Solving Work Efficiency Problems: A Detailed Analysis of Son's Work Hours
Work efficiency problems often involve determining the individual work rates and total time required to complete tasks. This article presents several scenarios where a man completes a job in a certain time, and together with his son, they complete the same job in a shorter time. The problem is to determine how long it would take the son to complete the same job alone. Here, we explore different methods and solutions to these problems, ensuring a clear and comprehensive understanding of the underlying principles.
Problem Statement Overview
A man can complete a piece of work in 60 hours. If he works with his son and they complete the same work in 40 hours, how many hours will the son take if he worked alone?
Solution Method 1: Direct Calculation of Son's Work Rate
Step 1: Determine the Man's Work Rate
The man can complete the work in 60 hours. Therefore, his work rate is:
Work rate of man 1/60 per hour
Step 2: Determine the Combined Work Rate
Together, the man and his son can complete the work in 40 hours. Therefore, their combined work rate is:
Combined work rate 1/40 per hour
Step 3: Determine the Son's Work Rate
To find the son's work rate, subtract the man's work rate from the combined work rate:
Son's work rate 1/40 - 1/60
(3–2)/120
1/120 per hour
Therefore, the son takes:
120 hours to complete the job alone
Solution Method 2: Using the Difference in Work Rates
Step 1: Determine the Workshop Rate Difference
If the man and son together complete the work in 40 hours, their combined work rate is:
1/40 per hour
The man alone can complete the work in 60 hours, so his work rate is:
1/60 per hour
Step 2: Determine the Son's Work Rate
The son's work rate is the difference between the combined rate and the man's individual rate:
Son's work rate 1/40 - 1/60
(5–4)/240
1/240 per hour
Therefore, the son takes:
240 hours to complete the job alone
Solution Method 3: Using Time Ratios
Step 1: Determine the Combined Work Rate
If the man and son together can do 1/48 of the work in 1 hour, the man alone can do 1/60 of the work in 1 hour. Therefore:
Combined work rate 1/48 per hour
Man's work rate 1/60 per hour
Step 2: Determine the Son's Work Rate
The son's work rate is:
Son's work rate 1/48 - 1/60
(5–4)/240
1/240 per hour
Therefore, the son takes:
240 hours to complete the job alone
Solution Method 4: Using a Reference Work Unit
Step 1: Determine the Man's and Son's Work Rates
The man and son together can complete the work in 20 hours, so their combined rate is:
1/20 per hour
The man alone can complete the work in 30 hours, so his work rate is:
1/30 per hour
Step 2: Determine the Son's Work Rate
The son's work rate is:
Son's work rate 1/20 - 1/30
(3–2)/60
1/60 per hour
Therefore, the son takes:
60 hours to complete the job alone
Solution Method 5: Using LCM for Unit Work
Step 1: Determine the Man's and Son's Work Rates
The LCM of 30 and 20 is 60 units. The man can do 2 units of work in 1 hour, and the man and son together can do 3 units in 1 hour. Therefore:
Man's work rate 2/60 1/30 per hour
Combined work rate 3/60 1/20 per hour
Step 2: Determine the Son's Work Rate
The son's work rate is:
Son's work rate 1/20 - 1/30
(3–2)/60
1/60 per hour
Therefore, the son takes:
60 hours to complete the job alone
Conclusion
Through various methods, we have determined that the son alone will take 60, 240, or 120 hours to complete the job, depending on the specific conditions and reference units used. These solutions illustrate the importance of accurate problem analysis and the application of proper mathematical techniques in solving work efficiency problems. It is crucial to carefully identify the given data and apply the correct formulae to derive the desired results.