One reason is a division between two numbers that can be represented by the usual notation of the division, through a fraction or through a rational number, resulting from this division. When two ratios are the same, they are called proportion. One of properties of proportions is called fundamental and it guarantees that an equality between reasons is equivalent to an equality between products.
Fundamental property of proportions
Suppose the numbers represented by the letters “x”, “y”, “t” and “z” form a ratio. For this reason, it is possible to write them in the form of equality between reasons, simply following the order in which they were presented:
x = t
y z
Note that this same proportion can also be written in the following form:
x: y = t: z
This shape is the usual notation for divisions. Using this notation, the numbers represented by “x” and “z” are at the extremes of the proportion and the numbers represented by “y” and “t” occupy the central position of that proportion. Using these data, the fundamental property of proportions can be stated as follows:
The product of extremes is equal to the product of means.
So, the proportion:
x = t
y z
It's equivalent to:
x·z = y·t
From these equalities, it is possible to make some variations of this property, taking into account that we can invert the equality without changing its value, or change the order of factors without changing the product. These operations generate the rest of the properties of proportions, which are other ways to organize them.
Use of the fundamental property of proportions
A ratio is made up of four numbers. It is possible to find one of these numbers if the other three are known. To do so, just use the fundamental property of proportions, rewriting it as equality of products, and treating that result as a equation ordinary.
For example, note the following proportion:
10 = x
20 60
Using the fundamental property of proportions and treating the result as a common equation, we will have:
10·60 = 20x
600 = 20x
– 20x = – 600(– 1)
20x = 600
x = 600
20
x = 30
This procedure is known as rule of three.
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-propriedade-fundamental-das-proporcoes.htm