I06-28

Question

Phyllis, Shania, and Taneisha are each working at a constant rate to fill envelopes for a local city council letter mailing informational campaign. Shania would take 3 hours to complete the job alone, and Taneisha can complete the job faster than Phyllis.
 
If Phyllis, Shania, and Taneisha, working together, can complete the job in 1 hour, select for  Phyllis  the time in hours she would take to complete the job alone, and select for  Taneisha  the time in hours she would take to complete the job alone that are jointly consistent with the information provided. Make only two selections, one in each column.

Bunuel · Accepted Answer

Since it takes Shania 3 hours to complete the job alone, in 1 hour she completes 1/3 of the job,  leaving 2/3 of the job to be completed by Phyllis and Taneisha together .

If both Phyllis and Taneisha required less than 3 hours to complete the job alone, each would complete more than 1/3 of the job in 1 hour, and  together they would complete  more  than 2/3 of the job in 1 hour.

If both Phyllis and Taneisha required more than 3 hours to complete the job alone, each would complete less than 1/3 of the job in 1 hour, and  together they would complete  less  than 2/3 of the job in 1 hour.

Since Taneisha can complete the job faster than Phyllis, Taneisha must need less than 3 hours, while Phyllis must need more than 3 hours.

Hope it's clear.

Bunuel · Answer

Official Solution:

Since it takes Shania 3 hours to complete the job alone, in 1 hour she'd complete  \frac{1}{3}  of the job, leaving  \frac{2}{3}  of the job to be completed by Phyllis and Taneisha. One being faster than the other implies that the faster one, Taneisha, can complete the job in less than 3 hours, and the slower one, Phyllis, can complete the job in more than 3 hours.
 
Among the options, the times less than 3 hours are 1 hour and 2 hours. If Taneisha took 1 hour, she would be able to finish the entire job alone in 1 hour, which cannot be true since the problem states all three together take 1 hour. Therefore, the only consistent option is that Taneisha takes 2 hours to complete the job alone.
 
Assuming Phyllis takes  p  hours to complete the job alone, we'd have (Taneisha's rate) + (Phyllis's rate)  = \frac{1}{2} + \frac{1}{p} = \frac{2}{3} , which gives  p = 6  hours.

Correct answer:

Phyllis  "6"

Taneisha  "2"

Bunuel · Answer

I have revised the question, solution, and formatting by adding more details to enhance clarity.

miag · Answer

hi  Bunuel ,

how did you come to the conclusion/inference that Taneisha will complete the job is <3 hours, and Phyllis will do so in >3 hours?

Thank you!

Edwin555 · Answer

I like the solution - it’s helpful. This one is good to learn the basics!

Kannikamalhotra21 · Answer