PlanetMath: proportion equation

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proportion equation

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The proportion equation, or usually simply proportion, is an equation whose both sides are ratios of (non-zero) numbers:

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

(1)

The numbers

are the members of the proportion;

and

are the extreme members and

and

are the middle members. The number

is called the fourth proportional of the numbers

and

Properties of proportions.

The product of the extreme members of the proportion is equal to the product of the middle members.
The proportion (1) is equivalent with the proportion
$\displaystyle \frac{a}{c} = \frac{b}{d},$
i.e., the middle members can be swapped.
The proportion (1) is equivalent with the proportion
$\displaystyle \frac{a+b}{a-b} = \frac{c+d}{c-d}$
if the divisors do not vanish.
If any three members of a proportion are known, then the fourth member may be determined (often by using the first property).
If the number satisfies the proportion

$\displaystyle \frac{a}{b} = \frac{b}{c},$ (2)

then is called the central proportional of and . We have
$\displaystyle b = \sqrt{ac},$
i.e., the central proportional of two real numbers (of same sign) equals to their geometric mean.
In (2), the number is called the third proportional of and .

"proportion equation" is owned by pahio.

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Other names:

proportion

Also defines:

proportion, extreme members, middle members, fourth proportional, central proportional, third proportional

Keywords:

ratio

This object's parent.

Attachments:: construction of central proportional (Algorithm) by pahio
contraharmonic proportion (Definition) by pahio

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Cross-references: geometric mean, real numbers, Proportion, property, vanish, product, equation
There are 7 references to this entry.

This is version 6 of proportion equation, born on 2004-12-16, modified 2007-10-04.
Object id is 6585, canonical name is ProportionEquation.
Accessed 10263 times total.

Classification:

AMS MSC:	12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
	97U99 (Mathematics education :: Educational material and media. Educational technology :: Miscellaneous)

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