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:
+ =
+ = 1
Solve that and get x = 24 hours. So Manny takes 24 hours
to do one job working alone.
Edwin