SOLUTION: Mark can overhaul an engine in 20 hours and Phil can do the same job by himself in 30 hours. If they both work together for a time and then Mark finishes the job by himself in 5 ho

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Question 122701This question is from textbook College Algebra
: Mark can overhaul an engine in 20 hours and Phil can do the same job by himself in 30 hours. If they both work together for a time and then Mark finishes the job by himself in 5 hours, how long did they work together?
Here is the work I've tried on this problem, for some reason I can't figure out the last part, the 5 hours part. Here's what I've got so far:
T(1/30 + 1/20) = 1
T(2 + 3) = 60
5T = 60
T = 12
So when they work together, it would take them 12 hours total, right?
Then I need the last part:
Would it be 12 hours minus the rate of Mark's last 5 hours?
12 - 5(1/20)= T ? T = 11.75? They worked together for 11 hours and 45 minutes?
Please let me know if I'm on the right track or if I'm way off. I hope I haven't confused you as much as I've confused myself!
Thanks!
 This question is from textbook College Algebra

Found 2 solutions by ankor@dixie-net.com, bucky:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
Mark can overhaul an engine in 20 hours and Phil can do the same job by himself in 30 hours. If they both work together for a time and then Mark finishes the job by himself in 5 hours, how long did they work together?
:
I will work it out then see how it compares to your procedure:
:
Let x = time that M & P work together
Let the completed job = 1
:
M & P working together + M alone for 5 hrs = completed job
x%2F20 + x%2F30 + 5%2F20 = 1
:
Get rid of the denominators; multiply equation by 60
3x + 2x + 3(5) = 60
5x + 2x + 15 = 60
5x = 60 - 15
5x = 45
x = 45%2F5
x = 9 hrs
:
To explain: It helps to write the problem as 3 different workers. I think
you over complicated the problem. What do you think?
:
Another way you can think of it,
In 5 hrs, M would do 5/20 or 1/4 of the job, leaving 75% of the job remaining:
x%2F20 + x%2F30 = .75
3x + 5y = 60(.75)
5x = 45
x = 9 hrs
:
You can always check you solution by seeing if all the fractions add up to 1
9%2F20 + 9%2F30 + 5%2F20 =
using calc and decimals
.45 + .3 + .25 = 1.

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
What you have done is correct, but is not the way to work the problem.
.
Your approach was correct. Since Mark takes 20 hours to overhaul and engine, in one hour he
can do one-twentieth of the entire job. Similarly, since Phil can do an overhaul in 30 hours,
he can do one-thirtieth of the job in an hour. When they work together, each hour that passes
allows them to do (1/20 + 1/30) of the job.
.
They start out working the job together, and they work for some unknown time T. Then Phil stops
and Mark works by himself for 5 hours to complete the job.
.
In that 5 hour period Mark does 5 times the 1/20 of the job that he can do each hour. So in
that 5 hours he does 5/20 or one-quarter of the job. So that means when both Mark and Phil
were working together they did three-quarters of the job.
.
That being the case, working in combination their combined rate was (1/20 + 1/30) times
the time T that they took to complete 3/4 of the job. In equation form this is
.
(1/20 + 1/30)*T = 3/4
.
Just as you did when you worked the problem, you can get rid of the denominators by multiplying
all terms on both sides by 60 to get:
.
(3 + 2)*T = 45
.
This becomes:
.
5T = 45
.
Solve for T by dividing both sides by 5 to get:
.
T = 9
.
So they work together for 9 hours. Then Mark works by himself for 5 more hours and completes
the job. The total time to rebuild the engine is therefore 9 + 5 or 14 hours.
.
This certainly seems about right because, as you found, when they work together for the entire
job it takes them 12 hours, and that would be the fastest they could do it.
.
If one of them quits, the other one will need more time to complete the job by himself, so
it should take longer that the fastest time ... in this case the fasted time is 12 hours.
.
Hope this helps you to see how to incorporate the change in plans that occurred when Phil
stopped working.
.