When do the Cars Meet?

Date: 01/23/97 at 12:55:34
From: Jennifer
Subject: Math

Write an equation that describes this information:

Two cars, an Edsel and a Studebaker, are 635 kilometers apart. They 
start at the same time and drive toward each other. The Edsel travels 
at a rate of 70 kilometers per hour and the Studebaker travels 57 
kilometers per hour. In how many hours will the two cars meet?

I'm stuck on the whole thing.  I don't understand this kind of 
problem.

Date: 01/23/97 at 14:52:10
From: Doctor Wilkinson
Subject: Re: Math

I suppose you know the basic formula for doing this kind of problem 
(D is distance, R is rate, T is time):

   D = RT

I'll give you two ways to look at this problem.  You have two rates 
given.  The rate of the Edsel is 70 and the rate of the Studebaker 
is 57.  So the distance traveled by the Edsel in time T is 70T and 
the distance traveled by the Studebaker in time T is 57T.  

Now the question is, when do they meet?  The other piece of 
information is that they started 635 kilometers apart.  So they're 
going to meet when the distance traveled by the Edsel and the distance 
traveled by the Studebaker adds up to 635.  This gives us an equation 
which you can solve for T:

   70T + 57T = 635

Another way of looking at it is to just look at the distance between 
the two cars.  It's decreasing because the cars are driving towards 
each other.  How fast is it decreasing?  The rate of decrease is the 
sum of the speeds of the two cars.  So we get the following equation 
for the time it takes to reduce the distance from 635 to 0:

 (57 + 70)T = 635

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

Date: 01/23/97 at 20:43:39
From: Doctor Wallace
Subject: Re: Math

Hi Jennifer!

I bet you're surprised to get two answers to this one.  I originally 
composed this one, but it got munched by my computer, and before I 
could send you a new one, my colleague Dr. Wilkinson sent you his 
thoughts on the problem.  I hope my original answer helps you out too.  
Here goes:

The first thing I do when faced with any math problem is to draw a 
diagram.  Here is the diagram I drew for this problem:

E -> 70km/h                                               57km/h <- S
----------------------------------------------------------------------      
                             635 km

The E stands for Edsel, and the S for Studebaker. I listed the speeds 
of the cars, and their direction of travel. The line represents their 
path, which is 635 km long  We need to understand exactly what is 
happening in this problem. Pretend we have a stopwatch which reads 0.  
When we press the button to start the stopwatch, the cars will begin 
moving toward each other. At some point, they have to meet, and we 
will then press the button to stop the stopwatch. What we want to 
find in this problem is what time the stopwatch reads when we stop it.  
In other words, how long does it take for the two cars to meet?

A helpful relation for any problem of this type (where you have 
something like a car moving at a constant speed) is:

                     D = R x T          
                  
              Distance = Rate x Time

For example, if a car is traveling at 60 km per hour for 2 hours, it 
will have traveled 60 x 2 or 120 km at the end of the 2 hours.  (The 
rate is 60 and the time is 2.)

Now how do we solve your problem?  Well, there are two ways to go 
about it. One way uses algebra, and the other doesn't.  Since I don't 
know how old you are, or what grade you're in, or whether you know 
algebra, I'll show you both ways.  If you're not in algebra, you can 
save the other way until you get there.

Way No. 1: Make a table (without algebra)

Since your problem seems to hint that the answer will be a nice, round 
number of hours, a table seems like a good way to solve the problem.  
It also helps in understanding.  We'll make a table of the distance 
values of each car after various hours of time.  We'll start with time 
zero as our first entry.  At time zero, the cars haven't gone 
anywhere, so their distance is, of course, zero.  Notice that, 
throughout the whole table, the rate of each car stays the same.  
This is because the cars do not change speed during the trip.  

Here are the first three entries for our table:
                                                                            
                   Edsel                           Studebaker
Time (hours)   Rate       Distance             Rate           Distance           
   0           70              0                 57                 0
   1           70              70                57                57
   2           70             140                57               114

Do you see how we get each entry in the table?  We just multiply the 
rate by the time.  

Now, the big question: How do we know when the two cars meet?  Well, 
let's take a look at our diagram again.  (You can plot the cars' 
progress on the line if it helps you.)  

Pick a place on the line where you think the cars will meet.  It 
doesn't matter where.  Call it point A.  Now, the Edsel will have 
traveled from the left to point A, and the Studebaker from the right 
to point A.  Notice that wherever you put point A, if we add up the 
distance that both cars have traveled, we always get 635 km.  This is 
very important.  When the cars finally do meet, together they must 
have traveled the length of the whole path.  Individually, each car 
will only have traveled part of the whole path.  But if we add them 
up, it must equal the whole 635 km.  Okay?

Now why is this important?  Well, it means that all we have to do to 
find out whether or not the cars have met is to add up their distances 
in our table.  Have they met after 1 hour?  Well, after 1 hour, the 
Edsel has gone 70 km and the Studebaker 57 km.  70 + 57 = 127 km, so 
they have not met after 1 hour.  How about after 2 hours?  Well, after 
2 hours, the Edsel has gone 140 km and the Studebaker 114 km.  
140 + 114 = 254 km, so they have not met after 2 hours either.  You 
can finish the table and check each hour and you'll easily find out 
when the two cars meet.

Way No. 2: Algebra

To solve any algebra word problem, you need three things.  First, you 
have to understand the relations between the elements in the problem.  
Second, you have to be able to translate those relations into 
equations using symbols.  Third, you need to have the skills to 
solve those equations. Let's look at how these apply to this problem.

1. Relations

We've already seen the two important relations for this problem.  
The first is that Distance = Rate times Time.  The second is that the 
total distance traveled by both cars together once they meet is 635km.

2. Equations

This part is harder.  We have to choose some variables.  First, we'll 
choose one for the quantity we're looking for.  This is the time at 
which the two cars meet.  We'll call this t.  What are some other 
things we don't know?  Well, when the two cars meet, we don't know how 
far each has traveled individually.  So we'll call the total distance 
the Edsel travels x, and the total distance the Studebaker travels y.  
So now we can use our relation D = R x T and write:

   x = 70t   and   y = 57t

Why is this true?  Well, after t hours, each car will have traveled a 
distance equal to its rate times the time spent traveling.  Now, we 
have 3 variables here, and we can't solve our equations until we only 
have 1 variable.  So we have to rewrite our equations in terms of only 
one variable, the one we want to solve for; in this case, t, the time.  
We do that by using our other relation.

Our other relation is that the total distance traveled by both 
cars is 635 km.  Because x and y are the total distances traveled by 
each car, the total distance traveled by both cars expressed in terms 
of our variables is:

   x + y = 635       

Now notice that we already have two expressions for x and y.  In other 
words, we already know what x and y are.  As above, x is 70t and y is 
57t.  So we can substitute these into x + y = 635 and we get:

   70t + 57t = 635

Now we have an equation in one variable!

3. Solve

Now all we have to do is solve this equation and we'll get t, which 
will be the time that passed before both cars to met.  I'll leave that 
to you.  It works out to the same answer as using the table.

After you become skilled at algebra, you'll want to use this method 
instead of constructing a table.  This is because your table entries 
have to get more and more "fine-tuned" if the answer isn't a round 
number.  The cars could meet after, say 2.776 hours.  It would take a 
lot longer to "guess and check" using table entries to get that 
precise.  The algebra method would give you this precision with less 
work.

Well, that was probably a longer answer than you were expecting, but I 
hope it helps you get an understanding of this and problems like it.  
If you have any more questions, or need further help, please feel free 
to write back!

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