Hey everyone! Let's dive into a problem about two runners diligently saving up for a marathon. It’s not just about their physical endurance; they're also flexing their financial muscles! We've got two runners with different savings strategies, and our mission is to figure out which equation will help us track their progress. This involves a bit of math, but don't worry, we'll break it down step by step. So, buckle up, mathletes, and let's get started!
The Savings Scenario
In this savings scenario, we have two dedicated runners, let's call them Runner A and Runner B, who are both determined to participate in a marathon. Marathons, as we know, involve not just physical preparation but also financial planning. There are entry fees, travel costs, accommodation, and other expenses to consider. Our runners are smart; they understand the importance of saving money well in advance. Now, here’s the breakdown of their current financial situations and their savings plans:
Runner A's Financial Strategy
Runner A is off to a great start. This runner already has $112 tucked away in their savings account. That’s a solid foundation! But wait, there’s more good news. A generous friend, impressed by Runner A’s dedication, gifted them $45. How awesome is that? So, before even lifting a finger to save this month, Runner A has a total of $112 + $45 = $157. Now, for the ongoing savings plan: Runner A is committed to saving $25 each month. This consistent saving habit is crucial for reaching their financial goal for the marathon. The key here is the consistent monthly savings, which will add up over time. Think of it like running consistent miles each week – it’s the steady effort that leads to success.
Runner B's Financial Strategy
Runner B is starting with a smaller initial amount but has an ambitious monthly savings goal. This runner currently has $50 in savings. It’s a modest start, but every great journey begins with a single step, or in this case, a single saved dollar! Runner B doesn't have the advantage of a gift, but they have a powerful savings strategy. They are determined to save $60 each month. That’s a significant monthly contribution, showing serious dedication to their marathon dream. Runner B's strategy is like a high-intensity training plan – it requires more effort upfront but can lead to rapid progress. The higher monthly savings rate means that Runner B will catch up over time, and we need to figure out how long that will take. This is where the beauty of mathematical equations comes in – they help us predict and plan!
Crafting the Equation: The Key to Unlocking Their Financial Progress
The core of our task is to pinpoint the equation that accurately models the runners' savings progress. An equation, in this context, is a mathematical statement showing the equality between two expressions. In our case, the expressions will represent the total savings of each runner over time. We need to create an equation that captures the initial savings, the gift (if any), and the monthly savings contribution. Let's break down the components we need to build this equation. This is where we put on our mathematical hats and think about how to translate the word problem into a symbolic representation.
Identifying the Variables
First, we need to identify the key variables in our problem. Variables are the symbols that represent quantities that can change or vary. In this scenario, the most important variable is the number of months, which we can represent with the letter 'm'. The total savings for each runner will depend on how many months they save. Another crucial variable is the total amount saved, which we can represent with 'S'. The total amount saved is what we ultimately want to calculate and compare for both runners. Identifying these variables is like setting the stage for our mathematical drama – it gives us the characters we need to tell the story.
Building the Expression for Runner A
Now, let’s construct the expression that represents Runner A's total savings. We know Runner A starts with $112 and receives a $45 gift, giving them a head start of $157. They then save $25 each month. So, after 'm' months, they will have saved an additional $25 * m. Putting it all together, the total savings (S) for Runner A can be expressed as:
S = 157 + 25m
This equation is like a financial blueprint for Runner A. It tells us exactly how their savings will grow over time. The initial amount of $157 is the foundation, and the $25m represents the monthly additions. This linear equation perfectly models the steady increase in Runner A's savings.
Building the Expression for Runner B
Next, we need to create the expression for Runner B's savings. Runner B starts with $50 and saves $60 each month. So, after 'm' months, they will have saved an additional $60 * m. The total savings (S) for Runner B can be expressed as:
S = 50 + 60m
This equation shows Runner B's savings growing at a faster rate than Runner A's, thanks to their higher monthly savings. The initial $50 is a smaller starting point, but the $60m shows the power of consistent, higher contributions. This equation is Runner B's financial roadmap, guiding them toward their marathon savings goal.
Finding the Right Equation: Connecting Savings and Time
Our main goal is to find the equation that can be used to determine when the two runners will have saved the same amount of money. To do this, we need to set the two expressions equal to each other. This is where the magic happens – we're essentially asking,
