proportionality of numbers

The nonzero numbers a1,a2,,an are (directly) proportional to the nonzero numbers b1,b2,,bn if

a1:a2::an=b1:b2::bn, (1)

which special notation means the simultaneous validity of the proportion equations

a1:a2=b1:b2,a2:a3=b2:b3,,an-1:an=bn-1:bn. (2)

It follows however that

ai:aj=bi:bjfor alli,j. (3)

In fact, if one multiplies the left hand sides of e.g. two first equations (2) and similarly their right hand sides, then one obtains  a1:a3=b1:b3.

Swapping the middle members of the proportions (2), which by the parent entry ( is allowable, one gets the system of equations

a1b1=a2b2==anbn (4)

which is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath ( with (1) and (2).

The numbers a1,a2,,an are inversely proportional to the numbers b1,b2,,bn if


Then we have

ai:aj=bj:bifor alli,j.

Note.  The notation  a1:a2::an  expressing the “ratio of several numbers” is, of course, , but it behaves as the ratio (= the quotient) of two numbers in the sense that all of its members ai may be multiplied by a nonzero number without injuring the validity of (1).

Example.  Let  a:b=2:3  and  b:c=4:5.  Determine the least positive integers to which the numbers a,b,c are  a) directly,  b) inversely proportional.
a) The least common multipleMathworldPlanetmathPlanetmath of 3 and 4, the members corresponding the members b in the given proportions, is 12.  Thus we must multiply the right hand sides of these proportions respectively by  123=4  and  124=3:

a:b= 2:3= 8:12,b:c= 4:5= 12:15.


a:b:c= 8:12:15.

b) We may write


where the denominators of the right hand sides have been multiplied by 5 and 2, respectively.  Consequently,


i.e. the required integers are 15, 10, 8.


  • 1 K. Väisälä: Geometria.  Reprint of the tenth edition.  Werner Söderström Osakeyhtiö, Porvoo & Helsinki (1971).
Title proportionality of numbers
Canonical name ProportionalityOfNumbers
Date of creation 2014-02-23 21:26:01
Last modified on 2014-02-23 21:26:01
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Definition
Classification msc 97U99
Classification msc 12D99
Synonym proportionality
Related topic Variation
Related topic KalleVaisala
Defines proportional
Defines directly proportional
Defines inversely proportional