SOLUTION: Mutt and Jeff need to paint a fence. Mutt can do the job alone 4 hours faster than Jeff. If together they work for 18 hours and finish only half of the job, how long would Jeff nee

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Question 335414: Mutt and Jeff need to paint a fence. Mutt can do the job alone 4 hours faster than Jeff. If together they work for 18 hours and finish only half of the job, how long would Jeff need to do the job alone?
Found 2 solutions by Edwin McCravy, magar8:
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Mutt and Jeff need to paint a fence. Mutt can do the job alone 4 hours faster than Jeff. If together they work for 18 hours and finish only half of the job, how long would Jeff need to do the job alone?

Make this chart


                       | Jobs done | Time spent | Rate in jobs/hour
-------------------------------------------------------------------
Mutt working alone     |           |            |
Jeff working alone     |           |            | 
Both working together  |           |            |

Let x be the number of hours it takes Jeff to do 1 job, so fill
in x for Jeff's time spent, and 1 for the number of jobs:

                       | Jobs done | Time spent | Rate in jobs/hour
-------------------------------------------------------------------
Mutt working alone     |           |            |
Jeff working alone     |     1     |     x      | 
Both working together  |           |            |


Since Mutt can do the job alone 4 hours faster than Jeff, his time
is 4 hours less, so put x-4 for Mutt's time to finish 1 job, and 1
for the number of jobs he can finish in that time:

                       | Jobs done | Time spent | Rate in jobs/hour
-------------------------------------------------------------------
Mutt working alone     |     1     |    x-4     |
Jeff working alone     |     1     |     x      | 
Both working together  |           |            |

Since if they work together for 18 hours they finish only half of the job,
we put 18 for their time working together and 1/2 for the jobs
done.

                       | Jobs done | Time spent | Rate in jobs/hour
-------------------------------------------------------------------
Mutt working alone     |     1     |    x-4     |
Jeff working alone     |     1     |     x      | 
Both working together  |    1/2    |    18      |

Now we fill in the rates by dividing jobs done by time spent:

                       | Jobs done | Time spent | Rate in jobs/hour
-------------------------------------------------------------------
Mutt working alone     |     1     |    x-4     |    1/(x-4)
Jeff working alone     |     1     |     x      |     1/x
Both working together  |    1/2    |    18      |  (1/2)/18

1%2F%28x-4%29%2B1%2Fx=%281%2F2%29%2F18

Multiply both side by 18 to clear the main denominator on the right

18%2F%28x-4%29%2B18%2Fx=1%2F2

Multiply through by 2x%28x-4%29

36x%2B36%28x-4%29=x%28x-4%29

36x%2B36x-144=x%5E2-4x

72x-144=x%5E2-4x

0=x%5E2-76x%2B144

x%5E2-76x%2B144=0


x+=+%28-%28-76%29+%2B-+sqrt%28%28-76%29%5E2-4%2A1%2A144+%29%29%2F%282%2A1%29+

x+=+%2876+%2B-+sqrt%285776-576%29%29%2F2+

x+=+%2876+%2B-+sqrt%285200%29%29%2F2+ 

x+=+%2876+%2B-+sqrt%285200%29%29%2F2+

x+=+%2876+%2B-+sqrt%28400%2A13%29%29%2F2+

x+=+%2876+%2B-+20sqrt%2813%29%29%2F2+

x+=+%284%2819+%2B-+5sqrt%2813%29%29%29%2F2+

x+=+2%2819+%2B-+5sqrt%2813%29%29+

That has solutions

x+=+2%2819+%2B+5sqrt%2813%29%29+ and x+=+2%2819+-+5sqrt%2813%29%29+

x = 74.05551275 and x = 1.944487245

We ignore the smaller solution since it makes no sense that Jeff
could finish the job in less than 2 hours since together it takes them
18 hours to complete only half the job.  So we discard that answer and
conclude that it takes Jeff 74.05551275 hours to do the job by
himself.

Edwin

Answer by magar8(1) About Me  (Show Source):
You can put this solution on YOUR website!
Mutt and Jeff need to paint a fence. Mutt can do the job alone 4 hours faster than Jeff. If together they work for 18 hours and finish only half of the job, how long would Jeff need to do the job alone?