Work-Rate Problem: Calculating Construction Time For 120 Workers

Hey guys! Today, we're diving into a classic work-rate problem that might seem a bit daunting at first glance, but trust me, we'll break it down into easily digestible steps. This is the kind of problem you might encounter in math classes or even in real-world project management scenarios. So, let's put on our thinking caps and get started!

Understanding the Problem: The Foundation of Our Solution

The problem presents a scenario where we have a group of workers constructing a certain area of a building in a specific timeframe. The core of the problem lies in understanding the relationship between the number of workers, the hours they work each day, the total area constructed, and the number of days it takes. To truly grasp the problem, we need to identify what's given and what we're trying to find. We know that 80 workers, toiling for 8 hours each day, can build 480 square meters of a structure in 15 days. The big question looming over us is: How many days will it take 120 workers, putting in 10 hours each day, to construct 960 square meters? This is where the fun begins!

Identifying Key Variables and Their Relationships

Before we jump into calculations, let's pinpoint the key players in this mathematical drama. We have the number of workers, the hours they dedicate each day, the area they construct, and the time it takes them to complete the task. These variables aren't just floating around independently; they're all interconnected. The number of workers directly impacts the pace of construction – more workers generally mean faster progress. Similarly, the hours worked per day influence the overall output. The area to be constructed sets the magnitude of the task, and the number of days is the time we're trying to determine. Understanding how these variables interact is crucial for solving the problem. For instance, if we increase the number of workers, we'd expect the number of days required to decrease, assuming everything else remains constant. This kind of logical reasoning will guide us toward the correct solution.

Setting Up the Problem: The Proportionality Principle

Now that we've dissected the problem, it's time to set up a framework for solving it. The key here is the concept of proportionality. We need to recognize that the work done is directly proportional to the number of workers, the hours they work, and the number of days they work. On the flip side, the number of days is inversely proportional to the number of workers and the hours they put in each day, but directly proportional to the area to be constructed. This might sound like a mouthful, but it's a fundamental principle that will help us untangle the problem. We can express this relationship mathematically, which will allow us to calculate the unknown number of days. Think of it like this: if you double the number of workers, you'd expect to halve the time it takes to complete the same amount of work, assuming the working hours remain the same. This intuitive understanding of proportionality is our secret weapon in solving this problem. Next, we'll translate this understanding into a concrete mathematical equation.

The Mathematical Setup: Building the Equation

Alright, let's get down to the nitty-gritty of the equation. Remember how we talked about proportionality? We can express the relationship between workers, hours, area, and days in a neat mathematical form. The core idea is that the total work done is constant, regardless of how many workers are involved or how many hours they work. This constant work can be represented as a ratio between the area constructed and the product of workers, hours, and days. So, for the first scenario, we have 480 square meters constructed by 80 workers working 8 hours a day for 15 days. This gives us our first ratio. For the second scenario, we have 960 square meters to be constructed by 120 workers working 10 hours a day, and we're trying to find the number of days. This gives us our second ratio, which includes our unknown variable. Now, since the total work done in both scenarios is essentially the same (just a different scale), we can equate these two ratios. This is the key step in setting up our equation.

Constructing the Proportion: A Step-by-Step Guide

Let's break down how to build the proportion. On one side of the equation, we'll have the ratio for the first scenario: 480 / (80 * 8 * 15). This represents the area constructed divided by the product of workers, hours, and days. On the other side, we'll have the ratio for the second scenario: 960 / (120 * 10 * x), where 'x' is the number of days we're trying to find. Now, we simply equate these two ratios: 480 / (80 * 8 * 15) = 960 / (120 * 10 * x). This equation is the heart of our solution. It captures the proportional relationship between the variables and sets the stage for us to solve for 'x'. It might look a bit intimidating at first, but don't worry, we'll simplify it step by step. The next step is to actually solve this equation, which involves some algebraic manipulation. We'll isolate 'x' on one side of the equation, which will reveal the number of days required in the second scenario. So, let's roll up our sleeves and dive into the algebra!

