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# proportion equation

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

$\displaystyle\frac{a}{b}\;=\;\frac{c}{d}\;\quad\mbox{or}\;\quad a:b\;=\;c:d$ | (1) |

The numbers $a$, $b$, $c$, $d$ are the members of the proportion; $a$ 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 $c$.

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

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

- •
The proportion (1) is equivalent to the proportion

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

## Mathematics Subject Classification

97U99*no label found*12D99

*no label found*

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