# proportionality of numbers

The nonzero numbers $a_{1},\,a_{2},\,\ldots,\,a_{n}$ are (directly) proportional to the nonzero numbers $b_{1},\,b_{2},\,\ldots,\,b_{n}$ if

 $\displaystyle a_{1}\!:\!a_{2}\!:\ldots:\!a_{n}\;=\;b_{1}\!:\!b_{2}\!:\ldots:\!% b_{n},$ (1)

which special notation means the simultaneous validity of the proportion equations

 $\displaystyle a_{1}\!:\!a_{2}\;=\;b_{1}\!:\!b_{2},\;\quad a_{2}\!:\!a_{3}\;=\;% b_{2}\!:\!b_{3},\;\quad\ldots,\;\quad a_{n-1}\!:\!a_{n}\;=\;b_{n-1}\!:\!b_{n}.$ (2)

It follows however that

 $\displaystyle a_{i}\!:\!a_{j}\;=\;b_{i}\!:\!b_{j}\quad\mbox{for all}\;\;i,\,j.$ (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  $a_{1}\!:\!a_{3}\;=\;b_{1}\!:\!b_{3}$.

Swapping the middle members of the proportions (2), which by the parent entry (http://planetmath.org/ProportionEquation) is allowable, one gets the system of equations

 $\displaystyle\frac{a_{1}}{b_{1}}\;=\;\frac{a_{2}}{b_{2}}\;=\;\ldots\;=\;\frac{% a_{n}}{b_{n}}$ (4)

which is equivalent (http://planetmath.org/Equivalent3) with (1) and (2).

The numbers $a_{1},\,a_{2},\,\ldots,\,a_{n}$ are inversely proportional to the numbers $b_{1},\,b_{2},\,\ldots,\,b_{n}$ if

 $a_{1}\!:\!a_{2}\!:\ldots:\!a_{n}\;=\;\frac{1}{b_{1}}\!:\!\frac{1}{b_{2}}\!:% \ldots:\!\frac{1}{b_{n}}.$

Then we have

 $a_{i}\!:\!a_{j}\;=\;b_{j}\!:\!b_{i}\quad\mbox{for all}\;\;i,\,j.$

Note.  The notation  $a_{1}\!:\!a_{2}\!:\ldots:\!a_{n}$  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 $a_{i}$ 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 multiple 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  $\frac{12}{3}=4$  and  $\frac{12}{4}=3$:

 $a\!:\!b\;=\;2\!:\!3\;=\;8\!:\!12,\quad b\!:\!c\;=\;4\!:\!5\;=\;12\!:\!15.$

Accordingly,

 $a\!:\!b\!:\!c\;=\;8\!:\!12\!:\!15.$

b) We may write

 $a\!:\!b\;=\;\frac{1}{3}\!:\!\frac{1}{2}\;=\;\frac{1}{15}\!:\!\frac{1}{10},% \quad b\!:\!c\;=\;\frac{1}{5}\!:\!\frac{1}{4}\;=\;\frac{1}{10}\!:\!\frac{1}{8},$

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

 $a\!:\!b\!:\!c\;=\;\frac{1}{15}\!:\!\frac{1}{10}\!:\!\frac{1}{8},$

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

## References

• 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