2nd – Exercices – Fractions

Fractions

Exercice 1    Somme

Calculer les nombres suivants en fournissant le résultat sous la forme d’une fraction simplifiée.

  1. $\quad$ $\dfrac{2}{3}+\dfrac{5}{6}$
    $\quad$
  2. $\quad$ $\dfrac{4}{5}-\dfrac{5}{3}$
    $\quad$
  3. $\quad$ $\dfrac{2}{7}-\dfrac{1}{4}$
    $\quad$
  4. $\quad$ $-\dfrac{5}{6}+\dfrac{3}{4}$
    $\quad$
  5. $\quad$ $-\dfrac{2}{5}-\dfrac{3}{11}$
    $\quad$
  6. $\quad$ $3+\dfrac{2}{5}$
    $\quad$
  7. $\quad$ $1-\dfrac{5}{4}$
    $\quad$
  8. $\quad$ $\dfrac{1}{3}-2$
    $\quad$
  9. $\quad$ $-\dfrac{2}{7}-5$
    $\quad$
Correction Exercice 1

  1. $\quad$ $\dfrac{2}{3}+\dfrac{5}{6}=\dfrac{2\times 2}{3\times 2}+\dfrac{5}{6}=\dfrac{4}{6}+\dfrac{5}{6}=\dfrac{9}{6}=\dfrac{3}{2}$
    $\quad$
  2. $\quad$ $\dfrac{4}{5}-\dfrac{5}{3}=\dfrac{12}{15}-\dfrac{25}{15}=-\dfrac{13}{15}$
    $\quad$
  3. $\quad$ $\dfrac{2}{7}-\dfrac{1}{4}=\dfrac{8}{28}-\dfrac{7}{28}=\dfrac{1}{28}$
    $\quad$
  4. $\quad$ $-\dfrac{5}{6}+\dfrac{3}{4}=-\dfrac{20}{24}+\dfrac{18}{24}=-\dfrac{2}{24}=-\dfrac{1}{12}$
    $\quad$
  5. $\quad$ $-\dfrac{2}{5}-\dfrac{3}{11}=-\dfrac{22}{55}-\dfrac{15}{55}=-\dfrac{37}{55}$
    $\quad$
  6. $\quad$ $3+\dfrac{2}{5}=\dfrac{15}{5}+\dfrac{2}{5}=\dfrac{17}{5}$
    $\quad$
  7. $\quad$ $1-\dfrac{5}{4}=\dfrac{4}{4}-\dfrac{5}{4}=-\dfrac{1}{4}$
    $\quad$
  8. $\quad$ $\dfrac{1}{3}-2=\dfrac{1}{3}-\dfrac{6}{3}=-\dfrac{5}{3}$
    $\quad$
  9. $\quad$ $-\dfrac{2}{7}-5=-\dfrac{2}{7}-\dfrac{35}{7}=-\dfrac{37}{7}$
    $\quad$

[collapse]

$\quad$

Exercice 2    Produit

Calculer les nombres suivants en fournissant le résultat sous la forme d’une fraction simplifiée.

  1. $\quad$ $2\times \dfrac{4}{5}$
    $\quad$
  2. $\quad$ $6\times \dfrac{8}{15}$
    $\quad$
  3. $\quad$ $\dfrac{2}{3}\times \dfrac{5}{7}$
    $\quad$
  4. $\quad$ $\dfrac{3}{4}\times \left(-\dfrac{7}{2}\right)$
    $\quad$
  5. $\quad$ $\dfrac{5}{6}\times \dfrac{3}{5}$
    $\quad$
  6. $\quad$ $-\dfrac{5}{9}\times \dfrac{3}{10}$
    $\quad$
  7. $\quad$ $\dfrac{8}{7}\times \dfrac{14}{3}$
    $\quad$
  8. $\quad$ $-\dfrac{3}{4}\times \left(-\dfrac{10}{9}\right)$
    $\quad$
  9. $\quad$ $\dfrac{8}{5}\times \dfrac{15}{2}$
    $\quad$
Correction Exercice 2

