Optimizing Workforce Allocation for Rapid Job Completion

Successfully completing a job within the shortest possible time frame requires efficient workforce allocation. This article will walk you through a specific scenario where a team of individuals, including a man, a woman, and a boy, can complete a job in varying time frames. The objective is to determine the optimal number of boys needed to assist a man and a woman in completing the job in one-fourth of a day.

Introduction to the Problem

The scenario states that a man, a woman, and a boy can complete a job in 3, 4, and 12 days respectively. This problem necessitates understanding their individual and collective work rates, and determining the number of boys required to speed up the completion of the job to one-fourth of a day. This requires an in-depth exploration of work rates and algebraic manipulation.

Understanding the Work Rates

To begin with, let's define the work rates for each individual based on the given information:

The man: 1 job in 3 days -> 1/3 of the job per day. The woman: 1 job in 4 days -> 1/4 of the job per day. The boy: 1 job in 12 days -> 1/12 of the job per day.

Combined Work Rate of a Man and a Woman

The combined work rate of a man and a woman can be calculated as:

1/3 1/4 7/12

This means that together, a man and a woman can complete 7/12 of the job per day.

Determining the Number of Boys Required

Let x represent the number of boys needed to assist the man and the woman to complete the job in one-fourth of a day. The total work rate, including the boys, can be expressed as:

7/12 x/12 7/12(1 x/7) 1/4

Simplifying this equation, we get:

7 x 48

Solving for x, we get:

x 41

Therefore, 41 boys are required to assist 1 man and 1 woman to complete the job in one-fourth of a day.

Conclusion and Practical Application

The optimal workforce allocation is crucial for efficient job completion, especially in critical scenarios where time is of the essence. By understanding the individual work rates and their combined effect, one can determine the necessary number of additional workers. In this case, it was found that 41 boys, when combined with the efforts of 1 man and 1 woman, can complete the job in one-fourth of a day.

For employers and project managers, this method provides a practical approach to optimizing labor resources. By analyzing each worker's contribution, one can allocate personnel more effectively, leading to increased productivity and timely project completion.