Introduction

The Relevance of the Topic

Ratio is one of the fundamental pillars of Mathematics, serving as a foundation for many subsequent topics, such as proportion, percentage, and rates. Its use permeates numerous practical situations in everyday life, from calculating speeds to determining discounts on purchases. Mastering the idea of ratio is therefore essential for a deep and applied understanding of Mathematics.

The Context of the Discipline

In the 2nd year of High School, the Mathematics discipline operates on the border between more abstract concepts and their application in the real world, making the understanding of the fundamentals of ratio even more vital. After the initial exploration of real numbers and their operations, understanding ratio allows for a smooth transition to the exploration of topics such as functions, analytical geometry, and trigonometry. Furthermore, mastering ratio is a transversal skill that enriches numerical perception and problem-solving in all spheres of life and broader thinking.

Theoretical Development

Components:

• Terms of the Ratio:

  • Understanding ratio requires understanding its structure, which is formed by two numbers: the antecedent and the consequent. The antecedent is the first number of the ratio, and the consequent is the second.

• Ratio and Proportion:

  • The ratio relationship allows us to establish a connection between measures of the same kind. In a proportion, the ratios between the first terms are equal to the ratios between the second terms.

• Operations with Ratios:

  • Ratios can be added, subtracted, multiplied, and divided just like integers.

• Inverse Ratio:

  • The inverse ratio of a given ratio k is the ratio that results when the terms of the ratio are inverted.

Key Terms:

• Ratio:

  • Ratio is a comparison or relationship between two quantities. It represents the quotient between two numbers.

• Antecedent and Consequent:

  • These are the terms used in the description of a ratio. The antecedent is the first term of the ratio, and the consequent is the second term of the ratio.

• Proportion:

  • Proportion is an equality that involves two ratios. The ratios that form a proportion are called terms of the proportion.

Examples and Cases:

1. Ratios in a Proportion:

   • Consider the following proportion: 2:5 = 8:20. Here, the ratio between 2 and 5 is the same as the ratio between 8 and 20.

2. Inverse Ratios:

   • If the ratio between two numbers is 3:2, its inverse ratio will be 2:3, where the terms were simply inverted.

3. Operations with Ratios:

   • Let A:B = 2:3 be a ratio. If we multiply both terms of this ratio by 2, we get the new ratio A:B = 4:6. Note that the resulting ratio is equivalent to the original ratio, only the terms were multiplied by 2.

Detailed Summary

Key Points:

• Definition of Ratio:

  • Ratio is a comparison between quantities. It is the quotient between two quantities of the same type. Understanding this definition is the first step to mastering the concept of ratio.

• Terms of the Ratio:

  • A ratio is formed by two numbers, the antecedent and the consequent. The antecedent is the first number, and the consequent is the second number of the ratio.

• Ratio and Proportion:

  • A proportion is an equality between two ratios. However, it is crucial to understand that not every equality between two ratios is a proportion.

• Operations with Ratios:

  • Ratios can undergo arithmetic operations such as addition, subtraction, multiplication, and division. However, it is important to keep in mind that the same operation must be applied to both terms of the ratio.

• Inverse Ratio:

  • The inverse ratio of a given ratio is obtained by reversing the terms of the original ratio.

Conclusions:

• Applicability of Ratio:

  • Ratio is more than an abstract concept - it is a tool that can be applied in real-world situations. Understanding ratio not only enhances mathematical skills but also translates into a better numerical perception and intuitive problem-solving.

• Foundations for Future Concepts:

  • Mastering ratio constitutes a solid foundation for numerous future concepts, such as proportion, percentage, and rates.

• Relevance of In-depth Understanding:

  • Delving into the logic and applications of ratio is a fundamental step in building a solid understanding in Mathematics, not only during High School but also in higher education courses that heavily rely on mathematical skills.

Exercises:

1. Describing Ratios:

   • Write three different ratios using the numbers 4, 7, and 11. Then, indicate what the antecedent and consequent represent in each ratio.

2. Adding Ratios:

   • Given the ratios A:B = 1:3 and C:D = 2:5, add the two ratios and write the result in ratio form.

3. Inverse Ratios:

   • If the ratio between a and b is 2:3, what is the inverse ratio of a:b?