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John works twice as fast as Peter, but John takes a half hour break af

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John works twice as fast as Peter, but John takes a half hour break af  [#permalink]

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New post 31 May 2017, 07:46
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Question Stats:

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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

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John works twice as fast as Peter, but John takes a half hour break af  [#permalink]

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New post 31 May 2017, 09:18
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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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Re: John works twice as fast as Peter, but John takes a half hour break af  [#permalink]

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New post 10 Aug 2018, 18:40
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


The rate of John is 1/5 and the rate of Peter is 1/10.

Their combined rate, with no breaks is, 1/5 + 1/10 = 2/10 + 1/10 = 3/10.

For the first 3 hours of working together, we see that each would have taken a one-hour break and thus each worked only 2 hours. Thus, they finished 2 x 3/10 = 6/10 of the job.

During the next hour, hour 4, they both worked for the full hour; thus, they finished another 3/10 of the job and so far they finished 6/10 + 3/10 = 9/10 of the job.

During the following hour, hour 5, John worked only half an hour (since he took a half-hour break) while Peter worked the full hour; thus, they would have completed another ½(1/5) + 1/10 = 2/10 of the job. However, by the end of hour 5, we see that they would have completed 9/10 + 2/10 = 11/10 or more than 1 entire job. Thus we need to push the time back.

So let’s only focus on the first 30 minutes of hour 5; John would not be working since he’s on his half-hour break, while Peter worked the entire 30 minutes. Thus, Peter would have completed another ½(1/10) = 1/20 of the job. By the end of the first 30 minutes of hour 5, we see that they would have completed 9/10 + 1/20 = 19/20 of the job.

We see that it takes more than 4 hour 30 minutes but less than 5 hours to complete this job. We also see that there are two answer choices that are between these two times. Let’s analyze choice C, 4 hours and 40 minutes, first. In other words, let’s see how much more work they complete in the extra 10 minutes.

In the extra 10 minutes, or ⅙ of an hour, they were both working; thus, they completed ⅙(1/5) + ⅙(1/10) = 1/30 + 1/60 = 3/60 = 1/20 of the job. Add this to the 19/20 of the job they completed in 4 hours 30 minutes, and we see that they would have completed exactly one entire job.

Answer: C
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Re: John works twice as fast as Peter, but John takes a half hour break af   [#permalink] 10 Aug 2018, 18:40
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