)-(a-2)(5-\frac{3}{2}b) $$

$$ T = (\frac{3}{2}a - \frac{3}{2}ab - 5 + 5b) - (5a - \frac{3}{2}ab - 10 + 3b) $$

$$ T = \frac{3}{2}a - \frac{3}{2}ab - 5 + 5b - 5a + \frac{3}{2}ab + 10 - 3b $$

$$ T = (\frac{3}{2}a - 5a) + (-\frac{3}{2}ab + \frac{3}{2}ab) + (5b - 3b) + (-5 + 10) $$

$$ T = (\frac{3}{2}a - \frac{10}{2}a) + 0 + 2b + 5 $$

$$ T = -\frac{7}{2}a + 2b + 5 $$

W:

$$ W = (\frac{3}{4}a-\frac{1}{2})(-\frac{4}{3}b+2)-(1.5+b)(a-1) $$

$$ W = (-\frac{4}{4}ab + \frac{6}{4}a + \frac{2}{3}b - 1) - (1.5a - 1.5 + ab - b) $$

$$ W = (-ab + \frac{3}{2}a + \frac{2}{3}b - 1) - (\frac{3}{2}a - \frac{3}{2} + ab - b) $$

$$ W = -ab + \frac{3}{2}a + \frac{2}{3}b - 1 - \frac{3}{2}a + \frac{3}{2} - ab + b $$

$$ W = (-ab - ab) + (\frac{3}{2}a - \frac{3}{2}a) + (\frac{2}{3}b + b) + (-1 + \frac{3}{2}) $$

$$ W = -2ab + \frac{5}{3}b + \frac{1}{2} $$

U:

$$ U = (\frac{5}{6}-b)(a+\frac{3}{5})+(\frac{5}{4}-\frac{6}{5}b)(-\frac{2}{3}a-\frac{2}{5}) $$

$$ U = (\frac{5}{6}a + \frac{15}{30} - ab - \frac{3}{5}b) + (-\frac{10}{12}a - \frac{10}{20} - \frac{12}{15}ab - \frac{12}{25}b) $$

$$ U = (\frac{5}{6}a + \frac{1}{2} - ab - \frac{3}{5}b) + (-\frac{5}{6}a - \frac{1}{2} - \frac{4}{5}ab - \frac{12}{25}b) $$

$$ U = (\frac{5}{6}a - \frac{5}{6}a) + (\frac{1}{2} - \frac{1}{2}) + (-ab - \frac{4}{5}ab) + (-\frac{3}{5}b - \frac{12}{25}b) $$

$$ U = 0 + 0 + (-\frac{5}{5}ab - \frac{4}{5}ab) + (-\frac{15}{25}b - \frac{12}{25}b) $$

$$ U = -\frac{9}{5}ab - \frac{27}{25}b $$

تفكيك إلى جداء عوامل:

A:

$$ A = \frac{35}{32}bc-\frac{49}{40}ac = 7c(\frac{5}{32}b - \frac{7}{40}a) $$

B:

$$ B = 2a-\frac{5}{4}b+\frac{5}{2}c $$

لا يوجد عامل مشترك واضح.

C:

$$ C = \frac{4}{3}a^{2}+2ab-1.5a = \frac{4}{3}a^{2}+2ab-\frac{3}{2}a $$

$$ C = a(\frac{4}{3}a+2b-\frac{3}{2}) $$

D:

$$ D = -\frac{6}{7}ab+\frac{3}{7}ac-\frac{9}{14}a $$

$$ D = a(-\frac{6}{7}b+\frac{3}{7}c-\frac{9}{14}) $$

$$ D = a \times \frac{3}{14}(-4b+2c-3) $$

E:

$$ E = \frac{5}{6}ab-\frac{10}{3}b^{2}+\frac{25}{9}bc $$

$$ E = b(\frac{5}{6}a-\frac{10}{3}b+\frac{25}{9}c) $$

F:

$$ F = -1.6ac+\frac{4}{3}ac-2abc = -\frac{8}{5}ac+\frac{4}{3}ac-2abc $$

$$ F = ac(-\frac{8}{5}+\frac{4}{3}-2b) = ac(-\frac{24}{15}+\frac{20}{15}-2b) $$

$$ F = ac(-\frac{4}{15}-2b) = -ac(\frac{4}{15}+2b) $$

G:

