Harvesting Wheat A Work Rate Problem And Solution

Introduction

In this article, we will delve into a classic work rate problem involving a group of men and women harvesting wheat in a village. This type of problem often appears in national exams and tests a candidate's ability to understand and apply concepts of work rate, proportionality, and combined effort. The scenario presented involves a workforce of 15 men and 10 women initially tasked with harvesting 20 hectares of wheat in 40 days. However, after 10 days, the dynamics change as five couples leave the workforce. The central question we aim to answer is: how many days will the project be delayed due to this reduction in manpower, considering the difference in work rate between men and women?

To tackle this problem effectively, we'll break it down into smaller, manageable steps. First, we'll establish the initial work rate of the group, taking into account that the work rate of a man is equivalent to that of two women. This conversion is crucial as it allows us to express the entire workforce in terms of a single unit, making calculations easier. Next, we'll determine the amount of work completed in the first 10 days before the workforce reduction. This step helps us understand how much work remains and how the reduced workforce will impact the remaining time required to complete the project. Finally, we'll calculate the new work rate after the five couples leave and use this information to estimate the additional days required to finish the job. By carefully analyzing each stage of the problem, we'll arrive at the solution and gain a deeper understanding of the principles governing work rate problems.

Understanding work rate problems is not just about finding the correct answer; it's about developing logical thinking and problem-solving skills. These skills are valuable in various real-life scenarios, from project management to resource allocation. Therefore, as we go through the solution, we'll also highlight the underlying concepts and strategies that can be applied to similar problems. So, let's embark on this journey of problem-solving and uncover the intricacies of this harvesting wheat problem.

Problem Statement

Our problem statement is as follows: In a village, 15 men and 10 women can harvest 20 hectares of wheat in 40 days. After 10 days of work, 5 couples (5 men and 5 women) leave. Knowing that the work rate of one man is equivalent to that of two women, by how many days will the completion of the work be delayed?

This problem presents several key pieces of information that we need to consider. First, we have the initial workforce composition: 15 men and 10 women. Second, we know the total work to be done: harvesting 20 hectares of wheat. Third, we are given the initial timeframe for completing the work: 40 days. Fourth, a significant event occurs after 10 days: 5 couples leave the workforce, reducing the manpower. Fifth, we have a crucial piece of information about the relative work rates of men and women: one man's work is equivalent to that of two women. This conversion factor will be essential in standardizing our calculations.

The core of the problem lies in determining the impact of the workforce reduction on the project's timeline. We need to figure out how much work was completed in the first 10 days and how much remains. Then, we need to calculate the new work rate of the reduced workforce and estimate how long it will take them to complete the remaining work. Finally, we can compare this new completion time with the original schedule to determine the delay.

To solve this problem effectively, we need to carefully analyze each piece of information and use it to build a step-by-step solution. We'll start by establishing a common unit of work based on the man-to-woman work rate ratio. This will allow us to express the entire workforce in terms of a single unit, simplifying our calculations. Then, we'll calculate the work done in the first 10 days and the remaining work. Finally, we'll determine the new work rate and the time required to complete the remaining work. By following this structured approach, we can systematically solve the problem and arrive at the correct answer.

Solution

To solve this work rate problem, we will follow a step-by-step approach, carefully considering each piece of information provided. The key to solving this problem lies in converting the workforce into a common unit based on the work rate ratio between men and women.

Step 1: Convert the workforce to a common unit

Since one man's work is equivalent to that of two women, we can express the entire workforce in terms of women. This conversion will simplify our calculations and allow us to treat the workforce as a single unit.

• 15 men are equivalent to 15 * 2 = 30 women.
• Therefore, the initial workforce of 15 men and 10 women is equivalent to 30 + 10 = 40 women.

This means that the initial workforce has a work capacity equivalent to 40 women working together. This conversion is crucial because it allows us to compare the work rate of the initial workforce with the reduced workforce later on.

Step 2: Calculate the total work done in woman-days

We know that 40 women can harvest 20 hectares of wheat in 40 days. We can use this information to calculate the total work done in woman-days. Woman-days represent the amount of work done by one woman in one day. It's a useful unit for measuring the total work required for the project.

