Savings Showdown: Inequality For Molly Vs. Lynn
Have you ever wondered how long it takes to catch up to someone who has a head start in savings? Let's dive into a fascinating scenario involving Molly and Lynn, who are both diligently saving money each week. Molly and Lynn have different starting amounts and weekly contributions, and we want to figure out when Molly's savings will surpass Lynn's. This involves setting up and understanding inequalities, a fundamental concept in mathematics that helps us compare quantities. Let's break down their savings plans and see how we can use math to predict their financial futures.
Understanding Molly and Lynn's Savings Plans
To really understand how their savings compare over time, we need to look at the details. Molly begins her savings journey with a solid $650 already set aside. That's a great start! On top of that, she's committed to adding $35 to her savings each week. This consistent weekly contribution is key to her growing savings. On the other hand, Lynn has an even more substantial head start, with $825 already saved. However, she adds a smaller amount each week, only $15. The question we're trying to answer is: How many weeks will it take for Molly, with her lower starting amount but higher weekly savings, to have more money than Lynn? This is where setting up an inequality becomes incredibly useful.
Think of it like a race. Lynn has a significant lead, but Molly is gaining ground faster each week. The inequality will help us pinpoint the exact moment when Molly overtakes Lynn. To visualize this, imagine two lines on a graph, one representing Molly's savings and the other representing Lynn's. Molly's line starts lower but has a steeper slope (because she saves more each week). We want to find the point where Molly's line crosses above Lynn's line. This intersection point represents the number of weeks it takes for Molly's savings to exceed Lynn's. Understanding this concept is crucial for setting up the correct inequality.
Now, let's translate these savings plans into mathematical expressions. Molly's total savings after w weeks can be represented as 650 + 35w. This means her initial $650 plus $35 for each week. Similarly, Lynn's total savings after w weeks can be represented as 825 + 15w. This is her initial $825 plus $15 for each week. To determine when Molly's savings exceed Lynn's, we need to create an inequality that compares these two expressions. This inequality will form the foundation of our solution and help us answer the core question: When will Molly's diligence pay off and her savings surpass Lynn's?
Formulating the Inequality
To formulate the inequality, we need to express the condition that Molly's savings are greater than Lynn's savings. Remember, Molly's savings after w weeks is 650 + 35w, and Lynn's savings after w weeks is 825 + 15w. So, we want to find the number of weeks (w) when:
650 + 35w > 825 + 15w
This inequality is the key to solving our problem. It represents the relationship between Molly's and Lynn's savings over time. The left side of the inequality represents Molly's total savings, while the right side represents Lynn's total savings. The "greater than" symbol (>) indicates that we're looking for the number of weeks when Molly's savings amount is larger than Lynn's savings amount. This step of translating the word problem into a mathematical statement is crucial in problem-solving.
Breaking down the inequality, we can see that it's a linear inequality, meaning it involves a variable (w) raised to the power of 1. Linear inequalities are relatively straightforward to solve using algebraic techniques. Our goal is to isolate the variable w on one side of the inequality. This will tell us the range of values for w that satisfy the condition that Molly's savings are greater than Lynn's. Before we start solving, it's important to understand what the solution will represent. The solution will give us the minimum number of weeks it will take for Molly's savings to overtake Lynn's. It's like figuring out when one runner will finally pass another in a race.
This inequality provides a concise way to represent the problem mathematically. It allows us to use the tools of algebra to find a solution. Without this inequality, we would have to resort to trial and error, which would be much less efficient. Now that we have the inequality, the next step is to solve it for w. This will involve using algebraic manipulations to isolate w on one side of the inequality. The solution will then tell us how many weeks it will take for Molly's savings to exceed Lynn's.