  1. $\quad$ $2\times \dfrac{4}{5}=\dfrac{2\times 4}{5}=\dfrac{8}{5}$
    $\quad$
  2. $\quad$ $6\times \dfrac{8}{15}=\dfrac{3\times 2\times 8}{3\times 5}=\dfrac{16}{5}$
    $\quad$
  3. $\quad$ $\dfrac{2}{3}\times \dfrac{5}{7}=\dfrac{2\times 5}{3\times 7}=\dfrac{10}{21}$
    $\quad$
  4. $\quad$ $\dfrac{3}{4}\times \left(-\dfrac{7}{2}\right)=-\dfrac{21}{8}$
    $\quad$
  5. $\quad$ $\dfrac{5}{6}\times \dfrac{3}{5}=\dfrac{5\times 3}{6\times 5}=\dfrac{3}{6}=\dfrac{1}{2}$
    $\quad$
  6. $\quad$ $-\dfrac{5}{9}\times \dfrac{3}{10}=\dfrac{5\times 3}{3\times 3\times 2 \times 5}=\dfrac{1}{6}$
    $\quad$
  7. $\quad$ $\dfrac{8}{7}\times \dfrac{14}{3}=\dfrac{8\times 2 \times 7}{7 \times 3}=\dfrac{16}{3}$
    $\quad$
  8. $\quad$ $-\dfrac{3}{4}\times \left(-\dfrac{10}{9}\right)=\dfrac{3\times 2 \times 5}{2\times 2\times 3\times 3}=\dfrac{5}{6}$
    $\quad$
  9. $\quad$ $\dfrac{8}{5}\times \dfrac{15}{2}=\dfrac{2\times 4\times 3\times 5}{5\times 2}=12$
    $\quad$

[collapse]

$\quad$

Exercice 3    Quotient

Calculer les nombres suivants en fournissant le résultat sous la forme d’une fraction simplifiée.

  1. $\quad$ $\dfrac{~\dfrac{2}{5}~}{\dfrac{7}{3}}$
    $\quad$
  2. $\quad$ $\dfrac{~\dfrac{5}{6}~}{\dfrac{3}{4}}$
    $\quad$
  3. $\quad$ $\dfrac{-\dfrac{3}{4}~~}{\dfrac{5}{8}}$
    $\quad$
  4. $\quad$ $\dfrac{2}{~\dfrac{5}{4}~}$
    $\quad$
  5. $\quad$ $\dfrac{1}{~\dfrac{1}{4}~}$
    $\quad$
  6. $\quad$ $\dfrac{~\dfrac{2}{5}~}{6}$
    $\quad$
Correction Exercice 3

  1. $\quad$ $\dfrac{~\dfrac{2}{5}~}{\dfrac{7}{3}}=\dfrac{2}{5}\times \dfrac{3}{7}=\dfrac{6}{35}$
    $\quad$
  2. $\quad$ $\dfrac{~\dfrac{5}{6}~}{\dfrac{3}{4}}=\dfrac{5}{6}\times \dfrac{4}{3}=\dfrac{5\times 2\times 2}{3\times 2\times 3}=\dfrac{10}{9}$
    $\quad$
  3. $\quad$ $\dfrac{-\dfrac{3}{4}~~}{\dfrac{5}{8}}=-\dfrac{3}{4}\times \dfrac{8}{5}=-\dfrac{3\times 4\times 2}{4\times 5}=-\dfrac{6}{5}$
    $\quad$
  4. $\quad$ $\dfrac{2}{~\dfrac{5}{4}~}=2\times \dfrac{4}{5}=\dfrac{8}{5}$
    $\quad$
  5. $\quad$ $\dfrac{1}{~\dfrac{1}{4}~}=1\times \dfrac{4}{1}=4$
    $\quad$
  6. $\quad$ $\dfrac{~\dfrac{2}{5}~}{6}=\dfrac{2}{5}\times \dfrac{1}{6}=\dfrac{2}{5\times 2\times 3}=\dfrac{1}{15}$
    $\quad$

[collapse]

$\quad$

Exercice 4    Mélange

Calculer les nombres suivants en fournissant le résultat sous la forme d’une fraction simplifiée.

