Savings Inequalities Determining When Molly's Savings Exceed Lynn's

In this article, we will explore a problem involving Molly and Lynn, who are both diligently saving money. This is a mathematical problem centered around setting up an inequality. To truly understand the situation and formulate the correct inequality, we need to break down the information provided step by step. Our main keywords are savings inequalities, Molly's savings, and Lynn's savings. Molly's savings is a good starting point. Molly already has a substantial $650 set aside. This is her initial savings, her starting point in this savings journey. What's crucial here is that Molly consistently adds $35 to her savings each week. This regular addition forms a pattern, a linear progression of her savings over time. So, we know Molly’s savings can be represented as $650 plus $35 multiplied by the number of weeks. Let's consider Lynn’s savings as the next key aspect. Lynn begins with a higher initial amount, a solid $825 already tucked away. This gives her a head start compared to Molly. However, Lynn's weekly contribution is less; she adds only $15 each week. This means her savings will grow at a slower pace than Molly’s. Again, we see a linear pattern here, with Lynn's savings being $825 plus $15 multiplied by the number of weeks. The core question we need to address is, which inequality can be used to determine when Molly's savings will exceed Lynn's savings? This is where the power of inequalities comes into play. We are not looking for an exact point where their savings are equal; we are interested in the range of weeks when Molly's savings surpasses Lynn's. This “exceed” is a key term here, indicating a “greater than” relationship. Now, let’s translate these observations into a mathematical inequality. We need to express Molly’s savings as greater than Lynn’s savings. Molly's savings can be mathematically represented as 650 + 35w, where ‘w’ represents the number of weeks. Lynn’s savings, similarly, can be written as 825 + 15w. Therefore, the inequality we are seeking will have the form 650 + 35w > 825 + 15w. This inequality is the heart of the problem. It captures the dynamic relationship between their savings plans. It allows us to calculate the number of weeks ('w') it will take for Molly’s savings to overtake Lynn’s. This problem underscores the practical application of inequalities in understanding real-world scenarios. Savings, investments, and financial planning often involve comparing different growth rates and determining when one option becomes more beneficial than another. Mastering the ability to set up and solve inequalities is a valuable skill in various contexts. Let's delve a bit deeper into the implications of this inequality. Solving this inequality will not give us a single answer but a range of weeks. It will tell us, after a certain number of weeks, Molly's savings will consistently be greater than Lynn's. This is crucial for making informed financial decisions. Understanding this range allows for better planning and strategizing. It’s not just about knowing when Molly's savings exceed Lynn's but also about understanding the rate at which Molly's savings are catching up. This involves comparing the weekly contributions ($35 for Molly and $15 for Lynn) and their initial savings differences ($825 for Lynn and $650 for Molly). The inequality encapsulates all these factors, providing a comprehensive view of their savings progression. In conclusion, the problem we've analyzed highlights the importance of mathematical modeling in financial literacy. The ability to translate a real-world scenario into a mathematical expression, specifically an inequality, is essential for making sound financial decisions. By understanding the relationships between different savings plans, we can effectively plan for our financial future. The inequality 650 + 35w > 825 + 15w is the key to unlocking the answer to our initial question, demonstrating the power and applicability of mathematical concepts in everyday life.

Setting Up the Inequality: A Step-by-Step Guide

To correctly set up the inequality, we need to represent Molly and Lynn's savings mathematically. This involves understanding the components of their savings plans and translating them into algebraic expressions. Here, the keywords are setting up inequalities, mathematical representation, and algebraic expressions. Let's begin with Molly's savings. As we established earlier, Molly starts with $650. This is a constant value, her initial investment. Now, she adds $35 each week. This weekly addition can be represented as 35w, where ‘w’ stands for the number of weeks. The ‘w’ here is a variable, as the number of weeks can change. Molly's total savings can be expressed as the sum of her initial savings and the total amount she saves weekly. So, Molly’s savings after ‘w’ weeks can be represented by the algebraic expression 650 + 35w. This expression is a linear equation, indicating a steady increase in savings over time. The slope of this line is 35, representing her weekly contribution. The y-intercept is 650, showing her initial savings. Now, let’s shift our focus to Lynn's savings. She begins with a higher amount, $825. This is her initial advantage in this savings race. However, she adds a smaller amount each week, only $15. Like Molly’s savings, Lynn's weekly savings can be expressed as 15w. Therefore, Lynn's total savings after ‘w’ weeks can be represented by the expression 825 + 15w. Similar to Molly's savings, this expression is also a linear equation, but with a different slope and y-intercept. The slope here is 15, reflecting her smaller weekly contribution. The y-intercept is 825, showing her higher initial savings. With both savings plans represented as algebraic expressions, we can now focus on the core of the problem: the inequality. The question asks us to determine when Molly’s savings will exceed Lynn’s savings. This