CALCUL NUMERIQUE
Correction des exercices ***
|
Exercice 1 (France juin 2012)
1)
\(\displaystyle \frac{10^{5}+1}{10^{5}}\)
\(\displaystyle \quad = \frac{100\text{ }000+1}{100\text{ }000}\)
\(\displaystyle \quad = \frac{100\text{ }001}{100\text{ }000}\)
\(\quad 1.00001\)
L'écriture décimale de ce nombre est 1.00001.
2) Antoine a raison. En effet, ce nombre est très proche de 1 mais
n'est pas égal à 1. la calculatrice a en effet arrondi le résultat :
\(\displaystyle \frac{10^{15}+1}{10^{15}}\)
\(\displaystyle \quad = \frac{1\text{ }000\text{ }000\text{ }000\text{ }000\text{ }000+1}{1\text{ }000\text{ }000\text{ }000\text{ }000\text{ }000}\)
\(\displaystyle \quad = \frac{1\text{ }000\text{ }000\text{ }000\text{ }000\text{ }001}{1\text{ }000\text{ }000\text{ }000\text{ }000\text{ }000}\)
\( \quad \neq 1 \)
Exercice 2 (QCM des brevets de 2012)
| N° |
Réponse
A |
Réponse
B |
Réponse
C |
Réponse
D |
| 1 |
|
x
|
|
|
| 2 |
x |
|
|
|
| 3 |
x |
|
|
|
| 4 |
|
|
x |
|
| 5 |
x |
|
|
|
| 6 |
|
|
x |
|
| 7 |
x |
|
|
|
Aide
:
3) \(\; \displaystyle 2+\frac{2}{3}\times \frac{1}{4}\)
\(\displaystyle \quad = 2+\frac{2\times 1}{3 \times 4}\)
\(\displaystyle \quad = 2+\frac{{\color{red}2}\times 1}{3 \times 2 \color{red} \times {\color{red}2}}\)
\(\displaystyle \quad = 2+\frac{1}{6}\)
\(\displaystyle \quad = \frac{12}{6}+\frac{1}{6}\)
\(\displaystyle \quad = \frac{13}{6}\)
4) \(\; 10^{2}\times 21 \times 10^{-7}\)
\(\quad = 2.1\times 10^{1}\times 10^{2}\times 10^{-7}\)
\(\quad = 2.1\times 10^{1+2+(-7)}\)
\(\quad = 2.1\times 10^{-4}\)
6) \(\; \displaystyle \frac{12}{25}\times \frac{7}{10}\)
\(\displaystyle \quad = \frac{12\times 7}{25\times 10}\)
\(\displaystyle \quad = \frac{84}{250}\)
7) \(\; \left(4\times 10^{-3}\right)^{2}\)
\(\quad = 4^{2}\times \left(10^{-3}\right)^{2}\)
\(\quad = 16\times 10^{-3\times 2}\)
\(\quad = 16\times 10^{-6}\)
\(\quad = 1.6\times 10^{1}\times 10^{-6}\)
\(\quad = 1.6\times 10^{1+(-6)}\)
\(\quad = 1.6\times 10^{-5}\)
Exercice 3 (QCM des brevets de 2011)
| N° |
Réponse
A |
Réponse
B |
Réponse
C |
Réponse
D |
| 1 |
|
|
x |
|
| 2 |
|
x |
|
|
| 3 |
|
|
x |
|
| 4 |
|
|
x |
|
| 5 |
x |
|
|
|
| 6 |
|
|
x |
|
| 7 |
|
x |
|
|
| 8 |
|
|
|
x |
| 9 |
|
|
x |
|
| 10 |
|
x |
|
|
Aide
:
1) \(\displaystyle \; \frac{\left(10^{-3}\right)^{2}\times 10^{4}}{10^{-5}}\)
\(\displaystyle \quad\frac{10^{-3\times 2}\times 10^{4}}{10^{-5}}\)
\(\displaystyle \quad\frac{10^{-6}\times 10^{4}}{10^{-5}}\)
\(\quad =10^{-6+4-(-5)}\)
\(\quad =10^{3}\)
3) \(\displaystyle \; \frac{7}{3}-\frac{4}{3}\div \frac{5}{2}\)
\(\displaystyle \quad =\frac{7}{3}-\frac{4}{3}\times \frac{2}{5}\)
\(\displaystyle \quad =\frac{7}{3}-\frac{4\times 2}{3\times 5}\)
\(\displaystyle \quad =\frac{7}{3}-\frac{8}{15}\)
