Math Problem: Workers, Time, And Workload

Hey guys! Today we're diving into a classic math problem that really makes you think about how different factors interact when it comes to getting a job done. We're talking about mathematicas, specifically a word problem that involves workers, the time they take, and the amount of work they complete. You know, the kind of stuff that seems tricky at first but is super satisfying to figure out once you break it down. Let's tackle this: Una obra la pueden hacer 34 obreros en cierto tiempo. ¿cuantos obreros se necesitaran para hacer la quinta parte de la obra en 3/4 del tiempo anterior, trabajando la mitad de las horas diarias iniciales? That's a mouthful, right? In plain English, it asks how many workers we'll need to do a fifth of the job, in three-quarters of the original time, all while they're working only half the daily hours. Sounds like a puzzle, and it is! But don't sweat it, we're going to unravel this step by step.

Breaking Down the Variables

Alright, before we jump into calculations, let's get a handle on what we're dealing with. In problems like these, we've got a few key players: the number of workers (obreros), the time it takes (tiempo), and the amount of work (obra). The fundamental idea here is that the total amount of work done is directly proportional to the number of workers and the time they spend working. Think about it: more workers mean more hands on deck, so the job gets done faster. More time means more hours to chip away at the task, so again, more work gets done. So, we can express this relationship in a simple way: Work = Workers × Time × Efficiency. For this particular problem, the 'efficiency' part is linked to the 'hours worked daily'. Let's set up some initial conditions. We have 34 workers (let's call this $W_1$) who complete a whole obra (let's call this $O_1 = 1$) in a certain amount of time (let's call this $T_1$). We want to find the new number of workers ($W_2$) needed to complete a fifth of the obra ($O_2 = 1/5$) in three-quarters of the original time ($T_2 = 3/4 imes T_1$), with the condition that they work half the daily hours compared to the initial setup. This last part is crucial because it means their individual efficiency or output per hour might change, or rather, the total hours they contribute per day is reduced. So, the equation becomes: Total Work = (Number of Workers) × (Total Time Worked) × (Hours Worked Per Day). Keeping these variables clear is the first big step to solving this kind of word problem. It's all about identifying what's given and what needs to be found, and how they relate to each other.

Setting Up the Equations

Okay, math enthusiasts, let's get down to business with the actual equations. We know that the amount of work done is proportional to the number of workers and the time they put in. So, for the first scenario, we have: $O_1 = k imes W_1 imes T_1 imes H_1$, where $O_1$ is the total work (1 obra), $W_1$ is the number of workers (34), $T_1$ is the total time, and $H_1$ is the initial number of hours worked per day. Since we're dealing with a proportion and comparing two scenarios, we can often simplify by assuming $k$ (a constant of proportionality) is 1, or by setting up a ratio. Let's use the ratio approach, as it's often cleaner. We can say that the total 'work units' done in the first case is $W_1 imes T_1 imes H_1$. Now, for the second scenario, we want to do $O_2$ amount of work, with $W_2$ workers, in time $T_2$, and with $H_2$ hours per day. So, the work units for the second case would be $W_2 imes T_2 imes H_2$. The key insight is that the total amount of work ($O$) must be proportional to these work units. Therefore, we can set up a relationship like this: rac{O_1}{W_1 imes T_1 imes H_1} = rac{O_2}{W_2 imes T_2 imes H_2}. This equation is gold, guys! It allows us to relate all the variables. Let's plug in what we know from the problem. We know $O_1 = 1$ (one whole obra), $W_1 = 34$, and $T_1$ is our baseline time. For the second scenario, $O_2 = 1/5$ (one-fifth of the obra), $T_2 = 3/4 imes T_1$, and $H_2 = 1/2 imes H_1$ (half the initial daily hours). Our goal is to find $W_2$. So, we substitute these values into our ratio equation: rac{1}{34 imes T_1 imes H_1} = rac{1/5}{W_2 imes (3/4 imes T_1) imes (1/2 imes H_1)}. See how $T_1$ and $H_1$ appear on both sides? This is where the simplification happens, and why we often don't need to know the exact initial time or hours per day. It's all about the relationships!

