# Fraction/Exercices/Sujet de brevet
## Exercice 1

Calculer et donner le résultat sous la forme d'une fraction la plus simple possible :
$A={\frac {4}{3}}-{\frac {1}{3}}\times \left(3+{\frac {1}{2}}\right)$
A = ${\frac {4}{3}}-{\frac {1}{3}}\times \left(3+{\frac {1}{2}}\right)$
= ${\frac {4}{3}}-{\frac {1}{3}}\times \left({\frac {6}{2}}+{\frac {1}{2}}\right)$
= ${\frac {4}{3}}-{\frac {1}{3}}\times \left({\frac {6+1}{2}}\right)$
= ${\frac {4}{3}}-{\frac {1}{3}}\times {\frac {7}{2}}$
= ${\frac {4}{3}}-{\frac {7}{6}}$
= ${\frac {8}{6}}-{\frac {7}{6}}$

## Exercice 2

Donner ces résultats sous forme de fractions irréductibles :
$A=\left({\frac {5}{7}}\right)^{2}-{\frac {2}{7}}$
$A=\left({\frac {5}{7}}\right)^{2}-{\frac {2}{7}}$
= $\left({\frac {25}{49}}\right)-{\frac {2}{7}}$
= $\left({\frac {25}{49}}\right)-{\frac {2\times 7}{7\times 7}}$
= ${\frac {25}{49}}-{\frac {14}{49}}$
= ${\frac {25-14}{49}}$
$B={\frac {1}{9}}+{\frac {1}{12}}$
$B={\frac {1}{9}}+{\frac {1}{12}}$
$={\frac {1\times 12}{9\times 12}}+{\frac {1\times 9}{12\times 9}}$
$={\frac {12}{108}}+{\frac {9}{108}}$
$={\frac {12+9}{108}}$
$={\frac {7*3}{36*3}}$
En électricité, pour calculer des valeurs de résistances, on utilise la formule: ${\frac {1}{R}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}$
Sachant que $R_{1}$=9 ohms et $R_{2}$= 12 ohms, déterminer la valeur exacte de R.
${\frac {1}{R}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}$
= ${\frac {1}{R}}={\frac {1}{9}}+{\frac {1}{12}}$
= ${\frac {1}{R}}={\frac {21}{108}}$

## Exercice 3

Mettre sous la forme de fraction irréductible ${\frac {{\frac {11}{3}}-7}{\frac {25}{6}}}$
${\frac {{\frac {11}{3}}-7}{\frac {25}{6}}}$
= ${\frac {{\frac {11\times 2}{3\times 2}}-{\frac {7\times 6}{6}}}{\frac {25}{6}}}$
= ${\frac {{\frac {22}{6}}-{\frac {42}{6}}}{\frac {25}{6}}}$
= ${\frac {22-42}{25}}$
= $-{\frac {20}{25}}$
Mettre sous la forme de fraction irréductible ${\frac {3}{7}}-{\frac {2}{5}}\times {\frac {15}{4}}$
${\frac {3}{7}}-{\frac {2}{5}}\times {\frac {15}{4}}$
= ${\frac {3}{7}}-{\frac {2\times 15}{5\times 4}}$
= ${\frac {3\times 20}{7\times 20}}-{\frac {2\times 15\times 7}{5\times 4\times 7}}$
= ${\frac {60}{140}}-{\frac {210}{140}}$
= ${\frac {60-210}{140}}$
= $-{\frac {150}{140}}$

## Exercice 4

calculer $A=\left({\frac {1}{3}}-{\frac {1}{5}}\right)\div {\frac {2}{5}}$
$A=\left({\frac {1}{3}}-{\frac {1}{5}}\right)\div {\frac {2}{5}}$
=$\left({\frac {1\times 5}{3\times 5}}-{\frac {1\times 3}{5\times 3}}\right)\div {\frac {2}{5}}$
=$\left({\frac {5}{15}}-{\frac {3}{15}}\right)\div {\frac {2}{5}}$
=${\frac {5-3}{15}}\div {\frac {2}{5}}$
=${\frac {\frac {2}{15}}{\frac {2}{5}}}$
=${\frac {2}{15}}\times {\frac {5}{2}}$
=${\frac {2\times 5}{15\times 2}}$
= ${\frac {10}{30}}$

