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Rule of Three Calculator

Reviewed by BS Román Ramos

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Type of proportionality

Table of Contents

How to Use the Calculator

To use our online simulator, you will need to enter 3 different data:

  1. A figure that corresponds to the first variable which we will call A
  2. A figure that corresponds to the second variable which we will call B
  3. The number X that establishes a direct or inverse relationship with A.

You can also change the type of proportionality from direct to inverse.

After entering these three data, we press the “Calculate” button and our simulator will return the result.


If 12 newspapers cost me 8 €, how much will 15 cost me?

In this case, the unknown A would be the number of newspapers, and we should enter 12 in the first box, 8 in the second, and 15 in the third.

In this third box, we enter the number 15 because it is directly related to the unknown A and, therefore, it will help us find its correlative of the unknown B, which in our example are the 8 €.

Once you have entered the 3 variables, click on “Calculate” and our tool will give you the result in a matter of seconds and automatically, in the case of the example the solution is 10 €, which would be the amount you would have to pay for the 15 newspapers.

If you want to learn how to perform this process manually to understand all the mechanisms that compose it, keep reading.

What is the Rule of Three

It is a method for solving problems of proportionality and that is frequently used in the day-to-day life of any student.

It is an arithmetic calculation in which two different variables are related that manifest some type of proportion between them. This relationship can be either directly proportional (as one increases, the rest increase) or inversely proportional (as one increases, the other decreases).

To find the solution to these mathematical problems, we will need to calculate the unknown, which in this case will be the fourth variable, an unknown data that we will discover thanks to the proportionality relationship it maintains with the rest.

What is Needed to Solve It

To get started and solve a problem with this rule, we will need to know a series of figures and factors.

First, you must ask yourself what type of proportion unites them, whether it is directly or inversely proportional.

Second, you must know the relationship they maintain the variables that maintain a proportional relationship and one to which we want to apply such relationship to obtain the unknown.

How to Calculate a Direct Rule of Three (Simple)

First Example

Juan has just been shown the grades for a university subject, which was divided into two blocks: the theoretical exam that counts for 60% of the total grade and the practicals, which count for 40% of the total.

If Juan has an 8 in the practicals, but only a 6 in the theoretical exam, what will be Juan's total grade?

The first step we will always take in this type of problems is to establish whether the two variables maintain direct proportionality or not. To do this, you have to answer the question "If the grade of the exam (or the practicals) increases, will the total grade increase?"

In this case, the answer is yes, therefore, we can affirm that the relationship between them is directly proportional.

The second step we will carry out will be to calculate the proportional part that the theoretical exam will contribute to the grade. Considering that it represents 60% of that grade and that Juan scored a 6, it would be as follows:

$$6 \longrightarrow 100$$

$$x \longrightarrow 60$$


$$x=\frac{60\times 6}{100} = \frac{360}{100} = 3.60$$

That is, Juan will have a 3.6 plus the (proportional) grade of the practical part.

It will be in this third step where we will calculate the proportion of the practical part in the total grade. Considering that this part represents 40% and that Juan scored an 8, the calculations to be made would be the following:

$$8 \longrightarrow 100$$

$$x \longrightarrow 40$$


$$\frac{8 \times 40}{100} = \frac{320}{100} = 3.20 $$

This is the amount that we must add to the previous one to know the total result (out of 10) that Juan will obtain in this subject:

$$3.6 + 3.2 = 6.8$$

This will be the final grade that will appear in Juan's record, to calculate it we have used the rule of 3, using it twice and then adding the results.

Second Example

Let's put another example, this time where only one calculation needs to be made.

Let's imagine that we find a vaporizer supplier that for larger orders of 50 units gives us a discount that remains, even if we order 500.

The only problem is that the information comes to us through a contact who bought 60 units and paid a total of 1260 €, but it turns out that you want to make an order of 80, how much will all these vaporizers cost you?

Well, we simply have to apply the previous formula changing the data, it would be as follows:

$$ \quad 6 \longrightarrow 1260$$

$$80 \longleftarrow \quad x $$

In this case, the operations to be performed would be the following:

$$x=\frac{80 \times 1260} {60} = \frac{100800}{60} = 1680 €$$

and this would be the total price that you would have to pay the supplier for the 80 vaporizers.

How to Calculate an Inverse Rule of Three

Something inversely proportional is nothing other than the non-symmetrical relationship between two values, that is, when one increases the other decreases.

Perhaps with an example it becomes clearer, the faster a car goes the less time it will take to cover a specific distance.

That is, both unknowns (speed and time used) maintain an inverse relationship, in that the faster the car goes the less time it will take to cover a specific distance.

There are more inversely proportional relationships than it might seem, let's pose a practical problem to solve it step by step and make everything much clearer.


6 workers in a factory take 30 days to build a car, how long would they need to build it if instead of 6 there were 10 workers?

First, we must clarify if both variables maintain an inversely proportional relationship, in this case, they do since the greater the number of workers the less time they will need to build a car.

The formula, however, is practically the same:

$$30 \longrightarrow 6$$

$$x \longrightarrow 10$$

$$x=\frac{30 \times 10}{6} = \frac{40}{6} = 6.6 \textrm{days}$$

This is how long 10 workers in the factory would take to build a car.

And this is all you need to know to be able to understand and apply the rule of three in mathematical problems, in this article you have learned what it is, what two types there are, what differentiates them, and how to calculate both always guided by examples.

We hope you have enjoyed the article, liked it, and it has answered all the questions you had before starting to read it.

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Finally, we would like to ask you that if you find errors in form both in the calculator and in the article, do not hesitate to let us know through the contact page, thus you will help us solve them as soon as possible.

BS Román Ramos
Bachelor of Mathematics and Master in Operations Research from UCV.
Román Ramos holds a Bachelor's degree in Mathematics from the Central University of ... Read more »

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