when two reasons have the same result, we say they are proportional. If these reasons represent measures of any greatness, we also say that they are proportional.
In other words, this equality means that the variations that occur in a greatness influence – or are influenced – by variations of the second.
Proportion example
Imagine that a car moves at 100 km/h and, in a certain period of time, travels a distance of 200 km. In this example, we have two greatnesses: speed and distance.
These magnitudes, in the same time interval, are dependent and influence each other, so that, if the car moves at a lower speed, it will not be able to cover the same distance. In fact, it is possible to say with certainty that, moving at half the speed, the car will cover half the distance and, therefore, in that period of time, it will reach 100 km.
From this example, you can write the reasons:
2 = 200 = 100 = Speed
100 50 distance
Concept formalization
Formally, a proportion it is an equality between reasons. Usually this equality is represented by fractions, as in the previous example. So, we say that A, B, C and D are proportional if the statement below is true:
THE = Ç = L
BD
In the chain of equalities above, the two fractions are called the proportion, and L is the proportionality constant. In the case of the previous example, the proportionality constant is 2.
How to identify proportional quantities
To identify proportional quantities, try to assemble one proportion between them. If possible, they will be proportionate; otherwise, no.
Example:
If a car travels 80 km at a speed of 40 km/h, then it will travel 160 km at a speed of 80 km/h. Note that the ratios between speed and distance have the same result:
40 = 80 = 1
80 160 2
A good example for non-proportional quantities is the weight and height ratio. It is evident that one size does not depend on the other, as there are thousands of people with different heights and weights.
Directly proportional quantities
Whenever an increase in one quantity results in an increase in another quantity proportional to it, we say they are directly proportional.
Imagine that a company works with assembling computer mice on several assembly lines. One of these lines is responsible for placing the central pulley, usually used to scroll the page accessed.
Suppose this company has 10 employees and they manage to assemble 380 mice per workday. If the company doubles the number of employees, will it also double the number of mounted mice? If the answer is yes, then we say that these quantities are directly proportional.
Inversely proportional quantities
Whenever the increase of one magnitude provides the reduction of another proportional to the first, we say that they are inversely proportional.
Imagine a trip made at 50 km/h in 2 hours. If we double the speed to 100 km/h, we will spend half the time, that is, just 1 hour. Therefore, by increasing the “speed” quantity, we decrease the “time” quantity.
Fundamental property of proportions
This property is the result of applying equations in proportionalities. Imagine that a, b, c and d are measures of two proportional quantities and respect the following proportion:
The = ç
b d
So, the above equality can also be written as follows:
ad = bc
This property is known as follows: The product of the means is equal to the product of the extremes.
Rule of three
The previous property is what makes it possible to find one of the measures of the magnitudes from the other three. This procedure is known as rule of three.
For example: In the company that assembles mice shown in the previous examples, 10 employees assemble 380 mice per workday. If it is necessary to assemble 1000 mice, how many employees must be hired at least?
Note that the number of mice produced divided by the number of employees must equal the same ratio in the second situation. This will need to have the employee number represented by some letter, as we don't know this number.
380 = 1000
10x
Using the fundamental property, we will have:
380x = 10·1000
380x = 10000
x = 10000
380
x = 26.3
As it is not possible to hire 0.3 employees, we know that the company will need 27 to meet the new target. Therefore, 17 more will be needed.
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-proporcao.htm