Feat: bilan sur coef mult et évo succ
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@ -57,7 +57,83 @@
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\end{exercise}
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\begin{solution}
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\section*{Lien entre taux d'évolution et coefficient multiplicateur}
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\begin{propriete}
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Soit $t$ le taux d'évolution qui fait évolution $v_i$ vers $v_f$ alors
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\[
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v_f = (1+t)v_i
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\]
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\end{propriete}
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\begin{definition}
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$1+t$ est appelé \textbf{coefficient multiplicateur}, noté $CM$, associé au taux d'évolution $t$.
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\noindent
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On a ainsi
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\[
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CM = 1+t \qquad \mbox{ou encore} \qquad t = CM - 1
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\]
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et
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\[
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v_f = CM \times v_i \qquad \mbox{ ou encore } \qquad CM = \frac{v_f}{v_i}
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\]
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\end{definition}
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\paragraph{Exemples}
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\begin{itemize}
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\item
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\begin{tikzpicture}[
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baseline,
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roundnode/.style={circle, draw=highlightbg, fill=green!5, very thick, minimum size=3mm},
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]
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%Nodes
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\node[roundnode] (leftterme) {\makebox[0.5cm]{120}};
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\node[roundnode] (centerterm) [right=2cm of leftterme] {\makebox[0.5cm]{...}};
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%Lines
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\path[->] (leftterme.north) edge [bend left]
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node [pos=0.5, above] {+25\%}
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node [pos=0.5, below] {\times CM = ...}
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(centerterm.north)
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;
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\end{tikzpicture}
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Coefficient multiplicateur: \hfill Valeur finale: \hfill.
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\item
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\begin{tikzpicture}[
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baseline,
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roundnode/.style={circle, draw=highlightbg, fill=green!5, very thick, minimum size=3mm},
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]
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%Nodes
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\node[roundnode] (leftterme) {\makebox[0.5cm]{55}};
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\node[roundnode] (centerterm) [right=2cm of leftterme] {\makebox[0.5cm]{...}};
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%Lines
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\path[->] (leftterme.north) edge [bend left]
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node [pos=0.5, above] {+...}
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node [pos=0.5, below] {\times 0.12}
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(centerterm.north)
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;
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\end{tikzpicture}
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Taux d'évolution: \hfill Valeur finale: \hfill.
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\item
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\begin{tikzpicture}[
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baseline,
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roundnode/.style={circle, draw=highlightbg, fill=green!5, very thick, minimum size=3mm},
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]
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%Nodes
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\node[roundnode] (leftterme) {\makebox[0.5cm]{34}};
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\node[roundnode] (centerterm) [right=2cm of leftterme] {\makebox[0.5cm]{123}};
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%Lines
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\path[->] (leftterme.north) edge [bend left]
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node [pos=0.5, above] {+...\%}
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node [pos=0.5, below] {\times ...}
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(centerterm.north)
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;
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\end{tikzpicture}
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Taux d'évolution: \hfill Valeur finale: \hfill.
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\end{itemize}
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\afaire{Compléter les calculs}
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\end{solution}
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\begin{exercise}[subtitle={Questions divers}, step={2}, origin={Création}, topics={ Information Chiffrée 2 }, tags={ Information chiffrée }, mode={\faIcon{tools}}]
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@ -108,7 +184,7 @@
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\end{exercise}
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\begin{exercise}[subtitle={Évolutions successives - bilan}, step={3}, origin={Création}, topics={ Information Chiffrée 2 }, tags={ Information chiffrée }, mode={\faIcon{users}}]
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Expliquer votre/vos méthode(s) pour calculer la taux d'évolution d'une transformation composé de plusieurs évolution. Vous illustrerez votre méthode en calculant le taux d'évolution global composé de
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Expliquer votre/vos méthode(s) pour calculer la taux d'évolution d'une transformation composé de plusieurs évolution. Vous illustrerez votre méthode en calculant le taux d'évolution global des trois situations suivantes:
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\begin{enumerate}
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\item 5 augmentation de 4\%.
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\item 100 diminution de 5\%.
