Question

How do you convert 0.12345 (12345 repeating) to a fraction?

Answer 1

`0.bar(12345) = 4115/33333`

Explanation:

Multiply by `100000-1` to get an integer:

`(100000-1) 0.bar(12345) = 12345.bar(12345)-0.bar(12345) = 12345`

Then divide both sides by `100000-1`:

`0.bar(12345) = 12345/(100000-1) = 12345/99999 = (color(red)(cancel(color(black)(3)))*4115)/(color(red)(cancel(color(black)(3)))*33333) = 4115/33333`

Answer 2

`4115/33333`

Explanation:

Require to obtain 2 equations with the same repeating part and subtract them to eliminate the repeating part.

Begin by letting x = 0.1234512345................... (A)

To obtain the same repeating part after the decimal point need to multiply by 100000

hence : 100000x = 12345.1234512345............... (B)

It is important to obtain 2 equations in x , where the recurring part after the decimal point are exactly the same.

Subtracting (A) from (B) will eliminate the repeating part.

(B) - (A) gives : 99999x = 12345 `rArr x = 12345/99999 = 4115/33333`