  1. $\dfrac{1}{3}+\dfrac{3}{4}\times \dfrac{2}{5}$
    $\quad$
  2. $\dfrac{5}{4}-\dfrac{1}{4}\times \dfrac{5}{2}$
    $\quad$
  3. $\dfrac{\dfrac{1}{2}+\dfrac{4}{3}}{\dfrac{3}{5}-\dfrac{2}{7}}$
    $\quad$
  4. $\dfrac{\dfrac{3}{4}-\dfrac{5}{3}}{\dfrac{3}{4}+\dfrac{5}{3}}$
    $\quad$
  5. $5-\dfrac{2}{3}\times \dfrac{7}{2}$
    $\quad$
  6. $\dfrac{1+\dfrac{3}{5}}{4-\dfrac{1}{2}}$
    $\quad$
  7. $\dfrac{\dfrac{2}{5}\times \dfrac{3}{4}}{\dfrac{2}{5}-\dfrac{5}{4}}$
    $\quad$
Correction Exercice 4

  1. $\quad$ $\dfrac{1}{3}+\dfrac{3}{4}\times \dfrac{2}{5}=\dfrac{1}{3}+\dfrac{3}{10}=\dfrac{10}{30}+\dfrac{9}{30}=\dfrac{19}{30}$
    $\quad$
  2. $\quad$ $\dfrac{5}{4}-\dfrac{1}{4}\times \dfrac{5}{2}=\dfrac{5}{4}-\dfrac{5}{8}=\dfrac{10}{8}-\dfrac{5}{8}=\dfrac{5}{8}$
    $\quad$
  3. $\quad$ $\dfrac{\dfrac{1}{2}+\dfrac{4}{3}}{\dfrac{3}{5}-\dfrac{2}{7}}=\dfrac{\dfrac{3}{6}+\dfrac{8}{6}}{\dfrac{21}{35}-\dfrac{10}{35}}=\dfrac{~\dfrac{11}{6}~}{\dfrac{11}{35}}=\dfrac{11}{6}\times \dfrac{35}{11}=\dfrac{35}{6}$
    $\quad$
  4. $\quad$ $\dfrac{\dfrac{3}{4}-\dfrac{5}{3}}{\dfrac{3}{4}+\dfrac{5}{3}}=\dfrac{\dfrac{9}{12}-\dfrac{20}{12}}{\dfrac{9}{12}+\dfrac{20}{12}}=\dfrac{-\dfrac{11}{12}~~}{\dfrac{29}{12}}=-\dfrac{11}{12}\times \dfrac{12}{29}=-\dfrac{11}{29}$
    $\quad$
  5. $\quad$ $5-\dfrac{2}{3}\times \dfrac{7}{2}=5-\dfrac{7}{3}=\dfrac{15}{3}-\dfrac{7}{3}=\dfrac{8}{3}$
    $\quad$
  6. $\quad$ $\dfrac{1+\dfrac{3}{5}}{4-\dfrac{1}{2}}=\dfrac{\dfrac{5}{5}+\dfrac{3}{5}}{\dfrac{8}{2}-\dfrac{1}{2}}=\dfrac{~\dfrac{8}{5}~}{\dfrac{7}{2}}=\dfrac{8}{5}\times \dfrac{2}{7}=\dfrac{16}{35}$
    $\quad$
  7. $\quad$ $\dfrac{\dfrac{2}{5}\times \dfrac{3}{4}}{\dfrac{2}{5}-\dfrac{5}{4}}=\dfrac{\dfrac{3}{10}}{\dfrac{8}{20}-\dfrac{25}{20}}=\dfrac{~\dfrac{3}{10}~}{-\dfrac{17}{20}}=-\dfrac{3}{10}\times \dfrac{20}{17}=-\dfrac{6}{17}$
    $\quad$

[collapse]

$\quad$

Exercice 5    Calcul littéral

Donner l’expression la plus simple des expressions suivantes.

  1. $\quad$ $3+\dfrac{5}{2+x}$
    $\quad$
  2. $\quad$ $2-\dfrac{4}{1-x}$
    $\quad$
  3. $\quad$ $\dfrac{3}{5}+\dfrac{2}{3x+1}$
    $\quad$
  4. $\quad$ $5+\dfrac{x+1}{x-2}$
    $\quad$
  5. $\quad$ $\dfrac{2x+5}{3x-1}+1$
    $\quad$
  6. $\quad$ $\dfrac{3x+2}{5x+3}-1$
    $\quad$
  7. $\quad$ $\dfrac{2x-1}{4x+2}+4$
    $\quad$
  8. $\quad$ $\dfrac{5x-3}{2x+3}-5$
    $\quad$
  9. $\quad$ $\dfrac{6x-2}{3-4x}-3$
    $\quad$
  10. $\quad$ $\dfrac{x-3}{2-3x}+6$
    $\quad$
  11. $\quad$ $\dfrac{2x+5}{3x-1}-\dfrac{3x-2}{5x-3}$
    $\quad$
  12. $\quad$ $\dfrac{3x-2}{2x+3}+\dfrac{7x-1}{2x+1}$
    $\quad$
  13. $\quad$ $\dfrac{x-2}{4x+2}-\dfrac{4x-1}{3x-2}$
    $\quad$
  14. $\quad$ $\dfrac{-2x+3}{2x-3}+\dfrac{3x+7}{4x+5}$
    $\quad$
Correction Exercice 5

