Classic Rate, Time, Distance Problem

Date: 09/30/2004 at 00:43:03
From: Cindy
Subject: rate, time and distance

If Joe travels at 50 mph, he arrives at a certain destination 20 
minutes faster than if he travels at 45 mph.  How far does he travel?

I know that rate times time equals distance.  I even know that the 
answer is 150 miles, but I cannot figure out how to write an equation
to make it work.  Please help!!!

Date: 09/30/2004 at 07:27:53
From: Doctor Wilko
Subject: Re: rate, time and distance

Hi Cindy,

Thanks for writing to Dr. Math!

I found a nice way to approach these kinds of problems.  First, I try
to draw a picture of what is going on, and then I try to organize the
data in a way that makes it easier to understand.

As you suggested, we are going to be using the equation:
 
  Distance = Rate * Time
      D    =   R  *   T

Since you are given the rate in 'miles per hour' (mph), then your 
time should be in 'hours' so that the units are all the same.  One key
to solving this problem is to note that 20 minutes is the same as
20/60 of an hour which reduces to 1/3 of an hour.

First, draw a picture:

  Joe drives at 45 mph:

  ---------------> [drives this distance in x hours]
  

  Joe drives at 50 mph:

  ---------------> [drives the same distance in (x - 1/3) hours]


The pictures help me organize my thoughts and visualize the situation 
better.

Now let's organize the data:

 - You know two rates at which Joe can drive the unknown distance:
   45 mph and 50 mph.

 - You don't know what time it takes to drive the distance at 45 mph,  
   but you DO know that it takes 1/3 hour less to drive the same 
   distance at 50 mph.

   Let's call x the time in hours it takes to drive the unknown 
   distance at 45 mph.  From this, if it takes x hours to drive the 
   unknown distance at 45 mph, you can conclude that it takes
   (x - 1/3) hours to drive the distance at 50 mph.

 - You want to know the distance in miles that Joe travels.

I'll put this into an organized format below.  I get distance from 
the formula that D = R*T.

 ---------------------------------------------------------------
|          |     Time            Rate          Distance         |
 ---------------------------------------------------------------|
| 45 mph   |     x hours          45 mph        45x miles       |
|----------|----------------------------------------------------|
| 50 mph   |   (x - 1/3) hours    50 mph      50(x - 1/3) miles |
 ---------------------------------------------------------------

To find x, note that Joe is only driving one distance, but varying his
speed (rate) each time, so you know the distances above are equal.  If
you set them equal you can find x, which is the time in hours to drive
the unknown distance at 45 mph.  Once you get x, plug it back into 
either of the distances above to get the total distance in miles 
that Joe travels.

Does this help?  Please write back if you need anything else.

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