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Capacité 10 a

Effectuer des opérations entre des fractions simples.

A retenir

\frac{a}{p} +\frac{b}{q} = \frac{aq}{pq} +\frac{bp}{qp} = \frac{aq+bp}{pq}

\frac{a}{p} -\frac{b}{q} = \frac{aq}{pq} -\frac{bp}{qp} = \frac{aq-bp}{pq}

\frac{a}{p} \times \frac{b}{q} = \frac{ab}{pq}

\frac{ak}{pk} = \frac{a}{p}

\frac{\ \frac{a}{p}\ }{\frac{b}{q}} = \frac{a}{p} \times \frac{q}{b} = \frac{aq}{pb}

\frac{\ \frac{a}{p}\ }{c} = \frac{a}{p} \times \frac{1}{c} = \frac{a}{pc}

Question

Ecrire \frac{5}{3} + \frac{1}{2} sous la forme n+ \frac{a}{b}n est entier et \frac{a}{b} est un rationnel entre 0 et 1.

Solution

\frac{5}{3} + \frac{1}{2} = \frac{5\times 2 + 1\times 3}{3\times 2} = \frac{13}{6} = \frac{12+1}{6} = 2+\frac{1}{6}

Question

Calculer \frac{1}{6} + \frac{1}{3} +\frac{1}{2}

Solution

\frac{1}{6} + \frac{1}{3} +\frac{1}{2} = \frac{1}{6} + \frac{2}{6} +\frac{3}{6} = \frac{6}{6} = 1

Question

Calculer \frac{1}{2} \times \left( 1 - \frac{1}{3} \right)

Solution

\frac{1}{2} \times \left( 1 - \frac{1}{3} \right) = \frac{1}{2} \times \frac{2}{3} = \frac{1}{3}

Question

Calculer \frac{12}{5} \times \frac{10}{3} - \frac{1}{2}

Solution

\frac{12}{5} \times \frac{10}{3} - \frac{1}{2} = \frac{12\times 10}{5\times 3} - \frac{1}{2} = \frac{4\times 3\times 2\times 5}{5\times 3} - \frac{1}{2}

soit {4\times 2 } - \frac{1}{2} = 7,5 = \frac{15}{2}

Question

Écrire sous forme de fraction irréductible les nombres: A = \frac{2}{7} \times \frac{1}{6}, B = \frac{3}{4} + 7, C = \frac{\frac{12}{5}}{\frac{1}{25}}.

Solution

A = \frac{2}{7} \times \frac{1}{6} = \frac{2}{7} \times \frac{1}{2\times 3} = \frac{1}{7\times 3}= \frac{1}{21}.

B = \frac{3}{4} + 7 = \frac{3}{4} + \frac{7\times 4}{4} = \frac{3 + 28}{4} = \frac{31}{4}.

C = \frac{\frac{12}{5}}{\frac{1}{25}} =\frac{12}{5} \times 25 = 12\times 5 = 60.

Question

Ecrire sous forme de fraction irréductible: A = 14\times \frac{2}{7} + \frac{1}{6} - \frac{\frac{1}{3}}{5}.

Solution

A = 14\times \frac{2}{7} + \frac{1}{6} - \frac{\frac{1}{3}}{5} = 2\times {2} + \frac{1}{6} - \frac{1}{3}\times \frac{1}{5}

soit A = 4 + \frac{1}{6} - \frac{1}{15} = 4 + \frac{5}{30} - \frac{2}{30} = 4 + \frac{3}{30} = 4 + \frac{1}{10},

d'où A = \frac{40}{10} + \frac{1}{10} = \frac{41}{10}.

Question

Ecrire sous forme irréductible:

a = \frac{\ \ \frac{4}{3}\ \ }{12}, b = \frac{4}{\ \frac{3}{12}\ }, c = \frac{\ \ \frac{5}{8}\ \ }{\frac{3}{4}}

Solution

a = \frac{\ \ \frac{4}{3}\ \ }{12} = \frac{\ \ \frac{4}{3}\ \ }{\frac{12}{1}} = \frac{4}{3}\times \frac{1}{12}= \frac{4\times 1}{3\times (4\times 3)}, soit a = \frac{1}{9}.

b = \frac{4}{\ \frac{3}{12}\ } = \frac{4}{\ \frac{1}{4}\ } = 4\times \frac{4}{1} = 16

c = \frac{\ \ \frac{5}{8}\ \ }{\frac{3}{4}} = \frac{5}{8}\times \frac{4}{3} = \frac{5\times 4}{(2\times 4)\times 3}, soit c = \frac{5}{6}.

