duahsolo wrote:
John works twice as fast as Peter, but John takes a half hour break after every one hour worked while Peter takes an hour break after every two hours worked. If John can complete the task in 5 hours working alone with no breaks, how long will it take both to complete the task if they start working together while maintaining their break habits?
A) 3 hours and 20 minutes
B) 4 hours and 30 minutes
C) 4 hours and 40 minutes
D) 4 hours and 45 minutes
E) 5 hours
If \(x\) is the performance each hour of Peter then \(2x\) is the performance each hour of John.
John takes a half hour break after every one hour worked. This mean John took \(2x\) work for every one and a half hour, or John took \(4x\) work for every 3 hours.
Peter takes an hour break after every two hours worked. This mean Peter took \(2x\) work for every 3 hours.
Hence, John and Peter took \(6x\) work for every 3 hours.
Without break, John can complete the task in 5 hours. This means \(2x=\frac{1}{5} \implies x=\frac{1}{10}\). This means John could complete \(\frac{1}{5}\) of the task each hour and Peter could complete \(\frac{1}{10}\) of the task each hour
Hence, John and Peter took \(\frac{6}{10}\) of the work for every 3 hours.
In the first 3 hours, they completed \(\frac{6}{10}\) of the task.
In the next hour, the 4th hour, they completed \(\frac{6}{10}+\frac{1}{5}+\frac{1}{10}=\frac{9}{10}\) of the task. The remaining of the task is \(\frac{1}{10}\)
After that, John took a break after working for a hour, left Peter worked alone.
In the next half hour, the 4.5th hour, they completed \(\frac{9}{10} +\frac{1}{2} \times \frac{1}{10}=\frac{19}{20}\). The reamining of the task is \(\frac{1}{20}\).
After that, John went back to work. The performance of them now is \(\frac{1}{5} +\frac{1}{10}=\frac{3}{10}\).
The remaining time they needed to completed the task is: \(\frac{1}{20} : \frac{3}{10}=\frac{10}{60}\) hour or 10 minutes < 0.5 hour.
Hence, the total time they needed to completed the task is 4 hours 40 minutes. The answer is C.
Here is the image illustrates the process
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