Decoding Nina's Stitching Speed Finding The Equation For Dresses And Hours
Hey guys! Ever wondered how math can help us understand everyday tasks like sewing? Let's dive into a super interesting problem about Nina, who's a total pro at stitching dresses. We're going to break down her stitching speed using math and figure out the equation that perfectly describes her skills. So, grab your thinking caps, and let's get started!
Understanding the Problem: Nina's Stitching Prowess
Our main goal here is to figure out the equation that shows how many dresses Nina can stitch in a certain amount of time. We know that Nina can stitch 2/3 of a dress in 4 hours. This is our key piece of information, and we need to use it to find the relationship between the number of dresses (d) and the number of hours (h). To solve this, we need to understand the concept of proportional relationships, which is a cornerstone of many mathematical and real-world applications. Proportional relationships describe situations where two quantities vary directly with each other. In simpler terms, if one quantity increases, the other quantity increases proportionally, and if one decreases, the other decreases proportionally. The key here is that the ratio between the two quantities remains constant. Now, before we get into the nitty-gritty, let's make sure we're all on the same page. The problem tells us that Nina can complete a fraction of a dress in a given number of hours, and we're tasked with finding an equation that accurately represents this relationship. This equation will allow us to predict how many dresses Nina can make in any given number of hours, assuming she maintains a consistent stitching speed. To begin, we need to identify the constant of proportionality, which is the unchanging ratio between the two quantities in a proportional relationship. In our case, this constant represents Nina's stitching rate—how much of a dress she can complete per hour. Finding this constant is crucial because it will be the key to building our equation. We'll use the information provided in the problem to calculate this constant and then use it to form the equation that represents the relationship between the number of dresses stitched and the time spent stitching. Remember, the goal is to find an equation that not only fits the given information but also makes logical sense in the context of the problem. For example, if we double the number of hours Nina spends stitching, we should expect her to complete double the number of dresses. This is the essence of a proportional relationship, and our equation should reflect this principle. As we work through the problem, we'll explore different approaches to find the constant of proportionality and construct the equation. We'll also discuss how to verify our answer to ensure it accurately represents Nina's stitching speed. So, let's keep our eyes on the prize – finding the equation that unlocks the secret to Nina's stitching prowess!
Breaking Down the Options: Which Equation Fits?
Now, let's look at the answer options. We have five equations to consider, and only one will perfectly describe Nina's stitching speed. Here's a quick rundown:
- A. d = 6h
- B. 3d = 4h
- C. d = (1/6)h
- D. d = 4h
Each of these equations represents a different relationship between d (the number of dresses) and h (the number of hours). Our job is to figure out which one lines up with the information we have about Nina. The key to figuring this out lies in understanding what each part of the equation means. In a proportional relationship, the equation typically takes the form y = kx, where y and x are the two quantities that are proportional to each other, and k is the constant of proportionality. In our case, d (the number of dresses) can be considered y, and h (the number of hours) can be considered x. The constant k represents Nina's stitching rate – how much of a dress she can complete in one hour. So, we're essentially looking for the equation that correctly calculates how many dresses Nina can complete based on the number of hours she works. To approach this, we can use the information we have – Nina stitches 2/3 of a dress in 4 hours – and plug these values into each equation to see if it holds true. For example, in option A, d = 6h, if we substitute h = 4, we get d = 6 * 4 = 24. This means that according to this equation, Nina would stitch 24 dresses in 4 hours, which doesn't match our given information. We'll repeat this process for each equation, substituting h = 4 and checking if the resulting d value is 2/3. This method of substitution is a powerful way to verify if an equation accurately represents a given situation. It allows us to test the equation with known values and see if the equation produces the expected result. Alternatively, we can also manipulate the given information to find Nina's stitching rate per hour and then compare that rate to the constant of proportionality in each equation. This approach involves dividing the amount of dress stitched (2/3) by the time taken (4 hours) to find the fraction of a dress stitched per hour. The equation that matches this rate is our correct answer. Let's get into the details of each equation and see which one truly captures Nina's stitching magic.
