Efficiency and Work Rate Calculations: A Comprehensive Analysis of Two People Mowing a Lawn Together
In this article, we delve into a common problem often found in high school mathematics, where two individuals, X and Y, are tasked with mowing a lawn. We explore the underlying concepts of work rate and efficiency, and provide detailed step-by-step solutions to illustrate the process.
The Problem and Initial Considerations
The problem at hand is straightforward: it takes person X two hours to mow a lawn, while an equally capable person, Y, can mow the same lawn in four hours. Our goal is to calculate how long it will take if they work together. However, the efficiency of Y is half that of X.
Understanding Work Rates and Efficiencies
To approach this problem, we begin by calculating the work rates of both individuals. The work rate is a measure of the amount of work each person can accomplish in a specific time. Here, the work is mowing one lawn.
Calculating Individual Work Rates
We define the work rate as the number of lawns mowed per unit of time. Given:
- Person X mows one lawn in 2 hours.
Rate of X 1 lawn / 2 hours 0.5 lawns per hour.
- Person Y mows one lawn in 4 hours.
Rate of Y 1 lawn / 4 hours 0.25 lawns per hour.
Combining Work Rates
When X and Y work together, their combined work rate is the sum of their individual work rates.
Combined rate Rate of X Rate of Y 0.5 0.25 0.75 lawns per hour.
Calculating the Time to Mow One Lawn Together
The total work required is one lawn. Using the combined rate, we can calculate the time it takes for X and Y to mow the lawn together.
Time required 1 lawn / 0.75 lawns per hour 4/3 hours.
Converting this time into minutes:
(4/3 hours) * 60 minutes per hour 80 minutes.
Short and Long Answer Explanations
The short answer to the problem is that it will take X and Y 80 minutes to mow the lawn together.
The long answer involves a deeper understanding that, although it takes X only two hours (120 minutes) to mow the lawn and Y four hours (240 minutes), their combined effort still faces a practical limitation—the availability of a single lawnmower. This constraint means that X and Y would have to share the lawnmower, reducing the effective contribution of each when they work together.
If the work is divided into 4 units, X would complete 2 units in one hour, and Y would complete 1 unit in one hour. Together, they complete 3 units in one hour, leaving 1 unit remaining. The remaining 1 unit is completed by X in 20 minutes, yielding a total time of 80 minutes.
Alternatively, it's helpful to consider the total work rate in terms of the smallest common unit of time, where X mows 1/120th of the lawn in a minute and Y mows 1/240th of the lawn in a minute. Combined, they mow 1/120 1/240 1/80th of the lawn per minute.
The time taken for X and Y to mow one lawn together is therefore 1 ÷ (1/80) 80 minutes.
Conclusion
Working together, X and Y can mow the lawn in 80 minutes, demonstrating the importance of understanding work rates and efficiencies in collaborative efforts. This problem not only tests mathematical skills but also highlights real-world constraints.
Harnessing the wisdom provided by these calculations can be applied to numerous scenarios, from mutual projects in school to household tasks. The key takeaway is that, while individual contributions add up, practical constraints, such as limited resources, must also be considered.