Running Problem (Time and Distance)

Date: 8/31/96 at 18:53:20
From: Anonymous
Subject: Time and distance problems

If Lucy runs 3 mph and Jan runs 7 mph, when will Jan catch up with 
Lucy if she gives Lucy a head start?

I think I solved it with a proportion, but I wonder if this is the 
best approach.  Here is my "solution":

   3 mph : 7mph = x minutes : x + 10 minutes
3x + 30 minutes = 7x
     30 minutes = 4x
    7.5 minutes = x
   17.5 minutes = x + 10 minutes

Jan will catch up with Lucy 17.5 minutes after Lucy begins to run.

Date: 9/5/96 at 10:3:26
From: Doctor Jerry
Subject: Re: Time and distance problems

From your answer, the head start must be 10 minutes, or 1/6 hour.  
Right?  

Solving problems by proportion is a very nice method, but you must be 
certain that the method fits. It doesn't fit here, because of the head 
start. 

Imagine a number line, with both girls starting to run to the right 
from the 0 mark. At the time Jan starts, Lucy will be at the 
3*(10/60) = 1/2 mile mark. Let a clock start running just as Lucy 
reaches the 1/2 mile mark.  Let t be the clock time in hours.  Lucy 
will be at the 1/2 + 3t mark and Jan will be at the 7t mark when the 
clock reads t hours.  

To find out when Jan catches up, we look for the t for which

   1/2 + 3t = 7t.

I'm sure you can solve this equation for t.  Remember that t is 
measured in hours.

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