  1. $\quad$ $3+\dfrac{5}{2+x}=\dfrac{3(2+x)}{2+x}+\dfrac{5}{2+x}$
    $\quad$ $\phantom{3+\dfrac{5}{2+x}}=\dfrac{6+3x}{2+x}+\dfrac{5}{2+x}=\dfrac{11+3x}{2+x}$
    $\quad$
  2. $\quad$ $2-\dfrac{4}{1-x}=\dfrac{2(1-x)}{1-x}-\dfrac{4}{1-x}$
    $\quad$ $\phantom{2-\dfrac{4}{1-x}}=\dfrac{2-2x}{1-x}-\dfrac{4}{1-x}=\dfrac{-2-2x}{1-x}$
    $\quad$
  3. $\quad$ $\dfrac{3}{5}+\dfrac{2}{3x+1}=\dfrac{3(3x+1)}{5(3x+1)}+\dfrac{10}{5(3x+1)}$
    $\quad$ $\phantom{\dfrac{3}{5}+\dfrac{2}{3x+1}}=\dfrac{9x+3}{15x+5}+\dfrac{10}{15x+5}=\dfrac{9x+13}{15x+5}$
    $\quad$
  4. $\quad$ $5+\dfrac{x+1}{x-2}=\dfrac{5(x-2)}{x-2}+\dfrac{x+1}{x-2}$
    $\quad$ $\phantom{5+\dfrac{x+1}{x-2}}=\dfrac{5x-10}{x-2}+\dfrac{x+1}{x-2}=\dfrac{6x-9}{x-2}$
    $\quad$
  5. $\quad$ $\dfrac{2x+5}{3x-1}+1=\dfrac{2x+5}{3x-1}+\dfrac{3x-1}{3x-1}=\dfrac{5x+4}{3x-1}$
    $\quad$
  6. $\quad$ $\dfrac{3x+2}{5x+3}-1=\dfrac{3x+2}{5x+3}-\dfrac{5x+3}{5x+3}=\dfrac{-2x-1}{5x+3}$
    $\quad$
  7. $\quad$ $\dfrac{2x-1}{4x+2}+4=\dfrac{2x-1}{4x+2}+\dfrac{4(4x+2)}{4x+2}$
    $\quad$ $\phantom{\dfrac{2x-1}{4x+2}+4}=\dfrac{2x-1}{4x+2}+\dfrac{16x+8}{4x+2}=\dfrac{18x+7}{4x+2}$
    $\quad$
  8. $\quad$ $\dfrac{5x-3}{2x+3}-5=\dfrac{5x-3}{2x+3}-\dfrac{5(2x+3)}{2x+3}$
    $\quad$ $\phantom{\dfrac{5x-3}{2x+3}-5}=\dfrac{5x-3}{2x+3}-\dfrac{10x+15}{2x+3}=\dfrac{-5x-18}{2x+3}$
    $\quad$
  9. $\quad$ $\dfrac{6x-2}{3-4x}-3=\dfrac{6x-2}{3-4x}-\dfrac{3(3-4x)}{3-4x}=\dfrac{6x-2}{3-4x}-\dfrac{9-12x}{3-4x}$
    $\quad$ $\phantom{\dfrac{6x-2}{3-4x}-3}=\dfrac{6x-2-9+12x}{3-4x}=\dfrac{18x-11}{3-4x}$
    $\quad$
  10. $\quad$ $\dfrac{x-3}{2-3x}+6=\dfrac{x-3}{2-3x}+\dfrac{6(2-3x)}{2-3x}$
    $\quad$ $\phantom{\dfrac{x-3}{2-3x}+6}=\dfrac{x-3}{2-3x}+\dfrac{12-18x}{2-3x}=\dfrac{-17x+9}{2-3x}$
    $\quad$
  11. $\quad$ $\dfrac{2x+5}{3x-1}-\dfrac{3x-2}{5x-3}=\dfrac{(2x+5)(5x-3)}{(3x-1)(5x-3)}-\dfrac{(3x-2)(3x-1)}{(3x-1)(5x-3)}$
