Calculs avec des fractions
A = \(\dfrac{4}{5}-\dfrac{2}{5}\left(\dfrac{1}{3}-3\right)\)
Corrigé
\[\begin{aligned}
A&=\dfrac{4}{5}-\dfrac{2}{5}\left(\dfrac{1}{3}-3\right)\\
A&=\dfrac{4}{5}-\dfrac{2}{5}\times\left(-\dfrac{8}{3}\right)\\
A&=\dfrac{4}{5}+\dfrac{16}{15}\\
A&=\dfrac{28}{15}
\end{aligned}\]
B = \(\dfrac{2}{a}-\dfrac{7}{5}\times\dfrac{1}{\dfrac{a}{4}-\dfrac{a}{2}}\)
Corrigé
\[\begin{aligned}
B&=\dfrac{2}{a}-\dfrac{7}{5}\times\dfrac{1}{\dfrac{a}{4}-\dfrac{a}{2}}\\
B&=\dfrac{2}{a}-\dfrac{7}{5}\times\dfrac{1}{-\dfrac{a}{4}}\\
B&=\dfrac{2}{a}-\dfrac{7}{5}\times\left(-\dfrac{4}{a}\right)\\
B&=\dfrac{2}{a}+\dfrac{28}{5a}\\
B&=\dfrac{10}{5a}+\dfrac{28}{5a}\\
B&=\dfrac{38}{5a}\qquad(a\neq 0)
\end{aligned}\]
C = \(\dfrac{\dfrac{3}{2}-1}{1,5+1}\times\dfrac{5^{3}}{18}\)
Corrigé
\[\begin{aligned}
C&=\dfrac{\dfrac{3}{2}-1}{1,5+1}\times\dfrac{5^{3}}{18}\\
C&=\dfrac{\frac{1}{2}}{\frac{5}{2}}\times\dfrac{125}{18}\\
C&=\dfrac{1}{5}\times\dfrac{125}{18}\\
C&=\dfrac{25}{18}
\end{aligned}\]
D = \(a^{-2}+\dfrac{\left(\dfrac{1}{2a}+\dfrac{1}{a}\right)^{2}}{\dfrac{5}{4}}\)
Corrigé
\[\begin{aligned}
D&=a^{-2}+\dfrac{\left(\dfrac{1}{2a}+\dfrac{1}{a}\right)^{2}}{\dfrac{5}{4}}\\
D&=a^{-2}+\dfrac{\left(\dfrac{3}{2a}\right)^{2}}{\dfrac{5}{4}}\\
D&=a^{-2}+\dfrac{9}{4a^{2}}\times\dfrac{4}{5}\\
D&=a^{-2}+\dfrac{9}{5a^{2}}\\
D&=\dfrac{5}{5a^{2}}+\dfrac{9}{5a^{2}}\\
D&=\dfrac{14}{5a^{2}}\qquad(a\neq 0)
\end{aligned}\]
E = \(\dfrac{\dfrac{7}{6}-\dfrac{a}{3}}{1-\dfrac{4a}{14}}\times\dfrac{a^{2}}{7}\)
Corrigé
\[\begin{aligned}
E&=\dfrac{\dfrac{7}{6}-\dfrac{a}{3}}{1-\dfrac{4a}{14}}\times\dfrac{a^{2}}{7}\\
E&=\dfrac{\dfrac{7-2a}{6}}{\dfrac{7-2a}{7}}\times\dfrac{a^{2}}{7}\\
E&=\dfrac{7}{6}\times\dfrac{a^{2}}{7}\\
E&=\dfrac{a^{2}}{6}\qquad\left(a\neq \dfrac{7}{2}\right)
\end{aligned}\]
F = \(\dfrac{2+3}{2+7}\div\left(\dfrac{5}{3}\right)^{2}\)
Corrigé
\[\begin{aligned}
F&=\dfrac{2+3}{2+7}\div\left(\dfrac{5}{3}\right)^{2}\\
F&=\dfrac{5}{9}\div\dfrac{25}{9}\\
F&=\dfrac{5}{9}\times\dfrac{9}{25}\\
F&=\dfrac{1}{5}
\end{aligned}\]
G = \(\dfrac{7}{18}\times\dfrac{2}{7}-\left(\dfrac{5}{3}-1\right)^{2}+1\)
Corrigé
\[\begin{aligned}
G&=\dfrac{7}{18}\times\dfrac{2}{7}-\left(\dfrac{5}{3}-1\right)^{2}+1\\
