Fabric Problem: Find The Initial Length!

by TextBrain Team 41 views

Let's dive into solving this fabric length problem! It's a classic example of working backward to find the initial amount. We know that after selling half the fabric and then half of what was left, there's 0.5 meters remaining. Our mission is to figure out how many meters there were to begin with. So, how do we unravel this puzzle? Let's break it down step by step and make it super easy to understand.

Understanding the Problem

Alright, guys, let's get real for a sec. Word problems can sometimes feel like a sneaky maze, right? This fabric conundrum is no different, but don't sweat it! The key here is to extract the vital information and translate it into something we can actually work with. In this case, the core details are the fractions of fabric sold and the final remaining amount. We gotta visualize the whole process, like picturing a bolt of fabric getting smaller with each sale. What makes this problem tick is the fact that we're dealing with fractions of a changing quantity. After the first sale, we're not selling half of the original amount, but half of the remaining amount. That's the tricky part we need to keep in mind. So, stay focused, and let's tackle this step-by-step!

Working Backwards

To solve this, we're going to reverse the operations. We know that 0.5 meters is what's left after selling half of the remainder. That means 0.5 meters represents the other half. So, before the second sale, there must have been 0.5 meters * 2 = 1 meter. Okay, we're making progress! Now, that 1 meter represents what's left after the first sale. Since half of the original amount was sold, the 1 meter represents the other half. Therefore, the original amount of fabric was 1 meter * 2 = 2 meters. Woo-hoo! We found our answer! So, to recap, we started with the final amount and worked our way back, doubling the amount at each step to undo the "selling half" operation. This approach is super useful for problems where you know the end result and need to find the starting point. Just remember to reverse the operations and you'll be golden!

Step-by-Step Solution

Let's formalize this a bit, shall we? Sometimes seeing the steps laid out clearly can really solidify our understanding. The main goal is to make sure you can solve these types of problems with confidence. Here's the step-by-step breakdown:

  1. Remaining Fabric: We know that 0.5 meters of fabric are left.
  2. Before the Second Sale: This 0.5 meters represents half of the fabric before the second sale. To find the amount before the second sale, we multiply by 2: 0. 5 m * 2 = 1 m
  3. Before the First Sale: The 1 meter represents half of the original amount of fabric. To find the original amount, we multiply by 2 again: 1 m * 2 = 2 m
  4. Answer: Therefore, the initial amount of fabric was 2 meters.

So, finally, we arrive at the answer: D) 2M. This step-by-step approach not only helps us find the solution but also provides a clear and organized way to understand the logic behind it. Remember, breaking down the problem into smaller, manageable steps is a fantastic strategy for tackling any math challenge. And don't be afraid to draw diagrams or use visual aids to help you visualize the problem – whatever works best for you! Just keep practicing, and you'll become a pro at solving these types of problems in no time!

Why Other Options are Wrong

It's super important to understand why the incorrect options are wrong. This can help prevent similar mistakes in the future. Sometimes this can give you a deeper understanding of the problem itself. Let's break down why options A, B, C, and E are not the correct answers:

  • A) 4M: If we started with 4 meters, selling half would leave 2 meters. Selling half of that would leave 1 meter, which is not 0.5 meters. So, 4 meters is too high.
  • B) 5M: If we started with 5 meters, selling half would leave 2.5 meters. Selling half of that would leave 1.25 meters, which is not 0.5 meters. This is also too high.
  • C) 3.5m: If we started with 3.5 meters, selling half would leave 1.75 meters. Selling half of that would leave 0.875 meters, which is not 0.5 meters. Again, too high.
  • E) 2.5M: If we started with 2.5 meters, selling half would leave 1.25 meters. Selling half of that would leave 0.625 meters, which is not 0.5 meters. And yep, this one is also too high.

By testing each of these options, we can clearly see that none of them lead to the final remaining amount of 0.5 meters. This process of elimination reinforces the correctness of our answer, D) 2M, and helps us understand why the other options are incorrect. This is a good strategy for test-taking: if you're not sure how to solve the problem directly, try plugging in the answer choices and see which one works!

Key Takeaways

Alright, guys, let's wrap this up with some key takeaways! These are the golden nuggets of wisdom that you can carry with you to conquer similar problems in the future:

  • Work Backwards: When you know the end result and need to find the starting point, reverse the operations. This is a powerful technique for solving many types of problems.
  • Visualize the Problem: Draw diagrams or use visual aids to help you understand the problem. This can make it easier to see the relationships between the different quantities.
  • Step-by-Step Approach: Break down the problem into smaller, manageable steps. This makes the problem less daunting and easier to solve.
  • Check Your Answer: After you've found a solution, plug it back into the original problem to make sure it works. This helps you catch any errors you may have made.
  • Understand the Logic: Don't just memorize the steps. Make sure you understand the logic behind each step so you can apply the same principles to other problems.

By keeping these key takeaways in mind, you'll be well-equipped to tackle any fabric-related (or any other type of) math challenge that comes your way. Keep practicing, stay curious, and never stop learning!

Practice Problems

To really nail down your understanding, let's try a couple of practice problems! These will help you solidify the concepts we've covered and build your confidence. Remember, practice makes perfect, so don't be afraid to give them a shot! Grab a pen and paper, and let's get started!

  1. Problem 1: John spent half of his money on a new video game. Then, he spent half of the remaining money on pizza. If he has $15 left, how much money did he start with?
  2. Problem 2: Sarah ate half of her cookies. Then, her friend ate half of the remaining cookies. If there are 3 cookies left, how many cookies did Sarah start with?

Try solving these problems on your own. Once you've given them a good attempt, you can check your answers below. The main goal is not just to get the right answer but to understand the process and the reasoning behind it.

Answers to Practice Problems

Alright, let's see how you did on those practice problems! Here are the solutions:

  1. Problem 1 Answer: John started with $60.
  2. Problem 2 Answer: Sarah started with 12 cookies.

How did you do? If you got them right, awesome! You're well on your way to mastering these types of problems. If you struggled a bit, don't worry! Just review the steps and key takeaways we discussed earlier and try again. Remember, the more you practice, the better you'll become!