Rope Division Problem: Finding The Number Of Pieces

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Rope Division Problem: Finding the Number of Pieces

Hey guys! Let's dive into this interesting math problem together. It's all about figuring out how many pieces we get when we divide a rope, and it involves some cool prime factorization too. So, buckle up, and let's get started!

Understanding the Problem

Okay, so here's the deal: We have a rope that's 84 centimeters long. This rope is cut into equal pieces. Now, here's the tricky part: the length of each piece is equal to the sum of the prime factors of the total length of the rope (which is 84 cm). Our mission, should we choose to accept it, is to find out how many pieces we end up with. Sounds fun, right?

Let's break this down a bit further. First, we need to understand what prime factors are. Prime factors are prime numbers that divide exactly into another number. For example, the prime factors of 12 are 2 and 3 because 2 x 2 x 3 = 12, and 2 and 3 are prime numbers (they are only divisible by 1 and themselves). So, the core of this problem lies in finding the prime factors of 84 and then using that information to determine the number of pieces. This involves a bit of number theory, but don't worry, we'll take it step by step. We'll first figure out those prime factors, and then we'll see how they help us solve the main question. Remember, understanding the problem clearly is half the battle won! We need to find those prime factors because they are the key to unlocking the answer. Once we have them, calculating the length of each piece becomes straightforward, and from there, finding the total number of pieces is just a simple division problem. So, let's roll up our sleeves and get factoring!

Finding the Prime Factors of 84

Alright, so our first mission is to find the prime factors of 84. There are a couple of ways we can do this, but let's use the method of prime factorization, where we break down 84 into its prime number components. Think of it like dismantling a Lego castle into its individual bricks.

We start by dividing 84 by the smallest prime number, which is 2. 84 divided by 2 is 42. So, we have 84 = 2 x 42. Now, we look at 42. Can we divide it by 2 again? Yep! 42 divided by 2 is 21. So, now we have 84 = 2 x 2 x 21. Great! We've got two 2s in our prime factorization so far. Next up is 21. Can we divide 21 by 2? Nope, it doesn't go in evenly. So, we move on to the next prime number, which is 3. Can we divide 21 by 3? Absolutely! 21 divided by 3 is 7. So, we now have 84 = 2 x 2 x 3 x 7. We're almost there!

Now we look at 7. Is 7 a prime number? You betcha! 7 is only divisible by 1 and itself, so it's a prime number. And guess what? We can't break it down any further. This means we've reached the end of our prime factorization journey. We've successfully broken down 84 into its prime factors. So, the prime factors of 84 are 2, 2, 3, and 7. In other words, 84 can be expressed as 2 x 2 x 3 x 7. This is a crucial step because, as the problem stated, the sum of these prime factors will give us the length of each piece of rope. Remember, finding these prime factors is like finding the secret ingredients to our recipe. Without them, we can't figure out the length of each piece, and we definitely can't figure out how many pieces we have in total. So, now that we've got our prime factors, let's move on to the next step: calculating the length of each piece.

Calculating the Length of Each Piece

Okay, guys, we've successfully found the prime factors of 84, which are 2, 2, 3, and 7. Now comes the fun part: figuring out the length of each piece of rope. Remember, the problem tells us that the length of each piece is equal to the sum of these prime factors. So, we just need to add them up! This is like putting the ingredients of our recipe together to see what we get.

So, let's do it: 2 + 2 + 3 + 7. What does that equal? If you add those numbers together, you'll get 14. That's right! The sum of the prime factors of 84 is 14. This means that each piece of rope is 14 centimeters long. See? We're making progress! We've gone from finding the prime factors to calculating the length of each piece. This is a significant step because now we have a crucial piece of information that will help us solve the main problem: how many pieces are there in total? Think of it this way: we now know the size of each slice of our pizza (the rope), and we know the total size of the pizza. All that's left is to figure out how many slices we have. And to do that, we're going to use some simple division. But before we jump into the division, let's just take a moment to appreciate how far we've come. We've tackled prime factorization, addition, and now we're on the verge of solving the entire problem. So, let's keep that momentum going and move on to the final calculation: finding the number of pieces.

Finding the Total Number of Pieces

Alright, we're in the home stretch now! We know the total length of the rope is 84 centimeters, and we know each piece is 14 centimeters long. The question we're trying to answer is: how many pieces do we have? This is a classic division problem, guys. Think of it like this: we have a big rope, and we're cutting it into smaller, equal-sized pieces. To find out how many pieces we get, we need to divide the total length of the rope by the length of each piece.

So, we're going to divide 84 by 14. What do we get? If you do the math, you'll find that 84 divided by 14 is 6. That's it! We've found our answer. The total number of pieces is 6. Hooray! We've successfully navigated through the prime factorization, the addition, and the division, and we've arrived at the solution. We started with a rope of 84 centimeters, figured out the length of each piece by using prime factors, and then calculated the total number of pieces. This is a fantastic example of how different mathematical concepts can come together to solve a single problem. It's like a puzzle where you need to put all the pieces together in the right way to see the whole picture. And in this case, the picture is that we have 6 pieces of rope. Now, before we celebrate too much, let's just quickly recap the steps we took to make sure we've got everything crystal clear. This will help solidify our understanding and make sure we can tackle similar problems in the future. So, let's rewind a bit and go over the journey we took to find the total number of pieces.

Solution

So, to recap, here’s how we solved the problem:

  1. Found the prime factors of 84: We broke down 84 into its prime number components: 2, 2, 3, and 7.
  2. Calculated the length of each piece: We added the prime factors together (2 + 2 + 3 + 7) to get 14 centimeters.
  3. Found the total number of pieces: We divided the total length of the rope (84 cm) by the length of each piece (14 cm) to get 6 pieces.

Therefore, the answer is A) 6.

Isn't it amazing how we can solve such problems by breaking them down into smaller, manageable steps? Remember, math isn't about memorizing formulas; it's about understanding the underlying concepts and applying them in a logical way. We started with a seemingly complex problem, but by systematically working through each step, we were able to arrive at the solution. This is a valuable skill, not just in math but in many areas of life. So, the next time you encounter a challenging problem, remember this example and try breaking it down into smaller parts. You might be surprised at how much easier it becomes! And that's a wrap, guys! We successfully solved the rope division problem. Great job!