THE rule of three is one of the basic contents of Math most important to students. Most of the assessment exercises, such as Enem, entrance exams and competitions, can be solved using this knowledge, in addition, this rule can also be applied to questions of Physics, Chemistry and also to solve everyday problems.
Because it's so important, we bring together the threemistakescommittedmore often in the application of the ruleinthree to help students not to commit them anymore and also to clarify possible doubts about this content.
1 – Problem interpretation
This mistake is not committed only in the ruleinthree, but in mathematical content in general. It is very important to correctly interpret the text of the problems.
From the following example, observe how to proceed in this case: A car travels at 90 km/h and, in a certain period of time, manages to travel 270 km. If this same car were at 120 km/h, how many more kilometers would it travel than in the first situation?
The first step in solving such an exercise is to realize that the time period in question is irrelevant to the calculations. It only matters that it is the same period for both situations. Then, also realize that, in order to find the extra kilometers that were covered, we must, first, find the total kilometers traveled at 120 km/h, that is, the calculations need to be made in
twophases.It turns out that, at the end of the first stage, some students believe they have finished the problem and end up leaving the solution incomplete. Note the ruleinthree for the first step of the exercise:
90 = 270
120x
90x = 270·120
90x = 32400
x = 32400
90
x = 360 km
As we want to know how many more kilometers were covered, we must still calculate the difference between 360 and 270:
360 - 270 = 90 km
Thus, the car will have covered 90 km more, at 120 km/h, in the indicated period of time.
2 – Mounting the resolution
All ruleinthree can be understood as a proportion, that is, it is the equality between two reasons. These two reasons can be taken from geometric figures or situations like the one in the previous example and, for them to be really equal, they need to follow a certain order.
Example: A factory produces 150 units of an element a day and, for this, has 25 employees. Planning an expansion of production to 275 pieces per day, how many employees will be needed to produce them, considering the ideal working conditions?
The first reason that we will assemble will refer to the current situation of the industry. THE fraction will be formed by numerator = number of employees, and denominator = number of pieces.
25
150
The second reason that we will assemble refers to the situation intended by the company and must follow the same pattern as the initial: number of employees in the numerator and number of parts in the denominator.
x
275
like the two reasons were assembled following a (correct) pattern, we know that your results will be the same, so we can write:
25 = x
150 275
solving the ruleinthree, we have:
150x = 25·275
x = 6875
150
x = 45,833…
Thus, 46 employees will be needed.
3 – Directly or inversely proportional quantities
One of the mistakesmostfrequent in the resolution of ruleinthree it concerns not checking whether the quantities involved are direct or inversely proportional. In the first case, the rule of three is done as in the two previous examples. In the second case, no. Therefore, it is necessary to be very careful not to make this kind of mistake.
Therefore, to consider two quantities as directlyproportional, we must notice that, when increasing the values referring to one of them, the values referring to the other also increase. Otherwise, the two quantities are inverselyproportional.
Example: A car is traveling at a speed of 90 km/h and it takes 2 hours to cover a certain route. If this car were at 45 km/h, how many hours would it spend on the same route?
Note that, when decreasing the car's speed, the correct thing is to understand that the time spent on the same route should increase. Therefore, the magnitudes are inverselyproportional.
To solve this kind of rule of three, set the ratio normally and then reverse one of the reasons before proceeding:
90 = 2
45 x
90 = x
45 2
45x = 90·2
45x = 180
x = 180
45
x = 4 hours
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/matematica/tres-erros-mais-cometidos-no-uso-regra-tres.htm