Elementary arithmetic, with brief notices of its history... by Robert Potts.

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And, universally, if any three terms of a proportion be given, the
fourth can always be found.
3. If the product of any two numbers be equal to the prodzct of two other
numbers, the four nmmbers can form a proportion.
For example, the four numbers 3, 5, 12, 20, are so related that the
product of two of them is equal to the product of the other two,
Or 3 x 20 = 5 x 12.
Dividing these equals by 5 x 20,
3 _ 1h
And 3: 5:: 12: 20.
It may be noted that when the terms of one ratio are prime to each
other, as 3 and 5, they are the least numbers in the proportion.
Since the two equal ratios which constitute a proportion are
denoted by two fractions, all the properties of ratios and proportions
depend on the properties of fractions.
If the four terms of a proportion be abstract numbers, or concrete
numbers of the same hind, the extremes and means may be interchanged one with another, so that the four numbers shall still continue!
to form a proportion.
Thus the first term shall have the same ratio to the third as the
second to the fourth; and the second shall have the same ratio to the
first as the fourth to the third.
If, however, the terms of one ratio be either abstract numbers, or
concrete numbers of a different kind from the other ratio, this interchange is not possible.
The following properties of a proportion can be readily shewn to
be true by means of fractions. Sec. IX.
1. If any equimultiples, or equisubmultiples, of the first and second
terms of a proportion be taken, and the same or any other equimultiples or equisubmultiples of the third and fourth be taken, the
four numbers will form a proportion.
term of the first ratio in questions of proportion. As for example-If 10 pounds of
tea cost 25 shillings, what is the cost of 100 pounds?
Here 10 pounds cost 25 shillings.
Let 100,,,, x
Here the number x bears the same ratio to 25 as 100 bears to 10.
And X 100 and 100 X25 ~12 10s.
2-5-O and x =z —
25 1' 10
But when the proportion is inverse, it appears more convenient to adopt the form in
Art. 1 p. 2, as in the following example:If 5 men perform a work in 12 days, in how many days can 4 men perform ar
equal work?
Here the effects are the same, and the effective causes must be equal.
If x denote the unknown number of men,
The first effective cause is 5X 12,, second,,,, 4x
5X1215
and 4Xx=5X12, x.. 15 men.
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