\(\displaystyle \quad =\frac{7\times 5}{3 \times 5}-\frac{8}{15}\)
\(\displaystyle \quad =\frac{35}{15}-\frac{8}{15}\)
\(\displaystyle \quad =\frac{35-8}{15}\)
\(\displaystyle \quad =\frac{27}{15}\)
5) \(\displaystyle \; 3^{-2}\times 3^{3}-3\)
\(\quad =3^{-2+3}-3\)
\(\quad =3^{1}-3\)
\(\quad =3-3\)
\(\quad =0 \)
6) \(\displaystyle \; \frac{1}{9}+\frac{1}{6}\)
\(\displaystyle \quad =\frac{1\times 2}{9 \times 2}+\frac{1 \times 3}{6 \times 3}\)
\(\displaystyle \quad =\frac{2}{18}+\frac{3}{18}\)
\(\displaystyle \quad =\frac{2+3}{18}\)
\(\displaystyle \quad =\frac{5}{18}\)
7) \(\displaystyle \; 2\times 10^{-3}\times 10^{5}\)
\(\quad =2\times 10^{-3+5}\)
\(\quad =2\times 10^{2}\)
8) \(\displaystyle \; \frac{6\times 10^{3}\times 28 \times 10^{-2}}{14\times 10^{-3}}\)
\(\displaystyle \quad \frac{6\times 28}{14}\times\frac{10^{3} \times 10^{-2}}{10^{-3}}\)
\(\quad =12\times 10^{3+(-2)-(-3)}\)
\(\quad =12\times 10^{4}\)
9) \(\displaystyle \; \frac{4}{3}-\frac{4}{3}\times \frac{27}{24}\)
\(\displaystyle \quad = \frac{4}{3}-\frac{4\times 27}{3 \times 24}\)
\(\displaystyle \quad = \frac{4}{3}-\frac{{\color{red}4 \color{red}\times \color{red}3 \color{red} \times \color{red} 3} \times 3}{{\color{red}3 \color{red}\times \color{red}3 \color{red}\times \color{red} 4}\times 2}\)
\(\displaystyle \quad = \frac{4}{3}-\frac{3}{2}\)
\(\displaystyle \quad = \frac{4\times 2}{3 \times 2}-\frac{3\times 3}{2 \times 3}\)
\(\displaystyle \quad = \frac{8}{6}-\frac{9}{6}\)
\(\displaystyle \quad = \frac{8-9}{6}\)
\(\displaystyle \quad =-\frac{1}{6}\)
10)\(\displaystyle \; \frac{5}{3}-\frac{6}{5}\)
\(\displaystyle \quad =\frac{5\times 5}{3 \times 5}-\frac{6 \times 3}{5 \times 3}\)
\(\displaystyle \quad =\frac{25}{15}-\frac{18}{15}\)
\(\displaystyle \quad =\frac{25-18}{15}\)
\(\displaystyle \quad =\frac{7}{15}\)
Exercice 4
\(\displaystyle A=\frac{3}{4}-\frac{2}{3}\div \frac{8}{15}\)
\(\displaystyle \quad =\frac{3}{4}-\frac{2}{3}\times \frac{15}{8}\)
\(\displaystyle \quad =\frac{3}{4}-\frac{2\times 15}{3 \times 8}\)
\(\displaystyle \quad =\frac{3}{4}-\frac{{\color{red}2 \color{red}\times \color{red}3} \times 5}{{\color{red}3 \color{red}\times \color{red}2} \times 4}\)
\(\displaystyle \quad =\frac{3}{4}-\frac{5}{4}\)
\(\displaystyle \quad =\frac{3-5}{4}\)
\(\displaystyle \quad =-\frac{2}{4}\)
\(\displaystyle \quad =-\frac{1}{2}\)
\(\displaystyle B=\frac{6}{5}-\frac{17}{4}\div \frac{5}{7}\)
\(\displaystyle \quad =\frac{6}{5}-\frac{17}{4}\times \frac{7}{5}\)
\(\displaystyle \quad =\frac{6}{5}-\frac{17\times 7}{4 \times 5}\)
\(\displaystyle \quad =\frac{6}{5}-\frac{119}{20}\)
\(\displaystyle \quad =\frac{6\times 4}{5 \times 4}-\frac{119}{20}\)
