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?
<pre><b>
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/x}}} 
B working alone         1              y          {{{1/y}}}
C working alone         1              z          {{{1/z}}}
A, B & C together       1              6          {{{1/6}}}
B and C together        1              9          {{{1/9}}} 
A and C together        1              8          {{{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:

{{{system(1/x+1/y+1/z=1/6, 1/y+1/z=1/9, 1/x+1/z=1/8)}}}
 

Subtract the second equation from the first equation:

{{{(1/x+1/y+1/z)-(1/y+1/z)=1/6-1/9}}}
{{{1/x+1/y+1/z-1/y-1/z=3/18-2/18}}}
{{{1/x=1/18}}}
{{{x=18}}}

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


Subtract the third equation from the first equation:

{{{(1/x+1/y+1/z)-(1/x+1/z)=1/6-1/8}}}
{{{1/x+1/y+1/z-1/x-1/z=4/24-3/24}}}
{{{1/y=1/24}}}
{{{y=24}}}

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

To find z, substitute for x and y

{{{1/x+1/z=1/8)}}}
{{{1/18+1/z=1/8}}}
Multiply every term by 72z to clear of fractions
{{{(72z)(1/18)+(72z)(1/z)=(72z)(1/8)}}}
{{{4z+72=9z}}}
{{{72=5z}}}
{{{72/5=z}}}
{{{14.4=z}}}

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

Edwin</pre>