Bunuel wrote:
At 8am on Thursday, two workers, A and B, each start working independently to build identical decorative lamps. Worker A completes her lamp at 5pm on Friday, while Worker B completes her lamp sometime during the morning on Friday. If both workers adhere to working hours of 8am to 12pm and 1pm to 5pm each day, at which of the following times might the two workers have completed a single lamp had they worked together at their respective constant rates?
A. Thursday, 1:30pm
B. Thursday, 2:15pm
C. Thursday, 3:00pm
D. Thursday, 4:15pm
E. Friday, 12:00pm
Kudos for a correct solution.
MANHATTAN GMAT OFFICIAL SOLUTION:This is a combined work problem. The required work is one lamp and the question asks for the time taken, so you need to determine something about the rates of the workers.
Worker A’s rate can be calculated. She works for 2 full days, or 16 hours, to complete one lamp. Therefore, she completes 1/16 lamp per hour.
However, Worker B’s rate is trickier to determine. She takes one full day and some part of the next morning. The first day is 8 hours and the morning shift is 4 hours, so she works on the lamp for more than 8 hours and less than 12 hours. Therefore, her rate is greater than 1/12 lot per hour but less than 1/8 lot per hour.
Add this information to an RTW chart.
Remember that RT = W!
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Because the work quantity is 1 lamp, express the time, t, as 1/Rate (that is, the reciprocal of the rate). The range of possible values for the time is 16/3 < t < 48/7. This translates to the following:
5 and 1/3 hours < t < 6 and 6/7 hours
1/3 of an hour is 20 minutes, so the lower boundary is 5 hours 20 minutes. 6/7 of an hour is a bit more annoying to calculate; check the answers first to see whether this is really necessary.
The workers start at 8am on Thursday and take a 1-hour break between 12 and 1pm, so if they worked together, they would complete the job after 2:20pm but a little before 4:00pm on Thursday. Only one answer choice fits: 3pm.
Alternatively, you could use estimation, though recognize that you might not always be able to narrow all the way down to one answer choice. In many cases, though, you can (including this case!). It’s surprising how often you can get to the right answer this way on rate questions.
When working together, the two workers must use at least the entire 4-hour shift on Thursday morning because the first answer choice is Thursday afternoon.
Worker A takes 16 hours to complete one lamp. Therefore, she completes 1/4 of a lamp during a 4-hour shift.
Worker B takes between 8 and 12 hours to complete one lamp. Therefore, she completes between 1/3 and 1/2 of a lamp during a 4-hour shift.
At the end of the Thursday morning shift, then, the workers have finished somewhere between (1/4 + 1/3) and (1/4 + 1/2) of a lamp, or between 7/12 and 3/4 of a lamp. They’ve completed more than half of the lamp during the first 4-hour shift, so they won’t need another complete 4-hour shift to finish; eliminate answer (E).
If they’ve completed 3/4 of the lamp, then they only have 1/4 to go. 1/4 of a lamp represents one-third of the work they did during the first 4 hours, so they’ll need another 4(1/3) hours to finish, or 1 hour 20 minutes. Therefore, they need at least until 2:20pm on Thursday afternoon. Eliminate answers (A) and (B).
One end of the possible time range is 2:20pm; the other end must include one of the two remaining answers but exclude the other one. Answer (C) must be it. If you want to prove it to yourself, finish off the estimation (but note that logic tells you that you can’t include D in the final range without including C!).
If the workers finish 7/12 of the lamp in 4 hours, then they need a bit less than the 4 hours of the second shift to finish off the other 5/12 of the lamp. Specifically, they need 5/7 of that second shift. That’s an annoying number. 5/8 is 62.5%. 5/6 is about 83%. So let’s call 5/7 about 70% (note that this is a bit of an underestimate).
75% of 4 hours is 3 hours, so the workers need a bit less than 3 hours after lunch to finish off the lamp. They’ll be done by 4p; eliminate answer (D).
The correct answer is (C).
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