$$ G = \frac{3}{4}x-1=\frac{3}{4}(x-\frac{4}{3}) $$

H:

$$ H = x+\frac{6}{5}=\frac{6}{5}(\frac{5}{6}x+1) $$

I:

$$ I = -4x+3=-4(x-\frac{3}{4}) $$

J:

$$ J = 2x-\frac{5}{2}=2(x-\frac{5}{4}) $$

X:

$$ X = \frac{3}{5}(y-x)-(y-x) = (y-x)(\frac{3}{5}-1) = (y-x)(-\frac{2}{5}) = -\frac{2}{5}(y-x) $$

Y:

$$ Y = (\frac{3}{4}x-1)(x+y)-\frac{1}{2}(x+y) = (x+y)(\frac{3}{4}x-1-\frac{1}{2}) $$

$$ Y = (x+y)(\frac{3}{4}x-\frac{3}{2}) = (x+y)\frac{3}{4}(x-2) $$

Z:

$$ Z = (\frac{1}{2}x-\frac{6}{5})(3x-\frac{1}{3})-(x-0.2)(\frac{1}{3}-3x) $$

نلاحظ أن $ (3x-\frac{1}{3}) = -( \frac{1}{3}-3x) $ و $ (x-0.2) = (x-\frac{1}{5}) $.

$$ Z = (\frac{1}{2}x-\frac{6}{5})(3x-\frac{1}{3}) - (x-\frac{1}{5})(-(3x-\frac{1}{3})) $$

$$ Z = (\frac{1}{2}x-\frac{6}{5})(3x-\frac{1}{3}) + (x-\frac{1}{5})(3x-\frac{1}{3}) $$

$$ Z = (3x-\frac{1}{3})[(\frac{1}{2}x-\frac{6}{5}) + (x-\frac{1}{5})] $$

$$ Z = (3x-\frac{1}{3})[\frac{1}{2}x-\frac{6}{5} + x-\frac{1}{5}] $$

$$ Z = (3x-\frac{1}{3})[\frac{3}{2}x-\frac{7}{5}] $$

تصحيح التمرين رقم 3:

A:

$$ A = \frac{12}{42} \div \frac{6}{7} = \frac{2}{7} \times \frac{7}{6} = \frac{2}{6} = \frac{1}{3} $$

B:

$$ B = \frac{\frac{2}{3}}{\frac{4}{9}} = \frac{2}{3} \times \frac{9}{4} = \frac{18}{12} = \frac{3}{2} $$

C:

$$ C = \frac{2-\frac{2}{3}}{1+\frac{1}{6}} = \frac{\frac{6-2}{3}}{\frac{6+1}{6}} = \frac{\frac{4}{3}}{\frac{7}{6}} = \frac{4}{3} \times \frac{6}{7} = \frac{24}{21} = \frac{8}{7} $$

D:

$$ D = \frac{4+\frac{5}{6}+\frac{3}{4}}{4-\frac{5}{6}+\frac{3}{4}} $$

$$ \text{Numerator: } 4+\frac{5}{6}+\frac{3}{4} = \frac{48+10+9}{12} = \frac{67}{12} $$

$$ \text{Denominator: } 4-\frac{5}{6}+\frac{3}{4} = \frac{48-10+9}{12} = \frac{47}{12} $$

$$ D = \frac{\frac{67}{12}}{\frac{47}{12}} = \frac{67}{47} $$

E:

$$ E = 3+2\times\frac{2-\frac{1}{5}}{2+\frac{1}{5}} = 3+2\times\frac{\frac{10-1}{5}}{\frac{10+1}{5}} = 3+2\times\frac{\frac{9}{5}}{\frac{11}{5}} $$

$$ E = 3+2\times\frac{9}{11} = 3+\frac{18}{11} = \frac{33+18}{11} = \frac{51}{11} $$

F:

$$ F = \frac{\frac{16}{5}}{\frac{32}{15}} = \frac{16}{5} \times \frac{15}{32} = \frac{16 \times 15}{5 \times 32} = \frac{1 \times 3}{1 \times 2} = \frac{3}{2} $$

