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
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
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:
Subtract the second equation from the first equation:
So it will take A 18 days to do the job working alone.
Subtract the third equation from the first equation:
So it will take B 24 days to do the job working alone.
To find z, substitute for x and y
Multiply every term by 72z to clear of fractions
So it will take C 14.4 days to finish the job working alone.
Edwin