CALCUL NUMERIQUE
Correction des exercices *

Exercice 1

\begin{align*} A&=\frac{48}{72}\\ &=\frac{6{\color{red}\times 8}}{9{\color{red}\times 8}} \\ &=\frac{6}{9}\\ &=\frac{2{\color{red}\times 3}}{3{\color{red}\times 3}}\\ &=\frac{2}{3} \end{align*}

\begin{align*} B&=\frac{80}{176}\\ & =\frac{40{\color{red}\times 2}}{88{\color{red}\times 2}}\\ &=\frac{40}{88}\\ &=\frac{5{\color{red}\times 8}}{11{\color{red}\times 8}}\\ &=\frac{5}{11} \end{align*}

\begin{align*} C&=\frac{2\times 5 \times 9}{2 \times 10 \times 3}\\ & =\frac{{\color{red}2}\times 5 \times 9}{{\color{red}2}\times 10 \times 3}\\ &=\frac{5\times 9}{10 \times 3}\\ &=\frac{{\color{red}5}\times 9}{2\times {\color{red}5}\times 3}\\ &=\frac{9}{2\times 3}\\ &=\frac{3{\color{red}\times 3}}{2{\color{red}\times 3}}\\ &=\frac{3}{2} \end{align*}

\begin{align*} D&=\frac{20}{60}\\ & =\frac{{\color{red}20}\times 1}{{\color{red}20}\times 3}\\ &=\frac{1}{3} \end{align*}

Exercice 2

\begin{align*} A&=\left(\frac{4}{5}\right)^{2}-\frac{2}{5}\\ &= \frac{4}{5}\times \frac{4}{5}-\frac{2}{5}\\ &=\frac{16}{25}-\frac{2}{5}\\ &=\frac{16}{25}-\frac{2{\color{red}\times 5}}{5{\color{red}\times 5}}\\ &=\frac{16}{25}-\frac{10}{25}\\ &=\frac{16-10}{25}\\ &=\frac{6}{25} \end{align*}

\begin{align*} B&=6-\frac{2}{3}\\ &=\frac{6{\color{red}\times 3}}{{\color{red}3}}-\frac{2}{3}\\ &=\frac{18}{3}-\frac{2}{3}\\ &=\frac{18-2}{3}\\ &=\frac{16}{3} \end{align*}

\begin{align*} C&=\left(\frac{1}{2}\right)^{2}+\frac{3}{8}\\ &=\frac{1}{2}\times \frac{1}{2}+\frac{3}{8}\\ &=\frac{1}{4}+\frac{3}{8}\\ &=\frac{1{\color{red}\times 2}}{4{\color{red}\times 2}}+\frac{3}{8}\\ &=\frac{2}{8}+\frac{3}{8}\\ &=\frac{2+3}{8}\\ &=\frac{5}{8} \end{align*}

\begin{align*} D&=3-\left(\frac{5}{2}\right)^{2}\\ &=3-\frac{5}{2}\times \frac{5}{2}\\ &=3-\frac{25}{4}\\ &=\frac{3{\color{red}\times 4}}{\color{red}4}-\frac{25}{4}\\ &=\frac{12}{4}-\frac{25}{4}\\ &=\frac{12-25}{4}\\ &=-\frac{13}{4} \end{align*}

Exercice 3

\begin{align*} A&=\frac{3}{7}+\frac{4}{7}\times \frac{2}{3}\\ &=\frac{3}{7}+\frac{4\times 2}{7\times 3}\\ &=\frac{3}{7}+\frac{8}{21}\\ &=\frac{3{\color{red}\times 3}}{7\color{red} \times 3}+\frac{8}{21}\\ &=\frac{9}{21}+\frac{8}{21}\\ &=\frac{9+8}{21}\\ &=\frac{17}{21} \end{align*}

\begin{align*} B&=\frac{\frac{5}{6}-2}{\frac{3}{7}+1}\\ &=\frac{\frac{5}{6}-\frac{2{\color{red}\times 6}}{\color{red}6}}{\frac{3}{7}+\frac{1{\color{red}\times 7}}{\color{red}7}}\\ &=\frac{\frac{5}{6}-\frac{12}{6}}{\frac{3}{7}+\frac{7}{7}}\\ &=\frac{\frac{5-12}{6}}{\frac{3+7}{7}}\\ &=\frac{-\frac{7}{6}}{\frac{10}{7}}\\ &=-\frac{7}{6}\times \frac{7}{10}\\ &=-\frac{7\times 7}{6\times 10}\\ &=-\frac{49}{60} \end{align*}

\begin{align*} C&=\frac{1+\frac{1}{2}}{3+\frac{3}{2}}\\ &=\frac{\frac{1{\color{red}\times 2}}{\color{red}2}+\frac{1}{2}}{\frac{3{\color{red}\times 2}}{\color{red}2}+\frac{3}{2}}\\ &=\frac{\frac{2}{2}+\frac{1}{2}}{\frac{6}{2}+\frac{3}{2}}\\ &=\frac{\frac{2+1}{2}}{\frac{6+3}{2}}\\ &=\frac{\frac{3}{2}}{\frac{9}{2}}\\ &=\frac{3}{2}\times \frac{2}{9}\\ &=\frac{3{\color{red}\times 2}}{{\color{red} 2}\times 9}\\ &=\frac{3}{9}\\ &=\frac{1}{3} \end{align*}

\begin{align*} D&=\frac{2}{3}\div \frac{5}{6} + \frac{3}{10}\\ &=\frac{2}{3}\times \frac{6}{5}+\frac{3}{10}\\ &=\frac{2\times 6}{3 \times 5}+\frac{3}{10}\\ &=\frac{2{\color{red}\times 3}\times 2}{{\color{red} 3}\times 5}+\frac{3}{10}\\ &=\frac{4}{5}+\frac{3}{10}\\ &=\frac{4{\color{red}\times 2}}{5{\color{red}\times 2}}+\frac{3}{10}\\ &=\frac{8}{10}+\frac{3}{10}\\ &=\frac{8+3}{10}\\ &=\frac{11}{10}\\ &=1.1 \end{align*}

