Landen Spent *L* Hours At The Beach Last Weekend. Matéo Spent 15% Fewer Hours At The Beach Than Landen. Which Two Expressions Represent The Hours Matéo Spent At The Beach?

This article explores a mathematical problem involving the calculation of time spent at the beach, focusing on percentage reductions. We'll break down the problem step-by-step, ensuring a clear understanding of the concepts involved. The core of the problem revolves around Landen and Matéo's beach visits, where Matéo spent 15% fewer hours at the beach than Landen. Our aim is to identify the expressions that accurately represent the time Matéo spent at the beach. This involves understanding how percentages translate into mathematical operations, particularly in the context of reductions. By the end of this article, you'll not only grasp the solution to this specific problem but also gain a broader understanding of how to approach similar percentage-based calculations. Understanding percentages is crucial in various real-life scenarios, from calculating discounts while shopping to understanding financial data. This article serves as a valuable resource for anyone looking to strengthen their mathematical skills and apply them to practical situations. We'll delve into the underlying principles of percentage calculations and how they relate to everyday experiences. The problem provides an excellent opportunity to enhance your analytical and problem-solving abilities, which are essential in both academic and professional settings. Through a detailed explanation and breakdown of the solution, we aim to empower you with the knowledge and confidence to tackle similar challenges with ease.

H2 Problem Statement

Landen spent L hours at the beach last weekend. Matéo spent 15% fewer hours at the beach than Landen did. Which of the following expressions could represent how many hours Matéo spent at the beach last weekend? We are given a scenario where we need to determine the expressions that accurately represent the time Matéo spent at the beach, considering he spent 15% less time than Landen. The key here is to translate the percentage reduction into a mathematical expression. To effectively solve this, we need to understand how to calculate a percentage of a given value and how to subtract it from the original value. The problem tests our understanding of percentage calculations and their practical application. It also encourages us to think critically about how different expressions can represent the same mathematical concept. The beauty of mathematics lies in its ability to express the same idea in multiple ways, and this problem beautifully illustrates this concept. We will explore different approaches to solve this problem, highlighting the flexibility and versatility of mathematical tools. This exercise not only helps us understand percentage reductions but also reinforces our understanding of algebraic expressions. By working through this problem, we enhance our ability to translate real-world scenarios into mathematical models, a crucial skill in many fields. The challenge lies in correctly interpreting the phrase "15% fewer hours" and converting it into an equivalent mathematical expression. This requires careful consideration of the base value (Landen's time) and the operation (reduction) involved. Let's dive into the different ways we can represent this situation mathematically and identify the correct expressions.

H3 Understanding Percentage Reduction

When we say Matéo spent 15% fewer hours at the beach than Landen, it means we need to subtract 15% of Landen's time from Landen's total time. The fundamental concept here is that percentages represent a proportion of a whole. In this case, 15% represents 15 out of 100 parts of Landen's time, denoted as L. To calculate 15% of L, we multiply L by 15/100 or its decimal equivalent, 0.15. This gives us 0.15L, which represents the amount of time Matéo spent less than Landen. Now, to find the actual time Matéo spent at the beach, we need to subtract this value from Landen's time, L. This is where the concept of subtraction comes into play, representing the reduction in time. The expression for the time Matéo spent at the beach becomes L - 0.15L. This expression clearly shows the reduction of 15% from Landen's time. However, this is not the only way to represent the situation. We can simplify this expression by factoring out L. This is a crucial algebraic technique that allows us to combine like terms and express the equation in a more concise form. Factoring out L helps us see the underlying proportion of time Matéo spent at the beach compared to Landen. By simplifying the expression, we gain a deeper understanding of the relationship between the two individuals' beach time. This process of simplification is a core skill in algebra and is vital for solving more complex mathematical problems. The ability to manipulate expressions and equations is a cornerstone of mathematical proficiency. Let's explore how factoring out L can lead to a different, yet equivalent, representation of the time Matéo spent at the beach. This alternative representation will offer a different perspective on the problem and highlight the versatility of mathematical expressions.

