Solving For The Original Account Balance After Withdrawals
Hey guys, ever stumbled upon a math problem that seems like a real brain-teaser? Well, today we're diving deep into one of those! Imagine a scenario where someone withdraws a sixth of their money from an account and then three-tenths of the remaining amount. After all that, they're left with 35,000 pesos. The big question is: how much money was in the account to begin with? Sounds like a classic riddle, right? But don't worry, we're going to break it down step by step, so it's as clear as day. We will explore the nuances of this problem to understand not just the solution but also the process of getting there, ensuring that anyone can tackle similar challenges with confidence. The beauty of mathematics lies in its ability to demystify complex situations by breaking them down into manageable parts, and that's exactly what we're going to do here. Letβs embark on this mathematical journey together and unravel this mystery! So, buckle up, and let's dive into the fascinating world of financial puzzles. Are you ready to put on your detective hats and solve this money mystery with me? Let's get started and see how we can crack this mathematical case wide open!
Understanding the Problem
Before we even think about crunching numbers, let's get crystal clear on what the problem is asking. We're not just looking for any number; we're trying to find the original amount of money in the account. This is crucial because it sets the stage for how we approach the solution. Imagine you're reading a novel β you need to know the premise before you can follow the plot, right? It's the same with math problems! We need to understand the starting point, the actions taken (the withdrawals), and the final result (35,000 pesos). Think of it like a detective story: we have clues, and we need to piece them together to solve the mystery. Each detail is a piece of the puzzle, and understanding their relationships is key to unlocking the answer. So, letβs break it down further: A certain amount was in the account initially; a sixth of it was taken out; then, three-tenths of what was left was withdrawn. Finally, 35,000 pesos remain. The mission? To rewind the story and find that original amount. By identifying the initial state, the changes, and the outcome, we lay a solid foundation for solving the problem. Let's keep this clear picture in our minds as we move forward to the next step: translating these words into a mathematical equation. This way, we're not just guessing; we're strategically working towards the solution. Remember, understanding the problem is half the battle won!
Setting Up the Equation
Okay, guys, now for the exciting part β let's turn this word problem into a mathematical equation! This is where we translate the language of the problem into the language of math. Think of it as learning a new language β we're taking the story and rewriting it in symbols and numbers. It might seem daunting, but trust me, it's like putting puzzle pieces together. We'll use 'x' to represent the unknown β the original amount of money. This is our mystery variable, the one we're hunting for. The first clue is that a sixth of the money was withdrawn. In math terms, that's (1/6)x. So, after this withdrawal, the remaining amount is x - (1/6)x. Make sense? We started with x, took away a sixth of it, and what's left is x minus that sixth. Next, three-tenths of the remaining amount was withdrawn. That's where we need to be careful! We're not taking three-tenths of the original amount, but three-tenths of what's left after the first withdrawal. Mathematically, this is (3/10) * [x - (1/6)x]. This part is super important because it captures the sequence of events. We first find the remainder, then calculate the second withdrawal based on that. Finally, we know that after both withdrawals, 35,000 pesos are left. So, we can set up our full equation: x - (1/6)x - (3/10) * [x - (1/6)x] = 35,000. This equation is like the DNA of our problem β it holds all the information we need to find the answer. It might look a bit complex, but don't worry! We're going to simplify it step by step. The key is to see how each part of the equation corresponds to the story we're trying to solve. Now, let's roll up our sleeves and simplify this equation to unveil the value of x!
