What is the Compound Rule of Three?

The Compound Rule of Three is a mathematical tool used to solve problems involving more than two related quantities. Unlike the Simple Rule of Three, which works with only two quantities, the compound one can involve three or more, simultaneously.

What is the Compound Rule of Three Used For?

It is fundamental for solving problems that have 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 proportionality relationships between the quantities and solve through a proportion. However, in the compound one, we must analyze each quantity in relation to the final result and verify if they are directly or inversely proportional.

Steps to Solve:

1. List all quantities involved.
2. Identify if each quantity is directly or inversely proportional to the result.
3. Set up the proportion considering the proportionality relationships.
4. 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 the 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 be produced by 6 machines 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, administration, and logistics, where several variables need to be considered simultaneously.
• Although it seems complex, it is just a logical extension of the Simple Rule of Three, requiring only an understanding of proportionality relationships.
• It is common for its application to occur automatically, such as when calculating travel time considering the number of vehicles and speed.

Conclusion

The Compound Rule of Three is a powerful tool that allows solving complex problems in a structured and logical way. Its mastery is essential for various professions and daily situations involving multiple variables.