proportion equationProportionEquation2004-12-16 13:19:032007-10-04 12:22:21Topicvariationproportionextreme membersmiddle membersfourth proportionalcentral proportionalthird proportionalratio% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe \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}.\, The number $d$ is called the {\em fourth proportional} of the numbers $a$, $b$ and $c$.
\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).
\item If the number $b$ satisfies the proportion
\begin{align}
\frac{a}{b} = \frac{b}{c},
\end{align}
then $b$ is called the {\em central proportional} of $a$ and $c$.\, We have
$$b = \sqrt{ac},$$
i.e., the central proportional of two real numbers (of same sign) equals to their geometric mean.
\item In (2), the number $c$ is called the {\em third proportional} of $a$ and $b$.
\end{itemize}