What is the Compound Rule of Three?
The Compound Rule of Three is a mathematical tool used to solve problems that
involve more than two related quantities. Unlike the Simple Rule of Three, which deals with only
two quantities, the compound rule can involve three or more simultaneously.
Para que Serve a Regra de Três Composta?
It is essential for solving problems with multiple variables that
influence each other. Common examples include:
- Production in factories: workers, time, and quantity of products.
- Civil construction: number of workers, days, and works.
- Transportation: quantity of vehicles, speed, and load.
How Does the Compound Rule of Three Work?
The principle is the same as the Simple Rule of Three: analyze the proportional relationships
between quantities and solve through a proportion. However, in the compound rule, we must analyze each
quantity in relation to the final result and check whether they are directly or inversely proportional.
Steps to Solve:
- List all quantities involved.
- Identify if each quantity is directly or inversely proportional to the result.
- Set up the proportion considering the proportionality relationships.
- Multiply the values and solve the equation.
General Formula of the Compound Rule of Three
For a problem with \( n \) quantities, the resolution structure is:
\[
x = \frac{(v_1^{\pm 1}) \times (v_2^{\pm 1}) \times \ldots \times (v_n^{\pm 1}) \times R}{1}
\]
Where:
- \( v_i \): value of quantity i
- O expoente "+1" indica relação direta e "-1" relação inversa.
- \( R \): result or final value to be determined
Cross Multiplication
In general, we multiply the values of directly proportional quantities and divide by the
inversely proportional ones:
\[
x = \frac{\text{Produto das grandezas diretas}}{\text{Produto das grandezas inversas}}
\]
Practical Example
Problem: If 4 machines produce 200 pieces in 5 hours, how many
pieces will 6 machines produce in 3 hours?
Solution:
- Machines → production: direct relationship (more machines, more production).
- Time → production: direct relationship (more time, more production).
We set up the proportion:
\[
\frac{x}{200} = \frac{6}{4} \times \frac{3}{5}
\]
We multiply:
\[
\frac{x}{200} = \frac{6 \times 3}{4 \times 5} = \frac{18}{20} = \frac{9}{10}
\]
Therefore:
\[
x = 200 \times \frac{9}{10}
\]
\[
x = 180 \text{ peças}
\]
Important Tips
- Always check if the relationship between the quantities is direct or inverse.
- Organize the data in a table to facilitate visualization.
- Simplify fractions when possible.
Fun Facts about the Compound Rule of Three
- The Compound Rule of Three is widely used in areas such as engineering, management,
and logistics, where multiple variables need to be considered simultaneously.
- Although it may seem complex, it is merely a logical extension of the Simple Rule of Three,
requiring only an understanding of proportional relationships.
- It is common for its application to occur automatically, such as when calculating the time of a
trip considering the number of vehicles and the speed.
Conclusion
The Compound Rule of Three is a powerful tool that allows solving complex problems
in a structured and logical manner. Mastery of it is essential for various professions and everyday situations that
involve multiple variables.