0.22 repeated = 22/100 + 22/10,000 + 22/1,000,000

Question

Express the repeating decimal as fraction in its lowest terms.

Answer 1

      Arthurs method is perfect and usually taught as the method in high school. Judging by the way you set this up, you're taking a higher level class that is asking you to do this as a geometric progression, which is another valid method. Recall the formula for the summation of an infinite geometric progression having the ratio, -1<r<1 and first term a..........is

 

S= a*1/(1-r)      so your first term is 22/100 and r =1/100

 

S= 22/100 * 1/(1-1/00)

S= 22/100 * 1divided by 99/100

S= 22/100 * 100/99          dividing by a fraction is the same as multiplying by its reciprocal 

S=22/99=2/9

 

In using this method, I would have done this problem using 2/10 + 2/100 +.....a=2/10 and r=1/10. You get the same answer. You use 2 digits if you have a 2 digit repitition like Arthur's second example. I'll do a 3 digit repitition.

 

.759759759 repeating is759/1000 + 759/1000000 + .......so a = 759/1000 and r=1/1000. so

S=759/1000 * 1divided by 999/1000

S=759/1000 * 1000/999 =759/999

Answer 2

let n=0.2222222222...

only one digit repeats in the repetend (the 2)

multiply both sides of the equation by 10 to the first power, or just 10 (because only 1 digit repeats)

10n=2.222222222...

subtract the original equation from the new equation

 

   10n=2.222222222...

-     n=0.222222222...

___________________

     9n=2 (the decimal parts are eliminated because 2-2=0,2-2=0,2-2=0, etc)

     9n/9=2/9

       n=2/9

 

here is another example:

 

0.72727272727272...

n=0.727272727272...

multiply both sides of the equation by 10 to the second power, or 10², or 100 because there are two digits in the repetend

 

100n=72.727272727272...

-

      n= 0.727272727272...

______________________

  99n=72

  99n/99=72/99

     n=72/99=8/11 in simplest form