Pedro, Maria, And Juan's Savings A Math Problem Solved

Introduction

Hey guys! Today, we're diving into a super interesting math problem that involves figuring out how much Maria saved, given that Pedro, Maria, and Juan's savings together total 200. This kind of problem is not only a great exercise for our brains but also a common type you might encounter in national exams. So, let's break it down step by step and make sure we understand every little detail. We will explore the problem statement, discuss the importance of understanding the relationships between the variables, and then walk through the algebraic solution to find out exactly how much Maria managed to save. These types of problems are crucial for developing problem-solving skills, which are essential not just for exams but also for real-life situations. Think about it – you're using math every day, whether you realize it or not, from budgeting your expenses to figuring out discounts at the store. So, understanding these concepts can really make a difference. Now, before we jump into the nitty-gritty, let’s make sure we’re all on the same page with the basics. Remember, the key to solving any word problem is to first understand what the problem is asking. Read it carefully, identify the knowns and unknowns, and then formulate a plan to tackle it. It’s like a detective solving a mystery – you gather the clues, analyze them, and then put the pieces together to find the solution. In this case, our clues are the relationships between Pedro, Maria, and Juan's savings, and our mystery is how much Maria saved. So, grab your thinking caps, and let's get started!

Understanding the Problem Statement

So, let's get this straight, Pedro, Maria, and Juan have combined their savings, and the grand total comes to 200. That's our big picture. But here's where it gets interesting: Juan didn't save the same amount as Maria, and Pedro's savings are also linked to Maria's. Specifically, Juan saved a third of what Maria did – that's like dividing Maria's savings into three parts and Juan only saving one of those parts. And Pedro? Well, he doubled Maria's savings, meaning he saved twice as much as she did. Now, our mission, should we choose to accept it, is to figure out how much Maria saved on her own. To solve this, we need to think about how these amounts relate to each other. It's like a puzzle where each person's savings is a piece, and we need to fit them together to see the whole picture. This isn't just about finding a number; it's about understanding the relationships between these numbers. Think of it like this: if we know Maria's savings, we can easily figure out Juan's and Pedro's because their savings are defined in terms of hers. This is a crucial insight because it allows us to use algebra to solve the problem. We can represent Maria's savings with a variable, and then express Juan's and Pedro's savings in terms of that same variable. This way, we can create an equation that represents the total savings of all three people, and then solve for Maria's savings. This approach not only helps us find the answer but also reinforces the idea that math is a powerful tool for modeling real-world situations. We're not just dealing with abstract numbers here; we're dealing with actual savings, actual people, and actual relationships between their savings. So, by understanding the problem statement thoroughly, we're setting ourselves up for success in solving the problem. It’s like laying the foundation for a building – if the foundation is strong, the building will stand tall. So, let's move on to the next step and see how we can translate these relationships into algebraic expressions.

Setting Up the Algebraic Equation

Alright, guys, let's put on our algebra hats! To crack this problem, we need to translate the word problem into the language of math. This means using variables and equations to represent the relationships between the savings of Pedro, Maria, and Juan. So, let's start with the most crucial piece of information: Maria's savings. Since we don't know how much Maria saved, let's represent it with a variable. The classic choice is 'x', so we'll say that Maria saved x amount of money. Now, remember what the problem tells us about Juan's savings? Juan saved a third of what Maria saved. In algebraic terms, that means Juan saved x / 3. See how we're using 'x' to express Juan's savings? This is the power of algebra – it allows us to represent unknown quantities with symbols and then manipulate those symbols to find the unknowns. And what about Pedro? The problem states that Pedro saved double what Maria saved. So, Pedro's savings can be represented as 2x. We've now expressed the savings of all three people in terms of the same variable, x. This is a huge step because it allows us to combine these expressions into a single equation. Now, we know that the total savings of Pedro, Maria, and Juan add up to 200. So, we can write this as an equation: x (Maria's savings) + x / 3 (Juan's savings) + 2x (Pedro's savings) = 200. This equation is the key to solving the problem. It represents the relationship between the savings of the three people and the total amount they saved. It's like a mathematical sentence that tells the story of their savings. Now, our goal is to solve this equation for x, which will give us Maria's savings. But before we start solving, let's take a moment to appreciate what we've done. We've taken a word problem, identified the key information, and translated it into a concise algebraic equation. This is a fundamental skill in problem-solving, and it's something that you'll use again and again in math and in life. So, with our equation in hand, let's move on to the next step and see how we can solve it.

