Subway Balance Showdown: When Do Kaisorn & Thom Have The Same?

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Hey guys! Let's dive into a fun math problem about Kaisorn and Thom and their public transportation cards. This is a classic problem that uses algebraic equations to figure out when two things are equal. So, buckle up and let’s break it down together! We will explore how to determine the number of subway trips it takes for Kaisorn and Thom to have the same amount of money left on their public transportation cards. It's a real-world scenario that highlights the practical application of math in our daily lives. So, if you've ever wondered how to calculate the best way to budget your own travel expenses, this is the perfect example to get you started.

Setting Up the Equations

Okay, first things first, let's get the facts straight. Kaisorn kicks things off by adding $50.00 to her card each month, and each subway ride sets her back $2.50. Thom, on the other hand, starts with a $40.00 deposit and pays $2.00 per ride. Our mission, should we choose to accept it, is to find out after how many trips, represented by x, they'll have the same amount of dough left on their cards. This is where the magic of algebra comes in! Remember, algebra is just a fancy way of using letters and symbols to represent numbers and relationships. We can use it to create equations that describe what’s happening with Kaisorn and Thom's money. Think of an equation as a mathematical sentence. It tells us that two things are equal. In our case, the amount of money Kaisorn has left on her card will be equal to the amount of money Thom has left on his card after a certain number of trips. To solve the problem, we need to translate the information we have into mathematical expressions. We'll start by representing Kaisorn's balance as an equation and then do the same for Thom. Once we have both equations, we can set them equal to each other and solve for x. This will give us the number of trips it takes for their balances to be the same. So, let's put on our math hats and get started!

To represent Kaisorn's situation, we start with her initial deposit of $50.00. Then, for each trip she takes, $2.50 is deducted from her balance. If x represents the number of trips, the total amount deducted is $2.50 multiplied by x, or 2.50x2.50x. So, the amount of money Kaisorn has left on her card can be represented by the equation:

  • Kaisorn's Balance = 50 - 2.50x

This equation tells us that Kaisorn's remaining balance is her initial deposit minus the total cost of her trips. Now, let's do the same for Thom.

Thom begins with a $40.00 deposit, and he spends $2.00 per trip. Using the same logic as before, we multiply the cost per trip by the number of trips, x, to get the total amount deducted, which is 2.00x2.00x. Therefore, Thom's remaining balance can be represented by the equation:

  • Thom's Balance = 40 - 2.00x

This equation shows that Thom's remaining balance is his initial deposit minus the total cost of his trips. Now that we have equations for both Kaisorn's and Thom's balances, we can move on to the next step: setting the equations equal to each other and solving for x.

Solving for the Number of Trips

Alright, now for the juicy part! We've got our equations all set up, and it's time to put on our detective hats and solve for x. Remember, x is the number of trips we're trying to find – the magic number that makes Kaisorn and Thom's card balances the same. To do this, we'll use a little bit of algebraic wizardry. Don’t worry, it’s not as scary as it sounds! We're essentially going to manipulate the equations to isolate x on one side, revealing its true value.

So, what do we do? We need to find the point where both of them have the same amount left. This means we need to equate the two expressions we derived for their balances. By setting the two equations equal to each other, we can create a new equation that relates the number of trips x to the condition where their balances are the same. The act of equating these expressions is a critical step in solving the problem, as it allows us to directly compare and relate Kaisorn's and Thom's spending habits.

Here's what the equation looks like:

  • 50 - 2.50x = 40 - 2.00x

See? Not so scary! This equation is our key to unlocking the value of x. It tells us that at some point, the amount of money Kaisorn has left will be the same as the amount of money Thom has left. The challenge now is to solve for x, which will reveal the number of trips it takes for this to happen. To solve this equation, we need to isolate x on one side of the equation. This involves rearranging the terms so that all the x terms are on one side and all the constant terms (the numbers without x) are on the other side. We can do this by adding or subtracting terms from both sides of the equation. Remember, whatever we do to one side of the equation, we must do to the other side to keep the equation balanced. Let's start by moving the x terms to one side and the constant terms to the other. This will bring us one step closer to finding the value of x. So, let's roll up our sleeves and get to work!

Our goal is to get all the x terms on one side and the numbers on the other. A neat trick here is to add 2.50x to both sides of the equation. Why? Because it cancels out the -2.50x on the left side, making our lives easier. It's like playing a strategic game where you're always thinking a few steps ahead to simplify things. This step is crucial because it begins the process of isolating x, which is our ultimate goal. By adding 2.50x to both sides, we're not changing the equation's balance; we're just rearranging the terms in a way that helps us solve for x. The idea is to make the equation simpler and more manageable. So, let's go ahead and add 2.50x to both sides and see what happens.

