Finding A Natural Number: A Math Puzzle

Hey guys! Let's dive into a fun little math puzzle. We're going to determine a natural number that fits some specific criteria. This isn't about complex equations or advanced calculus, it's about understanding the basics and using a bit of logical thinking. The problem is this: find a natural number where its half is less than 19 and its triple is greater than 108. Sounds intriguing, right? Let's break it down and see how we can solve this together. This is a classic example of how math can be both challenging and rewarding, kind of like leveling up in a game. We'll explore the steps, the logic, and the little tricks that will help us nail this problem.

Understanding the Problem: Breaking Down the Clues

Okay, so the first thing we need to do is really understand what the problem is asking. It’s like being a detective: we've got clues, and we need to piece them together to find the solution. The main goal here is to determine a natural number. Just a quick reminder, natural numbers are the positive whole numbers: 1, 2, 3, 4, and so on. We're not dealing with fractions, decimals, or negative numbers here. This keeps things nice and simple. The problem provides two key pieces of information, two constraints that our number must satisfy. The first clue is that the half of the number must be less than 19. Mathematically, if we call our number 'x', this can be written as x/2 < 19. The second clue is that the triple of the number is greater than 108. This translates to 3x > 108. So, basically, we need to find a number that fits both of these conditions. We're looking for a number that's not too big and not too small – a sort of Goldilocks number. This type of problem is great for building problem-solving skills, and once we get through this, we will be able to do other math problems easily.

Now, let’s get a bit more organized. We will translate each constraint into a more usable format. The first constraint is x/2 < 19. How do we get x by itself? Well, we multiply both sides of the inequality by 2. That gives us x < 38. This means our number must be less than 38. Now for the second constraint: 3x > 108. To isolate x here, we divide both sides of the inequality by 3. That leaves us with x > 36. So our number must be greater than 36. Now we have a clear range of possible values for our number. It has to be greater than 36 but less than 38. That means it must be 37.

Solving the Puzzle: Finding the Number

Now that we have a solid grasp of the problem and we've rephrased the constraints into something more manageable, let’s get to the solution. Remember, we have two key conditions: the number must be less than 38, and the number must be greater than 36. Think about the whole numbers that fit between 36 and 38. If you visualize a number line, you'll see that the only whole number that fits this bill is 37. So, that's it, that's our answer. Let's see how our number, which we've determined is 37, fits these conditions. First, let’s check if the first condition is met. Half of 37 is 18.5, which is indeed less than 19. Great. Now, let’s check the second condition. The triple of 37 is 111, which is greater than 108. Bingo! It fits perfectly. Therefore, the natural number we were looking for is 37. Pretty cool, huh? We started with a problem, broke it down, applied a bit of logic, and ended up with a neat, simple answer. That is the beauty of math, guys.

This kind of problem helps to build a foundational understanding of algebra and inequalities, which are essential for more advanced mathematical concepts. You’ll find that as you tackle more complex problems, the skills you develop from these basics will prove invaluable. Think of it like learning the alphabet before you write a novel. It's a stepping stone to bigger and better things. And you can pat yourselves on the back because we’ve successfully navigated this mathematical maze. We took the problem apart, put it back together, and now we have a clear, concise answer. Well done!

Checking the Solution and Understanding the Importance of the Solution

Always check your answers, guys! We have determined that the number is 37, but let's make sure. The first condition says that half the number must be less than 19. Half of 37 is 18.5, which is less than 19. Check. The second condition says that three times the number must be greater than 108. Three times 37 is 111, which is greater than 108. Check again. We are spot on! This process is super important; it builds trust in our skills and helps us avoid silly mistakes. It is also good practice for other complicated math problems. By checking the answer, we reinforce our understanding of the problem and the methods we used to solve it. This is like proofreading your work, except with numbers. This whole exercise of determining a natural number based on these specific conditions is more than just a math problem. It’s a showcase of how we can translate real-world scenarios into mathematical terms. The skill of breaking down a problem, identifying its components, and then using mathematical operations to solve it is applicable in many fields, from science and engineering to economics and even everyday life. From budgeting to calculating the best deals, mathematics is involved more than we think. This ability to reason logically and use numbers is not just a skill for exams or homework; it's a fundamental life skill.

In essence, the solution we've found is not just an answer, but a building block for future problem-solving. Each problem you conquer makes you more confident, more skilled, and better prepared to tackle the next mathematical hurdle. So, keep practicing, keep challenging yourself, and remember that every problem solved is a victory!