Rule of three
In mathematics, the rule of three consists of solving proportionality problems between three known values and an unknown. It establishes a relationship of linearity, proportionality, between the values.
- La Rule of three is the operation to find the fourth term of a proportion knowing the other three.
The most well-known rule of three is the direct simple rule of three, although there is also the inverse simple rule of three and the compound rule of three. The rule of three is very useful in mathematical cases due to its ease of use and understanding.
Simple rule of three
In the simple rule of three, the proportional relationship between two known values is established and knowing a third value 'X', we compute a fourth value Y.
The proportionality relationship can be direct or inverse. It will be direct when a greater value of A will have a greater value of B, and it will be inverse when a greater value of A corresponds to a lesser value. value of B.
Simple direct rule of three
The simple direct rule of three is based on a proportionality relationship, so it can be quickly observed that:
Where k is the constant of proportionality. For this proportionality to be fulfilled, an increase in A must correspond to an increase in B in the same proportion. It can be represented in the form:
It is then said that A is directly proportional to B, as X is to Y, where < b>AND
equals the product of B times X divided by A.
Imagine that we are asked the following:
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This problem is interpreted as follows: the relationship is direct, since the greater the number of rooms, the more paint will be needed, and we represent it like this:
Inverse Simple Rule of Three
In the simple inverse rule of three, in the relationship between the values it is true that:
where e is a constant product. For this constant to be preserved, an increase in A will require a decrease in B, so that their product remains constant. This relationship can be represented in the form:
and A is said to be inversely proportional to B, as X is to Y, where < b>Y equals the product of A times B divided by X.
If, for example, we have the problem:
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If the meaning of the statement is carefully observed, it becomes clear that the more workers work, the fewer hours they will need to build the same wall (assuming that they all work at the same pace).
The total number of working hours needed to build the wall is 120 hours, which can be contributed by a single worker who uses 120 hours, 2 workers in 60 hours, 3 workers who will do so in 40 hours, etc. In all cases the total number of hours remains constant.
We therefore have a relationship of inverse proportionality, and we must apply a simple inverse rule of three, in effect:
Compound Rule of Three
It is that mathematical operation, which is used when more than two magnitudes participate in the problem. Sometimes the problem posed involves more than three known quantities, in addition to the unknown. Let's look at the following example:
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In the problem presented, two proportionality relations appear at the same time. In addition, to complete the example, an inverse and a direct relationship have been included. Indeed, if a 100-meter wall is built by 12 workers, it is clear that fewer workers will be needed to build a 75-meter wall. The smaller the wall, the less number of workers we need: it is a relationship of direct proportionality. On the other hand, if we have 15 hours available for 12 workers to work, it is evident that having 26 hours we will need fewer workers. As one quantity increases, the other decreases: it is a relationship of inverse proportionality.
The problem would be stated as follows:
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The solution to the problem is to multiply 12 by 75 and by 15, and divide the result by the product of 100 by 26. Therefore, 13,500 divided by 2,600 is 5.19 (which by rounding turns out to be 6 workers since 5 workers would not be enough).
The problem looks like this:
- We leave the following table where we know that meters and workers are in direct proportionality while hours and workers are in reverse proportionality
- Our objective will be to transform in 4 steps the first row of the previous table in the second respecting the direct and reverse proportions for the relation meters-workers and hours-workers respectively
- We identify the unknown variable in the second row of the first table with the result obtained so that:
that is, we obtain the sought solution.
The problem can be posed with all the terms you want, whether they are all direct relationships, all inverse or mixed, as in the previous case. Each rule must be considered very carefully, taking into account whether it is inverse or direct, and taking into account (this is very important) not to repeat any term when joining each of the simple relations.
Examples
- To pass 60 degrees to radians we apply a three-direct rule:
We place the unknown in the first position:
This formalizes the question "How many radians are there in 60 degrees, given that π radians are 180 degrees?". Thus we have that:
A useful technique to remember how to find the solution of a rule of three is the following: X is equal to the product of the crossed terms (π and 60, in this case) divided by the term that is crossed with X.
- Calculate how many minutes there are in 7 hours. As there are 60 minutes in 1 hour, we apply a three-direct rule:
The result is:
- Calculate how long a group of 4 people takes to perform an activity if 3 people would take 2 hours to do it. The more people, the less time it takes, so we apply a three-verse rule:
The result is:
As a mnemonic, the inverse rule of three is calculated by multiplying the magnitudes horizontally and dividing by the isolated magnitude.