Exercice 4

\begin{align*} A&=\left(\frac{2}{5}-\frac{3}{2}\right)\div \left(\frac{5}{3}+\frac{2}{4}\right)\\ &=\left(\frac{2}{5}-\frac{3}{2}\right)\div \left(\frac{5}{3}+\frac{1}{2}\right)\\ &=\left(\frac{2\times 2}{5\times 2}-\frac{3\times 5}{2 \times 5}\right)\div \left(\frac{5\times 2}{3\times 2}+\frac{1\times 3}{2\times 3}\right)\\ &=\left(\frac{4}{10}-\frac{15}{10}\right)\div \left(\frac{10}{6}+\frac{3}{6}\right)\\ &=-\frac{11}{10} \div \frac{13}{6}\\ &=-\frac{11}{10} \times \frac{6}{13} \\ &=-\frac{66}{130}\\ &=-\frac{33\times 2}{65\times 2}\\ &=-\frac{33}{65} \end{align*}

\begin{align*} B&=\left(\frac{1}{2}-\frac{3}{4}\right)\div \frac{2}{3}\\ &=\left(\frac{1\times 2}{2\times 2}-\frac{3}{4}\right)\div \frac{2}{3}\\ &=\left(\frac{2}{4}-\frac{3}{4}\right)\div \frac{2}{3}\\ &=-\frac{1}{4} \div \frac{2}{3}\\ &=-\frac{1}{4} \times \frac{3}{2}\\ &=-\frac{3}{8} \end{align*}

\begin{align*} C&=\frac{1}{2}-\frac{3}{4} \div \frac{2}{3}\\ &=\frac{1}{2}-\frac{3}{4} \times \frac{3}{2}\\ &=\frac{1}{2}-\frac{9}{8}\\ &=\frac{1\times 4}{2\times 4} - \frac{9}{8}\\ &=\frac{4}{8}-\frac{9}{8}\\ &=\frac{4-9}{8}\\ &=-\frac{5}{8} \end{align*}

\begin{align*} D&=\left(2-4\times \frac{6}{7}\right)\div \frac{3}{14}\\ &=\left(2-\frac{24}{7}\right)\div \frac{3}{14}\\ &=\left(\frac{2\times 7}{1 \times 7}-\frac{24}{7}\right)\div \frac{3}{14}\\ &=\left(\frac{14}{7}-\frac{24}{7}\right)\div \frac{3}{14}\\ &=\left(\frac{14-24}{7}\right)\div \frac{3}{14}\\ &=-\frac{10}{7}\times \frac{14}{3}\\ &=-\frac{10\times 14}{7\times 3}\\ &=-\frac{10{\color{red}\times 7} \times 2}{{\color{red}7} \times 3}\\ &=-\frac{20}{3} \end{align*}

Exercice 5

\begin{align*} A&=12\times 10^{4}+5\times 10^{2}\\ &=120000+500\\ &=120500\\ &=1.205\times 10^{5} \end{align*}

\begin{align*} B&=13450\\ &=1.345\times 10^{4} \end{align*}

\begin{align*} C&=0.0079\\ &=7.9\times 10^{-3} \end{align*}

\begin{align*} D&=0.01\\ &=1\times 10^{-2} \end{align*}

\begin{align*} E&=0.1\times 10^{3}\times 10^{-2}\\ &=0.1\times 10^{3+(-2)}\\ &=0.1\times 10^{1}\\ &=1\\ &=1\times 10^{0} \end{align*}

Exercice 6

\begin{align*} A&=(10^{3})^{2}\times 15\\ &=10^{3\times 2}\times 15\\ &=10^{6}\times 15\\ &=10^{6}\times 1.5\times 10^{1}\\ &=1.5\times 10^{6+1}\\ &=1.5\times 10^{7} \end{align*}

\begin{align*} B&=3\times 10^{2}+5.2\times 10^{1}\\ &=300+52\\ &=352\\ &=3.52\times 10^{2} \end{align*}

\begin{align*} C&=\frac{2\times 10^{-2}\times 9\times 10^{4}}{3\times 10^{-1}\times 8\times 10^{2}}\\ &=\frac{2\times 9}{3\times 8}\times \frac{10^{-2}\times 10^{4}}{10^{-1}\times 10^{2}}\\ &=\frac{{\color{red}2\times 3}\times 3}{{\color{red}3\times 2}\times 4}\times \frac{10^{-2+4}}{10^{-1+2}}\\ &=\frac{3}{4}\times \frac{10^{2}}{10^{1}}\\ &=0.75\times 10^{2-1}\\ &=0.75\times 10\\ &=7.5\\ &=7.5\times 10^{0} \end{align*}

\begin{align*} D&=10^{1}\times 10^{-2}\times 10^{3}\times 10^{-4}\times 10^{5}\\ &=10^{1+(-2)+3+(-4)+5}\\ &=10^{3}\\ &=1\times 10^{3} \end{align*}

Correction des exercices d'entraînement sur le calcul numérique (révisions) pour la troisième (3ème)
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