Simple Rule of Three
Simple, fast and easy
What is the Rule of Three?
The Rule of Three is a mathematical method used to solve problems that involve two proportional quantities. It allows finding an unknown value when three related values are known.
Practically, it is a fundamental tool in daily life, used for proportion calculations, such as: converting recipes, calculating discounts, predicting costs, among others.
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 directly or inversely proportional quantities.
Example of Directly Proportional Quantities: The more hours worked, the higher the salary.
Example of Inversely Proportional Quantities: The more workers, the less time to complete the work.
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 realize it — such as when doubling a recipe or calculating travel time based on speed.
- In Professions: Widely used in engineering, architecture, economics, administration, 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. In how many days would 20 workers do the same job?
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.