Swimming Lengths

Date: 03/05/97 at 12:42:49
From: Sara Aldrich
Subject: whoba@webtv.net

Together, Raoul, Tammy, and Nathan swam 39 lengths of the pool. Each 
swam the same number of lengths. How many lengths did each swim?    

Dr. Math, how would I figure this problem out?  

Sara Aldrich

Date: 03/06/97 at 01:29:39
From: Doctor Mike
Subject: Re: whoba@webtv.net

Dear Sara,
  
This is a very nice beginning problem in algebra. There is an unknown 
number "N" that you must find which we will use to represent the 
number of laps that each person swam. We can use the same symbol (or 
variable, as it is usually called) to represent the lengths that each 
of the three people swam because we know that they all swam the same 
number of lengths.  We also know that the sum of the lengths that each 
person swam is 39. Here is what you know:
 
   Raoul swam N lengths
   Tammy swam N lengths
   Nathan swam N lengths
   N + N + N = 39

Do you see why these things are true?  They come directly from what we 
were told, but in order to write them down, we had to invent a name N 
for the number we don't know yet. This is a very important first step 
for an algebra problem: determine what you need to find and give it a 
name.
   
Notice that you could re-write the equation as 3*N = 39. This is 
because adding 3 N's together is the same as multiplying N by 3.  This 
brings me to another very important idea from algebra: If you have
an equation and you do **exactly** the same thing to both sides of the 
equation, then the thing you get as a result is also a true equation. 
Here is what I mean.  I choose to multiply both sides of the equation 
by the fraction 1/3 to get: 
   
     1               1 
    --- * 3 * N  =  --- * 39
     3               3 
  
Because 1/3 times 3 equals one, the left side is 1*N, or just N.  By 
arithmetic, 1/3 of 39 is 13. Putting these together, we get N = 13.  
This is a true equation because we got it by doing exactly the same to 
both sides of a true equation.  Thus we have the true result that we 
wanted, N = 13. This means that each person swam 13 lengths.  
  
There is another way to do this problem.  Once you get to the point 
where you have N + N + N = 39, you can just try some numbers to 
see if they work. If you try N = 10 you get N + N + N = 30, which is 
too small, but eventually by trial and error, you will get to N = 13 
and see that it works.  Now, this is fine for simple problems, but you 
need to practice algebra on the simple problems so you have the 
techniques well-learned for when you need them to do the very 
difficult problems.  The neat thing is that these simple little 
ideas apply very nicely to the tough problems, too.

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