Optimizing Work Efficiency for Painters P and Q: A Case Study
Painters P and Q have been working on several projects together, contributing to the success of each task. In this case study, we explore how their collaborative efforts on alternate days can optimize the time and resources required to complete a house painting project. Painters P and Q can paint a house in 16 and 12 days respectively. This article delves into the steps and calculations involved in determining how many days it will take for them to complete the work.
Introduction
The efficiency of painters in a collaborative environment can be significantly improved by optimizing their work cycles. This study focuses on the optimal way for Painters P and Q to work together, where P and Q work on alternate days to complete the painting job. By understanding their individual rates of work, we can calculate the total time needed for the project.
Calculating Work Efficiency
First, we need to determine how much work each painter can complete in a day. This is calculated based on their respective time to complete the job alone.
Painter P's Work Rate
Painter P can complete the job in 16 days, so in one day, painter P completes 1/16 of the work.
Painter Q's Work Rate
Similarly, painter Q can complete the job in 12 days, so in one day, painter Q completes 1/12 of the work.
Calculating Work Performed in Two Days
When working on alternate days, we need to find out how much work is completed in a cycle of two days.
First Day's Work
On the first day, P works, completing 1/16 of the work.
Second Day's Work
On the second day, Q works, completing 1/12 of the work.
Total Work in Two Days
The total work done in two days is the sum of the work done on each day, which requires finding a common denominator:
1/16 1/12 3/48 4/48 7/48
Calculating the Number of Cycles Required
To determine the total cycles needed to complete the work:
Since the total work required is 1, we need to find the number of 2-day cycles required:
7/48n 1 n > 48/7 ≈ 6.857
Therefore, it will take 7 cycles of 2 days to complete the work.
Total Days Worked
After 6 cycles (12 days), the work completed is:
6 * 7/48 42/48 7/8 1 - 7/8 1/8
Since Painters P and Q together do 7/48 of the work in each 2-day cycle, the remaining work after 12 days is 1/8.
Finalizing the Work
On the 13th day, Painter P works, completing 1/16 of the work. This leaves:
1/8 - 1/16 2/16 - 1/16 1/16
On the 14th day, Painter Q works, completing 1/12 of the work, which is more than enough to complete the remaining 1/16.
Thus, the total time required to complete the painting is 14 days.