Percentage: Discounts and Percentage Increases

Have you ever wondered why stores offer such attractive discounts at certain times of the year? During sales, it's common to see offers like '50% off' or 'buy 3, pay for 2'. These strategies are not just ways to attract customers, but they also reflect the practical use of percentages in our daily lives. During Black Friday 2022, many stores offered up to 70% off on their products, resulting in a significant increase in sales.

Think About: Have you ever stopped to think about how to calculate these discounts to know if you are really getting a good deal? And how do these calculations affect your purchasing decisions?

Percentage is a fundamental mathematical concept that we use in various situations in our daily lives, from calculating discounts in purchases to understanding price increases and interest rates. Understanding how percentage calculations work can help you make more informed financial decisions and better understand the offers and promotions you encounter in stores and supermarkets.

In the context of shopping, the ability to calculate percentage discounts can be extremely useful. For example, if you find a product with a 30% discount, knowing how to quickly calculate the discounted amount and the final price can help you decide if the deal is worth it. Similarly, understanding how percentage increases affect prices can be crucial for managing your budget and planning your purchases more effectively.

In this chapter, we will explore in detail how to calculate percentages, apply percentage discounts and increases to products and services, and solve practical problems that you may encounter in everyday situations. This knowledge will not only help you perform well on tests and exams but will also be valuable lifelong tools, allowing you to make smarter and more conscious financial choices.

Concept of Percentage

Percentage is a way of expressing a fraction of 100. This means that when we talk about a percentage, we are referring to a part of a total that has been divided into 100 equal parts. For example, when we say that 50% of something is complete, we are saying that 50 out of every 100 parts of that something are finished. This concept is extremely useful because it facilitates comparisons between different quantities and makes it more intuitive to understand fractions of a whole.

Percentage notation is represented by the symbol '%'. This symbol is essentially a shortcut to indicate 'per hundred'. For example, 25% is a shorthand way of saying '25 for every 100'. This concept is widely used in various fields of knowledge, including mathematics, statistics, economics, and social sciences, due to its simplicity and efficiency in representing parts of a whole.

To transform a fraction or decimal into a percentage, simply multiply the value by 100 and add the '%' symbol. For example, the fraction 1/4 can be converted into a percentage by multiplying by 100, resulting in 25%. Likewise, the decimal 0.75 becomes 75% when multiplied by 100. This reverse process is also valid: to convert a percentage to a decimal, simply divide by 100. For example, 40% becomes 0.40 when divided by 100.

Understanding the concept of percentage is crucial for solving practical problems that involve parts of a total. For example, when analyzing a survey, a percentage allows for a quick understanding of the proportion of people who agree or disagree with a statement. Additionally, percentage is widely used in finance to calculate interest, discounts, and price increases, making it an essential tool for informed decision-making in everyday life.

Calculating Percentage

Calculating percentages may seem complicated at first glance, but it is actually a fairly simple process once you understand the logic behind it. To calculate the percentage of a number, we multiply that number by the desired percentage expressed as a decimal. For example, to find 20% of 150, we convert 20% to 0.20 (by dividing 20 by 100) and then multiply 150 by 0.20. The result is 30, which represents 20% of 150.

Let's consider another practical example: calculating 15% of 200. First, we convert 15% to decimal, resulting in 0.15. Then, we multiply 200 by 0.15. The calculation is 200 x 0.15 = 30. This means that 15% of 200 is 30. These calculations are extremely useful for solving a variety of everyday problems, such as determining discounts on purchases or calculating the amount of an ingredient in a recipe.

Moreover, it's important to know how to calculate the total value from a percentage. For example, if you know that 40% of a value is 80, you can find the total value by dividing 80 by 0.40. The calculation is 80 ÷ 0.40 = 200. Thus, the total value is 200. This type of calculation is particularly useful in financial situations, such as when you know the value of a discount and need to determine the original price of a product.

To make understanding even easier, we can use a simple formula: 'Percentage = (Part / Total) x 100'. For example, if you have 25 apples out of a total of 100 fruits, the percentage of apples is (25 / 100) x 100 = 25%. This formula is a powerful tool for solving percentage problems efficiently and accurately, and it can be applied in various practical situations, from financial calculations to statistical analyses.

Percentage Discounts

Percentage discounts are a common practical application of the percentage concept, especially in commerce. A percentage discount represents a reduction in the original price of a product or service. To calculate a percentage discount, we multiply the original price by the percentage of the discount expressed as a decimal and subtract that value from the original price. For example, if a product costs R$ 200 and has a discount of 15%, we first convert 15% to 0.15 and then multiply 200 by 0.15, resulting in 30. The discounted price is then R$ 200 - R$ 30 = R$ 170.

