proportion equation
The proportion equation, or usually simply , is an equation whose both are ratios (http://planetmath.org/Division) of (nonzero) numbers:
$\frac{a}{b}}={\displaystyle \frac{c}{d}}\mathit{\hspace{1em}\hspace{0.25em}}\text{or}\mathit{\hspace{1em}\hspace{0.25em}}a:b=c:d$  (1) 
The numbers $a$, $b$, $c$, $d$ are the members of the ; $a$ and $d$ are the extreme members and $b$ and $c$ are the middle members. The number $d$ is called the fourth proportional of the numbers $a$, $b$ and $c$.
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The product of the extreme members of the is equal to the product of the middle members.

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The
$$\frac{a}{c}=\frac{b}{d},$$ i.e., the middle members can be swapped.

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The
$$\frac{a+b}{ab}=\frac{c+d}{cd}$$ if the do not vanish.

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If any three members of a are known, then the fourth member may be determined (often by using the first property).

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If the number $b$ satisfies the proportion
$\frac{a}{b}}={\displaystyle \frac{b}{c}$ (2) then $b$ is called the 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^{}.

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In (2), the number $c$ is called the third proportional of $a$ and $b$.
Title  proportion equation 
Canonical name  ProportionEquation 
Date of creation  20140223 21:37:10 
Last modified on  20140223 21:37:10 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  12 
Author  pahio (2872) 
Entry type  Topic 
Classification  msc 97U99 
Classification  msc 12D99 
Synonym  proportion 
Related topic  Equation 
Related topic  SimilarityInGeometry 
Related topic  GoldenRatio 
Related topic  ContraharmonicProportion 
Related topic  ConstructionOfFourthProportional 
Defines  proportion 
Defines  extreme members 
Defines  middle members 
Defines  fourth proportional 
Defines  central proportional 
Defines  third proportional 