Solving for Maria's Savings (x)

Okay, let's get down to business and solve for 'x', which represents Maria's savings. Remember our equation? It's x + x / 3 + 2x = 200. The first thing we need to do is simplify this equation. We have terms with 'x' and a fraction, so let's get rid of that fraction. To do that, we can multiply every term in the equation by 3. This will clear the fraction and make the equation easier to work with. So, multiplying each term by 3, we get: 3 * x + 3 * (x / 3) + 3 * 2x = 3 * 200. This simplifies to 3x + x + 6x = 600. See how the fraction is gone? Now, we can combine the 'x' terms on the left side of the equation. We have 3x + x + 6x, which adds up to 10x. So, our equation becomes 10x = 600. We're almost there! Now, to isolate 'x', we need to divide both sides of the equation by 10. This will undo the multiplication and leave us with 'x' on its own. So, dividing both sides by 10, we get: 10x / 10 = 600 / 10. This simplifies to x = 60. And there you have it! We've solved for x, which means we've found Maria's savings. Maria saved 60. But before we celebrate too much, let's make sure our answer makes sense. We can plug this value back into our original equation to check if it holds true. So, let's see if 60 + 60 / 3 + 2 * 60 equals 200. 60 / 3 is 20, and 2 * 60 is 120. So, we have 60 + 20 + 120, which indeed equals 200. So, our answer checks out! We've successfully solved the equation and found that Maria saved 60. This is a great example of how algebra can be used to solve real-world problems. We took a word problem, translated it into an equation, and then solved the equation to find the answer. This is a valuable skill that will serve you well in math and in life. So, let's move on to the next section and see how we can summarize our findings.

Verifying the Solution

Alright, let's double-check our work because in math, just like in life, it's always a good idea to be sure! We figured out that Maria saved 60, but let's make absolutely certain that this answer fits all the pieces of our puzzle. Remember, Juan saved a third of what Maria saved, and Pedro saved double. So, if Maria saved 60, then Juan saved 60 / 3 = 20, and Pedro saved 2 * 60 = 120. Now, let's add up their savings: Maria's 60 + Juan's 20 + Pedro's 120. What do we get? 60 + 20 + 120 = 200. Bingo! That's the total savings we were given in the problem. This confirms that our solution is correct. We've not only found the value of 'x' but also verified that it makes sense in the context of the problem. This is a crucial step in problem-solving because it helps us catch any errors we might have made along the way. It's like proofreading an essay before submitting it – you want to make sure everything is just right. In this case, we've proofread our math, and we've found that it all adds up. This gives us confidence that we've solved the problem correctly. But beyond just checking our work, verifying the solution also helps us deepen our understanding of the problem. It forces us to think about the relationships between the different quantities and how they fit together. It's like putting the final pieces of a jigsaw puzzle in place – you can see the whole picture more clearly. So, by verifying our solution, we've not only confirmed our answer but also strengthened our understanding of the problem-solving process. Now, let's move on to the final section and summarize our findings and takeaways.

Conclusion

Alright, guys, we've reached the finish line! Let's recap what we've done. We started with a word problem about Pedro, Maria, and Juan's savings. The challenge was to find out how much Maria saved, given that their total savings were 200, Juan saved a third of Maria's savings, and Pedro saved double. We tackled this problem using algebra, which is a powerful tool for solving these kinds of puzzles. We represented Maria's savings with the variable 'x', and then we expressed Juan's and Pedro's savings in terms of 'x'. This allowed us to create an equation that represented the total savings of all three people: x + x / 3 + 2x = 200. We then solved this equation for 'x', which involved simplifying the equation, clearing the fraction, and isolating 'x'. We found that x = 60, which means Maria saved 60. But we didn't stop there! We verified our solution by plugging it back into the original problem and making sure it made sense. We found that if Maria saved 60, then Juan saved 20, and Pedro saved 120, and these amounts added up to the total savings of 200. This gave us confidence that our solution was correct. So, what are the key takeaways from this problem-solving journey? First, understanding the problem statement is crucial. We need to read the problem carefully, identify the knowns and unknowns, and understand the relationships between them. Second, translating the word problem into an algebraic equation is a key step. This allows us to use the power of algebra to solve the problem. Third, solving the equation involves a series of steps, such as simplifying, clearing fractions, and isolating the variable. And finally, verifying the solution is essential to ensure that our answer is correct. This problem is a great example of how math can be used to solve real-world problems. We've not only found the answer but also developed our problem-solving skills along the way. So, keep practicing, keep thinking, and keep solving!

Keywords: Pedro, Maria, Juan, savings, algebra, equation, problem-solving