Solve: Number Plus Half Equals 18

Hey guys! Ever stumbled upon a math problem that seems a bit tricky at first glance? Well, let's dive into one together and break it down step-by-step. We're tackling a classic algebra problem today: “The sum of a number and its half equals 18. What is the number?” Don't worry; we'll make it super clear and easy to understand. So, grab your thinking caps, and let’s get started!

Understanding the Problem: Setting Up the Equation

So, when we approach these kinds of math problems, the first step is always understanding exactly what's being asked. Our key phrase here is: “The sum of a number and its half equals 18.” To solve this, we're going to translate these words into a mathematical equation. This is like turning a secret code into plain English, or, in our case, math!

First, we need to represent the unknown number. Since we don't know what it is yet, we'll use a variable. Let's go with the classic choice: x. This x is just standing in for the number we're trying to find. Now, what about "its half"? Well, the half of a number is that number divided by 2. So, the half of x is x / 2. Make sense so far?

Next, we have the word “sum.” In math language, “sum” means addition. So, “the sum of a number and its half” translates to x + (x / 2). And finally, we have “equals 18.” The equals sign (=) is pretty straightforward, so we just write = 18. Now, let’s put it all together. Our equation is:

x + (x / 2) = 18

See? We've taken a sentence and turned it into a neat little mathematical equation. This is the foundation for solving the problem. Now that we have our equation, the next step is to solve it. This involves a little bit of algebraic manipulation, but don’t worry, we’ll take it slowly. We're essentially going to rearrange things until we get x all by itself on one side of the equation. This will tell us what number x represents. Think of it like peeling away the layers of an onion until you get to the core – the value of x. So, let's move on to the next part and start solving this equation!

Solving the Equation: Step-by-Step

Alright, guys, now that we've got our equation, x + (x / 2) = 18, it's time to roll up our sleeves and solve it. Don't worry, it's not as scary as it might look! We're going to take it one step at a time. The main goal here is to isolate x on one side of the equation so we can figure out its value. To do that, we need to get rid of the fraction first. Fractions can sometimes make things look more complicated than they are, so let's simplify.

We have x / 2 in our equation. To eliminate the fraction, we can multiply every term in the equation by the denominator, which in this case is 2. Remember, whatever we do to one side of the equation, we have to do to the other side to keep things balanced. So, we multiply both sides of the equation by 2:

2 * [x + (x / 2)] = 2 * 18

Now, let's distribute that 2 on the left side. This means we multiply each term inside the brackets by 2:

2 * x + 2 * (x / 2) = 36

This simplifies to:

2x + x = 36

See how the fraction is gone now? Much cleaner, right? Now we have a simpler equation to work with. Next, we combine like terms. On the left side, we have 2x and x. If you think of x as one apple, then 2x is two apples. So, two apples plus one apple equals three apples, or in our case, 3x. So, we can rewrite the equation as:

3x = 36

We're getting closer! Now we just have one more step to isolate x. We have 3x, which means 3 times x. To undo multiplication, we use division. So, we divide both sides of the equation by 3:

(3x) / 3 = 36 / 3

This simplifies to:

x = 12

And there you have it! We've solved the equation. x equals 12. That means the number we were looking for is 12. But, hold on! We're not quite done yet. It's always a good idea to check our answer to make sure it makes sense in the original problem. So, let’s move on to the next section and verify our solution.

Verifying the Solution: Does It Add Up?

Okay, so we've arrived at our solution: x = 12. But in math, just like in life, it's always a good idea to double-check your work! We need to make sure that our answer actually fits the original problem. Remember, the problem stated that “the sum of a number and its half equals 18.” So, let’s plug our solution, 12, back into the original condition and see if it holds true.

The number is 12, and its half would be 12 divided by 2, which equals 6. Now, we add the number and its half together:

12 + 6 = ?

Simple addition tells us that:

12 + 6 = 18

Woo-hoo! It works! Our solution checks out. The sum of 12 and its half (which is 6) indeed equals 18. This confirms that our answer, x = 12, is correct. We've successfully solved the problem. This step of verification is super important because it helps you catch any mistakes you might have made along the way. It's like the final polish on a piece of work – it ensures everything is just right.

So, we’ve gone from understanding the problem, setting up the equation, solving it step-by-step, and finally, verifying our solution. That’s the whole process! Now that we’ve confirmed our answer, let’s wrap things up with a quick summary of what we’ve learned. This will help solidify your understanding and give you the confidence to tackle similar problems in the future.

Conclusion: The Magic Number Revealed

So, guys, we did it! We successfully cracked the problem: “The sum of a number and its half equals 18. What is the number?” Through our step-by-step journey, we discovered that the magic number is indeed 12. We started by translating the words into a mathematical equation: x + (x / 2) = 18. This is often the trickiest part, but breaking it down piece by piece makes it much easier.

Then, we rolled up our sleeves and solved the equation. We eliminated the fraction by multiplying every term by 2, simplified the equation, combined like terms, and finally, isolated x by dividing both sides by 3. This gave us our solution: x = 12. But we didn't stop there! We verified our solution by plugging 12 back into the original condition and confirmed that it checks out.

This whole process highlights some important problem-solving skills in mathematics. First, translating word problems into equations is a fundamental skill in algebra. It's like learning to speak a new language – the language of math! Second, solving equations involves manipulating them in a way that isolates the variable you're trying to find. This requires understanding the rules of algebra and applying them consistently. And third, verifying your solution is a crucial step that ensures accuracy and builds confidence.

Remember, math isn't just about finding the right answer; it's about understanding the process. By breaking down complex problems into smaller, manageable steps, you can tackle anything that comes your way. So, keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!