Getting the Job Done

Date: 12/17/96 at 14:59:02
From: Jerrad Evans
Subject: Solving a problem

Our teacher gave us a worksheet with a question on it that I don't 
seem to be able to solve. I've tried to make some equations out of it, 
but there doesn't seem to be one (at least nothing that I can see): 

Two typists share a job.  The second typist begins working one hour
after the first.  Three hours after the first typist has begun 
working, only 11/20 of the job has been completed.  When the job is 
finished, it turns out that each typist has done exactly half of the 
work.  How many hours would it take each typist, working alone, to 
complete the job?

Could you help?  Thanks!

Jerrad Evans

Date: 12/17/96 at 18:58:39
From: Doctor Wilkinson
Subject: Re: Solving a problem

What you're really being asked is how fast each typist types.  That
is, to answer the question "how long would this typist take to finish 
this job working alone" you need to know how much of this job the 
typist would get done in an hour.  For example, if he or she takes 
4 hours to finish the job, then in 1 hour he or she finishes 1/4 of 
the job.

These rates are easier to work with than the times.  In most problems
involving time, you may want to move back and forth between looking
at rates and looking at periods of time.

After that introduction, let's let:

   a = amount of job first typist can do in an hour
   b = amount of job second typist can do in an hour

Another number we don't know is T, the number of hours the typists
took to finish the job.

So we've got three unknowns, and we're going to need three equations.

Do we have three pieces of information we can get equations out of?

We have the curious piece of information that the first typist
started an hour ahead of the second typist.  What this means is that
after t hours, where t is at least 1, the amount that has been done
by the first typist is (at), and the amount that has been done by the
second typist is b(t-1), because the second typist has worked an hour
less than the first.

The next thing we know is that after three hours, 11/20 of the job
has been done.  At that time the first typist has done an amount 3a
and the second typist has done an amount 2b, so the total is

        3a + 2b = 11/20

That's one equation.  Two more to go!

We're also given that when the job is finished, each typist has done
exactly half of it. So this involves our unknown time T, the time
to finish the job. The first typist has done half of it, so

        aT = 1/2

and the second typist has also done half of it, so

        b(T-1) = 1/2.

And there we have the three equations.

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/