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@ -118,6 +194,71 @@
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\begin{solution}
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\section*{Taux d'évolution successifs}
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\begin{propriete}
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Quand une quantité subit des \textbf{évolution successives} $t_1, t_2, ...$, elle subit alors une \texbf{évolution globale}.
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Les taux d'évolution \textbf{ne peuvent pas} s'ajouter.
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\bigskip
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Il faut multiplier les \textbf{coefficient multiplicateur} entre eux.
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\begin{center}
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\begin{tikzpicture}[
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roundnode/.style={circle, draw=highlightbg, fill=green!5, very thick, minimum size=3mm},
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node distance=2cm and 2cm
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]
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%Nodes
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\node[roundnode] (termA) {\makebox[0.5cm]{}};
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\node[roundnode] (termB) [right=of termA] {\makebox[0.5cm]{}};
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\node[roundnode] (termC) [right=of termB] {\makebox[0.5cm]{}};
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\node[roundnode] (termD) [right=of termC] {\makebox[0.5cm]{}};
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\node (termE) [right=of termD] {\makebox[0.5cm]{...}};
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\node[roundnode] (termF) [right=of termE] {\makebox[0.5cm]{}};
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%Lines
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\path[->] (termA.north) edge [bend left=50] node [above] {$+t_1$} node [below] {$\times CM_1$} (termB.north) ;
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\path[->] (termB.north) edge [bend left=50] node [above] {$+t_2$} node [below] {$\times CM_2$} (termC.north) ;
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\path[->] (termC.north) edge [bend left=50] node [above] {$+t_3$} node [below] {$\times CM_3$} (termD.north) ;
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\path[->] (termA.south) edge [bend right=10] node [above] {Taux d'évolution global} node [below] {$\times CM_1 \times CM_2 \times CM_3 \times ...$} (termF.south);
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\end{tikzpicture}
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\end{center}
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\end{propriete}
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\paragraph{Exemples:}
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\begin{itemize}
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\item Une quantité a subit 5 augmentations de 10\%.
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\begin{center}
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\begin{tikzpicture}[
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roundnode/.style={circle, draw=highlightbg, fill=green!5, very thick, minimum size=3mm},
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node distance=2cm and 2cm
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]
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%Nodes
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\node[roundnode] (termA) {\makebox[0.5cm]{}};
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\node[roundnode] (termB) [right=of termA] {\makebox[0.5cm]{}};
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\node[roundnode] (termC) [right=of termB] {\makebox[0.5cm]{}};
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\node[roundnode] (termD) [right=of termC] {\makebox[0.5cm]{}};
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\node[roundnode] (termE) [right=of termD] {\makebox[0.5cm]{}};
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\node[roundnode] (termF) [right=of termE] {\makebox[0.5cm]{}};
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%Lines
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\path[->] (termA.north) edge [bend left=50] node [above] {$+10\%$} node [below] {$\times ...$} (termB.north) ;
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\path[->] (termB.north) edge [bend left=50] node [above] {$+10\%$} node [below] {$\times ...$} (termC.north) ;
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\path[->] (termC.north) edge [bend left=50] node [above] {$+10\%$} node [below] {$\times ...$} (termD.north) ;
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\path[->] (termD.north) edge [bend left=50] node [above] {$+10\%$} node [below] {$\times ...$} (termE.north) ;
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\path[->] (termE.north) edge [bend left=50] node [above] {$+10\%$} node [below] {$\times ...$} (termF.north) ;
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\path[->] (termA.south) edge [bend right=10] node [above] {Taux d'évolution global} node [below] {$\times ... \times ... \times ... \times ... \times ... = \times ...$} (termF.south);
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\end{tikzpicture}
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\end{center}
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Le coefficient global est donc de $CM = ...$
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On en déduit le \textbf{taux d'évolution global} $t = ...$
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\item Une quantité a subit une augmentation de 5\% puis un diminution de 10\% et enfin une autre augmentation de 5\%. Calculons le taux d'évolution global.
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\end{itemize}
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\afaire{Compléter les exemples}
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\end{solution}
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\begin{exercise}[subtitle={Techniques}, step={3}, origin={Création}, topics={ Information Chiffrée 2 }, tags={ Information chiffrée }, mode={\faIcon{tools}}]
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