How Many More Men Should Be Employed to Complete a Job 4 Days Earlier?

Problem Statement: If 21 men can do a job in 16 days, how many more men should be employed so that the same job is completed 4 days earlier?

Understanding the Problem

To solve this problem, we need to determine the number of additional men required to complete the job 4 days earlier. Initially, 21 men take 16 days to complete the job, and we need to find the number of men, n, such that the same job is done in 12 days.

Mathematical Interpretation

Let's denote the total amount of work as 1 job. If 21 men take 16 days to complete the job, the rate of work done by 21 men per day is 1/16 of the job.

Work Done by One Man Per Day

The work done by one man in one day is:

[ text{Work Done by One Man Per Day} frac{1}{21 times 16} frac{1}{336} ]

Additional Men Required

Let n be the number of additional men required. Then, the total number of men working for 12 days will be:

[ 21 n text{ men} ]

The work done by (21 n) men in 12 days should be equal to 1 job:

[ (21 n) times frac{1}{336} times 12 1 ]

Simplifying the equation:

[ (21 n) times frac{12}{336} 1 ]

[ frac{12}{336} times (21 n) 1 ]

[ frac{1}{28} times (21 n) 1 ]

[ 21 n 28 ]

[ n 7 ]

Conclusion

Therefore, 7 more men must be employed to ensure that the same job is completed 4 days earlier. This calculation is based on the assumption that the work done by each man is constant and that adding more men does not affect the work rate.

Discussion on Scalability and Productivity

It's important to note that adding more people to a project doesn't always lead to a proportionate increase in productivity. The amount of work done may not scale linearly with the number of employees, especially when:

New hires need time to become fully productive. Experienced employees may become less productive because of the need to train new employees. Miscommunication and chaos may arise from increased complexity.

Understanding these factors is crucial for effective project management and avoiding common misconceptions about the relationship between the number of employees and work completion times. As Brooks eloquently points out in The Mythical Man Month, the complexity of managing a project increases as the number of participants grows.