Question 582314
Mary and Jane could finish a job in 1 hour. After Mary worked in the job for 2 hours, Jane joined her and together they finished the job in 20 minutes. How long will each of them finish the job working alone?

*please answer the problem with 2 equations and 2 unknowns(x and y). thanks!
<pre>
Let x = the number of hours it would take Mary to finish 1 job working alone.
Let y = the number of hours it would take Jane to finish 1 job working alone.

20 minutes = 1/3 hour

Make this chart:
                          No. of jobs     No. of      Rate 
                          or fraction     hours        in
                          thereof         worker    jobs/hour
M. alone for x hours           1            x
J. alone for y hours           1            y
M. and J. together for 1 hr    1            1          1
M. alone for 2 hours                        2
M. and J. together for 1/3 hr              1/3

Fill in the rates in jobs/hour in the first two cases by dividing
jobs done by the number of hours:

                          No. of jobs     No. of      Rate 
                          or fraction     hours        in
                          thereof         worker    jobs/hour
M. alone for x hours           1            x         1/x
J. alone for y hours           1            y         1/y
M. and J. together for 1 hr    1            1          1
M. alone for 2 hours                        2
M. and J. together for 1/3 hr              1/3

We can now fill in M's rate working alone for 2 hours as the same
rate 1/x.  We can also fill in the rate of them working together for
1/3 hour as the same 1 job/hour rate as when they work for 1 hour.


                          No. of jobs     No. of      Rate 
                          or fraction     hours        in
                          thereof         worker    jobs/hour
M. alone for x hours           1            x         1/x
J. alone for y hours           1            y         1/y
M. and J. together for 1 hr    1            1          1
M. alone for 2 hours                        2         1/x
M. and J. together for 1/3 hr              1/3         1
  
Now we can fill in the no. of jobs or fraction thereof (actually fraction
tereof) that M. did alone for 2 hours by multiplying her rate times her
time (2 hours) getting 2/x.  Similar we can fill in the fraction of a 
job that they did together in 1/3 hour by multiplying their combined 
rate 1 job/hour by 1/3 hour.

                             No. of jobs     No. of      Rate 
                             or fraction     hours        in
                             thereof         worker    jobs/hour
M. alone for x hours              1            x         1/x
J. alone for y hours              1            y         1/y
M. and J. together for 1 hr       1            1          1
M. alone for 2 hours             2/x           2         1/x
M. and J. together for 1/3 hr    1/3          1/3         1

Our two equations come from:

             {{{(matrix(4,1,

"Mary's",rate,working,alone))}}} + {{{(matrix(4,1,

"Jane's",rate,working,alone))}}} = {{{(matrix(5,1,

Their,combined,rate,working,together))}}} 

             {{{(matrix(7,1,

fraction,of_a_job,Mary, did,working,alone, for_2_hours))}}} + {{{(matrix(7,1,

fraction,of_a_job,Mary_and_Jane, did,working,together_for, expr(1/3)_hour))}}} = {{{(matrix(3,1,

1,complete, job))}}} 

or

            {{{1/x}}} + {{{1/y}}} = {{{1}}} 
            {{{2/x}}} + {{{1/3}}} = {{{1}}}

Solve that system and get x = 3 and y = {{{3/2}}} or 1.5 hours

So Mary takes 3 hourse to do the job and Jane takes 1.5 hours

Edwin</pre>