Introduction to Percentage Problems

Relevance of the Topic

Percentage is one of the most widely applied mathematical concepts in everyday life. It helps us better understand the world around us, from financial calculations, weather forecasts to population health statistics. A solid understanding of percentages is therefore an important component of mathematical literacy.

Contextualization

In terms of the mathematics curriculum structure, the topic of percentage problems is the natural evolution of the study of proportion. As students progress from fractions to decimals and finally to percentages, the idea of a part of a whole takes on different forms. Percentage problems bring the dimension of a component of a whole to another level, where this component is expressed as a percentage of the whole. This additional complexity is effectively introduced in 7th-grade classes, preparing students for more advanced mathematical challenges.

Theoretical Development

Components

• Percentage: This is the main concept that allows us to express a part of the whole as a fraction of 100. Percentages are widely used in interest calculations, taxes, discounts, probabilities, among others.

• Percentage Problems: This section focuses on how to solve real-world problems using percentage as a tool. These include problems of increasing and decreasing values, discount and markup calculations, and comparing values using percentages.

Key Terms

• Percentage: A ratio denoting a part of each 100.

• Whole (Base): It is the total value or quantity of a group, considered as 100%.

• Part: It is an element or subgroup within the whole. The percentage is calculated based on the relationship between the part and the whole.

• Percentage Multiplication Factor: A value used to increase or decrease another quantity by a certain percentage. It is usually expressed as a decimal, for example, a 20% increase is represented by a multiplication factor of 1.2.

Examples and Cases

• Problem 1: Calculating a Discount: If an item costs R$ 80 and has a 20% discount, how to calculate the final price? In this case, the percentage (20%) is the multiplication factor. Therefore, the final price is obtained by multiplying R$ 80 by 0.8 (or 80%). The final price will be R$ 64.

• Problem 2: Increasing an Amount: If you have R$ 100 and want to increase this amount by 25%, what will be the new value? Here, the percentage is again the multiplication factor. The new value is obtained by multiplying R$ 100 by 1.25 (or 125%). The new value will be R$ 125.

• Problem 3: Comparing Values: A t-shirt costs R$ 40 and another one costs R$ 60. If you receive a 30% discount on the first t-shirt and a 40% discount on the second one, which one is cheaper? To solve this problem, we calculate the final price of each t-shirt after the discount. The first t-shirt will have a final price of R$ 28, while the second one will have a final price of R$ 36. Therefore, the first t-shirt is cheaper.

Detailed Summary

Key Points

• Definition of Percentage - Percentage represents the relationship of a part to a whole, with the part being expressed as a fraction of 100. This understanding is fundamental to solving percentage problems.

• Percentage Multiplication Factor - An essential concept for solving percentage problems. The multiplication factor is calculated by adding or subtracting the percentage (as a decimal) from 1. For example, a 20% percentage is represented as 0.2 in decimal. Therefore, a 20% increase is represented by a multiplication factor of 1.2 (1 + 0.2), while a 20% decrease is represented by a multiplication factor of 0.8 (1 - 0.2).

• Calculation of Markup and Discount - You have learned to calculate the price of an item after a discount or markup by multiplying the value by the corresponding multiplication factor for the percentage in question.

• Comparing Values using Percentage - To compare the value of two items after different discounts, it is necessary to calculate the final price of each item and then compare them.

• Practical Examples - Solving practical problems, such as calculating the final price of an item after a discount, increasing a value, and comparing prices after different discounts, helped solidify the concepts learned.

Conclusions

• Mastery of Percentage - A clear and comprehensive understanding of how percentage works is essential for solving percentage problems. The proper use of the percentage multiplication factor for markup and discount calculations is a fundamental skill.

• Importance of Percentage Problems - Percentage problems are a practical manifestation of the percentage concept and allow students to apply their mathematical knowledge in real-world contexts.

Suggested Exercises

1. A pair of shoes costs R$ 200. If a 15% discount is applied, what will be the new price? (Answer: R$ 170)

2. If you receive a 10% increase in your current salary of R$ 1500, what will be your new salary? (Answer: R$ 1650)

3. A bicycle costs R$ 800 and a scooter costs R$ 1000. If a store offers a 25% discount on the bicycle and a 15% discount on the scooter, which one is cheaper? (Answer: The bicycle, with a final price of R$ 600, is cheaper)