Proportion (mathematics) explained

A proportion is a mathematical statement expressing equality of two ratios.^{[1] [2]}

a:b = c:d

a and d are called extremes, b and c are called means.

Proportion can be written as

(a)/(= b) (c)/(d) , where ratios are expressed as fractions.

Such a proportion is known as geometrical proportion,^[3] not to be confused with arithmetical proportion and harmonic proportion.

Properties of proportions

• Fundamental rule of proportion. This rule is sometimes called Means‐Extremes Property.^[4] If the ratios are expressed as fractions, then the same rule can be phrased in terms of the equality of "cross-products"^[2] and is called Cross‐Products Property.^[4]

If

(ab = (cd)/())/() , then ad = bc

• If

(ab = (cd)/())/() , then (ba = (dc)/())/()

• If

(ab = (cd)/())/() , then(ac = (bd)/())/() ,(db = (ca)/())/() .

• If

(ab = (cd)/())/() , then\dfrac{a + b}{b} = \dfrac{c + d}{d} ,\dfrac{a - b}{b} = \dfrac{c - d}{d} .

• If

(ab = (cd)/())/() , then\dfrac{a + c}{b + d} = (ab)/(= (cd)/()) ,\dfrac{a - c}{b - d} = (ab)/(= (cd)/()) .

History

A Greek mathematician Eudoxus provided a definition for the meaning of the equality between two ratios. This definition of proportion forms the subject of Euclid's Book V, where we can read:

Later, the realization that ratios are numbers allowed to switch from solving proportions to equations, and from transformation of proportions to algebraic transformations.

Related concepts

Arithmetic proportion

An equation of the form

a - b = c - d is called arithmetic proportion or difference proportion.^[5]

Harmonic proportion

See main article: Golden ratio. If the means of the geometric proportion are equal, and the rightmost extreme is equal to the difference between the leftmost extreme and a mean, then such a proportion is called harmonic:^[6]

a : b = b : (a - b) . In this case the ratio a : b is called golden ratio.

See also

• Ratio
• Proportionality
• Correlation

Notes and References

1. Web site: Proportions: Introduction. Elizabeth. Stapel. www.purplemath.com.
2. Book: Intermediate Algebra: Identify Ratios, Rates, and Proportions. Alan S.. Tussy. R. David. Gustafson. January 2012 . 9781133714378 .
3. Web site: Geometrical proportion. oxforddictionaries.com.
4. Web site: Properties of Proportions. www.cliffsnotes.com.
5. Web site: Arithmetic proportion. encyclopediaofmath.org.
6. Web site: Harmonic Proportion in Architecture: Definition & Form. study.com.