CALCUL NUMERIQUE
Correction des exercices *

Exercice 1

\(\displaystyle A=\frac{48}{72}\)
\(\displaystyle \quad=\frac{6{\color{red}\times \color{red} 8}}{9{\color{red}\times \color{red} 8}}\)
\(\displaystyle \quad=\frac{6}{9}\)
\(\displaystyle \quad=\frac{2{\color{red}\times \color{red} 3}}{3{\color{red}\times \color{red} 3}}\)
\(\displaystyle \quad=\frac{2}{3}\)

\(\displaystyle B=\frac{80}{176}\)
\(\displaystyle \quad=\frac{40{\color{red}\times \color{red} 2}}{88{\color{red}\times \color{red} 2}}\)
\(\displaystyle \quad=\frac{40}{88}\)
\(\displaystyle \quad=\frac{5{\color{red}\times \color{red} 8}}{11{\color{red}\times \color{red} 8}}\)
\(\displaystyle \quad=\frac{5}{11}\)

\(\displaystyle C=\frac{2\times 5 \times 9}{2 \times 10 \times 3}\)
\(\displaystyle \quad=\frac{{\color{red}2}\times 5 \times 9}{{\color{red}2}\times 10 \times 3}\)
\(\displaystyle \quad=\frac{5\times 9}{10 \times 3}\)
\(\displaystyle \quad=\frac{{\color{red}5}\times 9}{2\times {\color{red}5}\times 3}\)
\(\displaystyle \quad=\frac{9}{2\times 3}\)
\(\displaystyle \quad=\frac{3{\color{red}\times \color{red} 3}}{2{\color{red}\times \color{red} 3}}\)
\(\displaystyle \quad=\frac{3}{2}\)

\(\displaystyle D=\frac{20}{60}\)
\(\displaystyle \quad=\frac{{\color{red} 2\color{red}0}\times 1}{{\color{red}2\color{red}0}\times 3}\)
\(\displaystyle \quad=\frac{1}{3}\)

Exercice 2

\(\displaystyle A=\left(\frac{4}{5}\right)^{2}-\frac{2}{5}\)
\(\displaystyle \quad= \frac{4}{5}\times \frac{4}{5}-\frac{2}{5}\)
\(\displaystyle \quad=\frac{16}{25}-\frac{2}{5}\)
\(\displaystyle \quad=\frac{16}{25}-\frac{2{\color{red}\times \color{red}5}}{5{\color{red}\times \color{red}5}}\)
\(\displaystyle \quad=\frac{16}{25}-\frac{10}{25}\)
\(\displaystyle \quad=\frac{16-10}{25}\)
\(\displaystyle \quad=\frac{6}{25}\)

\(\displaystyle B=6-\frac{2}{3}\)
\( \displaystyle \quad=\frac{6{\color{red}\times \color{red}3}}{{\color{red}3}}-\frac{2}{3}\)
\( \displaystyle \quad=\frac{18}{3}-\frac{2}{3}\)
\( \displaystyle \quad=\frac{18-2}{3}\)
\( \displaystyle \quad=\frac{16}{3}\)

\(\displaystyle C=\left(\frac{1}{2}\right)^{2}+\frac{3}{8}\)
\( \displaystyle \quad=\frac{1}{2}\times \frac{1}{2}+\frac{3}{8}\)
\( \displaystyle \quad=\frac{1}{4}+\frac{3}{8}\)
\( \displaystyle \quad=\frac{1{\color{red}\times\color{red} 2}}{4{\color{red}\times\color{red}2}}+\frac{3}{8}\)
\( \displaystyle \quad=\frac{2}{8}+\frac{3}{8}\)
\( \displaystyle \quad=\frac{2+3}{8}\)
\( \displaystyle \quad=\frac{5}{8}\)

\(\displaystyle D=3-\left(\frac{5}{2}\right)^{2}\)
\( \displaystyle \quad=3-\frac{5}{2}\times \frac{5}{2}\)
\( \displaystyle \quad=3-\frac{25}{4}\)
\( \displaystyle \quad=\frac{3{\color{red}\times \color{red}4}}{\color{red}4}-\frac{25}{4}\)
\( \displaystyle \quad=\frac{12}{4}-\frac{25}{4}\)
\( \displaystyle \quad=\frac{12-25}{4}\)
\( \displaystyle \quad=-\frac{13}{4}\)

