Fabric Roll Math Problem Step By Step Solution
Hey guys! Ever get stumped by a math problem that seems to have too many steps? Let's break down one of those problems today. We're diving into a fabric roll scenario where we need to figure out how much fabric was used for different items. It sounds a bit like a real-life sewing project, doesn't it? So, grab your thinking caps, and let's get started!
Understanding the Problem
Before we even think about numbers, it's super important to understand what the problem is asking. In this case, we have a roll of fabric that's being used to make duvet covers, sheets, and pillowcases. The tricky part? The fabric usage is described in fractions and leftovers. We need to carefully untangle each step to find our answer.
Our main goal here is to determine how much fabric was initially in the roll. This means we need to work backward, step by step, to account for all the fabric used for each item. We'll start by looking at the very end of the problem and work our way back to the beginning. This approach helps us to keep track of the fractions and leftovers effectively. Remember, each step depends on the previous one, so accuracy is key! We want to make sure we get the right amount of fabric, just like you would if you were actually planning a sewing project. Now, let's really dig into the details and pull out the key information. This is where we start turning the word problem into a solvable equation.
Identifying the Key Information
Okay, let's break it down like a detective solving a case! We need to find all the clues hidden in the problem statement. These clues are the specific amounts of fabric used and the order in which they were used.
First, the problem tells us that duvet covers were made using 3/5 of the fabric after 6 meters were removed. This is a crucial point – we need to remember that 6 meters were taken off before the fraction was applied. Next up, sheets were made using 3/5 of the remaining fabric after another 4 meters were removed. Notice the pattern? We're dealing with fractions of remainders, not the total amount. Finally, after making pillowcases, which used 3/4 of the new remainder minus 3 meters, there's some fabric left. This leftover amount is our starting point for working backward. It's like the last piece of a puzzle that helps us fit everything else together. We have three key stages to consider: duvet covers, sheets, and pillowcases. Each stage involves using a fraction of the fabric and then subtracting a certain amount. This step-by-step process is what we need to reverse to find the original length of the fabric roll. Now that we've identified all the important pieces, let's start thinking about how we can turn this information into a mathematical equation.
Setting Up the Equations
Alright, time to put on our math hats! This is where we translate the word problem into a language that numbers can understand. We're going to use variables and equations to represent the different stages of fabric usage. It might seem a little tricky at first, but don't worry, we'll take it step by step.
Let's start by assigning a variable. We'll use 'x' to represent the total length of the fabric roll – that's what we're trying to find! Now, let's think about the pillowcases. The problem says that after making the pillowcases, there were 3 meters left over after using 3/4 of the remaining fabric. This means that the remaining fabric before making pillowcases can be represented as a value that, when we take away 3/4 of it and then subtract 3 meters, we get a certain amount. Let’s call the fabric remaining before making pillowcases 'P'. So, we know that the pillowcases used (3/4)P - 3 meters of fabric. This is the first piece of our puzzle. Next, we need to think about how this 'P' relates to the fabric used for the sheets. Remember, the sheets used 3/5 of the fabric remaining after 4 meters were removed. This means that 'P' represents the fabric remaining after making the sheets. We're slowly working our way backward, unravelling the problem one step at a time. Now, let's see how we can connect the duvet covers to the equation. The duvet covers used 3/5 of the fabric remaining after 6 meters were removed. This is the very first step in our fabric usage timeline, so it's the step furthest away from the leftover amount. By carefully defining these relationships using variables, we can start building a series of equations that will lead us to the answer. It's like creating a roadmap that shows us exactly how to get from the end of the problem back to the beginning. So, let's keep going and build on these equations until we have a complete picture of the fabric usage.
Working Backwards: Pillowcases
Let’s get practical, guys! We're going to start at the end of the problem and work our way backward, like tracing our steps back to the starting point. Remember, after the pillowcases were made, there was a certain amount of fabric left. We need to figure out how much fabric there was before the pillowcases were made.
The problem tells us that 3 meters were left after using 3/4 of the fabric and then removing 3 meters. This means that the remaining 1/4 of the fabric must be equal to the 3 meters that were left over. Think of it like slicing a pie – if one slice (1/4) is 3 meters, we can figure out the whole pie (the fabric before the pillowcases). So, if 1/4 of the fabric equals 3 meters, then the whole amount of fabric before the pillowcases were made is 3 meters multiplied by 4, which equals 12 meters. That's our first milestone! We now know that there were 12 meters of fabric before the pillowcases were made. This is a crucial piece of information because it helps us link the pillowcase stage to the previous stage: the sheets. By working backward like this, we can unravel the problem step by step. It's like following a trail of breadcrumbs, with each step leading us closer to the original amount of fabric. Now that we know the fabric amount before the pillowcases, we can move on to the next step and figure out how much fabric there was before the sheets were made.
Unraveling the Sheets Stage
Okay, we've conquered the pillowcase stage! Now it's time to rewind a bit further and figure out the fabric situation before the sheets were made. This is where things get a little more interesting, but don't worry, we've got this!
