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From Infinite Decimals to Mixed Fractions

Date: 10/09/98 at 19:00:50
From: Jessi Michelle
Subject: A challenge from a teacher

I was given a challenging problem by my math teacher just to see if I 
could find a way to figure it out. The problem is to write this 
fraction as a mixed number:

   .131313... + .555...
   .161616... - .222...

I think that the top can first be simplified to .6868.... After that 
I'm stuck because of the 2 away from 1 thing in a repeating decimal.  
It seems like it would be easier to change the decimals to fractions, 
but I don't know how. Help!

Date: 10/13/98 at 11:10:28
From: Doctor Nick
Subject: Re: A challenge from a teacher

Hi Jessi -

Yes, converting the decimals to fraction is the way to go. There is a 
neat trick you can use to convert repeating fractions like this. I'll 
give you a couple of examples.

The main trick is to multiply the number by a power of 10 (10, 100, 
1000, 10000, etc.) that you pick to get the right effect. For 
instance, if x = 0.55555..., then 10x = 5.5555555..., and so 
10x - x = 5. That is, 9x = 5, so x = 5/9. The trick is to pick a power 
of 10 that makes the repeating pattern in the decimal expansion "line 
up" so it disappears when you subtract. Here is another example: if 
x = 0.1616161616..., then 100x = 16.16161616161616..., and so 
100x - x = 16, Then 99x = 16, which implies x = 16/99.

This works even with decimals that don't repeat right away:

         x =  0.71232323232323...
      100x = 71.23232323232323...
  100x - x = 99x = 71.23 - 0.71 = 70.52
         x = 70.52/99 = 7052/9900 = 1763/2475
That's a little more complicated than the other cases, but the method 
still works.

Now, convert all the decimals in your problem to fractions, and 
simplify to a single fraction, and you'll have it.

Have fun,

- Doctor Nick, The Math Forum   
Associated Topics:
High School Sequences, Series
Middle School Fractions

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