\(\displaystyle \quad =\frac{24}{20}-\frac{119}{20}\)
\(\displaystyle \quad =\frac{24-119}{20}\)
\(\displaystyle \quad =-\frac{95}{20}\)
\(\displaystyle \quad =-\frac{19 {\color{red}\times \color{red}5}}{4 {\color{red}\times \color{red}5}}\)
\(\displaystyle \quad =-\frac{19}{4} \)
\(\displaystyle C=\frac{5}{7}+\frac{1}{7}\times \frac{4}{3}\)
\(\displaystyle \quad =\frac{5}{7}+\frac{1\times 4}{7 \times 3}\)
\(\displaystyle \quad =\frac{5}{7}+\frac{4}{21}\)
\(\displaystyle \quad =\frac{5\times 3}{7\times 3}+\frac{4}{21}\)
\(\displaystyle \quad =\frac{15}{21}+\frac{4}{21}\)
\(\displaystyle \quad =\frac{15+4}{21}\)
\(\displaystyle \quad =\frac{19}{21} \)
\(\displaystyle D=\frac{7}{15}-\frac{4}{15}\times \frac{5}{8}\)
\(\displaystyle \quad =\frac{7}{15}-\frac{4\times 5}{15 \times 8}\)
\(\displaystyle \quad =\frac{7}{15}-\frac{4\times 5}{15\times 8}\)
\(\displaystyle \quad =\frac{7}{15}-\frac{\color{red}4 \color{red}\times \color{red} 5}{3\times {\color{red}5 \color{red}\times \color{red}4} \times 2}\)
\(\displaystyle \quad =\frac{7}{15}-\frac{1}{6}\)
\(\displaystyle \quad =\frac{7\times 2}{15 \times 2}-\frac{1\times 5}{6 \times 5}\)
\(\displaystyle \quad =\frac{14}{30}-\frac{5}{30}\)
\(\displaystyle \quad =\frac{14-5}{30}\)
\(\displaystyle \quad =\frac{9}{30}\)
\(\displaystyle \quad =\frac{{\color{red}3}\times 3}{{\color{red}3}\times 10}\)
\(\displaystyle \quad =\frac{3}{10} \)
Exercice 5
\(\displaystyle E=\frac{6\times 10^{-2}\times 5\times 10^{2}}{1.5\times 10^{-4}}\)
\(\displaystyle \quad \frac{6\times 5}{1.5}\times \frac{10^{-2}\times 10^{2}}{10^{-4}}\)
\(\quad =20\times 10^{-2+2-(-4)}\)
\(\quad =20\times 10^{4}\)
\(\quad =2\times 10^{1}\times 10^{4}\)
\(\quad =2\times 10^{1+4}\)
\(\quad =2\times 10^{5}\)
\(\displaystyle F=\frac{6\times 10^{12}\times 35\times 10^{-4}}{14\times 10^{3}}\)
\(\displaystyle \quad \frac{6\times 35}{14}\times \frac{10^{12}\times 10^{-4}}{10^{3}}\)
\(\quad =15\times 10^{12+(-4)-3}\)
\(\quad =15\times 10^{5}\)
\(\quad =1.5\times 10^{1}\times 10^{5}\)
\(\quad =1.5\times 10^{1+5}\)
\(\quad =1.5\times 10^{6}\)
\(\displaystyle G=\frac{8\times 10^{8}\times 1.6}{0.4\times 10^{-3}}\)
\(\displaystyle \quad \frac{8\times 1.6}{0.4}\times \frac{10^{8}}{10^{-3}}\)
\(\quad =32\times 10^{8-(-3)}\)
\(\quad =32\times 10^{11}\)
\(\quad =3.2\times 10^{1}\times 10^{11}\)
\(\quad =3.2\times 10^{1+11}\)
\(\quad =3.2\times 10^{12} \)
\(\displaystyle H=\frac{3\times 10^{-5}\times 6\times 10^{3}}{3\times 10^{11}}\)
\(\displaystyle \quad \frac{3\times 6}{3}\times \frac{10^{-5}\times 10^{3}}{10^{11}}\)
\(\quad =6\times 10^{-5+3-11}\)
\(\quad =6\times 10^{-13} \)
Correction des exercices de brevet sur le calcul numérique (révisions) pour la troisième (3ème)
© Planète Maths