Thread Reduction Problem Solving A Step By Step Guide

Hey guys! Let's dive into a math problem that involves some thread, a spool, and a bunch of students. This is the kind of stuff that might seem tricky at first, but once we break it down, it's totally manageable. We’ll go through each step together, making sure it’s crystal clear. So, grab your thinking caps, and let's get started!

The Problem: Thread Reduction

Our main task here is to figure out how much thread is removed from a spool when seven students each cut off a piece for their school project. The problem states that each student cuts $5 \frac{3}{4}$ inches of thread. To solve this, we need to determine the total amount of thread cut and represent that change. Let’s break it down step by step to make sure we understand exactly what’s going on. This problem revolves around understanding thread reduction in a practical, real-world scenario. We’re not just dealing with numbers; we’re looking at how these numbers translate into a physical quantity of thread being removed. The initial part of the problem gives us two key pieces of information: the number of students involved and the length of thread each student cuts. Knowing this, our primary goal is to combine these pieces of information to find the total length of thread that has been cut. This involves more than just simple arithmetic; it’s about visualizing the situation and understanding the operation we need to perform. So, let’s take a closer look at these key elements and see how they fit together.

Breaking Down the Thread Cutting Scenario

First, we need to really understand the scenario. We have seven students, and each one is cutting a piece of thread. The length of each piece is $5 \frac{3}{4}$ inches. The key here is to realize that we are dealing with repeated subtraction from the spool of thread. Each time a student cuts a piece, the total amount of thread on the spool decreases. This decrease is what we need to calculate. Think of it like this: if one student cuts $5 \frac{3}{4}$ inches, and then another student cuts the same amount, we're essentially adding these lengths together to find the total reduction. This understanding is crucial because it guides us to the correct mathematical operation: multiplication. We're not just dealing with whole numbers here, we've got a mixed fraction to contend with, $5 \frac{3}{4}$ inches. Mixed fractions can seem a bit daunting, but they’re just a combination of a whole number and a fraction. To make our calculations easier, we're going to convert this mixed fraction into an improper fraction. An improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This conversion will simplify the multiplication process and make sure we get an accurate final answer. Let's go ahead and convert that mixed fraction into its improper form.

Converting Mixed Fractions to Improper Fractions

Alright, let's talk about converting that mixed fraction, $5 \frac{3}{4}$, into an improper fraction. This is a super important step because it makes the math way easier. Remember, a mixed fraction has a whole number part and a fractional part. To convert it, we need to combine these into a single fraction. Here’s how we do it: First, we multiply the whole number (5) by the denominator of the fraction (4). So, 5 times 4 equals 20. Next, we add the numerator of the fraction (3) to the result we just got. So, 20 plus 3 equals 23. This new number, 23, becomes our new numerator. The denominator stays the same, which is 4. So, $5 \frac{3}{4}$ converted to an improper fraction is $\frac{23}{4}$. Now, why did we do this? Well, multiplying fractions is much simpler when they're in improper form. We can easily multiply the numerators and the denominators straight across. If we tried to multiply the mixed fraction directly, it would be much more complicated. By converting, we’ve set ourselves up for a smoother calculation. This conversion is a foundational skill in math, and it’s something you’ll use again and again, especially when you’re dealing with fractions in more complex problems. So, make sure you’re comfortable with this process. Now that we’ve got our improper fraction, we’re ready to move on to the next step: calculating the total thread cut by all the students.

Calculating Total Thread Reduction

Now that we've successfully converted our mixed fraction into an improper fraction, $\frac{23}{4}$, we can proceed with calculating the total thread reduction. Remember, we have seven students, and each of them is cutting $\frac{23}{4}$ inches of thread. To find the total amount of thread cut, we need to multiply the length of thread each student cuts by the number of students. Mathematically, this means we need to calculate 7 multiplied by $\frac{23}{4}$. Multiplying a whole number by a fraction is straightforward. We can think of the whole number 7 as a fraction itself, specifically $\frac{7}{1}$. So, our problem now looks like this: $\frac{7}{1} \times \frac{23}{4}$. To multiply fractions, we simply multiply the numerators together and the denominators together. The numerator will be 7 * 23, and the denominator will be 1 * 4. Let's take a closer look at how this multiplication works and what the resulting fraction represents. The total reduction in thread will give us a clear picture of how much thread has been removed from the spool. This is a critical step in solving the problem, as it provides the numerical answer we’re looking for. Once we have this total, we can then express it in the correct form and interpret it in the context of the original question. So, let’s dive into the multiplication and see what we get.

