Workforce Problem: Calculate The Extra Days To Finish A Job

Hey guys! Let's dive into a classic workforce problem. It's the kind of question that might pop up in a math class or even in real-life project management scenarios. So, buckle up, and let’s break it down! We're going to explore how to calculate the extra days needed to complete a task when the workforce changes mid-project. It's like a puzzle, and we're here to solve it together!

Understanding the Initial Work Rate

Okay, so the main idea here is to figure out how much work gets done each day. Let's start with the basics. We know that 6 workers can complete a job in 24 days. That's our starting point. Think of it like this: the job is a whole pie, and we need to figure out how many slices each worker takes each day. To do this, we need to determine the total work done and then break it down into daily contributions. The key concept here is the inverse relationship between the number of workers and the time it takes to complete a job. More workers mean less time, and vice versa. This is a fundamental principle in many real-world situations, from construction projects to software development.

First, let's find the total work done. We can think of the total work as the product of the number of workers and the number of days they work. So, in this case, we have 6 workers working for 24 days. This gives us a total of 6 * 24 = 144 worker-days. What does this 144 worker-days mean? It's a unit of measurement that represents the total effort required to complete the job. It's like saying the job requires 144 units of work. Each worker contributes one unit of work per day. Now, let's figure out the work rate per day. If 6 workers complete the job in 24 days, then each day, they complete 1/24 of the job. So, the combined work rate of the 6 workers is 1/24 of the job per day. This is crucial because it sets the baseline for our calculations. We know how much work is supposed to be done each day, and this will help us compare the progress when the workforce changes. The concept of work rate is essential in understanding how efficiently a task is being completed. It allows us to measure productivity and make informed decisions about resource allocation.

Calculating Work Done Before Reinforcements Arrive

Now, let's say that after 10 days, two more workers join the team. That's a game-changer, right? We need to figure out how much of the job was completed in those first 10 days before the extra hands showed up. This is like figuring out how many slices of the pie were eaten before more people came to the table. We know that the initial team of 6 workers was plugging away at the job. They were working at their steady pace, completing a certain amount of work each day. To calculate the work done, we simply multiply the number of days they worked by their daily work rate. This gives us a clear picture of the progress made before the workforce change.

Remember, the 6 workers complete 1/24 of the job each day. So, in 10 days, they would have completed 10 * (1/24) = 10/24 of the job. We can simplify this fraction to 5/12. This means that after 10 days, 5/12 of the job is done. Think of it as more than a third of the work is already completed. This is a significant milestone, and it's important to keep track of this progress. Now, we need to figure out how much work is left. If 5/12 of the job is done, then the remaining work is 1 - 5/12 = 7/12 of the job. This is the portion of the pie that still needs to be eaten. Knowing the remaining work is crucial because it will help us determine how much faster the job can be completed with the additional workers. This step is all about assessing the situation and understanding what's left to be done.

Adjusting for the New Workforce

Alright, the reinforcements have arrived! We now have 8 workers ready to tackle the remaining 7/12 of the job. This is where things get interesting. We need to figure out how this change in workforce affects the time it takes to complete the rest of the project. It's like having extra hands to help finish the task, but we need to calculate exactly how much faster things will go. The key here is to determine the new combined work rate of the team. With more workers, the job should get done quicker, but we need to quantify that improvement. This involves understanding how the work rate changes with the increased number of workers.

First, let's find the new work rate. If 6 workers complete 1/24 of the job per day, then one worker completes (1/24) / 6 = 1/144 of the job per day. Now, with 8 workers, the combined work rate is 8 * (1/144) = 8/144, which simplifies to 1/18 of the job per day. So, the new team is completing 1/18 of the job each day. This is faster than the original rate of 1/24, which makes sense since we have more workers. Now, let's calculate the time it takes to complete the remaining 7/12 of the job. We can use the formula: time = work / rate. In this case, time = (7/12) / (1/18). To divide fractions, we multiply by the reciprocal, so time = (7/12) * 18 = 10.5 days. This means it will take the new team 10.5 days to finish the job. This is a crucial piece of information because it tells us how long the remaining work will take with the increased workforce.

Calculating the Extra Days

Okay, we're in the home stretch now! We know the job was initially supposed to take 24 days. We've figured out that the first 6 workers worked for 10 days, and then the 8 workers finished the job in 10.5 days. So, the total time taken to complete the job is 10 + 10.5 = 20.5 days. But the big question is: how many days extra did it take compared to the original estimate? This is the final piece of the puzzle. We need to compare the actual time taken with the planned time to find the difference. This will give us the answer to how much longer the job took due to the workforce change.

To find the extra days, we subtract the actual time taken from the initially estimated time. So, extra days = 20.5 - 24 = -3.5 days. Wait a minute! A negative number? That means the job was actually completed faster than initially planned. This is a great outcome! The addition of the extra workers helped speed things up. The job was finished 3.5 days ahead of schedule. This highlights the importance of adjusting plans and resources based on changing circumstances. Sometimes, adding more people to a project can lead to significant time savings. This final calculation ties everything together and gives us a clear understanding of the impact of the workforce change on the project timeline.

Final Thoughts

So there you have it! We've walked through how to calculate the extra days (or in this case, fewer days!) needed to finish a job when the workforce changes. These kinds of problems might seem tricky at first, but breaking them down step-by-step makes them much more manageable. The key is to focus on understanding the work rates, calculating the work done, and adjusting for the changes in the team. Remember, math problems like this aren't just about numbers; they're about real-world scenarios and problem-solving. And who knows, maybe you'll use these skills in your own projects someday! Keep practicing, and you'll become a pro at these types of calculations in no time. You got this!