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16 Sep 2014, 00:14
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A qualified worker can dig a well in 5 hours. The worker invites 2 apprentices to help, each of whom works at a rate of \(\frac{3}{4}\) as fast as the qualified worker, and 2 trainees, each of whom works at a rate of \(\frac{1}{5}\) as fast as the qualified worker. If the five-person team works together to dig the same well, approximately how long will it take to complete the job?
A. 1 hour 29 minutes
B. 1 hour 39 minutes
C. 1 hour 44 minutes
D. 1 hour 49 minutes
E. 1 hours 54 minutes
A. 1 hour 29 minutes
B. 1 hour 39 minutes
C. 1 hour 44 minutes
D. 1 hour 49 minutes
E. 1 hours 54 minutes
16 Sep 2014, 00:14
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Official Solution:
A qualified worker can dig a well in 5 hours. The worker invites 2 apprentices to help, each of whom works at a rate of \(\frac{3}{4}\) as fast as the qualified worker, and 2 trainees, each of whom works at a rate of \(\frac{1}{5}\) as fast as the qualified worker. If the five-person team works together to dig the same well, approximately how long will it take to complete the job?
A. 1 hour 29 minutes
B. 1 hour 39 minutes
C. 1 hour 44 minutes
D. 1 hour 49 minutes
E. 1 hours 54 minutes
The qualified worker can dig the well in 5 hours, so his work rate is \(\frac{1}{5}\) of the well per hour.
Each apprentice works at \(\frac{3}{4}\) of the qualified worker's rate, so two apprentices together work at a rate of \(2*\frac{3}{4}* \frac{1}{5} = \frac{3}{10}\) of the well per hour.
Each trainee works at \(\frac{1}{5}\) of the qualified worker's rate, so two trainees together work at a rate of \(2*\frac{1}{5}* \frac{1}{5} = \frac{2}{25}\) of the well per hour.
Thus, the total work rate of the team is:
\(\frac{1}{5} + \frac{3}{10} + \frac{2}{25} = \frac{29}{50}\) of the well per hour.
Since the rate is reciprocal of time, the job will take \(\frac{50}{29}=1\frac{21}{29}\) hours to complete.
To approximate \(\frac{21}{29}\), we can notice that \(\frac{20}{30} < \frac{21}{29} < \frac{21}{28}\), which gives \(\frac{2}{3} < \frac{21}{29} < \frac{3}{4}\). Since \(\frac{2}{3}\) of an hour is 40 minutes and \(\frac{3}{4}\) of an hour is 45 minutes, the time required to complete the job is between 1 hour and 40 minutes and 1 hour and 45 minutes. The only option in this range is C.
Answer: C
A qualified worker can dig a well in 5 hours. The worker invites 2 apprentices to help, each of whom works at a rate of \(\frac{3}{4}\) as fast as the qualified worker, and 2 trainees, each of whom works at a rate of \(\frac{1}{5}\) as fast as the qualified worker. If the five-person team works together to dig the same well, approximately how long will it take to complete the job?
A. 1 hour 29 minutes
B. 1 hour 39 minutes
C. 1 hour 44 minutes
D. 1 hour 49 minutes
E. 1 hours 54 minutes
The qualified worker can dig the well in 5 hours, so his work rate is \(\frac{1}{5}\) of the well per hour.
Each apprentice works at \(\frac{3}{4}\) of the qualified worker's rate, so two apprentices together work at a rate of \(2*\frac{3}{4}* \frac{1}{5} = \frac{3}{10}\) of the well per hour.
Each trainee works at \(\frac{1}{5}\) of the qualified worker's rate, so two trainees together work at a rate of \(2*\frac{1}{5}* \frac{1}{5} = \frac{2}{25}\) of the well per hour.
Thus, the total work rate of the team is:
\(\frac{1}{5} + \frac{3}{10} + \frac{2}{25} = \frac{29}{50}\) of the well per hour.
Since the rate is reciprocal of time, the job will take \(\frac{50}{29}=1\frac{21}{29}\) hours to complete.
To approximate \(\frac{21}{29}\), we can notice that \(\frac{20}{30} < \frac{21}{29} < \frac{21}{28}\), which gives \(\frac{2}{3} < \frac{21}{29} < \frac{3}{4}\). Since \(\frac{2}{3}\) of an hour is 40 minutes and \(\frac{3}{4}\) of an hour is 45 minutes, the time required to complete the job is between 1 hour and 40 minutes and 1 hour and 45 minutes. The only option in this range is C.
Answer: C
General Discussion
04 Mar 2023, 06:00
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