We know that there were 12 meters of fabric before the pillowcases. The problem tells us that the sheets were made using 3/5 of the remaining fabric after 4 meters were removed. This means that the 12 meters we just calculated represent the 2/5 of the fabric that was left after making the sheets and subtracting those 4 meters. Think of it like this: if 12 meters is 2/5 of something, we can figure out what the whole 'something' is. To do this, we first need to find out what 1/5 is. If 2/5 is 12 meters, then 1/5 is half of 12 meters, which is 6 meters. Now, if 1/5 is 6 meters, then the whole amount (5/5) is 6 meters multiplied by 5, which equals 30 meters. But hold on! We're not quite there yet. Remember those 4 meters that were removed before using the fabric for the sheets? We need to add those back in. So, we add 4 meters to the 30 meters, giving us a total of 34 meters. This means that there were 34 meters of fabric before the sheets were made. We're making great progress! We've successfully untangled the pillowcase and sheet stages. Now, we have one more stage to conquer: the duvet covers. Let's keep the momentum going and see what we can discover.
Tackling the Duvet Covers
Alright, team! We're on the home stretch now. We've navigated the pillowcases and the sheets, and now it's time to tackle the final piece of the puzzle: the duvet covers. This is where we'll uncover the original length of the fabric roll, so let's give it our full attention.
We know that there were 34 meters of fabric before the sheets were made. The problem tells us that the duvet covers were made using 3/5 of the fabric after 6 meters were removed. This means that the 34 meters we calculated represent the 2/5 of the fabric that was left after making the duvet covers and subtracting those 6 meters. Just like we did with the sheets, we need to work backward to find the original amount. If 34 meters is 2/5 of something, we can figure out what 1/5 is. To do this, we divide 34 meters by 2, which gives us 17 meters. So, 1/5 of the fabric is 17 meters. Now, if 1/5 is 17 meters, then the whole amount (5/5) is 17 meters multiplied by 5, which equals 85 meters. But we're not done yet! Remember those 6 meters that were removed before using the fabric for the duvet covers? We need to add those back in to find the original length. So, we add 6 meters to the 85 meters, giving us a grand total of 91 meters. Woohoo! We've cracked the code. This means that the original length of the fabric roll was 91 meters. We've successfully worked our way backward through all the stages, untangling each step to reveal the final answer. Now, let's take a moment to double-check our work and make sure everything adds up.
Checking Our Work
Awesome job, guys! We've arrived at a solution, but before we celebrate, it's always smart to double-check our work. Think of it as proofreading a really important essay – we want to make sure we haven't made any sneaky mistakes.
We found that the original length of the fabric roll was 91 meters. To check this, we can go through each step forward, applying the fractions and subtractions to see if we end up with the correct leftover amount. First, 6 meters were removed, leaving 91 - 6 = 85 meters. Then, 3/5 of the remaining fabric was used for duvet covers, which means 2/5 was left. So, 2/5 of 85 meters is (2/5) * 85 = 34 meters. Next, 4 meters were removed, leaving 34 - 4 = 30 meters. Then, 3/5 of the remaining fabric was used for sheets, which means 2/5 was left. So, 2/5 of 30 meters is (2/5) * 30 = 12 meters. Finally, 3 meters were removed before making pillowcases, and then 3/4 of the remaining fabric was used. This means that 1/4 was left, and we know that the leftover amount was 3 meters. Let's see if this matches up. Before the pillowcases, there were 12 meters. If we used 3/4 of it, that's (3/4) * 12 = 9 meters. So, 12 meters - 9 meters = 3 meters left, which is exactly what the problem stated! We did it! Our calculations check out, and we can confidently say that the original length of the fabric roll was indeed 91 meters. Double-checking our work is not just about finding mistakes; it's also about solidifying our understanding of the problem and the solution. By going through the steps forward, we reinforce the logic and make sure everything fits together perfectly.
The Final Answer
Drumroll, please! After all our hard work, we've finally arrived at the final answer. We untangled the fractions, subtracted the leftovers, and worked our way backward to reveal the original length of the fabric roll.
The original length of the fabric roll was 91 meters.
Give yourselves a pat on the back, guys! You tackled a challenging problem with fractions and remainders, and you emerged victorious. This type of problem-solving is a valuable skill, not just in math class, but in real life too. Whether you're planning a sewing project, managing your budget, or figuring out how much pizza to order for a party, the ability to break down a problem into smaller steps and work through it methodically is super helpful. So, the next time you encounter a tricky problem, remember the strategies we used today: break it down, identify the key information, work backward if necessary, and always double-check your work. You've got this!
Why This Matters
So, we solved a fabric problem – great! But you might be wondering, why does this matter? Why spend time wrestling with fractions and remainders? Well, the skills we used to solve this problem are actually incredibly useful in everyday life.
Think about it: how often do you need to divide something up, calculate leftovers, or figure out a starting amount based on what's remaining? Maybe you're splitting a bill with friends, figuring out how much paint you need for a project, or even calculating discounts at the store. All of these situations involve similar thinking processes to what we used today. By practicing these types of problems, we're sharpening our logical thinking, problem-solving abilities, and mathematical skills. These are skills that will benefit you in all sorts of areas, from school and work to personal finances and DIY projects. Plus, there's a certain satisfaction in conquering a challenging problem, isn't there? It's like leveling up in a game – you feel a sense of accomplishment and confidence that you can tackle whatever comes your way. So, keep practicing, keep challenging yourself, and remember that every problem you solve is a step towards becoming a more skilled and confident problem-solver. And who knows, maybe you'll even be able to impress your friends and family with your amazing math skills!
