Work/Rate Problems

swerve wrote: ↑

Mon Feb 22, 2021 3:12 pm

At \(8\) am on Thursday, two workers, \(A\) and \(B\), each start working independently to build identical decorative lamps. Worker \(A\) completes her lamp at \(5\) pm on Friday, while Worker \(B\) completes her lamp sometime during the morning on Friday. If both workers adhere to working hours of \(8\) am to \(12\) pm and \(1\) pm to \(5\) pm 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, 1:30 pm
B. Thursday, 2:15 pm
C. Thursday, 3:00 pm
D. Thursday, 4:15 pm
E. Friday, 12:00 pm

The OA is C

Solution:

We see that a workday is 8 hours (4 hours in the morning and 4 hours in the afternoon). Thus, it takes 2 workdays, or 16 hours, for worker A to complete her lamp, and therefore, her rate is 1/16. We are given that worker B completes her lamp sometime during the morning on Friday. Thus, it takes her more than 1 workday (or 8 hours) and less than 1 ½ workdays (or 12 hours) to complete her lamp, and therefore, her rate is greater than 1/12 but less than 1/8. These two rates are the lower and upper bounds, respectively, of worker B’s rate.

At the upper bound of worker B’s rate, if the two workers work together, it will take 1/(1/16 + 1/8) = 1/(1/16 + 2/16) = 1/(3/16) = 16/3 = 5 ⅓ hr = 5 hr 20 min to complete one single lamp. Therefore, they will finish by 2:20 pm on Thursday (4 hours from 8 am to 12 pm and 1 hr 20 min after 1 pm).

At the lower bound of worker B’s rate, if the two workers work together, it will take 1/(1/16 + 1/12) = 1/(3/48 + 4/48) = 1/(7/48) = 48/7 = 6 6/7 hr ≈ 6 hr 51 min to complete one single lamp. Therefore, they will finish by 3:51 pm on Thursday (4 hours from 8 am to 12 pm and 2 hr 51 min after 1 pm).

Since worker B’s rate is between 1/12 and 1/8, the time when they work together will be between 2:20 pm and 3:51 pm on Thursday. The only time in the given answer choices is choice C: Thursday, 3:00 pm. Thus choice C is the correct answer.

Answer: C