Collaboration Efficiency: Analyzing Bricklayer Completion Time
Introduction
Collaboration among workers is a common practice in construction and various other industries. This article delves into a practical problem and its resolution, focusing on the individual and combined work rates of two bricklayers to determine the total time taken to complete a wall.
Problem Statement and Initial Determination of Work Rates
The first bricklayer can build a wall in 3 days, while the second bricklayer takes 4 days to complete the same task. The first bricklayer starts working alone but later joins the second bricklayer to finish the wall. The challenge is to determine the total time it takes to build the wall considering both scenarios.
Calculating Individual Work Rates
Let's begin by calculating the individual work rates of the bricklayers.
First Bricklayer's Work Rate: Since the first bricklayer can build the wall in 3 days, his work rate is: Rate of first bricklayer 1 wall / 3 days 1/3 walls per day Second Bricklayer's Work Rate: The second bricklayer can build the wall in 4 days, so his work rate is: Rate of second bricklayer 1 wall / 4 days 1/4 walls per dayWork Done by the First Bricklayer in One Day
In one day, the first bricklayer can complete:
Work done by first bricklayer in 1 day 1/3 wallsDetermining the Remaining Work
After the first bricklayer works alone for one day, the amount of work left to do is:
Remaining work 1 - 1/3 2/3 wallsCombined Work Rate of Both Bricklayers
The combined work rate of both bricklayers is:
Combined rate 1/3 1/4 4/12 3/12 7/12 walls per dayTime to Finish the Remaining Work Together
Let t be the time in days it takes for both bricklayers to finish the remaining 2/3 of the wall:
7/12 t 2/3Multiplying both sides by 12 to clear the fraction:
7t 8
Dividing both sides by 7:
t 8/7 days
Total Time to Build the Wall
The total time taken to build the wall is the 1 day the first bricklayer worked alone plus the time they worked together:
Total time 1 8/7 15/7 daysConverting this to a mixed number:
15/7 2 1/7 days
Approximately, this is 2 days and about 3 hours.
Alternative Method for Verification
Using the alternative method where the total work is considered as 12 units:
Work done by first bricklayer in a day: 12/3 4 units Work done by second bricklayer in a day: 12/4 3 units Work done by both bricklayers in a day when working together: 4 3 7 units Total work to complete: 12 units Time taken together to complete the remaining work: 7x 8 x 8/7 daysApproximately 1 day and 3 hours.
Conclusion
The total time taken to build the wall, considering both methods, is effectively the solution to the problem. This example demonstrates the importance of understanding individual and combined work rates in problem-solving, which is crucial in construction and project management.