## Exercice 5

Calculer $\left((-4+3)\times {\frac {2}{7}}\right)\div {\frac {3}{14}}$
$\left((-4+3)\times {\frac {2}{7}}\right)\div {\frac {3}{14}}$
= $\left(-1\times {\frac {2}{7}}\right)\div {\frac {3}{14}}$
= $\left(-{\frac {2}{7}}\right)\div {\frac {3}{14}}$
= $-{\frac {2}{7}}\times {\frac {14}{3}}$
= $-{\frac {2\times 14}{7\times 3}}$
= $-{\frac {2\times 2\times 7}{7\times 3}}$
= $-{\frac {2\times 2}{3}}$
Calculer ${\frac {4-\left(2-5\right)^{2}}{4+5}}$
${\frac {4-\left(2-5\right)^{2}}{4+5}}$
= ${\frac {4-\left(-3\right)^{2}}{9}}$
= ${\frac {4-9}{9}}$

## Exercice 6

Calculer $3-3\div {\frac {9}{2}}$
$3-3\div {\frac {9}{2}}$
= $3-3\times {\frac {2}{9}}$
= $3-{\frac {3\times 2}{9}}$
= $3-{\frac {2}{3}}$
= ${\frac {9}{3}}-{\frac {2}{3}}$
= ${\frac {9-2}{3}}$

## Exercice 7

Calculer $A={\frac {7}{18}}\times {\frac {2}{7}}-\left({\frac {5}{3}}-1\right)^{2}$
$A={\frac {7}{18}}\times {\frac {2}{7}}-\left({\frac {5}{3}}-1\right)^{2}$
= ${\frac {7\times 2}{18\times 7}}-\left({\frac {5}{3}}-{\frac {3}{3}}\right)^{2}$
= ${\frac {2}{18}}-\left({\frac {5-3}{3}}\right)^{2}$
= ${\frac {1}{9}}-\left({\frac {2}{3}}\right)^{2}$
= ${\frac {1}{9}}-\left({\frac {4}{9}}\right)$
= $-{\frac {3}{9}}$

## Exercice 8

$A={\frac {5}{7}}-{\frac {2}{7}}\times {\frac {4}{3}}$
$A={\frac {5}{7}}-{\frac {2}{7}}\times {\frac {4}{3}}$
= ${\frac {5\times 3}{7\times 3}}-{\frac {2\times 4}{7\times 3}}$
= ${\frac {15}{21}}-{\frac {8}{21}}$
= ${\frac {15-8}{21}}$
= ${\frac {7}{21}}$
$B=\left({\frac {1}{2}}-{\frac {1}{3}}\right)\times {\frac {3}{2}}+1$
$B=\left({\frac {1}{2}}-{\frac {1}{3}}\right)\times {\frac {3}{2}}+1$
= $\left({\frac {3}{6}}-{\frac {2}{6}}\right)\times {\frac {3}{2}}+1$
= $\left({\frac {1}{6}}\right)\times {\frac {3}{2}}+1$
= ${\frac {3}{12}}+1$
= ${\frac {15}{12}}$

## Exercice 9

$A={\frac {3}{2}}-{\frac {5}{2}}\times {\frac {3}{10}}$
$A={\frac {3}{2}}-{\frac {5}{2}}\times {\frac {3}{10}}$
= ${\frac {3}{2}}-{\frac {15}{20}}$
= ${\frac {30}{20}}-{\frac {15}{20}}$
= ${\frac {15}{20}}$
$B=\left({\frac {3}{5}}\right)^{2}\div {\frac {9}{20}}$
$B=\left({\frac {3}{5}}\right)^{2}\div {\frac {9}{20}}$
= $\left({\frac {9}{25}}\right)\times {\frac {20}{9}}$
= ${\frac {20}{25}}$

## Exercice 10

On pose $A=4-{\frac {3}{4}}\left({\frac {1}{3}}-{\frac {1}{6}}\right)$
En faisant apparaître les étapes de calcul, donner une écriture fractionnaire et une écriture décimale du nombre A.
$A=4-{\frac {3}{4}}\left({\frac {1}{3}}-{\frac {1}{6}}\right)$
$=4-{\frac {3}{4}}\left({\frac {2}{6}}-{\frac {1}{6}}\right)$
$=4-{\frac {3}{4}}\left({\frac {1}{6}}\right)$
$=4-{\frac {3}{24}}$
$=4-{\frac {1}{8}}$
$={\frac {32}{8}}-{\frac {1}{8}}$
$={\frac {31}{8}}$
On retire une petite partie de 4, donc on sait que le résultat est relativement proche de 4 au final.

