SOLUTION: It takes Frank 45 minutes to shelve the returned books at the library. It takes Erika 30 minutes to do the same job. if Frank and Erika work together, how quickly can they shelve t

Two ways:

First way:  In your head

The LCM of their times, 45 minutes and 30 minutes, is 90 minutes
So in 90 minutes Frank does 2 jobs, while Erika does 3 jobs.
So together they can do 2+3 or 5 jobs in 90 minutes
So they can do 1 job in 1/5 of 90 or 18 minutes. 

Second way.  By algebra:

Make this chart with x for the time for both to do 1 job.
Put in the times for the two workers to do the job separately
and put 1 for the number of jobs:

       number of jobs    time in minutes   rate in jobs/minute
Frank         1                45
Erika         1                30
Both          1                 x

Now fill in the column on the right by dividing number of jobs 
by the number of minutes

       number of jobs    time in minutes   rate in jobs/minute
Frank         1                45                 1/45
Erika         1                30                 1/30
Both          1                 x                 1/x

To get the equation:

           Frank's rate + Edith's rate = The combined rate for both
                 1/45   +    1/30      =       1/x

Clear of fractions by multiplying through by 90x

                               2x + 3x = 90    
                                    5x = 90
                                     x = 18

Edwin