How do you write the repeating decimal 0.5 as a fraction?

Question

15429 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-10-09T06:42:39

#0.55555.......=0.bar(5)=5/9#

There are two ways.

First consider #0.55555.......# as an infinite geometric series

#5/10+5/10^2+5/10^3+5/10^4+5/10^5+..................#

with first term #a+1=5/10# and common ratio #r=1/10#

Hence sum of the series is #a/(1-r)=(5/10)/(1-1/10)=(5/10)/(9/10)=5/9#

Second and easier is to consider #x=0.55555.......#

and hence #10x=5.55555..................#

and subtracting former from latter

#9x=5# i.e. #x=5/9#