proportion equation

The proportion equation, or usually simply , is an equation whose both are ratios (http://planetmath.org/Division) of (non-zero) numbers:

${\displaystyle \frac{a}{b}}={\displaystyle \frac{c}{d}}\mathit{\hspace{1em}\hspace{0.25em}}\text{or}\mathit{\hspace{1em}\hspace{0.25em}}a:b=c:d$ (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 $c$.

.

• The product of the extreme members of the is equal to the product of the middle members.

• The

  $$\frac{a}{c}=\frac{b}{d},$$

  i.e., the middle members can be swapped.

• The

  $$\frac{a+b}{a-b}=\frac{c+d}{c-d}$$

  if the do not vanish.

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

• If the number $b$ satisfies the proportion

  ${\displaystyle \frac{a}{b}}={\displaystyle \frac{b}{c}}$ (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.

• In (2), the number $c$ is called the third proportional of $a$ and $b$.

Title	proportion equation
Canonical name	ProportionEquation
Date of creation	2014-02-23 21:37:10
Last modified on	2014-02-23 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