The rule of three is a widely used mathematical tool in solving problems involving proportions. It is a technique that allows finding a **unknown value in a known ratio**, based on the direct or inverse relationship between the data provided. This is applied in various contexts, from business and financial calculations to everyday problems of daily life. In this article, we will explore what the rule of three in mathematics is and how it works, providing a clear and concise explanation of this fundamental tool in the world of proportions and percentages.

## What does the rule of three mean?

It is a mathematical technique used to compare quantities and **find an unknown value** based on the relationship of proportion between other known quantities. It can be applied in a variety of contexts, such as math, finance, business, and science problems.

It is generally used when three quantities are known and one wants to find **a fourth unknown quantity**. There are two types of rule of three: direct and inverse. In the direct rule of three, the quantities increase or decrease in the same proportion. In the inverse rule of three, when one quantity increases, the other decreases in proportion and vice versa.

It is a useful tool for solving practical problems, such as determining a quantity needed or getting percentages, and making decisions based on proportions. It is important to understand the basic concepts and how to apply it correctly in different situations.

### What is the inverse rule of three?

This is a way of solving math problems involving two quantities that vary in opposite directions. When two quantities** are inversely proportional**, means that each quantity increases, while the other decreases, and vice versa. It allows us to find the value of one of the quantities when we know the value of the other and the inverse relationship between them.

For example, suppose we are traveling in a car and we want to calculate **how long it will take to reach a given distance**. If we know that the speed of the car is inversely proportional to the travel time, that is, the higher the speed, the shorter the travel time, we can use the inverse rule of three to find the travel time needed to cover that distance.

For example, suppose a car travels a distance of 400 kilometers at a constant speed of 80 kilometers per hour. if we want to know **How long will it take **the car to travel a distance of 600 kilometers at the same speed, we can apply it as follows:

80 km ————— 1 hour

400 km ————– x hours (direct)

Dividing 400 by 80 you get 5:

80 km ————— 1 hour

400 km ————– 5 hours (direct)

Now, to find the time it takes the car to travel 600 km:

80 km ————— 1 hour

600 km ————– and hours (direct)

Dividing 600 by 80 gives 7.5:

80 km ————— 1 hour

600 km ————– 7.5 hours (direct)

Therefore, it would take the car 7.5 hours to travel a distance of 600 kilometers at a **constant velocity** of 80 kilometers per hour.

To apply the inverse rule of three, we would establish a proportion between the known velocity and the known travel time, and then solve for **unknown travel time**. Thus we will obtain the value of the travel time based on the known speed and the inverse relationship between them.

### What is the simple rule of three?

A quick and easy way to solve proportion problems. Imagine that you have three quantities: A, B and C, and you know the values of A and B, but** You want to find the value of C.**

If A and B have a constant relationship to each other, you can use the simple rule of three to find the value of C based on that relationship. If A increases, B also increases in the same proportion, and if A decreases, B t**also decreases in the same proportion**. This allows you to find the unknown quantity, C, based on this relationship.

To solve a problem, it is necessary to establish a proportion with the known values and the unknown quantity. Then you can use a **basic rule of multiplication or division** to find the value of the unknown quantity based on the established proportional relationship.

A common example of a simple rule of thumb is the following: if 3 apples cost $1, how much will 5 apples cost?

The solution can be found using a simple proportion. We can set it as follows:

3 apples———————- $1

5 apples———————– x

Where **x represents the unknown cost** of 5 apples.

To solve the equation, we can cross and multiply:

5 is multiplied by 1 and the result is divided by 3.

3x = 5($1)

3x = $5 x = $5 / 3

Therefore, the cost of 5 apples is** $1.67** (rounded to two decimal places).

It is important to remember that it only works when there is **a constant proportional relationship **between the known values and the unknown value. If the ratio is not constant, you may get incorrect results. However, in situations where the ratio is proportional, it can be a quick and useful tool for finding unknown values based on the stated ratio.

### What is the compound rule of three?

It is a way of solving mathematical problems involving three or more quantities that are related to each other. Imagine that you have three or more things that are related, such as time, distance, and the speed of three cars traveling together. This allows you to find the unknown value of one of these quantities, **as long as you know the values of the other two**.

For example, suppose you have a problem in which you are told that three workers can complete a job in a certain amount of time if they work together, but you are also given information about **how long would it take** to each of them to do the work separately. Using it, you can determine the length of time it would take each of them to complete the job if they worked alone.

