Passing Mr. Jones

Date: 09/30/2002 at 23:06:45
From: Maliakah
Subject: Distance and Time

Here's the problem. 

Mr. Jones lives 50 miles away from you. You both leave home at 5:00 
and drive toward each other. Mr. Jones drives at 35 mph and you drive 
at 40 mph. At what time will you pass Mr. Jones on the road? 

My daughter could not figure this out. We tried the rate times the 
time, which I thought was wrong. We weren't sure how to do this 
problem. I wanted to know the answer for myself, because this one had 
me puzzled.

Date: 10/01/2002 at 10:39:06
From: Doctor Ian
Subject: Re: Distance and Time

Hi Maliakah,

Suppose I have a REALLY big collapsible pole that reaches from your 
car to Jones's car. At 5:00, the pole is stretched to a distance of 
50 miles. 

Now, let's say that you drive for 12 minutes, or 1/5 of an hour. How 
long is the pole now? Well, you've been driving at 40 mph for 1/5 of 
an hour, which means you've gone 8 miles. And Jones has been driving 
at 35 mph for 1/5 of an hour, which means that he's gone 7 miles. 

So the pole is 15 miles shorter than it was at 5:00. 

Now, let's think about this for a moment. If we let the pole keep 
shrinking at that rate, after an hour it would shrink 5 times 15 
miles, or 75 miles. So the pole is shrinking at a 'speed' of 75 miles 
per hour... which is just the sum of your speed and Jones's speed.  

Does that make sense? 

Now, think about this:  As long as the pole shrinks at that same rate, 
you'll get the same answer, right?  That is, it will take the same 
amount of time for the cars to meet (which happens when the length of 
the pole becomes zero.)  

So all of these versions of the problem are the same:

     Your speed          Jones's speed
     ----------          -------------
        40                    35
        41                    34
        42                    33
        43                    32
         .                     .
         .                     .
        73                     2
        74                     1
        75                     0

(Note that if we change the speed, you and Jones will meet in 
different PLACES; but you'll always meet after the same amount of TIME 
has passed, and the problem is just asking about time.)

So instead of trying to solve the original problem, you can get the 
same answer by solving this simpler problem:

   How long would it take you to drive 50 miles at 75 miles per hour? 

Does this help? 

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