SOLUTION: A, B, and C can finish a job in 6 days. If B and C work together, the job will take 9 days; if A and C work together, the job will take 8 days. In how many days can each man work

Algebra ->  Rate-of-work-word-problems -> SOLUTION: A, B, and C can finish a job in 6 days. If B and C work together, the job will take 9 days; if A and C work together, the job will take 8 days. In how many days can each man work      Log On


   

Question 323669: A, B, and C can finish a job in 6 days. If B and C work together, the job will take 9 days; if A and C work together, the job will take 8 days. In how many days can each man working alone do the job?
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
A, B, and C can finish a job in 6 days. If B and C work together, the job will take 9 days; if A and C work together, the job will take 8 days. In how many days can each man working alone do the job?

Make this chart:

                   Number of jobs    Time in      Rate in
                    finished          days        jobs/day
A working alone
B working alone
C working alone
A, B & C together
B and C together
A and C together

Let A working alone be able to do 1 job in x days.
Let B working alone be able to do 1 job in y days.
Let C working alone be able to do 1 job in z days. 

So fill in x, y, and z for their times alone, and fill in 6, 9,
and 8 for the times for the various combinations or workers
given:


                   Number of jobs    Time in      Rate in
                    finished          days        jobs/day
A working alone                        x
B working alone                        y
C working alone                        z
A, B & C together                      6
B and C together                       9
A and C together                       8

In every case we are talking about doing exactly 1 job, so
fill in 1's for the number of jobs in every case:

                   Number of jobs    Time in      Rate in
                    finished          days        jobs/day
A working alone         1              x
B working alone         1              y
C working alone         1              z
A, B & C together       1              6
B and C together        1              9
A and C together        1              8

Fill in the rates by dividing the number of jobs by the number
of days:

                   Number of jobs    Time in      Rate in
                    finished          days        jobs/day
A working alone         1              x          1%2Fx 
B working alone         1              y          1%2Fy
C working alone         1              z          1%2Fz
A, B & C together       1              6          1%2F6
B and C together        1              9          1%2F9 
A and C together        1              8          1%2F8

Now form three equations from

       A's rate + B's rate + C's rate = A,B, and C's rate together

                  B's rate + C's rate = B and C's rate together
 
                  A's rate + C's rate = A and C's rate together                  

So we have this system:

system%281%2Fx%2B1%2Fy%2B1%2Fz=1%2F6%2C+1%2Fy%2B1%2Fz=1%2F9%2C+1%2Fx%2B1%2Fz=1%2F8%29
 

Subtract the second equation from the first equation:

%281%2Fx%2B1%2Fy%2B1%2Fz%29-%281%2Fy%2B1%2Fz%29=1%2F6-1%2F9
1%2Fx%2B1%2Fy%2B1%2Fz-1%2Fy-1%2Fz=3%2F18-2%2F18
1%2Fx=1%2F18
x=18

So it will take A 18 days to do the job working alone.


Subtract the third equation from the first equation:

%281%2Fx%2B1%2Fy%2B1%2Fz%29-%281%2Fx%2B1%2Fz%29=1%2F6-1%2F8
1%2Fx%2B1%2Fy%2B1%2Fz-1%2Fx-1%2Fz=4%2F24-3%2F24
1%2Fy=1%2F24
y=24

So it will take B 24 days to do the job working alone.

To find z, substitute for x and y

1%2Fx%2B1%2Fz=1%2F8%29
1%2F18%2B1%2Fz=1%2F8
Multiply every term by 72z to clear of fractions
%2872z%29%281%2F18%29%2B%2872z%29%281%2Fz%29=%2872z%29%281%2F8%29
4z%2B72=9z
72=5z
72%2F5=z
14.4=z

So it will take C 14.4 days to finish the job working alone.

Edwin