Solving For George's Original Rock Collection
Hey guys! Ever get stuck on a word problem and feel like your brain is doing the cha-cha with itself? Well, today we're diving into a super common type of math puzzle that involves figuring out a starting number after some stuff has happened. We've got our friend George here, and he's got a rock collection. Now, George is a generous dude, or maybe he just ran out of shelf space, but he decided to give half of his rocks to his pal Susan. The catch? He gave her 36 rocks. On top of that, before he even started giving rocks away, George got 5 new rocks. So, the big question is, which equation can we use to figure out how many rocks George had to begin with? Let's break it down and make this make sense, shall we? This is a classic algebra problem, and understanding how to set it up is half the battle, seriously. It's all about translating those words into mathematical symbols that our brains can process more easily. We're going to look at a few options and figure out the right way to represent George's rock-giving situation.
So, let's talk strategy, guys. When you're tackling a word problem like this, the first thing you want to do is identify what you don't know. In this case, it's the original number of rocks George had. That's our mystery, our unknown. In algebra, we usually represent unknowns with letters, and the most popular one is 'x'. So, let's say 'x' represents the total number of rocks George started with. Now, let's follow the story step-by-step. George first acquired 5 new rocks. So, after getting those new rocks, the number of rocks he had became 'x + 5'. This is a crucial step, folks. You can't just jump to giving rocks away without accounting for the ones he gained. It's like if you're baking cookies and you add flour before you divide them into batches, right? You gotta account for all the ingredients first. So, after this initial addition, George has 'x + 5' rocks. It's important to visualize this: imagine George's rock collection growing a bit. He's feeling good about his new additions.
Next up, George decides to give half of his current rock collection to Susan. Remember, he just got 5 new rocks, so he's giving away half of that new total. This is where a lot of people might trip up. They might think he's giving away half of his original 'x' rocks, but that's not what the problem says. It clearly states he gave half of his rock collection, implying the collection at that moment. So, if he has 'x + 5' rocks, giving away half of them means he's giving away rac{1}{2} (x+5) rocks. Now, we are told that the number of rocks he gave to Susan is 36. This is the key piece of information that allows us to set up an equation. So, the amount he gave away, which is rac{1}{2} (x+5), must be equal to 36. This gives us the equation: rac{1}{2} (x+5) = 36. This equation perfectly captures the scenario: George starts with 'x' rocks, adds 5, then gives half of that total away, and that amount equals 36. It’s like building a little mathematical story of what happened.
Let’s think about why the other options might be incorrect, guys. It's super important to understand why an answer is right, not just that it is right. Option A is rac{1}{2} x=36+5. This equation suggests that half of George's original collection (x) is equal to 36 plus 5. This doesn't make sense because George gained 5 rocks before giving any away. So, the 'x' in this equation would represent his original collection, but the addition of 5 seems to be tacked on after the division, which isn't the sequence of events. It's like saying you divided your pizza and then found 5 extra slices. Doesn't add up, does it? It ignores the crucial detail that the 5 rocks were added before the division. It's a common mistake to misinterpret the order of operations in word problems, and this equation falls into that trap.
Now let's look at option B: rac{1}{2}(x-5)=36. This equation implies that George lost 5 rocks before giving half away. The problem states George got 5 new rocks, not lost them. So, this equation flips the addition into a subtraction, which is a direct contradiction to the problem statement. If George started with 'x' rocks, and then lost 5, he'd have 'x-5' rocks. Then, giving half away would be rac{1}{2}(x-5). But that's not what happened; he gained rocks! So, this equation is fundamentally flawed because it misrepresents the initial change in George's rock collection. It's like saying you added sugar to your coffee and then someone told you to solve for the equation where you removed sugar. It just doesn't fit the narrative. You've got to stick to the story the word problem is telling you, step by step.
So, let's re-confirm our correct equation: rac{1}{2} (x+5) = 36. This equation correctly represents the situation. 'x' is the original number of rocks. George adds 5 rocks, making his collection 'x + 5'. Then, he gives half of that new total away, which is rac{1}{2} (x+5). And this amount equals the 36 rocks he gave to Susan. This is the most logical and accurate translation of the word problem into a mathematical expression. Now, just for fun, let's quickly solve it to see how many rocks George actually started with. If rac{1}{2} (x+5) = 36, we can multiply both sides by 2 to get rid of the fraction: . Then, to isolate 'x', we subtract 5 from both sides: , which means . So, George originally started with 67 rocks! Pretty cool, right? It feels good to crack the code and find the answer. So, the equation rac{1}{2} (x+5) = 36 is the one that accurately models the scenario described.
Understanding how to set up these algebraic equations is a seriously powerful skill, guys. It's not just for math class; it helps you break down complex problems in real life into manageable pieces. Think about budgeting, planning projects, or even figuring out recipes – it all involves a bit of algebraic thinking. The key is always to: 1. Identify the unknown (your 'x'). 2. Break down the problem into sequential steps. 3. Translate each step into mathematical operations (addition, subtraction, multiplication, division). 4. Set up an equation based on the final condition given. And always, always double-check if the equation makes sense with the story. Does it follow the order of events? Does it use the correct operations? If you can do that, you'll be crushing word problems like a boss! Remember, practice makes perfect. The more you tackle these types of problems, the more intuitive it becomes. Don't get discouraged if it takes a few tries. Every problem you solve is a win and builds your confidence. So, next time you see a word problem, give it a friendly nod, take a deep breath, and remember our friend George and his rock collection. You've got this!
Final Answer: The equation that could be used to determine how many rocks George started with is rac{1}{2}(x+5)=36. This equation correctly accounts for George gaining 5 rocks before giving half of his new total away to Susan, which amounted to 36 rocks. The other options presented either incorrectly alter the order of operations or misrepresent the initial change in the rock collection. Mastering this skill is all about careful reading and logical translation from words to symbols. Keep practicing, and you'll become a word problem whiz in no time!