Mark's Savings: Expression After N Weeks Explained

by Alex Johnson 51 views

In this article, we'll break down how to determine the correct mathematical expression that represents Mark's savings after working for a certain number of weeks. This is a common type of problem in algebra, and understanding the logic behind it can help you solve similar problems in the future. We'll go through the scenario step-by-step, analyze the given options, and arrive at the correct answer. So, let's dive in and figure out Mark's savings!

Understanding the Scenario: Mark's Savings Journey

To begin, let's clearly understand the scenario. Mark starts with an initial savings of $200. This is the amount he already has before he begins working. Then, Mark earns an additional $45 for each week he works. This weekly earning is crucial to understanding how his savings grow over time. Our goal is to create an expression that accurately calculates Mark's total savings after 'n' weeks of work. This means we need to incorporate both his initial savings and his weekly earnings into our equation. Think of it as building a financial picture, where the initial savings is the foundation and the weekly earnings are the bricks that add to it.

Now, let's consider how the number of weeks, represented by 'n', impacts Mark's total savings. For every week that Mark works, he adds $45 to his savings. So, if he works for 2 weeks, he'll add 2 * $45 to his initial savings. If he works for 10 weeks, he'll add 10 * $45. This pattern shows a direct relationship between the number of weeks worked and the amount earned. This understanding is key to formulating the correct expression. We're essentially looking for a mathematical formula that captures this relationship between weeks worked and savings accumulated. This formula will allow us to calculate Mark's savings for any given number of weeks.

Therefore, the core of the problem lies in correctly representing the combined effect of his initial savings and his weekly earnings. We need an expression that starts with his initial $200 and then adds the total amount he earns over 'n' weeks. This means we'll need to multiply his weekly earnings by the number of weeks he works. This is where the algebraic expression comes into play. It's a concise way to represent a mathematical relationship that can be applied to various scenarios. By carefully considering the scenario and the relationships between the different components, we can build the expression that accurately reflects Mark's savings after 'n' weeks. This expression will be a valuable tool for understanding and predicting Mark's financial progress.

Analyzing the Options: Which Expression Fits?

Let's examine the given options and see which one accurately represents Mark's savings:

A. 200+45n200 + 45n B. 200−45n200 - 45n C. 245n245n D. 45n−20045n - 200

Option A, 200+45n200 + 45n, suggests that Mark's savings start at $200 and increase by $45 for each week worked. This seems to align with our understanding of the scenario. The '45n45n' part of the expression represents the total earnings over 'n' weeks, and the addition of '$200' accounts for his initial savings. So, this option looks promising.

Option B, 200−45n200 - 45n, implies that Mark's savings decrease by $45 each week. This doesn't make sense in our scenario, as Mark is earning money, not losing it. The subtraction sign indicates a reduction in savings, which contradicts the problem statement. Therefore, we can confidently eliminate this option.

Option C, 245n245n, suggests that Mark's savings are simply $245 multiplied by the number of weeks worked. This option doesn't account for Mark's initial savings of $200. It implies that he started with zero savings, which is incorrect. While it does consider the weekly earnings, it fails to incorporate the crucial starting point.

Option D, 45n−20045n - 200, suggests that Mark's savings are calculated by subtracting $200 from his total weekly earnings. This would mean Mark starts with a debt of $200, which is not the case. This option also misrepresents the initial condition of the problem. It essentially flips the order of operations in a way that doesn't align with the scenario.

By carefully analyzing each option and comparing it to our understanding of the scenario, we can confidently narrow down the possibilities. Only one option correctly incorporates both the initial savings and the weekly earnings in a way that makes logical sense. This process of elimination and verification is a valuable strategy for solving mathematical problems.

The Correct Expression: 200+45n200 + 45n

After analyzing each option, it's clear that option A, 200+45n200 + 45n, is the correct expression to represent Mark's savings after working for 'n' weeks. This expression accurately captures the two key components of Mark's savings:

  • The initial savings of $200
  • The weekly earnings of $45, multiplied by the number of weeks worked ('n')

Let's break down why this expression works. The '45n45n' part represents the total amount Mark earns over 'n' weeks. For instance, if Mark works for 5 weeks, this part of the expression would be $45 * 5 = $225. This represents his total earnings from working. Then, we add the $200 to this amount, which represents his initial savings. So, after 5 weeks, Mark's total savings would be $200 + $225 = $425.

This expression is a powerful tool because it allows us to calculate Mark's savings for any number of weeks simply by plugging in the value of 'n'. Whether Mark works for 1 week, 10 weeks, or 100 weeks, we can use this expression to determine his total savings. This demonstrates the elegance and utility of algebraic expressions in representing real-world situations. They provide a concise and accurate way to model relationships and make predictions.

Therefore, the expression 200+45n200 + 45n perfectly encapsulates Mark's savings journey. It accounts for his starting point and his ongoing earnings, providing a clear and understandable representation of his financial progress. This problem highlights the importance of carefully considering all the components of a scenario and translating them into a mathematical expression.

Conclusion: Mastering Algebraic Expressions

In conclusion, we've successfully identified the correct expression, 200+45n200 + 45n, that represents Mark's savings after working for 'n' weeks. This problem demonstrates the importance of understanding how to translate real-world scenarios into mathematical expressions. By carefully analyzing the given information and considering the relationships between different components, we can create expressions that accurately model the situation.

Remember, the key to solving these types of problems is to break them down into smaller, more manageable parts. Identify the initial conditions, the rates of change, and the variables involved. Then, use this information to build an expression that captures the essence of the scenario. Practice is crucial for mastering this skill. The more you work with algebraic expressions, the more comfortable you'll become with translating real-world situations into mathematical models.

This skill is not only valuable in mathematics but also in various other fields, such as finance, science, and engineering. The ability to represent relationships mathematically allows us to make predictions, solve problems, and gain a deeper understanding of the world around us. So, continue practicing and exploring the world of algebraic expressions. You'll be amazed at the power and versatility they offer. For further learning and resources on algebraic expressions, consider visiting Khan Academy's Algebra 1 section.