SOLUTION: James can paint a room in 4 hours, while Mike can paint the same room in 5 hours. Working together, how many hours will it take them to paint the room if James only works for one

Algebra ->  Rate-of-work-word-problems -> SOLUTION: James can paint a room in 4 hours, while Mike can paint the same room in 5 hours. Working together, how many hours will it take them to paint the room if James only works for one       Log On


   

Question 1061794: James can paint a room in 4 hours, while Mike can paint the same room in 5 hours. Working together, how many hours will it take them to paint the room if James only works for one hour?
Dear tutor,
I solved this problem 3 ways, and I got the same answer, but I don't think I am understanding what is this problem really asking. Is it asking for the total time spent on the project (painting) OR is it asking for the time that Mike only spent working on the project?
Method 1:
J
1 job/4 hours
M
1 job/5 hours
Project Time
1 job/x hours
1 hour*(1 job/ 4 hours) + (x-1)*(1 job/5 hours) = 1 job
1/4 + (x-1)/5 = 1
5 + 4(x-1) = 20
x=19/4 = 4 3/4 hours = 4.75 hours
Based on my set-up (project time equal to 1 job/x hours), the x represents the project time. So, the whole project of painting the room is 4.75 hours. HOWEVER, THIS IS NOT LISTED AMONG ANSWER CHOICES!From this time, 1 hour is spent painting by James, and the rest by Mike. So Mike was painting for 3.75 hours. WHAT IS THE QUESTION REALLY ASKING, IS IT PAINTING BY MIKE OR PAINTING BY BOTH? IS THE ANSWER 3.75 HOURS OR 4.75 HOURS? I ANSWERED 4.75 HOURS, BUT IT'S THE WRONG ANSWER!
Method 2:
(3/4)job remaining and that job will be done now by M at the rate of 1 job/5 hours, therefore
(3/4)job/(1job/5hours) = 15hours/4=3 3/4 hours = 3.75 hours, so this is the time that M spent painting or that M spent on the project. Since J spent 1 hour on the project per givens in the question stem, then total time on the project is 4.75 hours. HOWEVER, 4.75 HOURS IS NOT AMONG ANSWER CHOICES, BUT 3.75 HOURS IS. I AM NOT SURE WHY.
Method 3:
1 hour*(1job/4hours) + x hours*(1job/5hours)=1 job
contribution of one person +contribution of second person gives me whole job
Since I don't know how many hours M worked on the project, I just set it to x hours.
1/4+x/5=1
x=15/4 hours = 3 3/4 hours=3.75 hours
This is the time that M spent on the project (painting). THE BOOK SAYS THAT 3.75 HOURS IS THE CORRECT ANSWER, BUT I AM NOT UNDERSTANDING WHY, SINCE THE QUESTION IS ASKING FOR TOTAL PROJECT TIME. ISN'T TOTAL PROJECT TIME 4.75 HOURS? WHY THEN THE ANSWER IS 3.75 HOURS?
Answer by josgarithmetic(39613) About Me  (Show Source):
You can put this solution on YOUR website!
PERSON/S           RATE rooms/hour

James                1%2F4

Mike                 1%2F5

Both Combined       1%2F4%2B1%2F5=%285%2B4%29%2F20=9%2F20

If James only works for 1 hour, assuming he is working together with Mike for this 1 hour time
then let x be the unknown time that Mike continues working alone, to finish this one job.

highlight_green%28%289%2F20%29%2A1%2B%281%2F5%29x=1%29
The question was, Working together, how many hours will it take them to paint the room if James only works for one hour?
The equation outlined in green seems to be what the description and question mean. THEY ARE ONLY working TOGETHER for 1 hour
othewise Mike is working alone for x hours.

20%289%2F20%2B%281%2F5%29x%29=20
9%2B4x=20
4x=20-9
4x=11
x=11%2F4
highlight%28x=2%263%2F4%29-----time Mike worked alone.

Putting all the time together, all this work time is 1%2B2%263%2F4 hours.
highlight%28%283%263%2F4%29hours%29-----total work time