Solving for 'x': Isolating the Unknown

Now comes the exciting part – solving for 'x'! We have the equation 480 / (80 * 8 * 15) = 960 / (120 * 10 * x). To isolate 'x', we'll use some basic algebraic principles. First, let's simplify both sides of the equation. On the left side, 80 * 8 * 15 equals 9600, so we have 480 / 9600. On the right side, 120 * 10 equals 1200, so we have 960 / (1200 * x). Now our equation looks like this: 480 / 9600 = 960 / (1200 * x). To get 'x' by itself, we can cross-multiply. This means multiplying 480 by (1200 * x) and setting it equal to 960 multiplied by 9600. This gives us 480 * 1200 * x = 960 * 9600. Now, let's simplify further. 480 * 1200 equals 576000, and 960 * 9600 equals 9216000. So, our equation is now 576000 * x = 9216000. Finally, to isolate 'x', we divide both sides of the equation by 576000: x = 9216000 / 576000. Calculating this division gives us x = 16. And there you have it! We've successfully solved for 'x', which represents the number of days required in the second scenario. The solution is 16 days. But let's not stop here; let's double-check our answer to make sure it makes sense in the context of the problem.

Verifying the Solution: Does It Make Sense?

Before we declare victory, it's crucial to verify our solution. We found that it would take 120 workers, working 10 hours a day, 16 days to construct 960 square meters. Does this answer logically fit the problem? Let's think it through. We increased both the number of workers and the hours they work each day compared to the first scenario. This should lead to a decrease in the number of days required, but we also doubled the area to be constructed, which would increase the number of days. So, we need to see if the effect of these changes balances out. In the first scenario, we had 80 workers * 8 hours/day * 15 days = 9600 worker-hours. In the second scenario, we have 120 workers * 10 hours/day * 16 days = 19200 worker-hours. We've doubled the worker-hours, and we've doubled the area, so it seems logical that the number of days would be in the same ballpark. Our answer of 16 days feels reasonable. To be even more certain, we could plug our answer back into the original proportion and see if both sides of the equation are equal. This would provide a definitive confirmation that our solution is correct. But for now, let's move on to discussing some alternative approaches to solving this type of problem.

Alternative Approaches: Exploring Different Paths

While we've successfully solved this problem using the proportionality method, it's always good to have other tools in our toolbox. There are alternative approaches we could have taken, which might be more intuitive for some people. One such approach is the unitary method. In this method, we first find the amount of work done by one worker in one hour, and then use this information to calculate the total time required for the second scenario. This involves a series of divisions and multiplications, but it breaks the problem down into smaller, more manageable steps. Another approach involves calculating the total work done in the first scenario in terms of worker-hours and then using this information to find the number of days required in the second scenario. This method highlights the relationship between total work, workers, hours, and days. The beauty of these alternative approaches is that they offer different perspectives on the problem and can help solidify our understanding of the underlying concepts. They also provide a way to double-check our answer, as arriving at the same solution using a different method increases our confidence in the result. So, let's briefly explore how one of these alternative methods, the unitary method, would work in this case.

The Unitary Method: A Different Perspective

Let's take a quick detour and see how we could tackle this problem using the unitary method. The core idea here is to break down the problem into smaller, more manageable chunks. First, we'll figure out how much area one worker can construct in one hour. In the first scenario, 80 workers working 8 hours a day for 15 days constructed 480 square meters. So, the total number of hours worked is 80 workers * 8 hours/day * 15 days = 9600 hours. This means that 9600 worker-hours resulted in 480 square meters of construction. To find the area constructed by one worker in one hour, we divide the total area by the total worker-hours: 480 square meters / 9600 worker-hours = 0.05 square meters per worker-hour. Now we know that one worker can construct 0.05 square meters in one hour. Next, we apply this knowledge to the second scenario. We have 120 workers working 10 hours a day, so the total worker-hours per day is 120 workers * 10 hours/day = 1200 worker-hours per day. To find out how many days it will take to construct 960 square meters, we first calculate the total worker-hours needed: 960 square meters / 0.05 square meters per worker-hour = 19200 worker-hours. Finally, we divide the total worker-hours needed by the worker-hours per day: 19200 worker-hours / 1200 worker-hours per day = 16 days. As you can see, we arrived at the same answer – 16 days – using a different approach. This reinforces our confidence in the solution and demonstrates the versatility of different problem-solving methods. Now, let's wrap things up with some key takeaways.

