proportionality of numbers
The nonzero numbers are (directly) proportional to the nonzero numbers if
which special notation means the simultaneous validity of the proportion equations
It follows however that
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 .
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
which is equivalent (http://planetmath.org/Equivalent3) with (1) and (2).
The numbers are inversely proportional to the numbers if
Then we have
Note. The notation 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 may be multiplied by a nonzero number without injuring the validity of (1).
Example. Let and . Determine the least positive integers to which the numbers are a) directly, b) inversely proportional.
a) The least common multiple of 3 and 4, the members corresponding the members in the given proportions, is 12. Thus we must multiply the right hand sides of these proportions respectively by and :
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|
|Date of creation||2014-02-23 21:26:01|
|Last modified on||2014-02-23 21:26:01|
|Last modified by||pahio (2872)|