Question

Ecrire sous forme irréductible:

d = \frac{2}{5} \times \left( \frac{-1}{2} +7 \times \frac{3}{14}\right)

Solution

d = \frac{2}{5} \times \left( \frac{-1}{2} +7 \times \frac{3}{14}\right) = \frac{2}{5} \times \left( \frac{-1}{2} + \frac{7}{1}\times\frac{3}{2\times 7}\right), soit d = \frac{2}{5} \times \left( \frac{-1}{2} + \frac{7\times 3}{1\times 2\times 7}\right),
d = \frac{2}{5} \times \left( \frac{-1}{2} + \frac{3}{2}\right)
d = \frac{2}{5} \times \left( \frac{2}{2} \right)
d = \frac{2}{5} \times \left( 1 \right)
d = \frac{2}{5}

Question

Ecrire sous forme irréductible:

k = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}.

Solution

k = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4} =\frac{2}{2}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4} = \frac{3}{2}+\frac{1}{3}+\frac{1}{4}
k = \frac{6}{4}+\frac{1}{3}+\frac{1}{4}
k = \frac{7}{4}+\frac{1}{3} (en regroupant le premier terme et le dernier)
k = \frac{7\times 3}{4\times 3}+\frac{1\times 4}{3\times 4}
k = \frac{21}{12}+\frac{4}{12}
k = \frac{25}{12}

Question

Ecrire sous forme irréductible:

j = \left(\frac{2}{3}-1\right)-\left(2+\frac{1}{3}\right).

Solution

j = \left(\frac{2}{3}-1\right)-\left(2+\frac{1}{3}\right)
j = \frac{2}{3}-1-2-\frac{1}{3}
j = \frac{1}{3}-3
j = \frac{1}{3}-\frac{9}{3}
j = \frac{-8}{3}

Question

Ecrire sous forme irréductible:

a = \frac{\ \ \frac{5}{3}\ \ }{20}

Solution

a = \frac{5}{3} \times \frac{1}{20}
a = \frac{5}{3} \times \frac{1}{5\times 4}
a = \frac{1}{3} \times \frac{1}{ 4}
a = \frac{1}{12}

Question

Ecrire sous forme irréductible:

b = \frac{15}{\ \frac{3}{20}\ }

Solution

b = {15}\times \frac{20}{3}
b = 5\times 3\times \frac{20}{3}
b = 5\times 20
b = 100

Question

Ecrire sous forme irréductible:

c = \frac{11}{5} \times \left( \frac{-1}{3} +7 \times \frac{4}{21}\right)

Solution

c = \frac{11}{5} \times \left( \frac{-1}{3} +7 \times \frac{4}{7\times 3}\right)
c = \frac{11}{5} \times \left( \frac{-1}{3} + \frac{4}{3}\right)
c = \frac{11}{5} \times \frac{3}{3}
c = \frac{11}{5}

Question

Ecrire sous forme irréductible:

d = 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}

Solution

d = \frac{1}{2}+\frac{1}{3}-\frac{1}{4}
d = \frac{3}{6}+\frac{2}{6}-\frac{1}{4}
d = \frac{5}{6} -\frac{1}{4}
d = \frac{10}{12} -\frac{3}{12}
d = \frac{7}{12}

Question

Ecrire sous forme irréductible:

u = \left(\frac{2}{7}-2\right)-\left(3+\frac{1}{7}\right)

Solution

u = \frac{2}{7}-2 - 3- \frac{1}{7}
u = \frac{1}{7}-2 - 3
u = \frac{1}{7}-5
u = \frac{1}{7}-\frac{35}{7}
u = \frac{-34}{7}