proportionality of numbers

The nonzero numbers $a_1,a_2,\mathrm{\dots },a_n$ are (directly) proportional to the nonzero numbers$b_1,b_2,\mathrm{\dots },b_n$ if

$a_1:a_2:\mathrm{\dots }:a_n=b_1:b_2:\mathrm{\dots }:b_n,$(1)

which special notation means the simultaneous validity of the proportion equations

$a_1:a_2=b_1:b_2,a_2:a_3=b_2:b_3,\mathrm{\dots },a_{n-1}:a_n=b_{n-1}:b_n.$(2)

It follows however that

$a_i:a_j=b_i:b_j\mathit{\hspace{1em}}\text{for all}i,j.$(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

${\displaystyle \frac{a_1}{b_1}}={\displaystyle \frac{a_2}{b_2}}=\mathrm{\dots }={\displaystyle \frac{a_n}{b_n}}$(4)

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

The numbers $a_1,a_2,\mathrm{\dots },a_n$ are inversely proportional to the numbers $b_1,b_2,\mathrm{\dots },b_n$ if

$$a_1:a_2:\mathrm{\dots }:a_n=\frac{1}{b_1}:\frac{1}{b_2}:\mathrm{\dots }:\frac{1}{b_n}.$$

Then we have

$$a_i:a_j=b_j:b_i\mathit{\hspace{1em}}\text{for all}i,j.$$

Note.  The notation  $a_1:a_2:\mathrm{\dots }:a_n$  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  $\frac{12}{4}=3$:

$$a:b=\mathrm{\hspace{0.33em}2}:3=\mathrm{\hspace{0.33em}8}:12,b:c=\mathrm{\hspace{0.33em}4}:5=\mathrm{\hspace{0.33em}12}:15.$$

Accordingly,

$$a:b:c=\mathrm{\hspace{0.33em}8}:12:15.$$

b) We may write

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

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.