Question

What is `0.5` with the 5 recurring as a fraction? `0.555555... = 0.bar5`

can the explanation please be simple

Answer 1

`5/9`

Explanation:

`"we require to create 2 equations with the recurring decimal"`

`"note that "0.5555-=0.bar(5)larrcolor(blue)"bar represents value recurring"`

`"let "x=0.bar(5)to(1)`

`"then "10x=5.bar(5)to(2)`

`"both equations have the recurring value after the decimal"`
`"point"`

`"subtracting "(1)" from "(2)" gives"`

`10x-x=5.bar(5)-0.bar(5)`

`rArr9x=5`

`rArrx=5/9larrcolor(blue)"required fraction"`

Answer 2

`0.bar5 = 5/9`

Explanation:

There is a nifty short cut method to change recurring decimals into fractions:

If all the digits recur

Write a fraction as :

`("the recurring digit(s)")/( 9" for each recurring digit")`

Then simplify if possible to get simplest form.

`0.55555..... = 0.bar5 = 5/9`

`0.272727... = 0.bar(27)= 27/99 = 3/11`

`3.bar(732) = 3 732/999= 3 244/333`

If only some digits recur

Write a fraction as:

`("all the digits - non-recurring digits")/(9 " for each recurring " and 0 " for each non-recurring digit")`

`0.654444... = 0.65bar4 = (654-65)/900 = 589/900`

`0.85bar(271) = (85271-85)/99900 = 85186/99900 = 42593/49950`

`4.167bar(4) = 4 (1673-167)/9000 = 4 1506/9000= 4 251/1500`