Key Takeaways: Mastering Work-Rate Problems

So, what have we learned on this mathematical adventure? First and foremost, we've conquered a work-rate problem by understanding the proportional relationships between workers, hours, area, and days. We successfully set up an equation, solved for the unknown variable, and even verified our solution to ensure it made sense in the real world. We also explored an alternative approach, the unitary method, which provided a different perspective on the problem and further solidified our understanding. But the key takeaway here is not just the solution to this specific problem; it's the process of problem-solving itself. By breaking down a complex problem into smaller, more manageable steps, we can tackle almost anything. Identifying key variables, understanding their relationships, setting up a mathematical framework, and verifying the solution are all crucial skills that extend far beyond the realm of mathematics. So, the next time you encounter a daunting problem, remember the lessons we've learned here. Embrace the challenge, break it down, and don't be afraid to explore different approaches. And remember, practice makes perfect! The more you work on these types of problems, the more comfortable and confident you'll become. Now, go forth and conquer those mathematical challenges!

Choosing the Correct Option: The Final Step

After all the calculations and verifications, we've confidently arrived at the answer: 16 days. Now, let's circle back to the options provided in the problem: a) 13, b) 14, c) 15, d) 16, and e) 17. It's clear that the correct option is d) 16. This final step is crucial in any problem-solving scenario. Always make sure you explicitly state your answer and select the corresponding option if one is provided. This leaves no room for ambiguity and ensures that you receive full credit for your hard work. So, congratulations! You've successfully navigated a complex work-rate problem and emerged victorious. Remember, the journey of problem-solving is just as important as the destination. Keep practicing, keep exploring, and keep challenging yourself. The world of mathematics is full of fascinating puzzles waiting to be solved!

Frequently Asked Questions (FAQ) on Work-Rate Problems

To further enhance your understanding of work-rate problems, let's address some frequently asked questions. These FAQs will provide additional insights and clarify any lingering doubts you might have.

Q1: What are the key factors that influence the time it takes to complete a task?

The time it takes to complete a task is primarily influenced by the amount of work to be done, the number of workers involved, and the rate at which each worker performs the work. This rate is often expressed in terms of hours worked per day or the amount of work completed per hour. Other factors, such as the efficiency of the workers and the resources available, can also play a role.

Q2: How do you identify a work-rate problem?

Work-rate problems typically involve scenarios where a task is completed by a group of individuals working at a certain rate. These problems often ask you to determine the time it takes to complete the task under different conditions, such as a change in the number of workers or the hours worked per day. Look for keywords like "workers," "hours," "days," "work done," and "time taken."

Q3: Can work-rate problems be solved using different methods?

Yes, as we've seen, work-rate problems can be solved using multiple methods, including the proportionality method and the unitary method. The best method to use depends on your personal preference and the specific details of the problem. It's beneficial to be familiar with different approaches to enhance your problem-solving skills.

Q4: What are some real-world applications of work-rate problems?

Work-rate problems have numerous real-world applications, particularly in project management, construction, and manufacturing. They can be used to estimate the time required to complete a project, allocate resources effectively, and optimize work schedules. Understanding work-rate principles is essential for efficient planning and execution in various industries.

Q5: How can I improve my skills in solving work-rate problems?

The best way to improve your skills in solving work-rate problems is through practice. Work through a variety of problems with different scenarios and complexities. Pay attention to the relationships between the variables, and try different problem-solving methods. Additionally, seek out resources such as textbooks, online tutorials, and practice quizzes to reinforce your understanding.

By addressing these frequently asked questions, we hope to have provided a more comprehensive understanding of work-rate problems and their applications. Remember, problem-solving is a skill that improves with practice, so keep challenging yourself and exploring new mathematical concepts.