72 [tex] \frac{1}{9} [/tex] 1÷9 = 111.... 2÷9 = 222.... and so on.
The proof is a little more difficult.
Think of all those repeating ones as a variable - let's call it n
so n= .111111......... repeating
How can we get a single one of those ones to jump across the decimal and be on the left side. We can multiply all of those ones by 10.
10n (ten times the original number) = 1.1111111 (ones still go on forever)
Now here is the interesting part. Let's take all the repeating ones in the first number we made away from the second number.
10n = 1. 1111111......
- n = . 1111111....
9n = 1 (all of the repeaters are gone and only the one we moved to the left
of the decimal is left)
Now let's divide by 9 to get n by itself
9n = 1
9 9
And voila! n = 1/9
So to repeat 72.111... written as a fraction is 72[tex] \frac{1}{9} [/tex]