## Exercice 11

Donner la valeur exacte la plus simple possible en indiquant le détail des calculs :
${\frac {34}{5}}\div \left({\frac {4}{5}}-{\frac {3}{8}}\right)$
$={\frac {34}{5}}\div \left({\frac {4}{5}}\times {\frac {8}{8}}-{\frac {3}{8}}\times {\frac {5}{5}}\right)$
$={\frac {34}{5}}\div \left({\frac {4}{5}}\times {\frac {8}{8}}-{\frac {3}{8}}\times {\frac {5}{5}}\right)$
$={\frac {34}{5}}\div \left({\frac {32-15}{5\times 8}}\right)$
$={\frac {34}{5}}\div \left({\frac {17}{5\times 8}}\right)$
$={\frac {34}{5}}\times \left({\frac {5\times 8}{17}}\right)$
$={\frac {17\times 2}{5}}\times \left({\frac {5\times 8}{17}}\right)$
${\frac {24\times 10^{-2}\times 3,5\times 10^{5}}{8\times 10^{-1}\times 21\times 10^{4}}}$
$={\frac {24\times 3,5}{8\times 21}}\times {\frac {10^{-2}\times 10^{5}}{10^{-1}\times 10^{4}}}$
$={\frac {8\times 3\times 3,5}{8\times 3\times 7}}\times {\frac {10^{3}}{10^{3}}}$
$={\frac {3,5}{7}}$

## Exercice 12

On considère les nombres :
* $A={\frac {6}{7}}-{\frac {4}{7}}\times {\frac {5}{2}}$
* $B={\frac {{\frac {3}{4}}-4}{{\frac {3}{4}}+{\frac {1}{3}}}}$
* $C=3^{2}\times 2-125\times 10^{-1}$
En précisant les différentes étapes des calculs
a) Écrire A sous la forme la plus simple possible et sans utiliser de valeur approchée
$A={\frac {6}{7}}-{\frac {4}{7}}\times {\frac {5}{2}}$
$A={\frac {6}{7}}-{\frac {4\times 5}{7\times 2}}$
$A={\frac {6}{7}}-{\frac {20}{14}}$
$A={\frac {12}{14}}-{\frac {20}{14}}$
$A={\frac {12-20}{14}}$
$A={\frac {-8}{14}}$
$A={\frac {-4}{7}}$
b) Écrire B sous la forme d'une nombre entier relatif
* $B={\frac {{\frac {3}{4}}-4}{{\frac {3}{4}}+{\frac {1}{3}}}}$
* $B={\frac {{\frac {3}{4}}-{\frac {16}{4}}}{{\frac {9}{12}}+{\frac {4}{12}}}}$
* $B={\frac {\frac {3-16}{4}}{\frac {9+4}{12}}}$
* $B={\frac {\frac {-13}{4}}{\frac {13}{12}}}$
* $B={\frac {-13}{4}}\times {\frac {12}{13}}$
* $B=-{\frac {12}{4}}$
* $B=-3$
c) Écrire C sous la forme d'un nombre décimal
* $C=3^{2}\times 2-125\times 10^{-1}$
* $C=9\times 2-12,5$
* $C=18-12,5$
* $C=5,5$

## Exercice 13

Sachant que $a={\frac {2}{3}}$, $b=-{\frac {1}{4}}$, $c={\frac {2}{5}}$ et $d=-{\frac {1}{2}}$,
a) Calculer $A=ab+cd$
$A=ab+cd$
$A={\frac {2}{3}}\times {\frac {-1}{4}}+{\frac {2}{5}}\times {\frac {-1}{2}}$
$A={\frac {-2}{12}}+{\frac {-2}{10}}$
$A={\frac {-10}{60}}+{\frac {-12}{60}}$
$A={\frac {-22}{60}}$
$A={\frac {-11}{30}}$
b) Calculer $B={\frac {a+d}{b+c}}$
$B={\frac {a+d}{b+c}}$
$B={\frac {{\frac {2}{3}}-{\frac {1}{2}}}{-{\frac {1}{4}}+{\frac {2}{5}}}}$
$B={\frac {{\frac {4}{6}}-{\frac {3}{6}}}{{\frac {-5}{20}}+{\frac {8}{20}}}}$
$B={\frac {\frac {1}{6}}{\frac {3}{20}}}$
$B={\frac {1\times 20}{6\times 3}}$
$B={\frac {20}{18}}$
$B={\frac {10}{9}}$

