Determining Work Completion Time with Different Numbers of Workers

When facing problems related to work completion and the number of workers involved, it can be challenging to calculate the exact duration. However, understanding the concept of man-days and worker efficiency can help solve such problems precisely. This article explores the relationship between the number of workers and the time taken to complete a job, providing examples and explanations to clarify the concept.

Problem Statement

Consider the scenario where 50 workers are required to complete a certain job in 30 days. Now, if the same job is to be completed by 40 workers, how many days will it take?

Understanding Man-Days and Worker Efficiency

Man-days refer to the total amount of work required to complete a task. A man-day is the amount of work that a single worker can accomplish in one day. If more workers are involved, the task is completed faster, provided each worker is equally efficient and there are no diminishing returns on adding more workers to the task.

Solved Example

Scenario: 50 workers complete a job in 30 days.

Question: How many days will it take for 40 workers to complete the same job?

Step-by-Step Solution:

The total work is 4500 man-days (50 workers x 30 days).

For 40 workers:

[text{Time taken} frac{text{Total Work (man-days)}}{text{Number of Workers}} frac{4500}{40} 112.5 , text{days}]

Mathematical Explanation

Alternatively, understanding the relationship between the number of workers and the time taken involves recognizing that the work required is constant. The work done is directly proportional to the number of workers and inversely proportional to the time taken.

Mathematical Representation:

[text{Time (days)} frac{text{Number of Workers}}{text{Original Number of Workers}} times text{Original Time (days)}]

Application:

[text{Time taken} frac{50}{40} times 30 37.5 , text{days}]

Common Misconceptions

Some might argue that the time required would decrease with an increase in the number of workers, which is true under ideal conditions. However, the problem must consider practical constraints such as space, resources, and the diminishing efficiency as more workers overlap.

Four More Examples Explained

Example 1: 40 workers take the same amount of time as 50 workers (30 days) to complete the job.

[text{Time taken} 30 , text{days}]

Example 2: 22.5 days (40 workers, assuming 7.5 days reduction for 10 more workers and a 20% increase in efficiency).

Example 3: Time taken remains the same (30 days) as the work does not scale linearly with the number of workers.

Example 4: Time taken is calculated as follows:

[text{Worker-days for 20 workers} 45 times 20 900 , text{worker-days}]

[text{Time taken by 30 workers} frac{900}{30} 30 , text{days}]

Example 5: Time taken calculated as:

[text{40 workers} times 30 , text{days} 1200 , text{worker-days}]

[text{Time taken to complete the work} frac{1200}{50} 24 , text{days}]

Example 6: Similar to Example 4, the work completed in 4500 man-days is divided by 30 workers:

[text{Time taken by 30 workers} frac{4500}{30} 150 , text{days}]

Conclusion: Understanding the balance between the number of workers and the time taken to complete a job is crucial for project management and resource allocation in industries.