Understanding Work and Time Relationships in Resource Allocation
Introduction
When it comes to resource allocation, especially in the context of labor, understanding how changes in the number of workers impact the completion time of a project is crucial. This article will delve into the relationship between the number of workers and the time they take to complete a job, using a specific example where 20 women complete a job in 18 days and asking how many days it would take 15 women to complete the same work.
The Mathematical Foundation
The core concept in this problem is the worker-days method. Worker-days represent the total amount of work that can be done in one day by one worker. In this context, the total amount of work is expressed in worker-days. This method allows us to compare the productivity of different numbers of workers and different time periods.
Work Done by 20 Women in 18 Days
A common scenario involves 20 women completing a job in 18 days. Each day, these 20 women collectively contribute to completing the job. Hence, the total worker-days required to complete the job is calculated as:
To find the total worker-days:
Total Worker-Days Number of Workers × Number of Days
Total Worker-Days 20 women × 18 days 360 worker-days
Calculating the Time for 15 Women
Now, if we have 15 women instead of 20, to find out how many days they would need to complete the same amount of work, we use the total worker-days calculated above. The equation to solve for the number of days, denoted as D, is:
Worker-Days for 15 Women Number of Women × Number of Days
360 worker-days 15 women × D days
Solving for D:
D 360 / 15 24 days
Multiple Methods to Solve the Problem
Several methods can be used to solve this problem:
Direct Calculation: Knowing that one woman would take 360 days to complete the job (20 workers × 18 days 360), dividing 360 days by 15 women gives us the answer of 24 days. Equation Setup: Setting up the equation 20 × 18 15 × D and solving for D also gives us the solution. Logic-based Approach: Considering the total work remains constant and that 15 women will be working at a speed that is 20/15 4/3 times slower than 20 women, the time taken will be 3/4 of the time taken by 20 women, i.e., 18 days × 3/4 24 days.Conclusion
In summary, it takes 24 days for 15 women to complete the same work that 20 women can complete in 18 days. This example clearly illustrates the principles of resource allocation and the direct relationship between the number of workers and the time required to complete a given task.