Solving for the Unknown Workers

Now comes the fun part – crunching the numbers and finding $W_2$! We have our equation set up: rac{1}{34 imes T_1 imes H_1} = rac{1/5}{W_2 imes (3/4 imes T_1) imes (1/2 imes H_1)}. Let's simplify the denominator of the right side. $(3/4 imes T_1) imes (1/2 imes H_1) = (3/4 imes 1/2) imes (T_1 imes H_1) = 3/8 imes (T_1 imes H_1)$. So, our equation becomes: rac{1}{34 imes T_1 imes H_1} = rac{1/5}{W_2 imes (3/8) imes T_1 imes H_1}. Notice that $T_1 imes H_1$ is present in both denominators. We can cancel it out! This is a huge simplification. Our equation now looks like: rac{1}{34} = rac{1/5}{W_2 imes (3/8)}. Let's clean up the right side further: rac{1/5}{W_2 imes (3/8)} = rac{1}{5} imes rac{1}{W_2 imes (3/8)} = rac{1}{5 imes W_2 imes (3/8)}. So, rac{1}{34} = rac{1}{5 imes W_2 imes (3/8)}. Since the numerators are both 1, the denominators must be equal. $34 = 5 imes W_2 imes (3/8)$. Now, we just need to isolate $W_2$. Let's multiply the constants on the right side: $5 imes (3/8) = 15/8$. So, $34 = W_2 imes (15/8)$. To find $W_2$, we multiply both sides by the reciprocal of $15/8$, which is $8/15$. $W_2 = 34 imes (8/15)$. Let's calculate this: $34 imes 8 = 272$. So, $W_2 = 272 / 15$. Now, we perform the division: 272 rdiv 15 rapprox 18.13. Wait a minute! We can't have a fraction of a worker, right? This means we need to round up to ensure the work gets done. If we only had 18 workers, they wouldn't quite finish the job within the specified time and conditions. Therefore, we need 19 workers. This is a common occurrence in these types of problems – you often have to round up to the nearest whole number because you can't have partial workers.

The Impact of Reduced Hours

Let's pause and reflect on the impact of working half the hours daily. This is a critical part of the problem that significantly influences the outcome. Initially, we assume a certain number of hours ($H_1$) are worked per day by the original 34 workers. When the problem states that the new group of workers ($W_2$) works half the initial hours daily ($H_2 = 1/2 imes H_1$), it means each worker contributes less productive time per day. If you think about it, if someone is only working half the day, they can accomplish only half the amount of work they would in a full day, assuming their work rate per hour remains constant. This reduction in daily working hours directly impacts the total work output. It's like trying to fill a bucket with water, but now you're only allowed to use half the time to pour. To compensate for this reduced efficiency per worker per day, you'd logically need more workers to achieve the same overall output within the given timeframe. In our calculation, this reduction was captured by $H_2 = 1/2 imes H_1$. Notice how it factored into the denominator of our work equation: $W_2 imes T_2 imes H_2$. The smaller $H_2$ makes the denominator smaller, which, in a proportion where the work amount ($O_2$) is also smaller, still leads to a need for more workers. Let's look at the equation again: rac{O_1}{W_1 imes T_1 imes H_1} = rac{O_2}{W_2 imes T_2 imes H_2}. When $O_2$ is smaller (1/5 of the original) and $H_2$ is smaller (1/2 of original), these two factors work in opposite directions on $W_2$. A smaller $O_2$ would reduce $W_2$, while a smaller $H_2$ would increase $W_2$. The actual value of $W_2$ depends on the interplay of all these changes. In our specific case, the requirement to complete only 1/5 of the obra and do it in 3/4 of the time, combined with working half the hours, meant that we still needed a substantial number of workers, resulting in the need to round up to 19. If the daily hours hadn't been reduced, the number of workers required would have been different. Understanding how each variable modification affects the final answer is key to mastering these problems. It’s not just about plugging numbers in; it’s about understanding why the numbers change the way they do.

The Final Answer and Takeaways

So, after all that number-crunching and variable juggling, we've arrived at our answer: we need 19 workers. Remember, we got $18.13$ and had to round up because you can't employ a fraction of a person on a construction site, and you need enough workers to complete the job. This problem beautifully illustrates the inverse relationship between the number of workers and the time needed, given a fixed amount of work. More workers mean less time, and less time means more workers. However, it also introduces complexity with changes in the amount of work and the daily working hours. The fact that they had to do only one-fifth of the work suggested fewer workers might be needed, but the constraints of three-quarters of the time and especially half the daily hours significantly increased the requirement. The reduction in daily hours is a major factor forcing us to bring in more hands. This kind of problem is super common in aptitude tests and real-world project planning. Whether you're a student tackling homework or a project manager estimating resources, understanding these proportional relationships is invaluable. The core takeaway here is to meticulously identify all variables, set up the correct proportional equation, and then carefully solve for the unknown. Don't shy away from rounding up when dealing with indivisible units like people or machines. It ensures the task is actually accomplished within the given parameters. Keep practicing these, guys, and soon you'll be solving complex word problems like a pro!