# proportion equation

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

 $\displaystyle\frac{a}{b}\;=\;\frac{c}{d}\;\quad\mbox{or}\;\quad a:b\;=\;c:d$ (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

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