How Can Division Result in an Increase?

Date: 10/29/2003 at 17:03:55
From: Jen
Subject: division

What is the logic behind dividing an integer by a decimal and getting 
a larger number?  For example, 

  100 / 0.9185 = 108.87

I just can't understand the logic!  It seems like the answer should be 
a smaller number.

Date: 10/29/2003 at 23:30:12
From: Doctor Ian
Subject: Re: division

Hi Jen,

Well, remember that a decimal is just a fraction.  For example, 

  0.23   = 23/100

  0.1542 = 1542/10,000

and so on.  So what happens when we divide by a fraction?  We multiply
by the reciprocal:

    8         3
  ----- = 8 * -
   2/3        2

So if we divide by something where the denominator is smaller than the
numerator, we'll end up multiplying by something where the numerator
is larger than the denominator.  Does that make sense? 

Here's another way to think about it, using the idea that division
means breaking things into pieces.  Suppose I have something 6 inches
long, and I divide it into pieces that are 2 inches long.  How many
pieces do I get? 

     1    
   +----+----+ +----+----+ +----+----+ 
   |    |    | |    |    | |    |    |         
   +----+----+ +----+----+ +----+----+ 

            6 / 2 = 3

Okay, now what if I divide it into pieces that are only 1/2 inch long?  
I'm going to end up with 12 pieces, right? 

    1
    -
    2
   +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+  
   |  | |  | |  | |  | |  | |  | |  | |  | |  | |  | |  | |  |
   +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+  
   
            6 / (1/2) = 12

So we divided by something less than one, and ended up with more than
our original result.  

But the clearest way to see the logic of this is to remember what we
_mean_ by division.  That is, a division is just another way of
representing a multiplication.  For example, when we say that

  3 * 4 = 12

two other ways to say exactly the same thing are

  12 / 3 = 4         and          12 / 4 = 3

So suppose we have something like 
 
  6 / (1/2) = ?

This is the same as saying that 

          6 = ? * (1/2)

Now, if this is true, then the value of '?' had better be larger than
6, right? 

Does this help?

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