SOLUTION: Chris and Manny work together for 6 hours to landscape a yard before Chris has to leave the job. Manny finishes the yard alone in 10 hours. If Chris can landscape a yard in 18 ho

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Question 694131: Chris and Manny work together for 6 hours to landscape a yard before Chris has to leave the job. Manny finishes the yard alone in 10 hours. If Chris can landscape a yard in 18 hours working alone, how long does it take Manny working alone to do the job?
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Chris and Manny work together for 6 hours to landscape a yard before Chris has to leave the job. Manny finishes the yard alone in 10 hours. If Chris can landscape a yard in 18 hours working alone, how long does it take Manny working alone to do the job?
Make this chart:

                        Number of 
                        Jobs or
                        Fraction      Time in      Rate in   
                        Thereof         hours      jobs/hour
------------------------------------------------------------               
Chris alone (1 job)            
Manny alone (1 job)    
C&M together (6 hrs)         
Manny alone (10 hrs) 

>>...how long does it take Manny working alone to do the job?...<<

Let the answer to that it takes Manny x hours to do 1 job. So
we fill in 1 for Manny's number of jobs and x hours for the time.

>>...Chris can landscape a yard in 18 hours working alone...<<

So we fill in 1 for Chris' number of jobs and 18 hours for the time.
And we fill in 6 and 10 for the other given times:

                        Number of 
                        Jobs or
                        Fraction      Time in      Rate in   
                        Thereof         hours      jobs/hour
------------------------------------------------------------
Chris alone (1 job)        1            18                                      
Manny alone (1 job)        1             x
C&M together (6 hrs)                     6   
Manny alone (10 hrs)                    10

Next we fill the rates in jobs/hour by dividing the number of jobs
by the number of hours in the first two lines:

                        Number of 
                        Jobs or
                        Fraction      Time in      Rate in   
                        Thereof         hours      jobs/hour
------------------------------------------------------------
Chris alone (1 job)        1            18           1/18
Manny alone (1 job)        1             x           1/x
C&M together (6 hrs)                     6   
Manny alone (10 hrs)                    10

Since Manny's rate alone for 10 hours is the same as his rate when
doing 1 whole job, we can also put 1/x for Manny's rate for the 10 hours

                        Number of 
                        Jobs or
                        Fraction      Time in      Rate in   
                        Thereof         hours      jobs/hour
------------------------------------------------------------
Chris alone (1 job)        1            18           1/18
Manny alone (1 job)        1             x           1/x
C&M together (6 hrs)                     6   
Manny alone (10 hrs)                    10           1/x

Chris and Manny's combined rate is the sum of their rates or 1/18 + 1/x,
So we fill in that for their combined rate (C&M together):

                        Number of 
                        Jobs or
                        Fraction      Time in      Rate in   
                        Thereof         hours      jobs/hour
------------------------------------------------------------
Chris alone (1 job)        1            18           1/18 
Manny alone (1 job)        1             x           1/x
C&M together (6 hrs)                     6       1/18 + 1/x
Manny alone (10 hrs)                    10           1/x

Next we get the fraction of a job that C&M did together in 6 hours,
by multiplying their combined rate by the time 6 hours.  We get the 
fraction of a job that Manny did alone in the 10 hours, by multiplying 
Manny's rate 1/x by the time 10 hours.  

                        Number of 
                        Jobs or
                        Fraction      Time in      Rate in   
                        Thereof         hours      jobs/hour
------------------------------------------------------------
Chris alone (1 job)        1            18           1/18 
Manny alone (1 job)        1             x           1/x
C&M together (6 hrs)  6(1/18 + 1/x)      6       1/18 + 1/x
Manny alone (10 hrs)     10/x           10           1/x

The equation comes from:
               %28matrix%287%2C1%2C%0D%0AFraction%2Cof%2Ca%2C+job%2Cthey%2Cdid%2Ctogether%29%29 + %28matrix%287%2C1%2C%0D%0AFraction%2Cof%2Ca%2C+job%2CManny%2Cdid%2Calone%29%29 = %28matrix%283%2C1%2C%0D%0A1%2Cwhole%2Cjob%29%29

               6%281%2F18%2B1%2Fx%29 + 10%2Fx = 1

Solve that and get x = 24 hours.  So Manny takes 24 hours
to do one job working alone.

  

Edwin