Amy, Marty, and Taylor working together can do a job in 2 hours. Amy and Marty working together can do the job in 4 hours. Marty and Taylor working together can do the job in 3 hours. How long does it take for each person to do the job working alone? How long does it take Amy and Taylor working together to do the job?
Make this chart:
No. of jobs done | time in hrs. | rate in jobs/hr
A. working alone | |
M. working alone | |
T. working alone | |
A., M & T working together | |
A. & M. working together | |
M. & T. working together | |
We look at the first question:
How long does it take for each person to do ONE job working alone?
Note that I change "a job" and "the job" to "ONE job".
Let their times working alone to do ONE job be x, y, and z respectively.
Fill those in along with 1 for the number of jobs.
No. of jobs done | time in hrs. | rate in jobs/hr
A. working alone 1 | x |
M. working alone 1 | y |
T. working alone 1 | z |
A., M & T working together | |
A. & M. working together | |
M. & T. working together | |
Amy, Marty, and Taylor working together can do ONE job in 2 hours.
So we fill in 1 for the number of jobs and 2 for the number of hours for
"A., M & T working together"
No. of jobs done | time in hrs. | rate in jobs/hr
A. working alone 1 | x |
M. working alone 1 | y |
T. working alone 1 | z |
A., M & T working together 1 | 2 |
A. & M. working together | |
M. & T. working together | |
Amy and Marty working together can do ONE job in 4 hours.
So we fill in 1 for the number of jobs and 4 for the number of hours for
"A. & M. working together"
How long does it take for each person to do the job working alone?
No. of jobs done | time in hrs. | rate in jobs/hr
A. working alone 1 | x |
M. working alone 1 | y |
T. working alone 1 | z |
A., M & T working together 1 | 2 |
A. & M. working together 1 | 4 |
M. & T. working together | |
Marty and Taylor working together can do ONE job in 3 hours.
So we fill in 1 for the number of jobs and 3 for the number of hours for
"M. & T. working together"
No. of jobs done | time in hrs. | rate in jobs/hr
A. working alone 1 | x |
M. working alone 1 | y |
T. working alone 1 | z |
A., M & T working together 1 | 2 |
A. & M. working together 1 | 4 |
M. & T. working together 1 | 3 |
Now we fill in the rates in jobs/hour by putting the number of jobs done,
whic is 1, over the time in hours:
No. of jobs done | time in hrs. | rate in jobs/hr
A. working alone 1 | x | 1/x
M. working alone 1 | y | 1/y
T. working alone 1 | z | 1/z
A., M & T working together 1 | 2 | 1/2
A. & M. working together 1 | 4 | 1/4
M. & T. working together 1 | 3 | 1/3
We form the first equation by
A's rate + M's rate + T's rate = A., M & T.'s rate working together.
1/x + 1/y + 1/z = 1/2
We form the second equation by
A's rate + M's rate = A. & M.'s rate working together.
1/x + 1/y = 1/4
We form the third equation by
M's rate + T's rate = M. & T.'s rate working together.
1/y + 1/z = 1/3
So we have this system:
1/x + 1/y + 1/z = 1/2
1/x + 1/y = 1/4
1/y + 1/z = 1/3
We do not clear of fractions for that would give us complicated
terms in xyz which we do want.
Since the middle equation has no term in z, we get another equation which
has no term in z by eliminating the 1/z terms from the 1st and 3rd
equations. So we multiply the 3rd equation by -1 and add it to the 1st
equation:
1/x + 1/y + 1/z = 1/2
-1/y - 1/z = -1/3
----------------------
1/x = 1/2 - 1/3
1/x = 3/6 - 2/6
1/x = 1/6
x = 6
Substitute 6 for x in:
1/x + 1/y = 1/4
1/6 + 1/y = 1/4
1/y = 1/4 - 1/6
1/y = 3/12 - 2/12
1/y = 1/12
y = 12
Substitute 12 for y in
1/y + 1/z = 1/3
1/12 + 1/z = 1/3
1/z = 1/3 - 1/12
1/z = 4/12 - 1/12
1/z = 3/12
1/z = 1/4
x = 4
So
It will take take Amy 4 hours to do ONE job.
It will take take Marty 12 hours to do ONE job.
It will take take Taylor 6 hours to do ONE job.
The second quetion is
How long does it take Amy and Taylor working together to do ONE job?
Their combined rate in jobs/hr. is the sum of their individual rates
Amy's rate is 1 job in 4 hours, or 1/4 job/hr.
Taylor's rate is 1 job in 6 hours, or 1/6 job/hr.
Their combined rate working together = 1/4 + 1/6 = 3/12 + 2/12 = 5/12 jobs/hr.
And since TIME = JOBS/RATE
TIME = 1/(5/12) = 12/5 = 2.4 hours or 2 hours and 24 minutes.
Edwin
