Solving Multi-Step Problems

Date: 11/01/2003 at 18:42:52
From: Andrea
Subject: Distance and time

Mary and Patty leave their house at 2 o'clock, following the same
route.  Patty walks at a steady pace of 4 mph.  Mary walks at a 
leisurely rate of 2 mph for 2 hours and then jogs at 6 mph.  When will 
Mary catch up with Patty?

I am trying to help my 6th grade son solve this and am stumped as to 
how to go about doing it, because it doesn't say how long Patty is
walking. 

Date: 11/02/2003 at 22:10:43
From: Doctor Ian
Subject: Re: Distance and time

Hi Andrea,

Let's see if a diagram can help. 

           4 mph
  ------------------------->M
  ---------->==============>P
   2 mph         6 mph
   2 hrs

If Patty walked at 2 mph for 2 hours, she walked 4 miles, right?  So
we can add that piece of information to our diagram:

           4 mph
  ------------------------->M
  ---------->==============>P
   2 mph         6 mph
   2 hrs
   4 mi

(What did the diagram do for us here?  It let us see that we had two
out of three quantities--distance, rate, and time--during one stretch 
of Patty's trip.  As soon as we have two of these, we can always get 
the third.  And having all three increases our flexibility, as we'll 
see in a moment.)

Now, during those two hours, how far did Mary walk?  8 miles, right? 
So let's divide Mary's trip into two parts:

   8 mi
   2 hrs
   4 mph            4 mph
  ---------------->-------->M
  ------->=================>P
   2 mph         6 mph
   2 hrs
   4 mi
         |--------|
          8-4=4 mi

Now we have a simpler problem!  What do I mean by that?  Well, if we
ignore the first two hours, what we have is this:

    Mary has a head start of 4 miles. 
    She walks at 4 miles per hour. 
    Patty jogs at 6 miles per hour.
    How long will it take Patty to catch up? 

Once you find the time Patty needs to catch up, add 2 hours to it. 
That will give you the total time that Mary walks.  Since you know her
speed the whole way, you can use that to find the total distance.

Does this make sense? 

The important lesson here is that as your son progresses in math, 
there will be fewer and fewer situations in which he'll be able to
solve a problem by writing down a single equation or formula and
plugging in the numbers given.  In general, the way he'll solve
problems is to try to reduce them to simpler problems, as we've done
here.  

It's a little like trying to win baseball games by hitting lots of
singles and doubles, instead of relying on home runs. 

I hope this helps.  Write back if you have more questions about this,
or anything else. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/