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what is this number as a fraction? show work

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1.2444444444444444444444444444444444444444444444444444444444 and so on

Oct 2nd, 2014

1.24444444444444444444444444444444444...

= 1.2 + 0.44444444444444444444444444444...

= 6/5 + 0.44444444444444444444444444444... (*)

Now 0.44444444444444444444444444444... = 4/10 + 4/ 100 + 4/1000 + ,,, is a geometric series with 

a= 0.4 and r = 0.1 and has sum to infinity = a/ (1 -r ) 

                                                                   = 0.4/ ( 1- 0.1)

                                                                   =  0.4/ 0.9

                                                                   = 4/9          

Hence the required fraction  from (*)   = 6/5 + 4/9 = 54/45 + 20/45 = 74/45       

Oct 2nd, 2014

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