2022-2023/2nd/14_Information_Chiffree_2/3B_successive.tex

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\documentclass[a4paper,12pt]{article}
\usepackage{myXsim}
\author{Benjamin Bertrand}
\title{Information chiffrée 2- Cours}
\date{Mars 2023}
\pagestyle{empty}
\begin{document}
\maketitle
\section*{Taux d'évolution successifs}
\begin{propriete}
Quand une quantité subit des \textbf{évolution successives} $t_1, t_2, ...$, elle subit alors une \textbf{évolution globale}.
Les taux d'évolution \textbf{ne peuvent pas} s'ajouter.
\bigskip
Il faut multiplier les \textbf{coefficient multiplicateur} entre eux.
\begin{center}
\begin{tikzpicture}[
roundnode/.style={circle, draw=highlightbg, fill=green!5, very thick, minimum size=3mm},
node distance=2cm and 2cm,
arrow/.style={->, shorten >=5pt, shorten <=5pt}
]
%Nodes
\node[roundnode] (termA) {\makebox[0.5cm]{}};
\node[roundnode] (termB) [right=of termA] {\makebox[0.5cm]{}};
\node[roundnode] (termC) [right=of termB] {\makebox[0.5cm]{}};
\node[roundnode] (termD) [right=of termC] {\makebox[0.5cm]{}};
\node (termE) [right=1cm of termD] {\makebox[0.5cm]{...}};
\node[roundnode] (termF) [right=1cm of termE] {\makebox[0.5cm]{}};
%Lines
\path[arrow] (termA.north) edge [bend left=50] node [above] {$+t_1$} node [below] {$\times CM_1$} (termB.north) ;
\path[arrow] (termB.north) edge [bend left=50] node [above] {$+t_2$} node [below] {$\times CM_2$} (termC.north) ;
\path[arrow] (termC.north) edge [bend left=50] node [above] {$+t_3$} node [below] {$\times CM_3$} (termD.north) ;
\path[arrow] (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);
\end{tikzpicture}
\end{center}
\end{propriete}
\paragraph{Exemples:}
\begin{itemize}
\item Une quantité a subit 5 augmentations de 10\%.
\begin{center}
\begin{tikzpicture}[
roundnode/.style={circle, draw=highlightbg, fill=green!5, very thick, minimum size=3mm},
arrow/.style={->, shorten >=5pt, shorten <=5pt}
]
%Nodes
\node[roundnode] (termA) {\makebox[0.5cm]{}};
\node[roundnode] (termB) [right=2cm of termA] {\makebox[0.5cm]{}};
\node[roundnode] (termC) [right=2cm of termB] {\makebox[0.5cm]{}};
\node[roundnode] (termD) [right=2cm of termC] {\makebox[0.5cm]{}};
\node[roundnode] (termE) [right=2cm of termD] {\makebox[0.5cm]{}};
\node[roundnode] (termF) [right=2cm of termE] {\makebox[0.5cm]{}};
%Lines
\path[arrow] (termA.north) edge [bend left=50] node [above] {$+10\%$} node [below] {$\times ...$} (termB.north) ;
\path[arrow] (termB.north) edge [bend left=50] node [above] {$+10\%$} node [below] {$\times ...$} (termC.north) ;
\path[arrow] (termC.north) edge [bend left=50] node [above] {$+10\%$} node [below] {$\times ...$} (termD.north) ;
\path[arrow] (termD.north) edge [bend left=50] node [above] {$+10\%$} node [below] {$\times ...$} (termE.north) ;
\path[arrow] (termE.north) edge [bend left=50] node [above] {$+10\%$} node [below] {$\times ...$} (termF.north) ;
\path[arrow] (termA.south) edge [bend right=10] node [above] {Taux d'évolution global} node [below] {$\times ... \times ... \times ... \times ... \times ... = \times ...$} (termF.south);
\end{tikzpicture}
\end{center}
Le coefficient global est donc de $CM = ...$
On en déduit le \textbf{taux d'évolution global} $t = ...$
\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.
\end{itemize}
\afaire{Compléter les exemples}
\end{document}