Cracking The Code: A Math Puzzle At Age 30

Hey math enthusiasts! Ever stumble upon a head-scratcher that just begs to be solved? Well, buckle up because we're diving into a classic word problem that's all about unraveling a numerical mystery. Let's break down a problem that involves adding, dividing, and a bit of algebraic thinking to find the original number. This is the kind of puzzle that’s perfect for anyone looking to sharpen their problem-solving skills. So, grab your pencils, and let's crack this math code together!

The Setup: Decoding the Problem

Alright, here's the deal: We're told a certain number exists. This mysterious number is our starting point, the foundation upon which the rest of the problem is built. Next, we're instructed to add 30 to this number. Think of it as increasing the original value by a significant amount. This addition is a key step that changes everything. After the addition, we're told to divide the entire sum by 6. This division is another operation that reshapes the initial number. Now, the real twist: the result of this division is claimed to be one more than half of the original number. This final piece of the puzzle sets up an equation, a mathematical balancing act where the left side (our additions and divisions) equals the right side (half the original number plus one). This one sentence is a compressed version of a mathematical puzzle. The beauty of this type of problem lies in its ability to be translated into a clear, easy-to-understand equation. To successfully resolve this problem, we must perform a careful translation of the sentences that makes up the problem into an algebraic equation. This requires a solid understanding of mathematical vocabulary and the ability to change words into their corresponding mathematical symbols. We'll start by giving the unknown quantity a variable. This variable will represent our unknown number. Let's use 'x' to represent the original number. With 'x' representing our starting number, we can start setting up the equation. Adding 30 to our original number gives us x + 30. Then, as instructed, we are going to divide that by 6, which would then give us (x + 30) / 6. Finally, the result is said to be one more than half the original number, which is represented by (x/2) + 1. Now, we have both sides of the equation, and we can set up the entire equation. So we put it all together, and the equation we are working with is (x + 30) / 6 = (x / 2) + 1. That is the equation that is needed to get the answer to our mathematical problem. With this, we are one step closer to our goal.

Translating Words into Equations

Okay, guys, let's break this down even further. The core of solving this kind of problem is translating words into mathematical symbols. It's like learning a secret language, and once you get the hang of it, you can decipher all sorts of puzzles. Let's get into the nitty-gritty of setting up the problem. Our first step is defining the unknown. The original number, which we are trying to find, is the unknown. We will call this number “x”. This is a super common move in algebra. Now, let's look at what the problem tells us to do with this number. We are told to add 30 to our mystery number. In math terms, that’s simply “x + 30.” Next, we are told to divide the sum by 6. This gives us “(x + 30) / 6.” Remember the parenthesis; they’re important! Now, on the other side of the equation, we have “the result is 1 more than half of the original number.” Half of the original number is x/2. One more than that is (x/2) + 1. And then the last step, we just bring it all together. The problem tells us that when you do the steps on the left, you get the expression on the right. So, the whole equation becomes: (x + 30) / 6 = (x / 2) + 1. Congrats! You have successfully translated the word problem into its mathematical form! Now, the solution is just a few steps away. You are now one step closer to finding the value of 'x', which is the answer to our question.

Solving the Equation: Step by Step

Alright, the equation is set up: (x + 30) / 6 = (x / 2) + 1. Now, let's roll up our sleeves and solve it. The goal here is to isolate 'x' and find its value. I'll show you how to walk through it step-by-step so you understand all the math we are doing. Step 1: Get rid of the fractions. To do this, we are going to multiply both sides of the equation by 6. This will cancel out the denominator on the left side. So we end up with: 6 * [(x + 30) / 6] = 6 * [(x / 2) + 1]. Which simplifies to: x + 30 = 3x + 6. Step 2: Simplify the equation. Here we are going to move all the 'x' terms to one side and the constants to the other side. Subtract x from both sides to get: 30 = 2x + 6. Step 3: Isolate 'x'. Subtract 6 from both sides to get: 24 = 2x. Step 4: Solve for 'x'. Divide both sides by 2: x = 12. We've got it! The original number, the one we were trying to find, is 12.

Checking Your Work: Does It Make Sense?

Great job, guys, we solved for x! But before we pop the champagne, let's check our answer. Always a smart move to make sure our solution actually works within the context of the original problem. Plugging it back into the initial word problem is the best way to do this. Remember what we started with: “If 30 is added to a certain number and the sum is divided by 6, the result is 1 more than $\frac{1}{2}$ of the original number.” If our original number is 12, then let's check it out. Step 1: Add 30 to the number. 12 + 30 = 42. Step 2: Divide by 6. 42 / 6 = 7. Step 3: Calculate $\frac{1}{2}$ of the original number and add 1. Half of 12 is 6. Add 1 and you get 7. Step 4: Compare the results. Is the result of the division (7) the same as half the original number plus 1? Yes! 7 = 7. Success! Our answer checks out, which means our solution of 12 is correct. This is the fun part where you get to pat yourself on the back and know that you've correctly solved the equation.

The Takeaway: Math is Your Friend

So, what did we learn today? We took a seemingly complex word problem and broke it down into manageable steps. We learned how to translate words into mathematical equations, how to solve those equations, and how to verify our answers. Solving this kind of problem is a fantastic way to boost your algebra skills and, more importantly, your confidence in tackling future challenges. The key is to take the problem one step at a time, and don’t be afraid to make mistakes. Everyone learns at their own pace, and the more you practice, the easier these problems become. The world is full of math puzzles, and now you've got a fresh set of skills to crack them. Keep practicing, keep learning, and remember: math doesn’t have to be scary. In fact, it can be pretty fun when you know how to approach it!