Maîtriser les fractions

Exercice 1

$D = \dfrac{5 - 8}{7 + 5} = \dfrac{-3}{12}$

$D = -\dfrac{3}{3 \times 4} = -\dfrac{1}{4}$

$E = \left(\dfrac{3}{4} - \dfrac{1}{2} \times \dfrac{2}{5}\right) \times \left(\dfrac{2}{7}\right)^2 = \left(\dfrac{3}{4} - \dfrac{1 \times 2}{2 \times 5}\right) \times \dfrac{2^2}{7^2}$

$E = \left(\dfrac{3}{4} - \dfrac{1}{5}\right) \times \dfrac{4}{49} = \left(\dfrac{3 \times 5}{4 \times 5} - \dfrac{1 \times 4}{5 \times 4}\right) \times \dfrac{4}{49}$

$E = \left(\dfrac{15}{20} - \dfrac{4}{20}\right) \times \dfrac{4}{49} = \dfrac{11}{20} \times \dfrac{4}{49}$

$E = \dfrac{11 \times 4}{20 \times 49} = \dfrac{11 \times 4}{5 \times 4 \times 49}$

$E = \dfrac{11}{5 \times 49} = \dfrac{11}{245}$

$F = \dfrac{\left(\dfrac{2}{3} - \dfrac{1}{15}\right)^2}{\dfrac{4}{9}} = \dfrac{\left(\dfrac{2 \times 5}{3 \times 5} - \dfrac{1}{15}\right)^2}{\dfrac{4}{9}}$

$F = \dfrac{\left(\dfrac{10}{15} - \dfrac{1}{15}\right)^2}{\dfrac{4}{9}} = \dfrac{\left(\dfrac{9}{15}\right)^2}{\dfrac{4}{9}}$

$F = \dfrac{9^2}{15^2} \times \dfrac{9}{4}$

$F = \dfrac{9 \times 9 \times 3 \times 3}{5 \times 3 \times 5 \times 3 \times 4}$

Exercice 2

$M = \dfrac{3}{14} - \dfrac{2}{21} = \dfrac{3 \times 3}{14 \times 3} - \dfrac{2 \times 2}{21 \times 2}$
$M = \dfrac{9}{42} - \dfrac{4}{42} = \dfrac{5}{42}$

$N = \dfrac{5}{18} + \dfrac{1}{27} = \dfrac{5 \times 3}{18 \times 3} + \dfrac{1 \times 2}{27 \times 2}$
$N = \dfrac{15}{54} + \dfrac{2}{54} = \dfrac{17}{54}$

$O = \dfrac{1}{24} - \dfrac{-1}{30} = \dfrac{1 \times 5}{24 \times 5} - \dfrac{-1 \times 4}{30 \times 4}$
$O = \dfrac{5}{120} + \dfrac{4}{120} = \dfrac{9}{120}$
$O = \dfrac{3}{40}$

$P = \dfrac{-1}{9} + \dfrac{1}{-15} = \dfrac{-1 \times 5}{9 \times 5} + \dfrac{1 \times 3}{-15 \times 3}$
$P = -\dfrac{5}{45} - \dfrac{3}{45} = -\dfrac{8}{45}$

$Q = \dfrac{19}{42} + \dfrac{2}{21} - \dfrac{1}{3} = \dfrac{19}{42} + \dfrac{2 \times 2}{21 \times 2} - \dfrac{1 \times 14}{3 \times 14}$
$Q = \dfrac{19}{42} + \dfrac{4}{42} - \dfrac{14}{42} = \dfrac{9}{42}$
$Q = \dfrac{3}{14}$

$R = \dfrac{17}{42} + \dfrac{2}{21} - \dfrac{1}{3} = \dfrac{17}{42} + \dfrac{4}{42} - \dfrac{14}{42}$
$R = \dfrac{7}{42} = \dfrac{1}{6}$

$S = \dfrac{8}{35} - \dfrac{-2}{35} = \dfrac{8}{35} + \dfrac{2}{35}$
$S = \dfrac{10}{35} = \dfrac{2}{7}$

$T = \dfrac{3}{8} - 2 - \dfrac{-1}{8} = \dfrac{3}{8} - \dfrac{2 \times 8}{8} + \dfrac{1}{8}$
$T = \dfrac{3}{8} - \dfrac{16}{8} + \dfrac{1}{8} = -\dfrac{12}{8}$
$T = -\dfrac{4 \times 3}{4 \times 2} = -\dfrac{3}{2}$

$U = \dfrac{7}{3} \times \dfrac{27}{14} = \dfrac{7 \times 9 \times 3}{3 \times 7 \times 2} = \dfrac{9}{2}$

