Fractions and Adding Zeros

Date: 08/26/2001 at 16:11:01
From: cassandra
Subject: Fractions

I am having problems with doing fractions on the calculator. Is there 
a way to do it? I can't do long division, like 23/34, and how to get 
the least common denominator.

Date: 09/04/2001 at 11:35:08
From: Doctor Ian
Subject: Re: Fractions

Hi Cassandra,

Let's look at the problem you mentioned, which is how to divide 23 by 
34 (or, to put it another way, how to divide 23 into 34 parts):

     ____
  34 ) 23

How many times does 34 go into 23?  Well, it doesn't go any times. 
However, it goes into 230 some number of times:

         ?
     _____
  34 ) 230

How many times? Well, the only way to find out is to guess. If you've 
memorized your times tables, you know that 3*8 is 24, so 30*8 would be 
240, so 8 is probably too high. What about 7?

  7 * 34 = 7 * (30 + 4)

         = 7*30 + 7*4

         = 210 + 28

         = 238

which is still too big.  How about 6?

  6 * 34 = 6*30 + 6*4              

         = 180 + 24

         = 204

So 34 goes into 230 at least 6 times:

         6
     _____
  34 ) 230
       204
       ---
        26        230/34 = 6 + 26/34

Now we're back in the same boat, trying to find out what 26/34 is.  

34 doesn't go into 26, but it will go into 260.  We already saw that 
7*34 is 238, so we know that it goes at least 7 times:

         67
     ______
  34 ) 2300
       204
       ---
        260
        238
        ---
         22

We can keep going on like this, and eventually, one of two things will 
happen. Either we'll end up with no remainder, 

       375
    ______
  8 ) 3000
      24
      --
       60
       56
       --
        40
        40
        --
         0        Done!

or we'll end up with a remainder that we've already seen,

         39
     _______
  33 ) 1300
        99
       ---
        310
        297
        ---
         13      We've already seen this!

in which case we can stop, because we know that we're just going to 
keep repeating the same calculations over and over again:

         39393...  (the sequence '39' repeats forever)
     _________
  33 ) 1300000
        99
       ---
        310
        297
        ---
         130
          99
         ---
          310
          297
          ---
           130
            99

Okay, but what's really going on here? Why can we just keep adding 
zeros? Don't we need a decimal point in here somewhere?

In fact, we do, but it certainly makes life easier if we forget about 
it until the end.  

Let's look at what happened with 3/8. Instead of taking 8 into 3, we 
changed the 3 to 3000.  And we found that 

  3000 / 8 = 375

How does this help?  Well, we can multiply both sides of an equation 
by the same thing without changing the truth of the equation, right?  
Let's try multiplying both sides of _this_ equation by 1/10.  The easy 
way to do that is to move the decimal point to the left by one place - 
for example, 375 multiplied by 1/10 is 37.5.  So

   3000
   ---- = 375
     8

  300.0
  ----- = 37.5             Multiply everything by 1/10.
    8


  30.00 
  ----- = 3.75             Again...
    8

  3.000
  ----- = 0.375            Again...
    8

Now we're back to our original problem, and we've found the answer. 

Now, there is a quicker way to do this, which is to leave the decimal 
point where it is, which gives us the answer including the decimal 
point without having to go through this final step:

      0.375
    _______
  8 ) 3.000
      24
      --
       60
       56
       --
        40
        40
        --
         0        

But if you don't see _why_ this 'adding more zeros' stuff works, then 
this looks pretty much like some kind of magic spell.  And the 
_reason_ that it works is that we're using the same multiplication 
trick to try to change our original problem into one that we can solve 
using only integers:

  3   30    1
  - = -- * --
  8    8   10

     
      300    1
    = --- * ---
       8    100

      3000     1
    = ---- * ----
       8     1000

       ^      ^_____________
       |                    |
    We can solve            |
    this problem 
    with integers... and then move the decimal point to
                     get the final answer.

    = 375 * (1/1000)

    = 0.375

In other words, we use multiplication on both sides of the equation to 
turn one messy problem into two simpler problems, whose solutions we 
can easily combine to get the solution to the original problem. 

As you take more mathematics, you'll see this pattern come up over and 
over again, because mathematicians love to solve hard problems by 
turning them into easy ones. In fact, when you get to algebra, _most_ 
of what you'll be learning is how to do this.

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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