Solving the Equation Step-by-Step
Alright, time to put on our math hats and dive into solving this equation! Don't worry, we'll take it slow and steady, step by step, like untangling a knot. The equation we're tackling is: x - (1/6)x - (3/10) * [x - (1/6)x] = 35,000. Our first mission is to simplify the expression inside the brackets. We have x - (1/6)x. To combine these terms, we need a common denominator. Think of 'x' as (6/6)x. So, (6/6)x - (1/6)x equals (5/6)x. Great! We've simplified the inside of the brackets. Now our equation looks like this: x - (1/6)x - (3/10) * (5/6)x = 35,000. Next up, let's deal with the multiplication. We need to multiply (3/10) by (5/6)x. When multiplying fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers). So, (3/10) * (5/6) becomes (35)/(106), which simplifies to 15/60. We can further simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 15. So, 15/60 simplifies to 1/4. Now, our equation is: x - (1/6)x - (1/4)x = 35,000. We're getting there! The next step is to combine the 'x' terms. Again, we need a common denominator. The least common multiple of 6 and 4 is 12. So, let's convert our fractions: (1/6)x becomes (2/12)x, and (1/4)x becomes (3/12)x. We also need to think of 'x' as (12/12)x. Now our equation looks like this: (12/12)x - (2/12)x - (3/12)x = 35,000. Let's combine those fractions: (12/12) - (2/12) - (3/12) equals (7/12). So, our equation simplifies to: (7/12)x = 35,000. We're in the home stretch! To isolate 'x', we need to get rid of the (7/12). We can do this by multiplying both sides of the equation by the reciprocal of (7/12), which is (12/7). So, [(12/7) * (7/12)x] = 35,000 * (12/7). The (12/7) and (7/12) on the left side cancel each other out, leaving us with 'x'. On the right side, we have 35,000 * (12/7). Let's simplify: 35,000 divided by 7 is 5,000. Then, 5,000 multiplied by 12 is 60,000. So, x = 60,000! We've done it! We've successfully solved the equation. But, the job isn't over yet. We need to make sure our answer makes sense in the context of the problem.
Verifying the Solution
Alright, we've arrived at a solution: x = 60,000 pesos. But before we pop the champagne, let's double-check to make sure our answer fits the story. This is a crucial step in problem-solving β it's like proofreading your work before you submit it. We're essentially going to rewind the scenario with our answer and see if it leads us back to the 35,000 pesos. So, let's start with the original amount: 60,000 pesos. First, a sixth of the money was withdrawn. A sixth of 60,000 is 60,000 / 6, which equals 10,000 pesos. After this withdrawal, the remaining amount is 60,000 - 10,000, which equals 50,000 pesos. Next, three-tenths of the remaining amount was withdrawn. Three-tenths of 50,000 is (3/10) * 50,000, which equals 15,000 pesos. After this second withdrawal, the amount left is 50,000 - 15,000, which equals 35,000 pesos. Bingo! That's exactly the amount we were told was left in the account. This confirms that our solution, 60,000 pesos, is correct. It's like finding the missing piece of a puzzle and seeing it fit perfectly β satisfying, right? By verifying our solution, we not only ensure accuracy but also deepen our understanding of the problem. We've essentially walked through the scenario, confirming that our answer makes logical sense within the given context. This is a fantastic habit to develop in problem-solving β always take that extra step to verify your solution. It's the final touch that turns a good solution into a great one. Now, we can confidently say that we've not only solved the problem but also proven our answer is correct. Pat yourselves on the back, guys β you've earned it!
Conclusion
So, guys, we've successfully navigated a tricky word problem and emerged victorious! We started with a seemingly complex scenario, but by breaking it down into manageable steps, we were able to unravel the mystery. We found that the original amount of money in the account was 60,000 pesos. How cool is that? This journey wasn't just about finding a number; it was about the process β understanding the problem, setting up the equation, solving it step-by-step, and verifying our solution. These are skills that go way beyond math class. They're about critical thinking, problem-solving, and approaching challenges with confidence. Remember, math problems are like puzzles β they might seem daunting at first, but with the right approach, they're incredibly satisfying to solve. And the best part? The more you practice, the better you get. So, don't shy away from challenges; embrace them! Each problem you solve is a step forward, a new skill learned, and a boost to your confidence. Keep practicing, keep exploring, and never stop asking questions. Math is a powerful tool, and with it, you can unlock all sorts of mysteries. Now, go forth and conquer those math challenges β you've got this! And remember, every problem solved is a victory earned. Keep up the fantastic work, and I can't wait to see what mathematical feats you accomplish next! You are now equipped not just with an answer, but with the skills and confidence to tackle similar problems in the future. Keep shining, mathletes! This is just the beginning of your problem-solving journey!