Introduction to Successive Percentage

Relevance of the Topic

Percentage is one of the most widely used mathematical tools in everyday situations, with applications ranging from discounts on purchases to interest rate calculations. Learning to work with successive percentages - that is, multiple percentages applied to an initial value - is a fundamental additional step in mastering this area, opening doors to more complex and in-depth calculations.

Contextualization

In the 9th-grade Mathematics curriculum, when introducing the topic of percentage, we naturally work on the initial concepts of simple percentages and their calculation. Now, we will delve into the world of successive percentages to further expand the understanding of how percentage principles influence a wide variety of real-world situations.

This topic is a stepping stone to the more in-depth study of Algebra and Calculus, as it involves manipulating numbers and expressions in ways that will prepare students for similar approaches in these fields. Therefore, mastering successive percentages is essential, as it is one of the foundations for algebraic abstraction.

Theoretical Development

Components

• Initial Value: Refers to the total quantity on which the percentages are being applied. This is the starting point for successive percentage calculations.

• Successive Percentages: Are the adjustments applied to the initial value. Each successive percentage is calculated from the resulting value of the previous percentage calculation. The result of one percentage becomes the next initial value.

• Final Value: Is the quantity obtained after applying all successive percentages. This final value represents the total result of the operation.

• Formulas and Methods: To calculate successive percentages, two commonly used approaches are: (1) multiplying the initial value by each successive percentage, and (2) applying the compound interest formula, where each successive percentage is treated as an interest rate to be applied to the resulting value from the previous iteration.

Key Terms

• Percentage: It is a fraction of 100 in which the percentage value represents a part of a whole. In the context of successive percentages, we are dealing with several fractions of 100.

• Successive: In this context, the word 'successive' refers to the fact that each percentage is applied to the resulting quantity from the previous operation. This differs from the application of independent percentages, where each percentage is applied to the initial value.

Examples and Cases

• Example 1: Successive Salary Increases: Let's assume an employee has their salary increased by 10% in the first year and then by another 5% in the following year. If the initial salary was $2,000, what will be the employee's salary after these successive increases? In this case, the first percentage (10%) is applied to the initial value ($2,000), resulting in $2,200. Then, the second percentage (5%) is applied to this value, resulting in $2,310. Therefore, the employee's salary after the successive increases will be $2,310.

• Example 2: Successive Discounts on Products: Suppose an item is initially sold for $500, but has a 20% discount. Additionally, customers can apply an additional 15% discount coupon on the already discounted price. How much will the customer pay for the item after these successive discounts? In this case, the first percentage (20%) is applied to the initial value ($500), resulting in $400. Then, the second percentage (15%) is applied to this value, resulting in $340. Therefore, the customer will pay $340 for the item after the successive discounts.

These examples illustrate the practical application of calculating successive percentages in real situations, demonstrating the usefulness of this mathematical topic in daily life and the need to master it.

Detailed Summary

Key Points

• Definition of Successive Percentages: This term refers to the series of percentages that are applied to an initial value, where each new percentage is calculated based on the result of the previous iteration.

• Initial Value: It is the starting point for successive percentage calculations. This value changes with each new applied percentage.

• Successive Percentages: These are the percentages that are applied one after the other to the resulting value from the previous iteration. Each of these percentages changes the value it is applied to.

• Final Value: It is the result obtained after applying all successive percentages. It is the resulting value of the operation.

• Formulas and Methods: There are several methods to calculate successive percentages, with multiplication being the most common. However, the compound interest formula can also be used.

• Practical Examples: The theoretical concepts of successive percentages become clearer when applied to real-life examples, such as consecutive salary increases or consecutive discounts on a product.

Conclusions

• Real-World Applications: Successive percentage is an extremely useful concept for everyday situations, such as calculating interest rates or recurring discounts.

• Importance of Order: The order in which percentages are applied is crucial for the final result. This concept reinforces the idea that mathematics is not just about numbers, but also about logical reasoning.

• Relation to Other Concepts: The study of successive percentages serves as a stepping stone for the development of more advanced mathematical skills - particularly the understanding of compound interest in Algebra and Calculus.

Exercises

1. The price of a product is increased by 15% and then, on the new value, a discount of 10% is applied. What is the total percentage variation?

2. An investment is made with an interest rate of 5% per month for 6 months. What is the equivalent annual interest rate, considering that the interest is compounded monthly?

3. The sides of a triangle measure 10 cm, 12 cm, and 15 cm. The sides of the triangle are reduced by 20%, 10%, and 12% respectively to form a new triangle. What is the percentage difference between the areas of the two triangles?

These exercises will help consolidate the understanding and practice of successive percentages.