Solving a Typical Rate-Time-Distance Problem

Date: 09/02/2005 at 20:14:45
From: Daniela
Subject: I need help with figuring out word problems

I need help with figuring out word problems.  I just don't understand
them!  I have this problem:
 
You are taking part in a charity walk-a-thon where you can either walk 
or run.  You walk at 4 km per hour and run 8 km per hour.  The
walk-a-thon lasts 3 hours.  Money is raised based on the total  
distance you travel in 3 hours.  Your sponsors donate $15 for each 
kilometer you travel.  Write an expression that gives the total amount 
of money raised.  Evaluate the expression if you walk for 2 hours and 
run for 1 hour.  

The most difficult part about problems like this is what do I do 
first?  When I read the question, I don't understand what they are 
asking me and from there I give up.  In this problem they ask me to
find the expression, but I don't know how to set it up and I don't 
understand what is going to be my variable in the expression. 

Well, if I walk for 2 hours, then I must travel 8 km because I 
multiply 4 km/h by 2 hrs.  Then, I did 8 km/h multiplied by 1 hour, 
and find that I will run 8 km.  After that, I added the 8's together
to find the whole distance and I got 16 km.  Then, I took that 16 km
and multiplied it by $15 to find the amount of money raised...I got 
$240.  I'm pretty sure that this is the right answer, but now I don't 
know how to make it into an expression, so that when I substitute 
2 hours and 1 hour for the variable I get $240.

Date: 09/02/2005 at 21:34:37
From: Doctor Peterson
Subject: Re: I need help with figuring out word problems

Hi, Daniela.

A lot of students have trouble with word problems. Let's see what I
can do for you.

Probably the first thing to do is just to make sure you understand the
problem--what is going on, and how are the various parts related?  I
like to write down the data in some orderly fashion, like this:

  Walk: 4 km/hr
  Run:  8 km/hr

  Time: 3 hr (total)

  Money earned: $15 per km

Next, look for the unknown(s).  What don't you know, that you would
need to know in order to answer the question (in this case, how much
money do you earn)?  In the first part of the problem, you don't know
how long you walked and how long you ran; you are asked to find an
expression that will tell you what you earned IF you know these 
things.  Now, we could either define TWO variables (for these two
numbers), or we can notice that the two times are related by the fact
that the total is 3 hr.  If you walk 2 hr, you HAVE TO run 1 hr.  So
let's arbitrarily choose the time you walk as the unknown.  (We could
have chosen the running time instead.)

  Let w = time walked (in hours)

I always state the unit used, because that saves a lot of trouble!

Now, you've already done one of the things I like to suggest: walk
through the calculation you'd do IF you knew the values.  Since they
gave you specific values to try, you tried them before making an
expression.  That's fine: it gives you a practice run before you 
actually start the algebra.

So what did you do?  You took the time walked and multiplied it by 4 
to get the distance walked, and did the same with the time and 
distance run.  Then you added those to get the total distance, and 
multiplied by 15 to get the money earned.  Great job there!  I'll put 
that down in an orderly way:

           time * speed  = distance
  walking  2 hr  4 km/hr   2*4 = 8 km
  running  1 hr  8 km/hr   1*8 = 8 km

  total                         16 km * $15/km = $240

Now all we have to do is to do just the same thing using variables.
Remember that the walking time is w hours; the running time is what's
left of the 3 total hours, or 3-w hours.

           time  *  speed  = distance
  walking   w hr   4 km/hr   4w km
  running  3-w hr  8 km/hr   8(3-w) km

Do you see what I've done so far?  I replaced your 2 hr and 1 hr with
the expressions w and (3-w).  Now I have to add those expressions, 
then multiply the product by 15.  Can you finish, and write up the 
final expression?

Once you've got that, try replacing w with 2 and see if you get 240.
That will be a good check, besides answering the last part of the
question.

If you have any further questions, feel free to write back.

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