Calculating Task Completion Time A Mathematical Approach

Have you ever wondered how changing the number of workers on a job affects the time it takes to finish? This is a classic problem in mathematics, and we're going to break it down step by step. Let's dive into a scenario where we figure out how long it takes a different number of workers to complete the same task.

The Problem: Two Workers, Twelve Hours

Task completion time is a common question in mathematics, especially when dealing with rates and proportions. Imagine this: Two workers are on a job, and they can complete the entire task in 12 hours. Now, what if we doubled the workforce? How would that change the amount of time it takes to finish the same job? This is the kind of problem we're tackling today. The question we need to answer is: If two workers take 12 hours to complete a task, how many hours will it take four workers, all working at the same efficiency, to complete the same task? We have a few options to choose from: A) 6 hours, B) 8 hours, C) 12 hours, or D) 24 hours. To solve this, we need to understand the relationship between the number of workers and the time it takes to complete the task.

When you increase the number of workers, the total time required to finish the job should decrease, assuming everyone works at the same pace. This is because more people are contributing to the effort, and the work can be divided more efficiently. Let's consider this intuitively before we jump into the math. If you have twice as many people working, it should take less time, but how much less? To figure this out, we need to establish the total amount of work that needs to be done, and then see how the increased workforce affects the time. This involves understanding the concept of work rate, which is the amount of work each worker can do in a certain amount of time. By calculating the total work and then dividing it by the new number of workers, we can find the new completion time. So, let's roll up our sleeves and get into the nitty-gritty of the calculations!

Breaking Down the Problem: Total Work

To really grasp this problem, we need to think about the concept of total work. Total work is the entire amount of effort needed to complete the task, regardless of how many people are working on it. We can think of this as a fixed quantity. To calculate total work, we often use the formula: Total Work = Number of Workers × Time Taken. In our scenario, we have two workers taking 12 hours. So, the total work can be calculated as follows: Total Work = 2 workers × 12 hours = 24 worker-hours. This means that the task requires 24 "worker-hours" of effort. It doesn't matter how many people are working; the total amount of work remains the same. Now that we know the total work, we can figure out how long it will take a different number of workers to complete the same job. This is a crucial step in solving the problem, because it gives us a baseline to compare the different scenarios. Understanding total work helps us see the inverse relationship between the number of workers and the time it takes to finish the job. More workers mean less time, and vice versa, but the total work stays constant. Let's move on to the next step, where we apply this knowledge to find out how long it takes four workers to complete the task.

Understanding the concept of 'worker-hours' is key here. It's a unit that represents the amount of effort one worker puts in over one hour. In our case, 24 worker-hours means that if one person were to do the job alone, it would take them 24 hours. Now, with this total work calculated, we can apply it to the situation with four workers. The total work of 24 worker-hours is a constant value, so we can use it to find the new time. This is where we start to see how the initial calculation of total work becomes super useful. It gives us a solid foundation to compare the two scenarios – two workers versus four workers – and helps us predict how much faster the job will get done with more hands on deck. By keeping the total work the same, we can directly see the impact of changing the workforce size. So, let's jump into the next section, where we actually use this 24 worker-hours figure to figure out the new completion time.

Calculating the New Time: Four Workers

Alright, now we get to the fun part: figuring out how long it takes four workers! We've already established that the total work required is 24 worker-hours. We can use the same formula we used before, but this time, we're solving for time. The formula is: Time = Total Work / Number of Workers. We know the total work is 24 worker-hours, and we now have four workers. Plugging these values into the formula, we get: Time = 24 worker-hours / 4 workers = 6 hours. So, four workers would take 6 hours to complete the same task. Looking back at our options, A) 6 hours is the correct answer. This makes sense intuitively as well. If you double the number of workers, you would expect the time to be halved, assuming everyone works at the same rate. Let's think about why this works mathematically. When you have more workers, the task can be divided into smaller portions, and each worker can focus on their part. This increased efficiency leads to a reduction in the overall time needed to complete the task.

Understanding this inverse relationship between the number of workers and the time it takes to complete a task is crucial in many real-world scenarios, from project management to everyday tasks. For example, if you're planning a group project, knowing how many people you need to finish it within a certain timeframe can be really helpful. This kind of problem also highlights the importance of teamwork and how efficiently distributing work can save time and effort. So, let's recap what we've done so far. We started with the initial scenario of two workers taking 12 hours. We calculated the total work required as 24 worker-hours. Then, we applied this total work to a new scenario with four workers, and we found that it would take them 6 hours to complete the same task. This demonstrates how understanding basic mathematical principles can help us solve practical problems. Now that we have our answer, let's take a moment to discuss why the other options are incorrect, reinforcing our understanding of the solution.

Why Other Options Are Incorrect

It's always a good idea to understand why the wrong answers are wrong. This helps solidify our understanding of the correct solution. In our problem, we had options B) 8 hours, C) 12 hours, and D) 24 hours. Let's break down why these aren't the right answers.

Option B) 8 hours: This might seem plausible at first glance, but it doesn't reflect the inverse relationship between the number of workers and the time taken. If we doubled the number of workers, we should expect the time to decrease by a proportional amount, not just a small amount. If it took two workers 12 hours, it logically should take four workers less than 12 hours, so 8 hours is too high.

Option C) 12 hours: This is the time it took two workers to complete the task. If we had more workers, it should definitely take less time. Keeping the time the same despite doubling the workforce doesn't make sense. This option doesn't account for the increased efficiency of having more people working on the task. It assumes that the time would remain constant, which contradicts the principles of work rate and efficiency.

Option D) 24 hours: This is the opposite of what we'd expect. This answer suggests that doubling the workforce would double the time taken, which is incorrect. Increasing the number of workers should decrease the time required, not increase it. This option misunderstands the core concept of the problem, which is the inverse relationship between workers and time.

By understanding why these options are incorrect, we reinforce our understanding of why A) 6 hours is the correct answer. It's not just about finding the right number; it's about understanding the logic and principles behind the problem. When we analyze the wrong answers, we can see the common mistakes people might make, and this helps us avoid those pitfalls in the future. Understanding the process of elimination is also a valuable skill in problem-solving. Now that we've thoroughly dissected this problem, let's wrap up with a final overview of our solution and the key takeaways.

Final Answer and Key Takeaways

So, to recap, if two workers take 12 hours to complete a task, four workers, working at the same efficiency, would take 6 hours to complete the same task. The correct answer is A) 6 hours.

The key takeaway from this problem is the understanding of the inverse relationship between the number of workers and the time it takes to complete a task. When you increase the number of workers, the time taken to complete the task decreases, assuming everyone works at the same rate. This is because the total work remains constant, and more workers can divide the work more efficiently.

We solved this problem by first calculating the total work required, which we found to be 24 worker-hours. Then, we used this total work to calculate the time it would take four workers to complete the task, which was 6 hours. We also discussed why the other options were incorrect, reinforcing our understanding of the solution.

This type of problem is a great example of how basic mathematical principles can be applied to real-world situations. Whether you're planning a project, managing a team, or just trying to estimate how long a task will take, understanding these concepts can be incredibly valuable. So, the next time you're faced with a similar situation, remember the principles we've discussed, and you'll be well-equipped to solve it! And that's a wrap, guys! Hope you found this breakdown helpful and insightful. Keep practicing, and you'll become a pro at these types of problems in no time!