Question

What is `0.3828282828...` as a fraction?

Answer 1

`0.3bar(82) = 379/990`

Explanation:

In case you have not encountered it, note that the repeating part of a decimal representation can be indicated by a bar over the digits.

In our example:

`0.382828282... = 0.3bar(82)`

To convert to a fraction we want to find an integer to multiply it by to get another integer.

Let's multiply by `10(100-1) = 1000-10`.

The first factor `10` is to shift the number one place to the left so that the repeating part starts immediately after the decimal point.

The `(100-1)` factor shifts the resulting number two places further to the left (i.e. the length of the repeating pattern) then subtracts the original to cancel the repeating tail...

`(1000-10)0.3bar(82) = 382.bar(82) - 3.bar(82)`

`color(white)((1000-10)0.3bar(82)) = 379`

Now divide both ends by `(1000-10)` and simplify:

`0.3bar(82) = 379/(1000-10) = 379/990`

This does not simplify any further since `379` and `990` have no common factor.

Answer 2

`379 /990`

Explanation:

`0.38282828282.... =0.3 + 0.082 + 0.00082 +0.0000082 + ...`

`= 3/10 + 0.082 + 0.00082 +0.0000082 + ....`

we can use geometric progression to solve where,
`a_1 = 0.082, r =0.00082/0.082 = 1/100`

`s_oo = a_1/(1-r) = 0.082/(1-1/100)`

`= 0.082/(99/100) = 8.2/99 =82/990`

therefore
`3/10 + 0.082 + 0.00082 +0.0000082 + .... = 3/10 + 82/990= 297/990 + 82/990 = 379 /990`