How do you convert 0.254 (4 repeating) as a fraction?

Let `a/b=0.25444444444444" "`first equation

Multiply both sides of the first equation by 100 and the result is

`100a/b=25.444444444444" "`second equation

Multiply both sides of the first equation by 1000 and the result is

`1000a/b=254.44444444444" "`third equation

Subtract second from the third

`1000a/b-100a/b=254.44444444444-25.444444444444`

`900a/b=229`

`a/b=229/900`

God bless....I hope the explanation is useful

There are easy rules to use for converting recurring decimals to fractions:
There are 2 types of recurring decimals - those where ALL the digits recur and those where SOME of the digits recur.

1. If ALL the digits recur, the fraction is formed from;

`"the digits which recur"/"a 9 for each digit"`

eg 0.777777...... = `7/9`

0.454545..... = `45/99 rArr "this can be simplified to" 5/11"`

5.714714714.... = `5 714/999 rArr 238/333`

`"2."` If only some digits recur:

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

eg 0.3544444..... = `(354-35)/900 = 319/900`

eg. 0.4565656... = `(456-4)/990 = 452/990= 226/495`

eg 4.62151515... = `4 6215-62/9900 = 4 6153/9900 = 4 2051/3300`

0.254444.... = `(254-25)/900 = 229/900`

These rules are short cuts for algebraic methods, but it is often useful to be able to get to answer immediately.