Distance/Time

Date: 12/8/95 at 15:39:32
From: Anonymous
Subject: word problems

Bernice is cycling around a track at 15 mph.  Betty starts at the same 
time, but only goes 12 mph.  How many minutes after they start will 
Bernice pass Betty if the track is 1/2 mile long?  They are moving in 
the same direction.  

Please include a variety of other d=rt to help us out on the process of 
using formulas to solve for the missing element.  Thank you for your 
time and your wisdom.

Mr. Ames

Date: 5/31/96 at 15:49:3
From: Doctor Gary
Subject: Re: word problems

Dear Mr. Ames;

Although distance is in fact the product of rate and time, I much prefer 
to be absolutely certain that students understand the DEFINITION of rate 
before encouraging (or even permitting) them to use any formulas.  A 
rate of speed, as the name implies, is equal to the quotient of distance 
divided by time.  Naturally, your equation of:

  d  =  rt

follows naturally from the definition of:

  r  =  d/t.

Now, on to your story problem, which is an excellent 
opportunity to get students talking and thinking about 
math, so that they can better understand that formulas 
are nothing more than ways of expressing what is 
already known to be true.  

I'd make a game of it, asking the following questions:

  If Bernice goes 15 mph, how many miles will she 
travel in an hour?

  How many times around the 1/2 mile track is that?

  If Betty goes 12 mph, how many miles will she travel in an hour?

  How many times around the 1/2 mile track is that?

  Once students understand that, they should 
appreciate that Bernice "laps" (i.e.  catches up to 
and passes, or "meets") Betty 6 times every hour, or 
once every 10 minutes.

  Here are two related questions:  How long does it take 
Bruce to circle the same track once at the rate of 
3 mph?   How fast would Dave have to walk around a 
one-mile track in order to circle it once every twenty 
minutes? 

-Doctor Gary,  The Math Forum