Express each of the following nonterminating decimals in the form p/q in lowest terms. Consider each as an infinite series

Question

1.  0.22222...

                   2.      0.55555...

                   3.      0.292929...

                   4.      0.343434...

                   5.      2.656565...

Answer 1

1) let x = 0.222222.....

then 10x = 2.222222

so, 10x - x = 2

9x =2

or x = 2/9

2) let x = 0.55555...

then 10x = 5.55555...

10x - x = 5

or 9x =5

or x = 5/9

3) x = 0.292929...

100x = 29.292929...

100x - x = 29

99x = 29

x = 29/99

4) let x = 0.343434...

100x = 34.343434...

100x - x = 34

99x = 34

x = 34/99

5) let x = 2.656565...

100x = 265.65656565...

100x - x = 263

99x = 263

or x = 263/99