## Exercice 14

Calculer et donner chaque résultat sous la forme d'une fraction aussi simple que possible
$A={\frac {3}{4}}-{\frac {5}{6}}\times {\frac {3}{2}}$
$A={\frac {3}{4}}-{\frac {5}{6}}\times {\frac {3}{2}}$
$A={\frac {3}{4}}-{\frac {5\times 3}{6\times 2}}$
$A={\frac {3}{4}}-{\frac {15}{12}}$
$A={\frac {9}{12}}-{\frac {15}{12}}$
$A={\frac {9-15}{12}}$
$A={\frac {-6}{12}}$
$A=-{\frac {1}{2}}$

## Exercice 15

Calculer :
${\frac {4}{5}}-2\times {\frac {6}{5}}$
${\frac {4}{5}}-2\times {\frac {6}{5}}$
${\frac {4}{5}}-{\frac {12}{5}}$
${\frac {4-12}{5}}$
${\frac {-8}{5}}$

## Exercice 16

Mettre sous la forme la plus simple le nombre ${\frac {4}{5}}\times {\frac {2}{3}}+{\frac {1}{2}}-{\frac {1}{30}}$
${\frac {4}{5}}\times {\frac {2}{3}}+{\frac {1}{2}}-{\frac {1}{30}}$
$={\frac {4\times 2}{5\times 3}}+{\frac {1}{2}}-{\frac {1}{30}}$
$={\frac {8}{15}}+{\frac {1}{2}}-{\frac {1}{30}}$
$={\frac {16}{30}}+{\frac {1}{2}}-{\frac {1}{30}}$
$={\frac {16}{30}}+{\frac {15}{30}}-{\frac {1}{30}}$
$={\frac {16+15-1}{30}}$
$={\frac {30}{30}}$
$=1$

## Exercice 17

Écrire le résultat sous la forme la plus simple $A={\frac {5}{7}}-{\frac {2}{7}}\times {\frac {4}{3}}$
$A={\frac {5}{7}}-{\frac {2}{7}}\times {\frac {4}{3}}$
$A={\frac {5}{7}}-{\frac {2\times 4}{7\times 3}}$
$A={\frac {5}{7}}-{\frac {8}{21}}$
$A={\frac {15}{21}}-{\frac {8}{21}}$
$A={\frac {15-8}{21}}$
$A={\frac {7}{21}}={\frac {1}{3}}$
$B=\left({\frac {1}{2}}-{\frac {1}{3}}\right)\div {\frac {2}{3}}+1$
$B=\left({\frac {1}{2}}-{\frac {1}{3}}\right)\div {\frac {2}{3}}+1$
$B=\left({\frac {3}{6}}-{\frac {2}{6}}\right)\div {\frac {2}{3}}+1$
$B=\left({\frac {3-2}{6}}\right)\div {\frac {2}{3}}+1$
$B=\left({\frac {1}{6}}\right)\div {\frac {2}{3}}+1$
$B={\frac {1}{6}}\div {\frac {2}{3}}+1$
$B={\frac {1}{6}}\times {\frac {3}{2}}+1$
$B={\frac {1\times 3}{6\times 2}}+1$
$B={\frac {3}{12}}+1$
$B={\frac {3}{12}}+{\frac {12}{12}}$
$B={\frac {3+12}{12}}$
$B={\frac {15}{12}}={\frac {5}{4}}$

## Exercice 18

Calculer A et B. On donnera les résultats sous la forme la plus simple possible $A={\frac {1}{3}}\times 4+{\frac {7}{6}}$
$A={\frac {1}{3}}\times 4+{\frac {7}{6}}$
$A={\frac {1\times 4}{3}}+{\frac {7}{6}}$
$A={\frac {4}{3}}+{\frac {7}{6}}$
$A={\frac {8}{6}}+{\frac {7}{6}}$
$A={\frac {8+7}{6}}$
$A={\frac {15}{6}}$
$B={\frac {2}{5}}\times {\frac {3}{4}}-{\frac {2}{1-{\frac {2}{7}}}}$
$B={\frac {2}{5}}\times {\frac {3}{4}}-{\frac {2}{1-{\frac {2}{7}}}}$
$B={\frac {2\times 3}{5\times 4}}-{\frac {2}{1-{\frac {2}{7}}}}$
$B={\frac {6}{20}}-{\frac {2}{1-{\frac {2}{7}}}}$
$B={\frac {6}{20}}-{\frac {2}{{\frac {7}{7}}-{\frac {2}{7}}}}$
$B={\frac {6}{20}}-{\frac {2}{\frac {7-2}{7}}}$
$B={\frac {6}{20}}-{\frac {2}{\frac {5}{7}}}$
$B={\frac {6}{20}}-2\times {\frac {7}{5}}$
$B={\frac {6}{20}}-{\frac {2\times 7}{5}}$
$B={\frac {6}{20}}-{\frac {14}{5}}$
$B={\frac {3}{10}}-{\frac {28}{10}}$
$B={\frac {3-28}{10}}$
$B={\frac {-25}{10}}$
$B={\frac {-5}{2}}$