Exercice 3

\(\displaystyle A=\frac{3}{7}+\frac{4}{7}\times \frac{2}{3}\)
\( \displaystyle \quad=\frac{3}{7}+\frac{4\times 2}{7\times 3}\)
\( \displaystyle \quad=\frac{3}{7}+\frac{8}{21}\)
\( \displaystyle \quad=\frac{3{\color{red}\times \color{red} 3}}{7\color{red} \times \color{red} 3}+\frac{8}{21}\)
\( \displaystyle \quad=\frac{9}{21}+\frac{8}{21}\)
\( \displaystyle \quad=\frac{9+8}{21}\)
\( \displaystyle \quad=\frac{17}{21}\)

\(\displaystyle B=\frac{\frac{5}{6}-2}{\frac{3}{7}+1}\)
\( \displaystyle \quad=\frac{\frac{5}{6}-\frac{2{\color{red}\times \color{red} 6}}{\color{red}6}}{\frac{3}{7}+\frac{1{\color{red}\times \color{red} 7}}{\color{red}7}}\)
\( \displaystyle \quad=\frac{\frac{5}{6}-\frac{12}{6}}{\frac{3}{7}+\frac{7}{7}}\)
\( \displaystyle \quad=\frac{\frac{5-12}{6}}{\frac{3+7}{7}}\)
\( \displaystyle \quad=\frac{-\frac{7}{6}}{\frac{10}{7}}\)
\( \displaystyle \quad=-\frac{7}{6}\times \frac{7}{10}\)
\( \displaystyle \quad=-\frac{7\times 7}{6\times 10}\)
\( \displaystyle \quad=-\frac{49}{60}\)

\(\displaystyle C=\frac{1+\frac{1}{2}}{3+\frac{3}{2}}\)
\( \displaystyle \quad=\frac{\frac{1{\color{red}\times \color{red}2}}{\color{red}2}+\frac{1}{2}}{\frac{3{\color{red}\times \color{red}2}}{\color{red}2}+\frac{3}{2}}\)
\( \displaystyle \quad=\frac{\frac{2}{2}+\frac{1}{2}}{\frac{6}{2}+\frac{3}{2}}\)
\( \displaystyle \quad=\frac{\frac{2+1}{2}}{\frac{6+3}{2}}\)
\( \displaystyle \quad=\frac{\frac{3}{2}}{\frac{9}{2}}\)
\( \displaystyle \quad=\frac{3}{2}\times \frac{2}{9}\)
\( \displaystyle \quad=\frac{3{\color{red}\times \color{red} 2}}{{\color{red}2}\times 9}\)
\( \displaystyle \quad=\frac{3}{9}\)
\( \displaystyle \quad=\frac{1}{3}\)

\(\displaystyle D=\frac{2}{3}\div \frac{5}{6} + \frac{3}{10}\)
\( \displaystyle \quad=\frac{2}{3}\times \frac{6}{5}+\frac{3}{10}\)
\( \displaystyle \quad=\frac{2\times 6}{3 \times 5}+\frac{3}{10}\)
\( \displaystyle \quad=\frac{2{\color{red}\times \color{red}3}\times 2}{{\color{red} 3}\times 5}+\frac{3}{10}\)
\( \displaystyle \quad=\frac{4}{5}+\frac{3}{10}\)
\( \displaystyle \quad=\frac{4{\color{red}\times \color{red}2}}{5{\color{red}\times \color{red}2}}+\frac{3}{10}\)
\( \displaystyle \quad=\frac{8}{10}+\frac{3}{10}\)
\( \displaystyle \quad=\frac{8+3}{10}\)
\( \displaystyle \quad=\frac{11}{10}\)
\( \displaystyle \quad=1.1\)

Exercice 4

\(\displaystyle A=\left(\frac{2}{5}-\frac{3}{2}\right)\div \left(\frac{5}{3}+\frac{2}{4}\right)\)
\( \displaystyle \quad=\left(\frac{2}{5}-\frac{3}{2}\right)\div \left(\frac{5}{3}+\frac{1}{2}\right)\)
\( \displaystyle \quad=\left(\frac{2\times 2}{5\times 2}-\frac{3\times 5}{2 \times 5}\right)\div \left(\frac{5\times 2}{3\times 2}+\frac{1\times 3}{2\times 3}\right)\)
\( \displaystyle \quad=\left(\frac{4}{10}-\frac{15}{10}\right)\div \left(\frac{10}{6}+\frac{3}{6}\right)\)
\( \displaystyle \quad=-\frac{11}{10} \div \frac{13}{6}\)
\( \displaystyle \quad=-\frac{11}{10} \times \frac{6}{13} \)
\( \displaystyle \quad=-\frac{66}{130}\)
\( \displaystyle \quad=-\frac{33 \color{red}\times \color{red}2}{65 \color{red}\times \color{red}2}\)
\( \displaystyle \quad=-\frac{33}{65}\)