Here’s how it looks:

  • 50 - 2.50x + 2.50x = 40 - 2.00x + 2.50x

  • 50 = 40 + 0.50x

Boom! We've already made progress. The -2.50x and +2.50x on the left side canceled each other out, leaving us with just 50. And on the right side, we combined the -2.00x and +2.50x to get 0.50x. Now, the equation looks much simpler. But we're not done yet. We still need to isolate x completely. This means getting rid of the 40 on the right side. So, what's the next step? Well, we can subtract 40 from both sides. This will cancel out the 40 on the right side and move us closer to our goal. Remember, the key to solving algebraic equations is to perform the same operation on both sides to maintain balance. Think of it like a seesaw: if you add or remove weight from one side, you need to do the same on the other side to keep it level. So, let's subtract 40 from both sides and see what we get.

Now, let’s subtract 40 from both sides:

  • 50 - 40 = 40 - 40 + 0.50x

  • 10 = 0.50x

We're on the home stretch now! We've simplified the equation to 10 = 0.50x. This means that 10 is equal to half of x. To find the full value of x, we need to get rid of the 0.50 that's multiplying it. How do we do that? By dividing both sides of the equation by 0.50, of course! This is the final step in isolating x. By dividing both sides by 0.50, we're essentially undoing the multiplication, leaving x all by itself on one side of the equation. This will reveal the value of x, which is the number of trips it takes for Kaisorn and Thom to have the same amount of money left on their cards. So, let's go ahead and divide both sides by 0.50 and see what we get. Get ready to celebrate, because we're about to solve the mystery!

To finally isolate x, we divide both sides by 0.50:

  • 10 / 0.50 = 0.50x / 0.50

  • x = 20

The Grand Finale: Interpreting the Result

Woohoo! We did it! We cracked the code and found that x equals 20. But hold on a second, guys. In math, it's not enough just to find the answer. We need to understand what it means. So, what does x = 20 actually tell us in the context of our problem? Well, remember that x represents the number of trips Kaisorn and Thom take on the subway. So, our answer means that after 20 trips, Kaisorn and Thom will have the same amount of money left on their public transportation cards. Isn't that neat? We used algebra to solve a real-world problem and find a concrete answer. This is why math is so cool – it helps us make sense of the world around us.

So, after 20 trips, both Kaisorn and Thom will have the same balance on their cards. This is super useful information! If they're trying to budget their travel expenses, they now know exactly when their spending will align. This kind of calculation can help them make informed decisions about whether to add more money to their cards, explore alternative transportation options, or even plan their trips more strategically. But it's not just about this specific scenario. The process we used to solve this problem can be applied to countless other situations in daily life. Whether you're comparing phone plans, calculating discounts, or figuring out how long it will take to save up for something you want, the ability to set up and solve equations is a valuable skill. So, the next time you're faced with a problem that seems a bit tricky, remember the steps we took to solve this one: identify the key information, define your variables, set up your equations, and solve for the unknown. You might be surprised at what you can achieve!

But wait, there's more to this story! While we've found the number of trips it takes for their balances to be the same, it might be interesting to know what that balance actually is. To find out, we can simply substitute x = 20 into either Kaisorn's or Thom's equation. Since we know their balances will be equal at this point, it doesn't matter which equation we use. Let's try Kaisorn's equation:

  • Kaisorn's Balance = 50 - 2.50x

  • Kaisorn's Balance = 50 - 2.50 * 20

  • Kaisorn's Balance = 50 - 50

  • Kaisorn's Balance = 0

So, after 20 trips, Kaisorn will have $0 left on her card. If we plug x = 20 into Thom’s equation:

  • Thom's Balance = 40 - 2.00x

  • Thom's Balance = 40 - 2.00 * 20

  • Thom's Balance = 40 - 40

  • Thom's Balance = 0

As we expected, Thom also has $0 left on his card. This adds another layer of understanding to our solution. Not only do we know when their balances will be the same, but we also know that they'll both have run out of money at that point! This could be a crucial piece of information for Kaisorn and Thom, as it helps them plan ahead and ensure they don't get stuck without enough money for their subway rides. See? Math isn't just about numbers; it's about real-life situations and making smart decisions. So, keep practicing, keep exploring, and keep using math to unlock the world around you!

In conclusion, we have successfully navigated the world of public transportation finances using the power of algebra! By setting up and solving equations, we determined that Kaisorn and Thom will have the same amount of money on their cards after 20 trips. This problem not only demonstrates the practical application of math but also highlights the importance of critical thinking and problem-solving skills. So, the next time you're faced with a similar challenge, remember the steps we took and embrace the power of math to find the solution. And who knows, maybe you'll even save some money along the way!