Solving the Puzzle: Finding the Right Equation
Okay, let's roll up our sleeves and get to the heart of the matter: finding the correct equation. We know Nina stitches 2/3 of a dress in 4 hours. To find the equation that represents this relationship, we need to determine her stitching rate per hour. This is a crucial step because it will give us the constant of proportionality, which is the key to unlocking the right answer. To find the stitching rate, we can divide the amount of dress stitched (2/3) by the time taken (4 hours). This gives us: (2/3) / 4 = 2/3 * (1/4) = 2/12 = 1/6. So, Nina stitches 1/6 of a dress per hour. Now that we know Nina's stitching rate, we can write the equation that represents the relationship between the number of dresses (d) and the number of hours (h). The equation will be in the form d = kh, where k is the constant of proportionality (Nina's stitching rate). Since Nina stitches 1/6 of a dress per hour, our equation becomes: d = (1/6)h. This equation tells us that the number of dresses Nina stitches is equal to 1/6 of the number of hours she spends stitching. Now, let's compare this equation to the answer options provided. We can see that option C, d = (1/6)h, matches the equation we derived. This suggests that option C is the correct answer. To confirm our answer, we can plug in the given values (h = 4) into the equation and see if we get the correct number of dresses (d = 2/3). Substituting h = 4 into d = (1/6)h, we get: d = (1/6) * 4 = 4/6 = 2/3. This confirms that our equation d = (1/6)h accurately represents Nina's stitching speed. But what about the other options? Let's quickly analyze why they are incorrect. Option A, d = 6h, suggests that Nina stitches 6 dresses per hour, which is much faster than her actual rate. Option B, 3d = 4h, can be rearranged to d = (4/3)h, which means Nina stitches 4/3 of a dress per hour, also incorrect. Option D, d = 4h, implies that Nina stitches 4 dresses per hour, which is far from her actual speed. Therefore, by carefully calculating Nina's stitching rate and comparing it to the given equations, we have confidently identified option C, d = (1/6)h, as the correct answer. This equation perfectly captures the proportional relationship between the number of dresses Nina stitches and the time she spends stitching. So, we've not only solved the problem but also deepened our understanding of proportional relationships in real-world scenarios.
The Verdict: Option C is the Champion!
Drumroll, please! After carefully analyzing each equation and comparing it to Nina's stitching speed, the winning equation is C. d = (1/6)h. This equation perfectly captures the proportional relationship between the number of dresses Nina stitches (d) and the number of hours she spends stitching (h). We arrived at this answer by first calculating Nina's stitching rate per hour, which we found to be 1/6 of a dress. This means that for every hour Nina works, she completes 1/6 of a dress. We then used this rate to construct the equation d = (1/6)h, which accurately reflects this relationship. We also verified our answer by plugging in the given values (h = 4) into the equation and confirming that it yields the correct number of dresses (d = 2/3). This process of verification is crucial in problem-solving, as it ensures that our solution is not only mathematically correct but also logically consistent with the given information. Now, let's take a moment to appreciate why the other options didn't make the cut. Option A, d = 6h, would mean Nina stitches a whopping 6 dresses every hour, which is way faster than her actual pace. Option B, 3d = 4h, translates to d = (4/3)h, suggesting Nina completes 4/3 of a dress per hour, again, not the right rate. And option D, d = 4h, would have Nina making 4 dresses an hour, which is also inaccurate. These incorrect options highlight the importance of carefully calculating the constant of proportionality and ensuring that the equation aligns with the given context. The equation d = (1/6)h not only solves the problem but also provides a clear and concise representation of Nina's stitching ability. It allows us to predict how many dresses she can complete in any given number of hours, assuming she maintains her consistent stitching speed. This is the power of mathematical equations – they can capture real-world relationships and allow us to make predictions and solve problems efficiently. So, hats off to option C for being the champion equation that unlocks the secret to Nina's stitching prowess!
Key Takeaways: Math in Action!
So, what have we learned from this awesome dress-stitching adventure? Firstly, we've seen how proportional relationships work in a real-world scenario. These relationships are all about how two things change together at a constant rate. In Nina's case, the number of dresses she stitches is directly proportional to the number of hours she works. This means that if she doubles her working hours, she'll double the number of dresses she completes. Understanding this concept is super helpful because proportional relationships pop up everywhere – from cooking recipes to calculating distances on a map. Secondly, we've mastered the art of finding the constant of proportionality. This magical number tells us the rate at which one quantity changes with respect to another. In our problem, the constant of proportionality is Nina's stitching rate, which is 1/6 of a dress per hour. Finding this constant is crucial because it allows us to build the equation that represents the proportional relationship. We found the constant by dividing the amount of dress stitched (2/3) by the time taken (4 hours). This simple calculation is a powerful tool for solving a wide range of problems involving proportions. Thirdly, we've learned how to construct and verify equations. We started with the basic form d = kh and plugged in the constant of proportionality to get d = (1/6)h. Then, we verified our equation by substituting the given values (h = 4) and checking if it yielded the correct result (d = 2/3). This process of verification is a crucial step in problem-solving, as it ensures that our answer is accurate and reliable. By going through these steps, we've not only solved the problem but also developed valuable problem-solving skills that can be applied to various situations. So, the next time you encounter a problem involving proportional relationships, remember Nina and her amazing stitching skills. Think about how the quantities are related, find the constant of proportionality, build the equation, and verify your answer. You'll be a math whiz in no time!
In conclusion, we successfully navigated through the problem of finding the equation that represents Nina's stitching speed. By understanding proportional relationships, calculating the constant of proportionality, and carefully analyzing the answer options, we confidently identified C. d = (1/6)h as the correct equation. This problem serves as a great example of how math can be used to model and understand real-world situations. So, keep those mathematical gears turning, and who knows, maybe you'll be the one to stitch together the next big mathematical breakthrough!
