Finding a Fraction that Equals an Unending Number

Date: 07/08/2003 at 10:21:40
From: Chase
Subject: Finding a fraction that equals an unending number

Is there a way to find if there is a fraction that equals 0.777777...?

Chase

Date: 07/08/2003 at 11:07:44
From: Doctor Mike
Subject: Re: Finding a fraction that equals an unending number

Hi Chase, 
  
There is a commonly used way to find out what the fraction form is. Do 
you remember that multiplying a number by 10 moves the decimal point 
one place to the right? If you call the number you are interested in 
N, then ten times N is 7.7777....  

Now, here is the clever part.  7.7777.... is your original number N 
plus 7. That is, 

   7.7777....  =  7 + .7777....  =  7+N  
  
What is important about this is it gives you an equation for N.
The equation is

   10*N = 7+N  

Probably you have not done much algebra equation solving yet in
school, but some of the most useful algebra is not that hard.

When you have an equation like this, it says that the "10 times N" on 
the left and the "7 plus N" on the right are EXACTLY the same thing.  
So, if they are the same thing, you should be able to DO the same to 
both sides and still wind up with a true equation. 
  
As you get more algebra experience, you learn the most useful things 
to do to help solve the equation. My experience tells me to subtract 
the unknown N from both sides.
  
   9*N = 7  

That is, 9 times N equals 7. Do you see how that happened on both 
sides of the equation? We are very Very VERY close to the answer to 
your question. Can you see how to get from the equation 9N=7 to an 
equation of the form N=something? That is what I want you to think 
about. Remember my hint about how you can do whatever you want to BOTH 
sides of an equation, and get another equation that is true. What 
could you do with each side of 9N=7 so that the left side is just N by 
itself? Then, the right side will be your answer. 
  
Good luck.  Write back if you need more help.
   
- Doctor Mike, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/08/2003 at 11:11:40
From: Doctor Sarah
Subject: Re: Finding a fraction that equals an unending number

Hi Chase - thanks for writing to Dr. Math.

For some more help, see the Dr. Math archives:

   Changing a Repeating Decimal to a Fraction
   http://mathforum.org/library/drmath/view/61257.html 

   Fractions and Repeating/Recurring Decimals
   http://mathforum.org/library/drmath/sets/select/
dm_repeat_decimal.html

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