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'proportion equation'
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| Title of object: |
proportion equation |
| Canonical Name: |
ProportionEquation |
| Type: |
Definition |
| Created on: |
2004-12-16 13:19:03 |
| Modified on: |
2004-12-16 13:34:08 |
| Classification: |
msc:12D99, msc:97U99 |
| Defines: |
extreme members, middle members |
| Synonyms: |
proportion equation=proportion |
Revision comment (for changes between this and next version):
| Added 4th proportional and central proportional |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
The \emph{proportion equation}, or usually simply \PMlinkescapetext{{\em proportion}}, is an equation whose both \PMlinkescapetext{sides} are \PMlinkname{ratios}{Division} of (non-zero) numbers:
\begin{align}
\frac{a}{b} = \frac{c}{d}
\end{align}
The numbers $a$, $b$, $c$, $d$ are the {\em members} of the \PMlinkescapetext{proportion}; $a$ and $d$ are the {\em extreme members} and $b$ and $c$ are the {\em middle members}.
\textbf{\PMlinkescapetext{Properties of proportions}}.
\begin{itemize}
\item The product of the extreme members of the \PMlinkescapetext{proportion} is equal to the product of the middle members.
\item The \PMlinkescapetext{proportion (1) is equivalent with the proportion}
$$\frac{a}{c} = \frac{b}{d},$$
i.e., the middle members can be swapped.
\item The \PMlinkescapetext{proportion (1) is equivalent with the proportion}
$$\frac{a+b}{a-b} = \frac{c+d}{c-d}$$
if the \PMlinkescapetext{divisors} do not vanish.
\item If any three members of a \PMlinkescapetext{proportion} are known, then the fourth member may be determined (often by using the first property).
\end{itemize} |
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