## Exercice 19

Les résultats seront donnés sous forme fractionnaire.
Que faut-il ajouter à ${\frac {3}{7}}$ pour obtenir 2 ?
$x+{\frac {3}{7}}=2$
$x=2-{\frac {3}{7}}$
$x={\frac {14}{7}}-{\frac {3}{7}}$
$x={\frac {14-3}{7}}$
$x={\frac {11}{7}}$
Que faut-il ajouter à ${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{6}}$ pour obtenir 1 ?
$x+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{6}}=1$
$x=1-{\frac {1}{2}}-{\frac {1}{4}}-{\frac {1}{6}}$
$x={\frac {12}{12}}-{\frac {6}{12}}-{\frac {3}{12}}-{\frac {2}{12}}$
$x={\frac {12-6-3-2}{12}}$
$x={\frac {1}{12}}$
À un nombre, j'ajoute ${\frac {7}{5}}$; je multiplie le résultat obtenu par ${\frac {3}{11}}$ et j’obtiens 1. Quel est ce nombre ?
$\left(x+{\frac {7}{5}}\right)\times {\frac {3}{11}}=1$
$\left(x+{\frac {7}{5}}\right)=1\div {\frac {3}{11}}$
$\left(x+{\frac {7}{5}}\right)=1\times {\frac {11}{3}}$
$\left(x+{\frac {7}{5}}\right)={\frac {3}{3}}\times {\frac {11}{3}}$
$\left(x+{\frac {7}{5}}\right)={\frac {3\times 11}{3\times 3}}$
$\left(x+{\frac {7}{5}}\right)={\frac {33}{9}}$
$x+{\frac {7}{5}}={\frac {33}{9}}$
$x={\frac {33}{9}}-{\frac {7}{5}}$
$x={\frac {33\times 5}{9\times 5}}-{\frac {7\times 9}{5\times 9}}$
$x={\frac {165}{45}}-{\frac {63}{45}}$
$x={\frac {165-63}{45}}$
$x={\frac {102}{45}}$
$x={\frac {34}{15}}$

## Exercice 20

Calculer puis simplifier $A={\frac {13}{14}}-{\frac {1}{15}}\times {\frac {10}{7}}$
$A={\frac {13}{14}}-{\frac {1}{15}}\times {\frac {10}{7}}$
$A={\frac {13}{14}}-{\frac {1\times 10}{15\times 7}}$
$A={\frac {13}{14}}-{\frac {10}{105}}$
$A={\frac {13}{14}}-{\frac {2}{21}}$
$A={\frac {13\times 21}{14\times 21}}-{\frac {2\times 14}{21\times 14}}$
$A={\frac {273}{294}}-{\frac {28}{294}}$
$A={\frac {273-28}{294}}$
$A={\frac {245}{294}}$
$A={\frac {35}{42}}$

## Exercice 21

Calculer et mettre le résultat sous la forme de fraction irréductible
$A={\frac {3}{14}}+{\frac {5}{21}}$
$A={\frac {3}{14}}+{\frac {5}{21}}$
$A={\frac {18}{84}}+{\frac {20}{84}}$
$A={\frac {18+20}{84}}$
$A={\frac {38}{84}}$
$A={\frac {19}{42}}$
$B={\frac {2}{3}}-{\frac {7}{3}}\times {\frac {9}{34}}$
$B={\frac {2}{3}}-{\frac {7}{3}}\times {\frac {9}{34}}$
$B={\frac {2}{3}}-{\frac {7\times 9}{3\times 34}}$
$B={\frac {2}{3}}-{\frac {63}{102}}$
$B={\frac {68}{102}}-{\frac {63}{102}}$
$B={\frac {68-63}{102}}$
$B={\frac {5}{102}}$
$C={\frac {2^{3}}{3^{2}}}\div {\frac {2^{4}}{3}}$
$C={\frac {2^{3}}{3^{2}}}\div {\frac {2^{4}}{3}}$
$C={\frac {2^{3}}{3^{2}}}\times {\frac {3}{2^{4}}}$
$C={\frac {2^{3}}{2^{4}}}\times {\frac {3}{3^{2}}}$
$C={\frac {1}{2}}\times {\frac {1}{3}}$
$C={\frac {1}{2\times 3}}$
$C={\frac {1}{6}}$
| Exercices simples | Sommaire |