To build a house, 4 workers are needed, who working 8 hours a day, take 20 days to finish it. If you wanted to build another house just like it, but in just 15 days, **How many workers would you need?**

In this problem, there are three magnitudes: the number of workers, the working time and the amount of work done. These magnitudes are related in a directly proportional way. So, we can raise the proportion:

4 workers x 8 hours x 20 days = x workers x 8 hours x 15 days

Where ‘x’ is the number of workers we need to build the house in 15 days.

Solving the proportion, we have:

X = 4 x 8 x 20 / 8 x 15

x = 640/120

x = **5.33 (rounding to a whole number)**

Therefore, we need to hire 6 workers to build the house in 15 days.

The compound rule of three has its application in **everyday life situations**, such as in buying and selling problems, mixing ingredients in cooking recipes, or calculating the relative speed of two objects moving in different directions. It is a useful tool for solving practical problems involving proportional relationships between three or more quantities.

## What is the rule of three used for in mathematics?

It is a way of solving problems in which one seeks to find an unknown value based on a proportion. It is like making a comparison between different quantities **for** **discover how they are related to each other**. It is used to answer questions such as ‘If 5 apples cost $10 or 1 apple costs 1 euro, how much will 8 apples cost?’, or ‘If 3 people can build a house in 10 days, how long will it take if 6 people are working? ‘.

To solve a problem, simple multiplication and division can be used to find the unknown value. Is a **practical and useful tool **which has its appearances in everyday situations that involve comparing proportions and calculating unknown values. The rule of three can be applied in different contexts, whether they are fractions, identifying an unknown physical quantity, calculating percentages, medication dosage, unit conversion, finance, among others. It is a useful tool and **widely used in various fields**such as finance, business, science and more.

However,** not applicable across the board**, with it it is not possible to determine non-proportional and non-linear relationships, if the complete data is not available it cannot be useful either, in addition it is not adequate to determine the value of some measure of area or volume such as M². And so with other mathematical applications or not that do not require its use.

## How does the rule of three work to get the percentage?

To get the percentage of an unknown value, you must follow these steps:

**Set the ratio:**You need to have two known quantities and one unknown quantity. For example, if you want to calculate 20% of 100, you have the original value (100) and the known percentage (20).**Write the equation:**Write the proportion in the form of an equation, using the known facts and the unknown quantity. In our example, the equation would be: ’20 is to 100 as x is to 100′, where ‘x’ represents the unknown quantity.**Solve the equation:**You can use the crossed rule of three to solve the equation. Cross multiply and divide by the amount you want to find. In our example, we would cross multiply like this: ’20 multiplied by 100 is equal to x multiplied by 100′.**Clear the unknown quantity:**To find the value of ‘x’ (which represents the percentage you’re looking for), divide both sides of the equation by the remaining known value (in this case, 100). This will give you the value of ‘x’, which is the percentage you were looking for.

## How to do the rule of three in mathematics?

To make a rule of three in mathematics:

- Read the problem carefully and identify the quantities you know and the one you want to find.
- Write the
**known and unknown quantities**in the form of a fraction, placing the unknown quantity on top (numerator) and the known quantities below (denominator). - Use the rule that ‘
**cross terms are multiplied**‘ to solve the fraction by multiplying the numerators and denominators together. - Clear the unknown quantity by solving the resulting equation for
**find its value.** - Check your answer by making sure it makes sense in the context of the problem.

Remember that the rule of three is just a mathematical tool that can help you solve proportion problems, and that it is important to understand the problem statement and** check your answer** to make sure it is correct.

## Examples of the use of the rule of three in mathematics

Some practical examples of how the rule of three can be used:

**When two quantities increase or decrease together:**Imagine that you need to paint a wall and you know that with 5 liters of paint you can cover an area of 100 square meters. But now you want to know how many liters you will need to paint a wall larger than 150 square meters. Using the rule of three, you can do a simple calculation to find that you will need 7.5 liters of paint.**When two quantities vary in opposite directions:**Suppose you are building a bridge and you know that with 6 workers you can finish it in 10 days. But now you wonder how many days it will take if you add 2 more workers. You can calculate that it will take 7.5 days in total.**When there is a combination of direct and inverse proportions:**Imagine that you are managing a factory and you need to know how many machines are required to produce 90 parts in 6 hours, whereas before you produced 60 parts in 8 hours with 4 machines. You can determine that you will need 5 machines in total.