G&=\dfrac{1}{9}-\left(\dfrac{2}{3}\right)^{2}+1\\
G&=\dfrac{1}{9}-\dfrac{4}{9}+1\\
G&=-\dfrac{3}{9}+1\\
G&=\dfrac{2}{3}
\end{aligned}\]
H = \(\left(\dfrac{2}{5}-\dfrac{3}{4}\right)^{2}\div\dfrac{5}{8}-\dfrac{8}{3}\)
Corrigé
\[\begin{aligned}
H&=\left(\dfrac{2}{5}-\dfrac{3}{4}\right)^{2}\div\dfrac{5}{8}-\dfrac{8}{3}\\
H&=\left(-\dfrac{7}{20}\right)^{2}\div\dfrac{5}{8}-\dfrac{8}{3}\\
H&=\dfrac{49}{400}\div\dfrac{5}{8}-\dfrac{8}{3}\\
H&=\dfrac{49}{400}\times\dfrac{8}{5}-\dfrac{8}{3}\\
H&=\dfrac{49}{250}-\dfrac{8}{3}\\
H&=\dfrac{147}{750}-\dfrac{2000}{750}\\
H&=-\dfrac{1853}{750}
\end{aligned}\]
I = \(\dfrac{a}{3}-\dfrac{a}{3}\times\dfrac{7}{21}-\dfrac{7}{21}\)
Corrigé
\[\begin{aligned}
I&=\dfrac{a}{3}-\dfrac{a}{3}\times\dfrac{7}{21}-\dfrac{7}{21}\\
I&=\dfrac{a}{3}-\dfrac{a}{3}\times\dfrac{1}{3}-\dfrac{1}{3}\\
I&=\dfrac{a}{3}-\dfrac{a}{9}-\dfrac{1}{3}\\
I&=\dfrac{3a}{9}-\dfrac{a}{9}-\dfrac{3}{9}\\
I&=\dfrac{2a-3}{9}
\end{aligned}\]
J = \(\dfrac{\dfrac{3}{4}-\dfrac{2}{3}}{\dfrac{3}{4}+\dfrac{2}{3}}\div\dfrac{\dfrac{4}{5}-\dfrac{3}{4}}{\dfrac{4}{5}+\dfrac{3}{4}}\)
Corrigé
\[\begin{aligned}
J&=\dfrac{\dfrac{3}{4}-\dfrac{2}{3}}{\dfrac{3}{4}+\dfrac{2}{3}}\div\dfrac{\dfrac{4}{5}-\dfrac{3}{4}}{\dfrac{4}{5}+\dfrac{3}{4}}\\
J&=\dfrac{\dfrac{1}{12}}{\dfrac{17}{12}}\div\dfrac{\dfrac{1}{20}}{\dfrac{31}{20}}\\
J&=\dfrac{1}{17}\div\dfrac{1}{31}\\
J&=\dfrac{1}{17}\times 31\\
J&=\dfrac{31}{17}
\end{aligned}\]
K = \(\dfrac{a+\dfrac{a}{3}-\dfrac{a}{2}}{2a+\dfrac{3a}{4}+\dfrac{a}{3}}\)
Corrigé
\[\begin{aligned}
K&=\dfrac{a+\dfrac{a}{3}-\dfrac{a}{2}}{2a+\dfrac{3a}{4}+\dfrac{a}{3}}\\
K&=\dfrac{\dfrac{5a}{6}}{\dfrac{37a}{12}}\\
K&=\dfrac{5a}{6}\times\dfrac{12}{37a}\\
K&=\dfrac{60a}{222a}\\
K&=\dfrac{10}{37}\qquad(a\neq 0)
\end{aligned}\]
L = \(\dfrac{-5+3^{2}\times 2+4}{12\times 2+10}\)
Corrigé
\[\begin{aligned}
L&=\dfrac{-5+3^{2}\times 2+4}{12\times 2+10}\\
L&=\dfrac{-5+18+4}{24+10}\\
L&=\dfrac{17}{34}\\
L&=\dfrac{1}{2}
\end{aligned}\]
M = \(\dfrac{2}{a+1}+\dfrac{1-\dfrac{1}{a}}{1+\dfrac{1}{a}}\)
Corrigé
\[\begin{aligned}
M&=\dfrac{2}{a+1}+\dfrac{1-\dfrac{1}{a}}{1+\dfrac{1}{a}}\\
M&=\dfrac{2}{a+1}+\dfrac{a-1}{a+1}\\
M&=\dfrac{2+a-1}{a+1}\\
M&=\dfrac{a+1}{a+1}\\
M&=1\qquad(a\neq 0,\ a\neq -1)
\end{aligned}\]