G:

$$ G = \frac{\frac{3}{5}}{9} = \frac{3}{5} \div 9 = \frac{3}{5} \times \frac{1}{9} = \frac{3}{45} = \frac{1}{15} $$

H:

$$ H = \frac{2-\frac{4}{5}+\frac{1}{2}}{2+\frac{4}{5}+\frac{1}{2}} $$

$$ \text{Numerator: } 2-\frac{4}{5}+\frac{1}{2} = \frac{20-8+5}{10} = \frac{17}{10} $$

$$ \text{Denominator: } 2+\frac{4}{5}+\frac{1}{2} = \frac{20+8+5}{10} = \frac{33}{10} $$

$$ H = \frac{\frac{17}{10}}{\frac{33}{10}} = \frac{17}{33} $$

I:

$$ I = 2+3\times\frac{2+\frac{3}{4}}{2-\frac{3}{4}} = 2+3\times\frac{\frac{8+3}{4}}{\frac{8-3}{4}} = 2+3\times\frac{\frac{11}{4}}{\frac{5}{4}} $$

$$ I = 2+3\times\frac{11}{5} = 2+\frac{33}{5} = \frac{10+33}{5} = \frac{43}{5} $$

J:

$$ J = \frac{\frac{1}{4}+\frac{-3}{2}}{\frac{-3}{5}-\frac{1}{2}} = \frac{\frac{1}{4}-\frac{6}{4}}{\frac{-6}{10}-\frac{5}{10}} = \frac{\frac{-5}{4}}{\frac{-11}{10}} $$

$$ J = \frac{-5}{4} \times \frac{10}{-11} = \frac{-50}{-44} = \frac{25}{22} $$

تصحيح التمرين رقم 10:

لدينا $ x = -\frac{1}{2} $ و $ y = -\frac{1}{3} $.

A:

$$ A = \frac{x}{y} = \frac{-\frac{1}{2}}{-\frac{1}{3}} = \frac{1}{2} \times \frac{3}{1} = \frac{3}{2} $$

B:

$$ B = \frac{2x}{-3y} = \frac{2(-\frac{1}{2})}{-3(-\frac{1}{3})} = \frac{-1}{1} = -1 $$

C:

$$ x-y = -\frac{1}{2} - (-\frac{1}{3}) = -\frac{1}{2} + \frac{1}{3} = \frac{-3+2}{6} = -\frac{1}{6} $$

$$ x+y = -\frac{1}{2} + (-\frac{1}{3}) = -\frac{1}{2} - \frac{1}{3} = \frac{-3-2}{6} = -\frac{5}{6} $$

$$ C = \frac{x-y}{x+y} = \frac{-\frac{1}{6}}{-\frac{5}{6}} = \frac{1}{6} \times \frac{6}{5} = \frac{1}{5} $$

D:

$$ y-x = -\frac{1}{3} - (-\frac{1}{2}) = -\frac{1}{3} + \frac{1}{2} = \frac{-2+3}{6} = \frac{1}{6} $$

$$ D = \frac{x}{y-x}-\frac{y}{x+y} = \frac{-\frac{1}{2}}{\frac{1}{6}} - \frac{-\frac{1}{3}}{-\frac{5}{6}} $$

$$ D = (-\frac{1}{2} \times 6) - (\frac{1}{3} \times \frac{6}{5}) $$

$$ D = -3 - \frac{2}{5} = \frac{-15-2}{5} = -\frac{17}{5} $$

E:

$$ x+1 = -\frac{1}{2} + 1 = \frac{1}{2} $$

$$ x(x+1) = (-\frac{1}{2})(\frac{1}{2}) = -\frac{1}{4} $$

$$ E = \frac{3y}{x(x+1)} = \frac{3(-\frac{1}{3})}{-\frac{1}{4}} = \frac{-1}{-\frac{1}{4}} = -1 \times -4 = 4 $$

تصحيح التمرين رقم 11:

1)

$$ \frac{3}{5}x=\frac{3}{8} $$

$$ x = \frac{3}{8} \div \frac{3}{5} = \frac{3}{8} \times \frac{5}{3} = \frac{5}{8} $$