Solving the Inequality Step-by-Step
Now, let's solve the inequality we formulated: 650 + 35w > 825 + 15w. Our goal is to isolate w on one side of the inequality to determine the number of weeks it will take for Molly's savings to exceed Lynn's. The first step in solving this inequality is to gather the w terms on one side and the constant terms on the other side. We can do this by subtracting 15w from both sides of the inequality:
650 + 35w - 15w > 825 + 15w - 15w
This simplifies to:
650 + 20w > 825
Next, we need to isolate the term with w by subtracting 650 from both sides of the inequality:
650 + 20w - 650 > 825 - 650
This gives us:
20w > 175
Finally, to solve for w, we divide both sides of the inequality by 20:
(20w) / 20 > 175 / 20
This simplifies to:
w > 8.75
This solution tells us that w must be greater than 8.75 weeks for Molly's savings to exceed Lynn's. However, since we can't have a fraction of a week in this context (we're dealing with whole weeks of savings), we need to round up to the next whole number. Therefore, it will take 9 weeks for Molly's savings to be greater than Lynn's savings.
By solving this inequality step-by-step, we have arrived at a concrete answer. This demonstrates the power of using inequalities to solve real-world problems involving comparisons and changes over time. The solution w > 8.75 is not just a number; it's a piece of information that helps us understand the dynamics of Molly and Lynn's savings plans.
Interpreting the Solution in Context
The solution to our inequality, w > 8.75, tells us something very specific about Molly and Lynn's savings. It means that after more than 8.75 weeks, Molly's savings will finally surpass Lynn's. But, in the real world, we don't usually talk about fractions of weeks when it comes to savings plans. We need a whole number of weeks. Since Molly's savings only exceed Lynn's after 8.75 weeks, we need to round up to the next whole number.
Therefore, it will take Molly 9 weeks to have more money saved than Lynn. This is a crucial step in problem-solving: interpreting the mathematical solution in the context of the original problem. The 8.75 is a mathematically correct answer to the inequality, but the real-world answer is 9 weeks. Think about it this way: at 8 weeks, Molly's savings are not yet greater than Lynn's. It's only sometime during the 9th week that the crossover happens. So, to be accurate, we need to say it takes a full 9 weeks.
This interpretation also highlights the importance of understanding the limitations of mathematical models. While the inequality provides a precise answer, the context of the problem dictates how we use that answer. In this case, the discrete nature of weeks (we can't save for half a week) forces us to round up. This rounding can have a significant impact, especially in financial contexts. For example, if we were calculating loan interest, rounding down might seem beneficial, but it could lead to underpayment and penalties. Understanding the nuances of interpreting solutions in context is a vital skill in mathematics and its applications.
Conclusion
In conclusion, we've successfully used an inequality to determine how many weeks it will take for Molly's savings to exceed Lynn's savings. By setting up the inequality 650 + 35w > 825 + 15w, solving for w, and interpreting the solution in context, we found that it will take Molly 9 weeks to save more money than Lynn. This problem demonstrates the power of using mathematical tools to analyze real-world situations involving comparisons and changes over time. Inequalities are not just abstract mathematical concepts; they are powerful tools for making predictions and understanding relationships between quantities.
This exercise highlights several key concepts in mathematics and problem-solving. First, it emphasizes the importance of translating word problems into mathematical expressions. The ability to represent a real-world scenario with an equation or inequality is a fundamental skill. Second, it showcases the steps involved in solving linear inequalities, including isolating the variable and interpreting the solution. Finally, it underscores the need to consider the context of the problem when interpreting mathematical results. In the case of Molly and Lynn's savings, rounding up to the next whole week was crucial for providing an accurate answer.
Understanding inequalities is a valuable skill that extends beyond the classroom. Inequalities are used in various fields, including finance, economics, engineering, and computer science. They help us make decisions in situations where we need to compare quantities, set limits, or optimize outcomes. Whether you're planning your own savings goals, analyzing business data, or designing a bridge, inequalities provide a powerful framework for problem-solving. To further explore the concept of inequalities and their applications, you might find resources and explanations on websites like Khan Academy's Algebra Resources.