H4 Calculating the Remaining Percentage

Instead of directly calculating the 15% reduction and subtracting it, we can think in terms of the remaining percentage. If Matéo spent 15% fewer hours, he effectively spent 100% - 15% = 85% of the time Landen spent. This is a crucial shift in perspective that simplifies the calculation. We are now dealing with the proportion of time Matéo did spend at the beach, rather than the amount he didn't. This approach often makes percentage problems more intuitive and easier to solve. To find 85% of Landen's time (L), we multiply L by 85/100 or its decimal equivalent, 0.85. This gives us the expression 0.85L, which directly represents the time Matéo spent at the beach. This expression is more concise and directly reflects the proportion of time Matéo spent compared to Landen. The use of 0.85L provides a clear and efficient way to calculate the time Matéo spent at the beach. This method avoids the intermediate step of calculating the reduction amount and directly gives us the final answer. This approach highlights the importance of understanding different ways to represent the same mathematical concept. The ability to think in terms of remaining percentages is a valuable skill in various contexts, including discounts, sales, and financial calculations. It simplifies the process and reduces the chances of errors. This method also reinforces the connection between percentages, decimals, and proportions. By understanding these relationships, we can approach percentage problems with greater confidence and flexibility. Let's now compare this expression with the one we derived earlier (L - 0.15L) and see how they are mathematically equivalent.

H5 Equivalent Expressions

We have two expressions representing the time Matéo spent at the beach: L - 0.15L and 0.85L. To confirm their equivalence, we can simplify the first expression. This is a fundamental step in verifying that different expressions represent the same value. Simplifying expressions not only confirms their equivalence but also often reveals underlying mathematical relationships. Factoring out L from the expression L - 0.15L, we get L(1 - 0.15). This step is crucial in bringing the expression to a form that is easily comparable with 0.85L. Now, we simplify the expression inside the parentheses: 1 - 0.15 = 0.85. This gives us L(0.85), which is the same as 0.85L. This equivalence demonstrates that both expressions are mathematically identical and accurately represent the time Matéo spent at the beach. This process of simplification and comparison is a core skill in algebra and is essential for solving equations and verifying solutions. It also highlights the flexibility of mathematical notation and the different ways we can represent the same concept. The ability to manipulate expressions and recognize equivalent forms is a hallmark of mathematical proficiency. This exercise reinforces our understanding of algebraic principles and their application in practical problem-solving. The fact that we arrived at the same result using two different approaches strengthens our confidence in the solution and our understanding of the underlying mathematical concepts. Now, let's summarize the key takeaways from this problem and discuss how these concepts can be applied in other situations.

H2 Conclusion

In conclusion, the problem of calculating the time Matéo spent at the beach, given a percentage reduction from Landen's time, highlights the importance of understanding percentage calculations and algebraic manipulations. We explored two main approaches: directly calculating the reduction and subtracting it, and calculating the remaining percentage. Both methods led to equivalent expressions, demonstrating the flexibility and versatility of mathematical representations. The key takeaway is that 15% fewer hours can be represented as both L - 0.15L and 0.85L. Understanding these equivalent expressions is crucial for solving similar problems involving percentage increases or decreases. This skill is applicable in various real-world scenarios, from calculating discounts and sales tax to understanding financial data and statistics. The ability to translate word problems into mathematical expressions is a fundamental skill in mathematics and problem-solving. This problem serves as a valuable exercise in developing this skill. Furthermore, the process of simplifying expressions and recognizing equivalent forms is a cornerstone of algebraic proficiency. By mastering these skills, we can approach more complex mathematical challenges with confidence and clarity. This article aimed to provide a comprehensive understanding of the problem and the underlying mathematical concepts. We encourage you to practice similar problems to solidify your understanding and enhance your problem-solving abilities. The more you engage with mathematical concepts, the more intuitive they become. Remember, mathematics is not just about memorizing formulas; it's about understanding the relationships between concepts and applying them creatively to solve problems. This exercise has hopefully reinforced your understanding of percentages, algebraic expressions, and problem-solving strategies. We encourage you to continue exploring mathematical concepts and applying them to real-world situations.