2)

$$ -\frac{15}{7}x=\frac{33}{14} $$

$$ x = \frac{33}{14} \div (-\frac{15}{7}) = \frac{33}{14} \times (-\frac{7}{15}) = -\frac{33 \times 7}{14 \times 15} = -\frac{3 \times 11 \times 7}{2 \times 7 \times 3 \times 5} = -\frac{11}{10} $$

3)

$$ -\frac{8}{9}x=-\frac{5}{27} $$

$$ x = -\frac{5}{27} \div (-\frac{8}{9}) = \frac{5}{27} \times \frac{9}{8} = \frac{5 \times 9}{27 \times 8} = \frac{5}{3 \times 8} = \frac{5}{24} $$

4)

$$ \frac{3}{11}x+\frac{2}{3}=\frac{-5}{4} $$

$$ \frac{3}{11}x = -\frac{5}{4} - \frac{2}{3} = \frac{-15-8}{12} = -\frac{23}{12} $$

$$ x = -\frac{23}{12} \div \frac{3}{11} = -\frac{23}{12} \times \frac{11}{3} = -\frac{253}{36} $$

5)

$$ -2x+\frac{1}{3}=\frac{5}{2} $$

$$ -2x = \frac{5}{2} - \frac{1}{3} = \frac{15-2}{6} = \frac{13}{6} $$

$$ x = \frac{13}{6} \div (-2) = \frac{13}{6} \times (-\frac{1}{2}) = -\frac{13}{12} $$

6)

$$ \frac{-2}{3}x+\frac{1}{2}=\frac{-7}{2} $$

$$ -\frac{2}{3}x = -\frac{7}{2} - \frac{1}{2} = -\frac{8}{2} = -4 $$

$$ x = -4 \div (-\frac{2}{3}) = -4 \times (-\frac{3}{2}) = \frac{12}{2} = 6 $$

7)

$$ \frac{2}{3}x-\frac{1}{4}=\frac{-3}{4} $$

$$ \frac{2}{3}x = -\frac{3}{4} + \frac{1}{4} = -\frac{2}{4} = -\frac{1}{2} $$

$$ x = -\frac{1}{2} \div \frac{2}{3} = -\frac{1}{2} \times \frac{3}{2} = -\frac{3}{4} $$

8)

$$ \frac{4}{3}x-\frac{2}{3}=\frac{2}{9} $$

$$ \frac{4}{3}x = \frac{2}{9} + \frac{2}{3} = \frac{2+6}{9} = \frac{8}{9} $$

$$ x = \frac{8}{9} \div \frac{4}{3} = \frac{8}{9} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{2}{3} $$

9)

$$ \frac{4}{3}x-\frac{3}{2}=\frac{1}{2} $$

$$ \frac{4}{3}x = \frac{1}{2} + \frac{3}{2} = \frac{4}{2} = 2 $$

$$ x = 2 \div \frac{4}{3} = 2 \times \frac{3}{4} = \frac{6}{4} = \frac{3}{2} $$

10)

$$ \frac{3}{4}x+\frac{4}{5}=\frac{2}{3} $$

$$ \frac{3}{4}x = \frac{2}{3} - \frac{4}{5} = \frac{10-12}{15} = -\frac{2}{15} $$

$$ x = -\frac{2}{15} \div \frac{3}{4} = -\frac{2}{15} \times \frac{4}{3} = -\frac{8}{45} $$

11)

$$ -\frac{5}{2}x+\frac{1}{4}=-\frac{7}{2} $$

$$ -\frac{5}{2}x = -\frac{7}{2} - \frac{1}{4} = -\frac{14+1}{4} = -\frac{15}{4} $$

$$ x = -\frac{15}{4} \div (-\frac{5}{2}) = \frac{15}{4} \times \frac{2}{5} = \frac{3 \times 2}{4} = \frac{6}{4} = \frac{3}{2} $$

12)

$$ -4x+\frac{1}{5}=2 $$

$$ -4x = 2 - \frac{1}{5} = \frac{10-1}{5} = \frac{9}{5} $$

$$ x = \frac{9}{5} \div (-4) = \frac{9}{5} \times (-\frac{1}{4}) = -\frac{9}{20} $$