• Total work = (Number of women) * (Number of days)
• Total work = 40 women * 40 days = 1600 woman-days

This calculation tells us that the total work required to harvest 20 hectares of wheat is equivalent to 1600 woman-days. This value will serve as our benchmark for measuring the progress of the work and the impact of the workforce reduction.

Step 3: Determine the work done in the first 10 days

The initial workforce of 40 women worked for 10 days before the reduction. We need to calculate the amount of work they completed during this period. This will help us determine how much work remains to be done.

• Work done in 10 days = (Number of women) * (Number of days)
• Work done in 10 days = 40 women * 10 days = 400 woman-days

So, in the first 10 days, the workforce completed 400 woman-days of work. This means that a significant portion of the work was already completed before the workforce reduction occurred.

Step 4: Calculate the remaining work

Now that we know the total work and the work done in the first 10 days, we can calculate the remaining work. This is the amount of work that needs to be completed by the reduced workforce.

• Remaining work = Total work - Work done in 10 days
• Remaining work = 1600 woman-days - 400 woman-days = 1200 woman-days

Therefore, 1200 woman-days of work remain to be completed after the first 10 days. This is the workload that the reduced workforce will need to handle.

Step 5: Determine the new workforce after the reduction

After 10 days, 5 couples (5 men and 5 women) left the workforce. We need to calculate the equivalent number of women in the reduced workforce.

• Men remaining = 15 men - 5 men = 10 men
• Women remaining = 10 women - 5 women = 5 women
• Convert the remaining men to women: 10 men * 2 = 20 women
• Total equivalent women in the reduced workforce = 20 women + 5 women = 25 women

So, the reduced workforce is equivalent to 25 women. This is a significant reduction from the initial workforce of 40 women, and it will undoubtedly impact the time required to complete the remaining work.

Step 6: Calculate the time required to complete the remaining work with the reduced workforce

Now that we know the remaining work and the size of the reduced workforce, we can calculate the time required to complete the remaining work.

• Time = (Remaining work) / (Number of women)
• Time = 1200 woman-days / 25 women = 48 days

This calculation tells us that the reduced workforce will take 48 days to complete the remaining work. This is a longer time than initially planned due to the reduction in manpower.

Step 7: Calculate the delay

Finally, we can calculate the delay by comparing the new completion time with the original schedule. The original schedule was for 40 days, and the workforce had already worked for 10 days, leaving 30 days remaining. However, with the reduced workforce, it will take 48 days to complete the remaining work. The delay is the difference between these two times.

• Original time remaining = 40 days - 10 days = 30 days
• Delay = New time to complete remaining work - Original time remaining
• Delay = 48 days - 30 days = 18 days

Therefore, the completion of the work will be delayed by 18 days due to the reduction in the workforce.

Final Answer

The final answer is that the work will be delayed by 18 days due to the departure of 5 couples after the first 10 days. This delay arises because the remaining workforce has a reduced capacity, and it takes them longer to complete the remaining work.

Conclusion

In this detailed walkthrough, we have successfully solved a work rate problem involving a scenario where a group of men and women are tasked with harvesting wheat, and a portion of the workforce leaves mid-project. By carefully breaking down the problem into smaller, manageable steps, we were able to determine the impact of the workforce reduction on the project's timeline.

The key to solving this problem was to convert the workforce into a common unit based on the work rate ratio between men and women. This allowed us to express the entire workforce in terms of a single unit, simplifying our calculations. We then calculated the total work, the work done in the first 10 days, and the remaining work. Next, we determined the new workforce size after the reduction and calculated the time required to complete the remaining work. Finally, we compared this new completion time with the original schedule to determine the delay.

This problem highlights the importance of understanding work rate concepts and how changes in manpower can impact project timelines. The principles and strategies used in solving this problem can be applied to various real-world scenarios, such as project management, resource allocation, and scheduling.

Furthermore, this exercise underscores the value of a systematic approach to problem-solving. By breaking down a complex problem into smaller, more manageable steps, we can make the solution process more transparent and less daunting. Each step builds upon the previous one, leading us to the final answer in a logical and coherent manner.

In conclusion, solving work rate problems is not just about finding the correct answer; it's about developing critical thinking and problem-solving skills. These skills are essential in various aspects of life, both personal and professional. Therefore, mastering the concepts and techniques presented in this article will undoubtedly be beneficial in your academic and professional endeavors.