\(\displaystyle B=\left(\frac{1}{2}-\frac{3}{4}\right)\div \frac{2}{3}\)
\( \displaystyle \quad=\left(\frac{1\times 2}{2\times 2}-\frac{3}{4}\right)\div \frac{2}{3}\)
\( \displaystyle \quad=\left(\frac{2}{4}-\frac{3}{4}\right)\div \frac{2}{3}\)
\( \displaystyle \quad=-\frac{1}{4} \div \frac{2}{3}\)
\( \displaystyle \quad=-\frac{1}{4} \times \frac{3}{2}\)
\( \displaystyle \quad=-\frac{3}{8}\)

\(\displaystyle C=\frac{1}{2}-\frac{3}{4} \div \frac{2}{3}\)
\( \displaystyle \quad=\frac{1}{2}-\frac{3}{4} \times \frac{3}{2}\)
\( \displaystyle \quad=\frac{1}{2}-\frac{9}{8}\)
\( \displaystyle \quad=\frac{1\times 4}{2\times 4} - \frac{9}{8}\)
\( \displaystyle \quad=\frac{4}{8}-\frac{9}{8}\)
\( \displaystyle \quad=\frac{4-9}{8}\)
\( \displaystyle \quad=-\frac{5}{8}\)

\(\displaystyle D=\left(2-4\times \frac{6}{7}\right)\div \frac{3}{14}\)
\( \displaystyle \quad=\left(2-\frac{24}{7}\right)\div \frac{3}{14}\)
\( \displaystyle \quad=\left(\frac{2\times 7}{1 \times 7}-\frac{24}{7}\right)\div \frac{3}{14}\)
\( \displaystyle \quad=\left(\frac{14}{7}-\frac{24}{7}\right)\div \frac{3}{14}\)
\( \displaystyle \quad=\left(\frac{14-24}{7}\right)\div \frac{3}{14}\)
\( \displaystyle \quad=-\frac{10}{7}\times \frac{14}{3}\)
\( \displaystyle \quad=-\frac{10\times 14}{7\times 3}\)
\( \displaystyle \quad=-\frac{10{\color{red}\times \color{red}7} \times 2}{{\color{red}7} \times 3}\)
\( \displaystyle \quad=-\frac{20}{3}\)

Exercice 5

\(A=12\times 10^{4}+5\times 10^{2}\)
\( \quad=120000+500\)
\( \quad=120500\)
\( \quad=1.205\times 10^{5}\)

\(B=13450\)
\( \quad=1.345\times 10^{4}\)

\(C=0.0079\)
\( \quad=7.9\times 10^{-3}\)

\(D=0.01\)
\( \quad=1\times 10^{-2}\)

\(E=0.1\times 10^{3}\times 10^{-2}\)
\( \quad=0.1\times 10^{3+(-2)}\)
\( \quad=0.1\times 10^{1}\)
\( \quad=1\)
\( \quad=1\times 10^{0}\)

Exercice 6

\(A=(10^{3})^{2}\times 15\)
\( \quad=10^{3\times 2}\times 15\)
\( \quad=10^{6}\times 15\)
\( \quad=10^{6}\times 1.5\times 10^{1}\)
\( \quad=1.5\times 10^{6+1}\)
\( \quad=1.5\times 10^{7}\)

\( B=3\times 10^{2}+5.2\times 10^{1}\)
\( \quad=300+52\)
\( \quad=352\)
\( \quad=3.52\times 10^{2}\)

\( C=\displaystyle \frac{2\times 10^{-2}\times 9\times 10^{4}}{3\times 10^{-1}\times 8\times 10^{2}}\)
\( \displaystyle \quad=\frac{2\times 9}{3\times 8}\times \frac{10^{-2}\times 10^{4}}{10^{-1}\times 10^{2}}\)
\( \displaystyle \quad=\frac{{\color{red}2 \color{red}\times \color{red}3}\times 3}{{\color{red}3 \color{red}\times \color{red}2}\times 4}\times \frac{10^{-2+4}}{10^{-1+2}}\)
\( \displaystyle \quad=\frac{3}{4}\times \frac{10^{2}}{10^{1}}\)
\( \quad=0.75\times 10^{2-1}\)
\( \quad=0.75\times 10\)
\( \quad=7.5\)
\( \quad=7.5\times 10^{0}\)

\( D=10^{1}\times 10^{-2}\times 10^{3}\times 10^{-4}\times 10^{5}\)
\( \quad=10^{1+(-2)+3+(-4)+5}\)
\( \quad=10^{3}\)
\( \quad=1\times 10^{3}\)