Performing the Multiplication

Okay, let's get into the nitty-gritty of the multiplication. We're multiplying $\frac{7}{1}$ by $\frac{23}{4}$. Remember, to multiply fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers) separately. So, we have 7 multiplied by 23 for the new numerator, and 1 multiplied by 4 for the new denominator. Let’s start with the numerator. 7 times 23 is 161. If you're not super quick with your multiplication facts, you can break it down: 7 times 20 is 140, and 7 times 3 is 21. Add those together, and you get 161. For the denominator, 1 times 4 is simply 4. So, when we multiply $\frac{7}{1}$ by $\frac{23}{4}$, we get $\frac{161}{4}$. This fraction, $\frac{161}{4}$, represents the total amount of thread cut by all seven students, measured in inches. But, this is an improper fraction, which means the numerator is larger than the denominator. While $\frac{161}{4}$ is a perfectly valid answer, it's not the most intuitive way to understand the quantity. We usually prefer to express such fractions as mixed numbers, which give us a better sense of the whole units involved. So, our next step is to convert this improper fraction back into a mixed number. This will help us see the total thread reduction in a more relatable way. Let's move on to that conversion now!

Converting Improper Fractions Back to Mixed Numbers

Alright, we've got the improper fraction $\frac{161}{4}$, and now we need to turn it back into a mixed number. This might sound like a complicated process, but it’s actually pretty straightforward once you get the hang of it. The basic idea is to figure out how many whole times the denominator (4) goes into the numerator (161). This will give us the whole number part of our mixed number. The remainder will become the numerator of the fractional part, and the denominator stays the same. So, let’s think about how many times 4 goes into 161. You might want to do a little long division here, or if you’re comfortable with your multiplication facts, you might be able to do it in your head. 4 goes into 16 four times (4 x 4 = 16), so 4 goes into 160 forty times (4 x 40 = 160). That leaves us with 1 left over. So, 4 goes into 161 a total of 40 times with a remainder of 1. This means our whole number part is 40, our new numerator is 1, and our denominator remains 4. Putting it all together, $\frac{161}{4}$ is equal to $40 \frac{1}{4}$. Now, let’s think about what this means in the context of our problem. $40 \frac{1}{4}$ inches is the total amount of thread that was cut by the seven students. We’re getting closer to our final answer, but there’s one more crucial step: considering the sign of this change.

Determining the Sign: Reduction Implies Negative Change

Okay, we’ve figured out that the total amount of thread cut is $40 \frac{1}{4}$ inches. Great job! But here’s a super important point: we need to think about what this means in terms of the change in the amount of thread on the spool. The problem asks for the number that represents the change. When thread is cut from the spool, the amount of thread decreases. A decrease means we’re dealing with a negative change. Think of it like this: if you have a certain amount of money and you spend some, the change in your money is negative because you have less than you started with. It’s the same with the thread. The spool started with a certain amount, and now it has less. So, the change in the amount of thread is negative. This is a critical concept in math and in real-life situations. We often use positive numbers to represent increases or additions, and negative numbers to represent decreases or subtractions. In our case, because the thread is being removed from the spool, we need to represent this change with a negative sign. So, the final answer isn’t just $40 \frac{1}{4}$ inches, it’s $-40 \frac{1}{4}$ inches. The negative sign tells us that this is a reduction, not an increase. Now, let's put it all together and see which of the answer choices matches our solution.

Final Answer

Alright, guys, we've done all the hard work, and now it's time to nail the final answer! We figured out that the total change in the amount of thread on the spool is $-40 \frac{1}{4}$ inches. The negative sign is super important because it tells us that the amount of thread decreased. Now, let's look at the answer choices: A. $-40 \frac{1}{4}$ inches B. $-35 \frac{3}{4}$ inches C. $-29 \frac{3}{4}$ inches The correct answer is A. $-40 \frac{1}{4}$ inches. We matched our calculated value perfectly! It's so satisfying when all the steps come together and you arrive at the right answer, isn't it? This problem was a great example of how math can be used to solve real-world scenarios. We took a word problem, broke it down into smaller, manageable steps, and used our math skills to find the solution. We converted a mixed fraction to an improper fraction, multiplied fractions, converted an improper fraction back to a mixed number, and even thought about the sign of the change. That’s a lot of math in one problem! But, by taking it one step at a time, we made it through. So, give yourself a pat on the back for tackling this problem with me. You're building some serious math skills, and that's something to be proud of. Keep practicing, keep asking questions, and keep challenging yourself. You've got this!

Final Answer: The final answer is $\boxed{A. -40 \frac{1}{4} inches}$