Repeating decimals

I consider myself to be pretty good at math...I have a thorough math background, I've homeschooled my kids in math (so it's currently in my thoughts!), I am a math tutor, and I am learning calculus through self-study.

SO -- I would've thought that I would know how to do this already, but it's new to me!  I learned it actually when I was tutoring a student (haha!!)--and then I came home and taught it to my kids :)

The question is how to convert a repeating decimal into its fraction. Now we probably all already "know" some repeatig decimal conversions (like .3333... and .6666....) but at least for me, I did not know how to convert them to their fraction--so here it is!

If you have a repeating decimal, we'll use .333... first (sorry I'm not writing it with the line over top, I can't figure out how to type it that way on blogger!)

We will set x=.333...

Okay, so you look at the decimal and see how many digits are repeating. In this case, it is one. You will now multiply both sides of the equation by a multiple of ten with that many zeroes after it (so for one repeat, you multiply it by 10).

so now we have

x=.333...
10x=3.333...

now we will subtract the original from the multiple of ten one.

          10x = 3.333...
    -         x= 0.333....
             9x=3

Okay, so now we solve for x by dividing both sides of the equation by 9, which gives us the fraction
3/9, which we reduce to 1/3!

Cool, huh?

Let's try a number with a longer repeat. How about .512512...

We'll  look at the repeat, and it is three digits, so our multiplier will be 1000.

so x=.512512... and 1000x=512.512...

subtract:

1000x=512.512...
 -      x=   0.512
   999x=512

divide both sides by 999 and you get 512/999. 

Pretty easy, huh? :) So there's your math lesson for the day :) :)