Question

Decimal to fraction ?

Answer 1

`=33/101`.

Explanation:

In general, when you have a number of the form `0.`[repeating part] you can convert it to the fraction

`n/(10^c-1)`

where `n` is the repeating part (the integer created by the decimal digits of one repetition, in our case it is `3267`) and `c` is the number of digits in the repeating part. It's equally easy with the form

`a.`[repeating part], you just get

`a+n/(10^c-1)`

The reason this works is because if you multiply that repeating decimal by `10^c`, you'll get `n` plus the repeating decimal. In our case, `c=4` and `n=3267`:

`0.3267...*10^4=3267.3267...=3267+0.3267...`

Therefore,

`0.3267...*10^4-0.3267...=3267`

`0.3267...*(10^4-1)=3267`

`0.3267...=3267/(10^4-1)=3267/9999=33/101`

It's also simple to find `10^c-1`, just write `c` nines.

Answer 2

`33/101`

Explanation:

`0. color(red)(3267)32673267....=0. bar(3267)`

`"the bar represents the digits being repeated"`

`"these form the terms of a "color(blue)"geometric sequence"`

`3267/10000,3267/100000000,...`

`"with "a=3267/10000`

`"and "r=a_2/a_1`

`rArrr=cancel(3267)/(10000cancel(0000))xx(1cancel(0000))/cancel(3267)=1/10000`

`•color(white)(x)S_oo=a/(1-r)`

`rArrS_oo=(3267/10000)/(1-1/10000)`

`=(3267/10000)/(9999/10000)`

`=3267/9999=33/101`