Proportion (mathematics)

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📅 22/10/2025

From Wikipedia, the free encyclopedia

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

a:b=c:d{\displaystyle a:b=c:d}

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

Proportion can be written as ab=cd{\displaystyle {\frac {a}{b}}={\frac {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.

  • 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

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

, then  ad=bc

{\displaystyle \ ad=bc}
  • If  ab=cd{\displaystyle \ {\frac {a}{b}}={\frac {c}{d}}}, then  ba=dc{\displaystyle \ {\frac {b}{a}}={\frac {d}{c}}}
  • If  ab=cd{\displaystyle \ {\frac {a}{b}}={\frac {c}{d}}}, then

 ac=bd

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

, db=ca

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

.

  • If  ab=cd{\displaystyle \ {\frac {a}{b}}={\frac {c}{d}}}, then

 a+bb=c+dd

{\displaystyle \ {\dfrac {a+b}{b}}={\dfrac {c+d}{d}}}

, a−bb=c−dd

{\displaystyle \ {\dfrac {a-b}{b}}={\dfrac {c-d}{d}}}

.

  • If  ab=cd{\displaystyle \ {\frac {a}{b}}={\frac {c}{d}}}, then

 a+cb+d=ab=cd

{\displaystyle \ {\dfrac {a+c}{b+d}}={\frac {a}{b}}={\frac {c}{d}}}

, a−cb−d=ab=cd

{\displaystyle \ {\dfrac {a-c}{b-d}}={\frac {a}{b}}={\frac {c}{d}}}

.