13)

$$ -\frac{3}{5}-x=-\frac{1}{10} $$

$$ -x = -\frac{1}{10} + \frac{3}{5} = -\frac{1}{10} + \frac{6}{10} = \frac{5}{10} = \frac{1}{2} $$

$$ x = -\frac{1}{2} $$

14)

$$ (1-\frac{2}{1+\frac{2}{3}})x=\frac{1}{5} $$

$$ 1+\frac{2}{3} = \frac{3+2}{3} = \frac{5}{3} $$

$$ \frac{2}{1+\frac{2}{3}} = \frac{2}{\frac{5}{3}} = 2 \times \frac{3}{5} = \frac{6}{5} $$

$$ 1-\frac{6}{5} = \frac{5-6}{5} = -\frac{1}{5} $$

$$ -\frac{1}{5}x = \frac{1}{5} $$

$$ x = -1 $$

15)

$$ \frac{\frac{5}{12}+\frac{3}{8}}{-\frac{4}{18}-\frac{1}{12}}x=-3 $$

$$ \text{Numerator of fraction: } \frac{5}{12}+\frac{3}{8} = \frac{10+9}{24} = \frac{19}{24} $$

$$ \text{Denominator of fraction: } -\frac{4}{18}-\frac{1}{12} = -\frac{2}{9}-\frac{1}{12} = \frac{-8-3}{36} = -\frac{11}{36} $$

$$ \text{Fraction value: } \frac{\frac{19}{24}}{-\frac{11}{36}} = \frac{19}{24} \times (-\frac{36}{11}) = -\frac{19 \times 36}{24 \times 11} = -\frac{19 \times 3}{2 \times 11} = -\frac{57}{22} $$

$$ -\frac{57}{22}x = -3 $$

$$ x = -3 \div (-\frac{57}{22}) = -3 \times (-\frac{22}{57}) = \frac{66}{57} = \frac{22}{19} $$

16)

$$ \frac{3}{4}(\frac{2}{3}-x)=1 $$

$$ \frac{2}{3}-x = 1 \div \frac{3}{4} = 1 \times \frac{4}{3} = \frac{4}{3} $$

$$ -x = \frac{4}{3} - \frac{2}{3} = \frac{2}{3} $$

$$ x = -\frac{2}{3} $$

17)

$$ -\frac{3}{5}(2x-\frac{1}{3})=\frac{4}{3} $$

$$ 2x-\frac{1}{3} = \frac{4}{3} \div (-\frac{3}{5}) = \frac{4}{3} \times (-\frac{5}{3}) = -\frac{20}{9} $$

$$ 2x = -\frac{20}{9} + \frac{1}{3} = -\frac{20}{9} + \frac{3}{9} = -\frac{17}{9} $$

$$ x = -\frac{17}{9} \div 2 = -\frac{17}{9} \times \frac{1}{2} = -\frac{17}{18} $$

18)

$$ 2(\frac{3}{4}x+\frac{1}{2})=-\frac{4}{5} $$

$$ \frac{3}{4}x+\frac{1}{2} = -\frac{4}{5} \div 2 = -\frac{4}{5} \times \frac{1}{2} = -\frac{2}{5} $$

$$ \frac{3}{4}x = -\frac{2}{5} - \frac{1}{2} = \frac{-4-5}{10} = -\frac{9}{10} $$

$$ x = -\frac{9}{10} \div \frac{3}{4} = -\frac{9}{10} \times \frac{4}{3} = -\frac{3 \times 2}{5} = -\frac{6}{5} $$

19)

$$ \frac{2}{3}(\frac{9}{8}x-\frac{6}{5})=-\frac{3}{4} $$

$$ \frac{9}{8}x-\frac{6}{5} = -\frac{3}{4} \div \frac{2}{3} = -\frac{3}{4} \times \frac{3}{2} = -\frac{9}{8} $$

$$ \frac{9}{8}x = -\frac{9}{8} + \frac{6}{5} = \frac{-45+48}{40} = \frac{3}{40} $$

$$ x = \frac{3}{40} \div \frac{9}{8} = \frac{3}{40} \times \frac{8}{9} = \frac{1 \times 1}{5 \times 3} = \frac{1}{15} $$