SOLUTION: Joe takes 30 minutes longer than Mary to do a task. If together they complete the same task in 8 minutes, then how long does it take for Mary to finish the task by herself?

Let m be the time for Mary to complete the job alone;

then the time for Joe is (x+30) minutes.


In each minute, Mary makes    part of the job, 

while Joe makes    part of the job,

so together they make   +   part of the job per minute.



From the other side, the problem says that their combined rate of work is    of the job per minute.


It gives us this equation

     +  = .


It is the basic equation for the problem.
As soon as you get it (and as soon as you do understand on how to get it),
the setup is completed.



To solve the equation, multiply both sides by 8m*(m+30).  You will get then

    8(m+30) + 8m = m*(m+30)

    8m + 240 + 8m = m^2 + 30m

    m^2 + 14m - 240 = 0

    (m+24)*(m-10) = 0.             (after factoring left side)


So, you get two roots,  m= -24  and  m= 10, but only the positive root m= 10 minutes is meaningful.


ANSWER.  Mary needs 10 minutes to complete the job working alone.