Let's consider another example: an item that costs R$ 80 and has a discount of 25%. First, we convert 25% to 0.25. Then, we multiply the original price of R$ 80 by 0.25, resulting in 20. We then subtract the discount amount from the original price: R$ 80 - R$ 20 = R$ 60. Therefore, the final price of the item after applying the 25% discount is R$ 60. This type of calculation is extremely useful for consumers who want to ensure they are taking full advantage of promotions and saving money.

In addition to calculating the discounted amount, it is also possible to directly calculate the final price. To do this, we multiply the original price by the complement of the percentage discount. For example, if a product has a discount of 30%, the complement is 70%, or 0.70. We multiply the original price by the complement: R$ 100 x 0.70 = R$ 70. This method can be faster and more useful in situations where we need to quickly calculate the final price during a purchase.

Understanding how to calculate percentage discounts is a valuable skill not just for consumers but also for professionals working in areas such as sales, marketing, and finance. Knowing how to calculate and apply discounts can help create more effective and attractive sales strategies, as well as allow for a better understanding of how promotions affect prices and profit margins. Additionally, informed consumers can make smarter purchasing decisions and take better advantage of the savings opportunities offered by discounts and promotions.

Percentage Increases

Percentage increases are the opposite of percentage discounts and represent an addition to the original price of a product or service. To calculate a percentage increase, we multiply the original price by the percentage of the increase expressed as a decimal and add that value to the original price. For example, if a product costs R$ 150 and has a 10% increase, we first convert 10% to 0.10. Then, we multiply 150 by 0.10, resulting in 15. The new price after the increase is R$ 150 + R$ 15 = R$ 165.

Let's consider another example: an item that costs R$ 120 and receives a 15% increase. First, we convert 15% to 0.15. Then, we multiply the original price of R$ 120 by 0.15, resulting in 18. We then add the increase amount to the original price: R$ 120 + R$ 18 = R$ 138. Therefore, the new price of the item after applying the 15% increase is R$ 138. This type of calculation is especially useful for understanding how price variations affect the cost of products and services over time.

In addition to calculating the increase amount separately, we can also calculate the new price directly. To do this, we multiply the original price by the total of the percentage increase added to 1. For example, if a product has a 20% increase, the total is 1 + 0.20 = 1.20. We multiply the original price by the total: R$ 200 x 1.20 = R$ 240. This method can be faster and more efficient, especially in situations that require quick calculations, such as negotiations and price adjustments.

Understanding how to calculate percentage increases is essential not only for consumers but also for professionals who work with pricing, financial management, and budget planning. Knowing how to calculate and apply increases allows for better cost and price management, as well as helping to predict the impact of price fluctuations on budgets and personal finances. For consumers, understanding percentage increases can help plan their budgets better and make more informed purchasing decisions, especially in times of inflation or price adjustments.

Reflect and Respond

• Think about how the knowledge of percentages can be applied in your everyday shopping. How can this help you save money?
• Reflect on the importance of understanding percentage increases in products and services. How can this influence your financial decisions?
• Consider other areas of your life where calculating percentages can be useful. How can you apply this knowledge in different contexts, such as studies or financial planning?

Assessing Your Understanding

• Explain how you would calculate the percentage discount of a product on sale and how this would impact your purchasing decision.
• Describe a situation where you had to deal with a percentage increase in a service or product. How did you handle it and what were your conclusions?
• Discuss the importance of knowing how to calculate percentages in financial life. Give examples of how this knowledge can influence your daily decisions.
• Analyze a supermarket flyer with various promotions and calculate the percentage discounts. Which promotion do you consider the most advantageous and why?
• Evaluate the importance of understanding both discounts and percentage increases. How can this knowledge help you make more informed decisions in the future?

Reflection and Final Thought

In this chapter, we explored the concept of percentage and its application in everyday situations such as percentage discounts and increases. We understood that percentage is a practical way of expressing a fraction of 100, facilitating data comparison and interpretation. We learned to calculate percentages, both to find a part of a total and to determine the total value from a known percentage.

We discussed in detail how to apply percentage discounts, understanding that this calculation involves multiplying the original price by the discount percentage and subtracting the result from the original price. Likewise, we saw how to calculate percentage increases by multiplying the original price by the increase percentage and adding the result to the original price. These calculations are essential for making informed financial decisions and taking full advantage of promotions and price adjustments in everyday life.

Understanding and applying the concepts of percentage, discounts, and percentage increases not only improves your academic performance but also provides practical skills for daily life. By mastering these calculations, you will be better prepared to manage your budget, make smarter purchases, and better understand price variations that occur in the market. Continue practicing these concepts and applying them in different contexts to strengthen your knowledge and ability to make conscious financial decisions.