What is the Rule of Three?
The Rule of Three is a mathematical method used to solve problems involving two proportional quantities. It allows you to find an unknown value when three related values are known.
Practically, it is an essential everyday tool, used for proportion calculations such as converting recipes, calculating discounts, estimating costs, and more.
What is the Rule of Three Used For?
The Rule of Three is applied in various situations, such as:
- Unit conversion (e.g., kilograms to grams).
- Calculation of proportional prices.
- Time and distance estimates.
- Adjustment of culinary recipes.
- Financial problems, such as yields and discounts.
- Civil construction, for scales and proportions.
In summary: whenever two quantities are proportionally related, we can use the Rule of Three.
How Does the Rule of Three Work?
There are two main types of Rule of Three:
1. Simple Rule of Three
It is used when there are only two quantities that are directly or inversely proportional.
Example of Directly Proportional Quantities: {% trans "Quanto mais horas
trabalhadas, maior será o salário." %}
Example of Inversely Proportional Quantities: {% trans "Quanto mais
operários, menos tempo para concluir a obra." %}
How to Solve:
- Identify the quantities involved.
- Determine if they are directly or inversely proportional.
- Set up the proportion, relating the known values with the value to be discovered.
- Solve the equation.
Formulas:
Directly Proportional Quantities:
\[
\frac{a}{b} = \frac{c}{x}
\]
\[
a \times x = b \times c
\]
\[
x = \frac{b \times c}{a}
\]
Inversely Proportional Quantities:
\[
a \times b = c \times x
\]
\[
x = \frac{a \times b}{c}
\]
2. Compound Rule of Three
Used when there are more than two quantities involved in the problem.
Example: If 4 machines produce 100 pieces in 5 hours, how many
pieces will 6 machines produce in 3 hours?
Here, there are more quantities: number of machines, time, and production.
How to Solve:
- Relate each quantity to the result.
- Identify if the relationship is direct or inverse.
- Set up the compound proportion.
- Multiply and divide according to the proportionality.
Fun Facts About the Rule of Three
- History: The origin of the Rule of Three dates back to Antiquity, used by Babylonians and
Egyptians in commercial and agricultural activities.
- In Education: It is one of the first mathematical techniques taught in school, as it
develops logical reasoning and the ability to solve practical problems.
- In Daily Life: It is applied automatically, even when we don't notice — such as when doubling
a recipe or calculating travel time based on speed.
- In Professions: Widely used in engineering, architecture, economics, management,
gastronomy, logistics, and more.
Practical Examples
Example 1:
A car travels 300 km with 30 liters of fuel. How many liters will be needed to
travel 500 km?
\[
\begin{array}{cc}
300 \text{ km} & \rightarrow & 30 \text{ L} \\
500 \text{ km} & \rightarrow & x \text{ L}
\end{array}
\]
\[
\frac{300}{500} = \frac{30}{x}
\]
\[
300x = 500 \times 30
\]
\[
x = \frac{15000}{300}
\]
\[
x = 50 \text{ L}
\]
Example 2: (Inversely Proportional)
10 workers build a wall in 8 days. How many days would 20 workers take to complete the same
work?
More workers → fewer days → inverse relationship.
\[
10 \times 8 = 20 \times x
\]
\[
80 = 20x
\]
\[
x = \frac{80}{20}
\]
\[
x = 4 \text{ dias}
\]
Important Tips
- Always analyze if the relationship is direct or inverse.
- Use cross-multiplication carefully.
- Check the coherence of the result: it must make sense in context.