Introduction to Compound Rule of Three Problems

Relevance of the Topic

Compound Rule of Three is one of the most used and practical tools in applied mathematics. It has a strong and immediate presence in our daily lives, especially when it comes to negotiations, financial calculations, statistics, complex proportionalities, among others.

The ability to solve compound rule of three problems is a reflection of an individual's logical-mathematical thinking and, therefore, is a central axis in the mathematics curriculum. By mastering this topic, you will be prepared to tackle more complex and interdisciplinary problems that require this type of analysis and reasoning.

Contextualization

Compound Rule of Three cannot be understood in isolation, it is part of a broader spectrum of mathematical knowledge that includes simple rule of three, ratios, proportions, and scales. Mastering this subject will allow you to progress to more complex topics, such as arithmetic and geometric progressions, logarithms, functions, and much more.

As a student in the 2nd year of High School, this is the ideal time to broaden your understanding of the rule of three, as the problems you will encounter now will be more comprehensive and challenging, requiring not only the application of the rule but also a deep understanding of how the different elements of a problem interact and influence the final solution.

Therefore, this theme is perfectly inserted within the broad panorama of mathematics and the development of analytical and problemsolving skills. Understanding and mastering the compound rule of three will be your passport to explore broader and more complex horizons within the mathematical sphere.

Theoretical Development: Compound Rule of Three Problems

Components

• Background of Compound Rule of Three: To understand the compound rule of three, it is essential to have a command of the simple rule of three. The simple rule of three compares two directly proportional quantities, while the compound rule compares three quantities. This is essential because understanding the concept of proportionality is the basis for all rule of three.

• Reading the Problem: A critical step in solving compound rule of three problems is to fully understand the statement. This includes identifying the quantities involved, understanding how they relate to each other, and what question the problem is asking.

• Division into Parts: Once you understand the relationships between the quantities, it is important to break down the problem into smaller and more manageable parts. This may involve creating a scheme or table to organize the information.

• Building Proportions: After dividing the problem into parts, the next step is to build the appropriate proportions. Remember, proportion is an equality of two ratios, and each ratio compares two quantities.

• Solving Proportions: At this stage, you will use your algebra skills to solve the proportions. This may involve cross-multiplication or the rule of null producer, depending on the situation.

• Verification and Interpretation of the Answer: Finally, check your answer and interpret the result in the context of the problem. This often involves going back to the problem statement and ensuring that your answer makes sense.

Key Terms

• Compound Rule of Three: Method used to solve proportion problems involving the comparison of three or more quantities. It is an extension of the simple rule of three for more than two quantities.

• Directly Proportional Quantities: Two quantities are directly proportional when the increase (or decrease) of one implies the increase (or decrease) of the other in the same proportion.

• Inversely Proportional Quantities: Two quantities are inversely proportional when the increase of one implies the decrease of the other in the same proportion, and vice versa.

Examples and Cases

• Example 1: Travel Time: If you travel at an average speed of 60 km/h and want to cover a distance of 240 km, how long will it take to complete the trip, considering that you need to stop for refueling every 200 km traveled, spending 15 minutes at each stop?

• Example 2: Workers' Production: If 5 workers produce 10 pieces in 2 hours, in how many hours will 8 workers produce 32 pieces, considering that the production is constant for each worker?

These examples illustrate the need to apply the compound rule of three to solve real-life problems and work-related scenarios. In addition to presenting the logic behind the compound rule of three, they highlight the importance of reading the problem carefully, identifying the relationships between the quantities, and interpreting the answer within the given context. These are the essential steps that must be followed to solve any type of compound rule of three problem.