Rule of Three

What is the Rule of Three?

The rule of three, also known as "proportion" or "rule of proportion," is a mathematical method used to solve problems of proportions and relationships between different quantities.

There are two types of rule of three: simple and compound. Below, I explain the difference between them:

Simple Rule of Three

The simple rule of three is used when there are three values and you want to find a fourth value that maintains the same proportion as the previous three. It is divided into two types: direct and inverse.

Direct Rule of Three

It is used when the two magnitudes vary in a directly proportional manner. That is, if one magnitude increases, the other also increases and vice versa. The direct rule of three is calculated with the following formula:

Example of Direct Rule of Three

If 5 apples cost 10 euros, how much will 8 apples cost?

Inverse Rule of Three

It is used when the two magnitudes vary in an inversely proportional manner. That is, if one magnitude increases, the other decreases and vice versa. The inverse rule of three is calculated with the following formula:

Example of Inverse Rule of Three

If 5 workers take 10 days to do a job, how long will 8 workers take?

Compound Rule of Three

The compound rule of three is used when more than two magnitudes are involved and you want to find a value that maintains the proportion with the others. It can be direct or inverse, depending on how the magnitudes are related.

Example of Compound Rule of Three

Statement

If 5 workers build 3 houses in 10 days, how many days will it take 8 workers to build 6 houses?

• Identify the magnitudes involved: workers, houses, and days.
• Establish the proportions so that one magnitude is constant and the others vary proportionally.
• Solve the compound proportion.

Solution

First, solve the relationship between the magnitudes:

• Relationship between workers and days: inversely proportional (more workers, fewer days).
• Relationship between houses and days: directly proportional (more houses, more days).

Then, calculate the days it would take 8 workers to build 3 houses using the inverse rule of three:

Since we now know that 8 workers take 6.25 days to build 3 houses, using the direct rule of three we find the days for 6 houses:

Therefore, 8 workers will need 12.5 days to build 6 houses.