How Long Does It Take for Pat to Mow 12 Lawns?

Introduction

Pat is a local lawn mowing service provider who has faced a unique challenge: he can only mow 1/4 of a lawn in 20 minutes. The question arises: how long will it take him to mow 12 lawns?

Mathematical Analysis

Standard Calculation:

Pat can mow 1/4 of a lawn in 20 minutes.

To find the time for a full lawn, we can use the proportion: 1/4 lawn in 20 minutes 1 lawn in X minutes.

Calculating the time for one full lawn:

( X frac{20 , text{minutes} times 4}{1} 80 , text{minutes} )

To mow 12 lawns, we multiply 80 minutes by 12:

( 12 times 80 , text{minutes} 960 , text{minutes} )

Converting 960 minutes to hours:

( frac{960 , text{minutes}}{60} 16 , text{hours} )

Based on the calculations, it takes Pat 16 hours to mow 12 full lawns.

Practical Considerations:

Assumption of Identical Lawns: It is assumed that all 12 lawns are of identical shape and area, and that the grass growth is uniform. This assumption simplifies the calculation.

Labor Considerations: Realistically, Pat is human, and factors such as fatigue, breaks, and efficiency might vary. However, for theoretical purposes, the above calculation holds.

Real-World Application

While the theoretical answer is 16 hours, this calculation does not factor in real-world elements such as Pat's fatigue, breaks, and potential robot-like consistency. A realistic approach would need to consider these elements:

Example Scenario:

Assuming Pat is a human and takes regular 15-minute breaks:

1. Breaks: Every 40 minutes of work, Pat takes a 15-minute break.

2. Work Cycle: Pat works for 40 minutes, takes a 15-minute break, and then works for 40 more minutes.

3. Time Calculation: For 12 hours of work (1200 minutes), Pat will work in 30-minute segments, including breaks. This would require 7 full cycles, totaling 105 minutes of work time and 105 minutes of break time (excluding the last 45 minutes of the day).

4. Final Calculation: Pat can work for 2100 minutes in 12-hour blocks, plus an additional 20 minutes on the next day. Therefore, it will take him 48 days to complete 12 lawns, with the last 20 minutes full work on the final day.

Conclusion

From a theoretical standpoint, it takes Pat 16 hours to mow 12 lawns. However, in a more practical setting, considering real-world factors, it could take him 48 days with regular breaks, plus an additional 20 minutes of work on the next day.