How Many Light Fixtures Can Sandy Install In 41 Hours? A Math Problem

Hey guys! Have you ever wondered how much work you can get done in a day? Let's dive into a fun math problem about Sandy, who's a pro at installing light fixtures. This problem is all about figuring out how many fixtures Sandy can install in 41 hours, considering the time it takes her to assemble, hang, and wire each one. We're going to break down each step and use some basic math to solve it. This is a great way to see how fractions and time calculations come into play in real-life scenarios. So, let's put on our thinking caps and get started!

Okay, so here’s the deal. Sandy is super efficient at installing these light fixtures, but each one takes a bit of time. First, she needs $\frac{5}{6}$ hour to assemble the fixture. That's almost a full hour just to put it together! Then, she spends $\frac{1}{3}$ hour hanging it up – imagine climbing up and down the ladder! And finally, she needs another $\frac{1}{5}$ hour to wire it, making sure everything is safe and sound. So, the big question is: how many of these fixtures can Sandy install in 41 hours? This isn't just about adding fractions; it's about figuring out how many complete sets of these tasks she can squeeze into her workday. To tackle this, we'll first need to find the total time it takes for one complete fixture installation. We're talking about assembling, hanging, and wiring – the whole shebang! Once we know that, we can divide the total time Sandy has (41 hours) by the time per fixture to get our answer. Are you ready to crunch some numbers? Let's get to it!

Alright, let's figure out how much time Sandy spends on each fixture from start to finish. We know she spends $\frac{5}{6}$ hour assembling, $\frac{1}{3}$ hour hanging, and $\frac{1}{5}$ hour wiring. To find the total time, we need to add these fractions together. But here's the catch: we can only add fractions if they have the same denominator. So, we need to find the least common multiple (LCM) of 6, 3, and 5. The LCM is the smallest number that all three of these numbers divide into evenly. In this case, the LCM of 6, 3, and 5 is 30. Now, we'll convert each fraction to have a denominator of 30. For $\frac{5}{6}$, we multiply both the numerator and the denominator by 5, giving us $\frac{25}{30}$. For $\frac{1}{3}$, we multiply both the numerator and the denominator by 10, giving us $\frac{10}{30}$. And for $\frac{1}{5}$, we multiply both the numerator and the denominator by 6, giving us $\frac{6}{30}$. Now we can easily add these fractions: $\frac{25}{30} + \frac{10}{30} + \frac{6}{30}$. This equals $\frac{41}{30}$ hours. So, it takes Sandy $\frac{41}{30}$ hours to install one complete light fixture. We're one step closer to solving the problem!

Okay, we know it takes Sandy $\frac{41}{30}$ hours to install one fixture, and she has a total of 41 hours to work with. To figure out how many fixtures she can install, we need to divide the total time (41 hours) by the time it takes for one fixture ($\frac{41}{30}$ hours). Dividing by a fraction can be a little tricky, but it's easier than it looks! Remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{41}{30}$ is $\frac{30}{41}$. So, we're going to multiply 41 by $\frac{30}{41}$. This looks like 41 * $\frac{30}{41}$. We can think of 41 as $\frac{41}{1}$, so the equation becomes $\frac{41}{1}$ * $\frac{30}{41}$. Now, we can multiply the numerators (41 * 30) and the denominators (1 * 41). This gives us $\frac{1230}{41}$. But wait! We can simplify this. Notice that 41 appears in both the numerator and the denominator. We can cancel them out, leaving us with $\frac{30}{1}$, which is simply 30. So, Sandy can install 30 light fixtures in 41 hours. How cool is that? We've solved the problem!

To double-check our answer, let’s think about it logically. If Sandy installs one fixture in $\frac{41}{30}$ hours, and she has 41 hours, then we should expect the number of fixtures she can install to be close to the denominator of the fraction, which is 30. Our calculation shows that she can install 30 fixtures, which makes sense! We can also multiply the time per fixture ($\frac{41}{30}$ hours) by the number of fixtures (30) to see if it equals the total time available (41 hours). So, $\frac{41}{30}$ * 30 = 41. This confirms that our answer is correct! So, in conclusion, Sandy can install 30 light fixtures in 41 hours. We've successfully solved this problem by breaking it down into smaller, manageable steps. We added fractions to find the total time per fixture, and then we divided the total available time by the time per fixture to find the number of fixtures installed. Math is awesome, isn't it? And you guys nailed it!

Alright, let's recap what we've learned! We tackled a real-world math problem involving fractions and time. We figured out how many light fixtures Sandy could install in 41 hours, considering the time it took her to assemble, hang, and wire each one. We learned how to add fractions with different denominators by finding the least common multiple. This is a super useful skill, not just for math problems, but for everyday situations too! We also practiced dividing by a fraction, which is the same as multiplying by its reciprocal. This might sound complicated, but we broke it down step by step, and you guys got it! Finally, we verified our answer to make sure it made sense. This is a crucial step in problem-solving – always double-check your work! This problem shows us how math can be used to solve practical problems. Whether you're planning a project, managing your time, or just curious about how things work, math is your friend. So, keep practicing, keep exploring, and keep having fun with math!