$V = \dfrac{-8}{5} \times \dfrac{-15}{16} = \dfrac{-8 \times (-15)}{5 \times 16}$
$V = \dfrac{2 \times 4 \times 5 \times 3}{5 \times 4 \times 2 \times 2} = \dfrac{3}{2}$

$W = \dfrac{3}{-4} : \dfrac{-9}{40} = \dfrac{3}{-4} \times \dfrac{40}{-9}$
$W = \dfrac{3 \times 4 \times 10}{4 \times 3 \times 3} = \dfrac{10}{3}$

$X = \dfrac{16}{9} \times \dfrac{3}{14} \times \dfrac{5}{8} = \dfrac{16 \times 3 \times 5}{9 \times 14 \times 8}$
$X = \dfrac{2 \times 2 \times 4 \times 3 \times 5}{3 \times 3 \times 7 \times 2 \times 4 \times 2} = \dfrac{5}{3 \times 7}$
$X = \dfrac{5}{21}$

Exercice 3

$M = \left(\dfrac{3}{4} \right)^2 = \dfrac{3^2}{4^2}$
$M = \dfrac{9}{16}$

$N = \dfrac{3^2}{4} = \dfrac{9}{4}$

$O = \dfrac{2}{3} + \dfrac{3}{10} \times \dfrac{35}{9} = \dfrac{2}{3} + \dfrac{3 \times 35}{10 \times 3 \times 3}$
$O = \dfrac{2}{3} + \dfrac{35}{30} = \dfrac{20}{30} + \dfrac{35}{30}$
$O = \dfrac{55}{30} = \dfrac{5 \times 11}{5 \times 6}$
$O = \dfrac{11}{6}$

$P = \left(\dfrac{2}{3} + \dfrac{3}{10} \right) \times \dfrac{35}{9} = \left(\dfrac{20}{30} + \dfrac{9}{30} \right) \times \dfrac{35}{9}$
$P = \dfrac{29}{30} \times \dfrac{35}{9} = \dfrac{29 \times 7 \times 5}{6 \times 5 \times 9}$
$P = \dfrac{203}{54}$

$Q = \left(1 - \dfrac{1}{2} \right) : \left(2 - \dfrac{1}{4} \right) = \left(\dfrac{2}{2} - \dfrac{1}{2} \right) : \left(\dfrac{8}{4} - \dfrac{1}{4} \right)$
$Q = \dfrac{1}{2} : \dfrac{7}{4} = \dfrac{1}{2} \times \dfrac{4}{7}$
$Q = \dfrac{1 \times 4}{2 \times 7} = \dfrac{2}{7}$

$R = 1 - \dfrac{1}{2} : 2 - \dfrac{1}{4} = 1 - \dfrac{1}{2} \times \dfrac{1}{2} - \dfrac{1}{4}$
$R = 1 - \dfrac{1}{4} - \dfrac{1}{4} = \dfrac{4}{4} - \dfrac{1}{4} - \dfrac{1}{4}$
$R = \dfrac{2}{4} = \dfrac{2}{2 \times 2}$
$R = \dfrac{1}{2}$

$S = \dfrac{\dfrac{3}{4} - \dfrac{5}{2}}{7} = \dfrac{\dfrac{3}{4} - \dfrac{10}{4}}{7}$
$S = \dfrac{-\dfrac{7}{4}}{7} = -\dfrac{7}{4} \times \dfrac{1}{7}$
$S = -\dfrac{7 \times 1}{4 \times 7} = -\dfrac{1}{4}$

$T = \left(\dfrac{1}{9} - \dfrac{1}{3}\right)^2 \times \dfrac{9}{2} = \left(\dfrac{1}{9} - \dfrac{3}{9}\right)^2 \times \dfrac{9}{2}$
$T = \left(-\dfrac{2}{9}\right)^2 \times \dfrac{9}{2} = \dfrac{4}{81} \times \dfrac{9}{2}$
$T = \dfrac{2 \times 2 \times 9}{9 \times 9 \times 2} = \dfrac{2}{9}$

$U = \dfrac{\dfrac{5}{3} - \dfrac{7}{9}}{\dfrac{1}{4} - \dfrac{1}{2}} = \dfrac{\dfrac{15}{9} - \dfrac{7}{9}}{\dfrac{1}{4} - \dfrac{2}{4}}$
$U = \dfrac{\dfrac{8}{9}}{-\dfrac{1}{4}} = -\dfrac{8}{9} \times 4$
$U = -\dfrac{32}{9}$

$V = \dfrac{1 - 5^2}{(1 - 5)^2} = \dfrac{1 - 25}{(-4)^2}$
$V = \dfrac{-24}{16} = \dfrac{-3 \times 8}{2 \times 8}$
$V = \dfrac{-3}{2}$

