28. Ratios and proportions

Ratios and proportions are fundamental concepts in mathematics, being widely used in several areas of knowledge, from physics to economics. In the context of preparing for the Enem, understanding these concepts is crucial, as they are often addressed in math issues and its technologies.

Ratio is a concept that expresses the relationship between two magnitudes of the same nature, that is, that can be measured in the same unit. Let a and b be two numbers, with b not zero, the ratio of a to b is the quotient a/b. For example, if in a classroom there are 28 students and 4 teachers, the ratio of students to teachers is 28/4, that is, 7. This means that for every teacher, there are 7 students.

Ratios are often used to express proportionality relationships, that is, to represent how one quantity varies in relation to another. In the previous example, the ratio of students to teachers allows us to say that the number of students is proportional to the number of teachers, with a ratio of 7 to 1.

A proportion, in turn, is an equality between two ratios. Let a, b, c and d be numbers, with b and d different from zero, we say that a/b = c/d is a proportion. In this case, a and d are called the extremes of the proportion, while b and c are the means. The fundamental property of proportions is that the product of the extremes is equal to the product of the means, that is, a*d = b*c.

Proportions are widely used in problems involving scale, such as maps and models. For example, on a map where 1 cm represents 100 km, the ratio between the distance on the map and the actual distance is 1/100. This means that if two cities are 2 cm apart on the map, the actual distance between them is 2*100 = 200 km.

In addition, proportions are also used in problems involving percentages, simple and compound interest, the rule of three, and more. For example, if an investment yields 5% per month, this means that the ratio between the yield and the amount invested is 5/100. Thus, if R$ 1000 were invested, the return after one month will be 1000*5/100 = R$ 50.

Therefore, the study of ratios and proportions is essential for solving many types of math problems. In addition, these concepts are the basis for understanding other more advanced topics, such as functions, arithmetic and geometric progressions, among others. Thus, mastering ratios and proportions is an important step towards achieving a good performance in the Enem math test.

In short, ratios and proportions are concepts that express relationships between quantities. The ratio is the quotient between two magnitudes of the same nature, while the proportion is an equality between two ratios. These concepts are widely used in math problems involving scales, percentages, interest, the rule of three, and more. Mastering ratios and proportions, therefore, is crucial for preparing for the Enem.

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29Rule of three simple and compound

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