    $\quad$ $\phantom{\dfrac{2x+5}{3x-1}-\dfrac{3x-2}{5x-3}} =\dfrac{10x^2-6x+25x-15}{(3x-1)(5x-3)}-\dfrac{9x^2-3x-6x+2}{(3x-1)(5x-3)}$
    $\quad$ $\phantom{\dfrac{2x+5}{3x-1}-\dfrac{3x-2}{5x-3}} = \dfrac{10x^2+19x-15}{(3x-1)(5x-3)}-\dfrac{9x^2-9x+2}{(3x-1)(5x-3)}$
    $\quad$ $\phantom{\dfrac{2x+5}{3x-1}-\dfrac{3x-2}{5x-3}} =\dfrac{x^2+28x-17}{(3x-1)(5x-3)}$
    $\quad$
  12. $\quad$ $\dfrac{3x-2}{2x+3}+\dfrac{7x-1}{2x+1}=\dfrac{(3x-2)(2x+1)}{(2x+3)(2x+1)}+\dfrac{(7x-1)(2x+3)}{(2x+3)(2x+1)}$
    $\quad$ $\phantom{\dfrac{3x-2}{2x+3}+\dfrac{7x-1}{2x+1}}=\dfrac{6x^2+3x-4x-2}{(2x+3)(2x+1)}+\dfrac{14x^2+21x-2x-3}{(2x+3)(2x+1)}$
    $\quad$ $\phantom{\dfrac{3x-2}{2x+3}+\dfrac{7x-1}{2x+1}}=\dfrac{6x^2-x-2}{(2x+3)(2x+1)}+\dfrac{14x^2+19x-3}{(2x+3)(2x+1)}$
    $\quad$ $\phantom{\dfrac{3x-2}{2x+3}+\dfrac{7x-1}{2x+1}}=\dfrac{20x^2+18x-5}{(2x+3)(2x+1)}$
    $\quad$
  13. $\quad$ $\dfrac{x-2}{4x+2}-\dfrac{4x-1}{3x-2}=\dfrac{(x-2)(3x-2)}{(4x+2)(3x-2)}-\dfrac{(4x-1)(4x+2)}{(4x+2)(3x-2)}$
    $\quad$ $\phantom{\dfrac{x-2}{4x+2}-\dfrac{4x-1}{3x-2}}=\dfrac{3x^2-2x-6x+4}{(4x+2)(3x-2)}-\dfrac{16x^2+8x-4x-2}{(4x+2)(3x-2)}$
    $\quad$ $\phantom{\dfrac{x-2}{4x+2}-\dfrac{4x-1}{3x-2}}=\dfrac{3x^2-8x+4}{(4x+2)(3x-2)}-\dfrac{16x^2+4x-2}{(4x+2)(3x-2)}$
    $\quad$ $\phantom{\dfrac{x-2}{4x+2}-\dfrac{4x-1}{3x-2}}=\dfrac{-13x^2-12x+6}{(4x+2)(3x-2)}$
    $\quad$
  14. $\quad$ $\dfrac{-2x+3}{2x-3}+\dfrac{3x+7}{4x+5}=\dfrac{(-2x+3)(4x+5)}{(2x-3)(4x+5)}+\dfrac{(3x+7)(2x-3)}{(2x-3)(4x+5)}$
    $\quad$ $\phantom{\dfrac{-2x+3}{2x-3}+\dfrac{3x+7}{4x+5}}=\dfrac{-8x^2-10x+12x+15}{(2x-3)(4x+5)}+\dfrac{6x^2-9x+14x-21}{(2x-3)(4x+5)}$
    $\quad$ $\phantom{\dfrac{-2x+3}{2x-3}+\dfrac{3x+7}{4x+5}}=\dfrac{-8x^2+2x+15}{(2x-3)(4x+5)}+\dfrac{6x^2+5x-21}{(2x-3)(4x+5)}$
    $\quad$ $\phantom{\dfrac{-2x+3}{2x-3}+\dfrac{3x+7}{4x+5}}=\dfrac{-2x^2+7x-6}{(2x-3)(4x+5)}$
    $\quad$

[collapse]