$W = \left(\dfrac{6}{-7}\right)^2 = \dfrac{6^2}{(-7)^2}$
$W = \dfrac{36}{49}$

$X = \dfrac{5^2}{-3} = -\dfrac{25}{3}$

$Y = \left(\dfrac{1}{10} - \dfrac{7}{15}\right) : \left(\dfrac{7}{6} + \dfrac{7}{16}\right) = \left(\dfrac{3}{30} - \dfrac{14}{30}\right) : \left(\dfrac{56}{48} + \dfrac{21}{48}\right)$
$Y = -\dfrac{11}{30} : \dfrac{77}{48} = -\dfrac{11}{30} \times \dfrac{48}{77}$
$Y = -\dfrac{11 \times 2 \times 3 \times 8}{2 \times 3 \times 5 \times 7 \times 11} = -\dfrac{8}{5 \times 7}$
$Y = -\dfrac{8}{35}$

$Z = \dfrac{1}{8} - \dfrac{7}{12} : \left(\dfrac{7}{6} + \dfrac{7}{16}\right) = \dfrac{1}{8} - \dfrac{7}{12} : \left(\dfrac{56}{48} + \dfrac{21}{48}\right)$
$Z = \dfrac{1}{8} - \dfrac{7}{12} : \dfrac{77}{48} = \dfrac{1}{8} - \dfrac{7}{12} \times \dfrac{48}{77}$
$Z = \dfrac{1}{8} - \dfrac{7 \times 12 \times 4}{12 \times 7 \times 11} = \dfrac{1}{8} - \dfrac{4}{11}$
$Z = \dfrac{11}{88} - \dfrac{32}{88} = -\dfrac{21}{88}$

$A' = \dfrac{\dfrac{1}{8} + \dfrac{7}{12}}{\dfrac{5}{6} - \dfrac{4}{15}} = \dfrac{\dfrac{3}{24} + \dfrac{14}{24}}{\dfrac{25}{30} - \dfrac{8}{30}}$
$A' = \dfrac{\dfrac{17}{24}}{\dfrac{17}{30}} = \dfrac{17}{24} \times \dfrac{30}{17}$
$A' = \dfrac{17 \times 2 \times 3 \times 5}{2 \times 4 \times 3 \times 17} = \dfrac{5}{4}$

$B' = \left(\dfrac{1}{8} + \dfrac{7}{12}\right) \times \left(\dfrac{6}{5} - \dfrac{15}{4}\right) = \left(\dfrac{3}{24} + \dfrac{14}{24}\right) \times \left(\dfrac{24}{20} - \dfrac{75}{20}\right)$
$B' = \dfrac{17}{24} \times \left(-\dfrac{51}{20}\right) = -\dfrac{17 \times 51}{24 \times 20}$

$B' = -\dfrac{17 \times 3 \times 17}{3 \times 8 \times 20} = -\dfrac{289}{160}$

Exercice 4

$1.$

$\text{A} = \dfrac{1}{5+3} = \dfrac{1}{8}$

$\text{B} = \dfrac{1}{6} + \dfrac{1}{2} = \dfrac{1}{6} + \dfrac{3}{6} = \dfrac{4}{6} = \dfrac{2}{3}$

$\text{C} = \dfrac{1}{\dfrac{1}{2} - \dfrac{1}{3}} = \dfrac{1}{\dfrac{3}{6} - \dfrac{2}{6}} = \dfrac{1}{\dfrac{1}{6}} = 6$

$\text{D} = \dfrac{1}{\dfrac{1}{3}}-\dfrac{1}{\dfrac{1}{5}} = 3 - 5 = -2$

$2.$ a) Prenons deux nombres : 2 et 3.

La somme des inverses de ces deux nombres est :
$\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}$

L'inverse de la somme de ces deux nombres est :
$\dfrac{1}{2 + 3} = \dfrac{1}{5}$

Et $\dfrac{5}{6} \neq \dfrac{1}{5}$, donc $\dfrac{1}{2} + \dfrac{1}{3} \neq \dfrac{1}{2 + 3}$.

Conclusion : La somme des inverses de deux nombres n'est pas égale à l'inverse de la somme de ces deux nombres.

$2.$ b) Prenons deux nombres : 2 et 3.

La différence des inverses de ces deux nombres est :
$\dfrac{1}{2} - \dfrac{1}{3} = \dfrac{3}{6} - \dfrac{2}{6} = \dfrac{1}{6}$

L'inverse de la différence de ces deux nombres est :
$\dfrac{1}{2-3} = -1$

Et $\dfrac{1}{6} \neq -1$, donc $\dfrac{1}{2} - \dfrac{1}{3} \neq \dfrac{1}{2-3}$.

Conclusion : La différence des inverses de deux nombres n'est pas égale à l'inverse de la différence de ces deux nombres.