Infinite Series (Geometric) Problem

[Ascendant0]

Homework Statement

Use $s = a/(1-r)$ to find the fraction that is equivalent to 0.678571428571428571...

Relevant Equations

Obviously $s = a/(1-r)$

As far as how far I've gotten, I split the non-repeating portion of the series apart from the repeating portion, set r as $10^{-6}$ and get this:

$0.65+285714/9999990$

From here though, I don't see how to simplify that fraction without something extremely tedious, like pulling out every single factor I can from trial and error (beyond the obvious "2" factor), and seeing which ones cancel. But sure enough, it's possible, because their answer in the book is $19/28$ and if I punch mine and theirs into a calculator to get the answer, I get the same answer for all 10 digits on the calculator. So, I know my answer is headed in the right direction, I just don't have a clue how you can simplify something like that without going through a bunch of trial and error factoring, or is that what you're expected to do?

 

[PeroK]

Why not separate the non-repeating part of the fraction. I.e. 0.67.

 

[Ascendant0]

Why not separate the non-repeating part of the fraction. I.e. 0.67.

I did, it's actually 0.65, hence the "$0.65 + 285714/9999990$". The first # is the non-repeating part, the second number (fraction) is the repeating part. So, essentially I'm trying to figure out the fastest, most effective way to simplify that fraction and get their $19/28$ answer out of it?

 

[PeroK]

What's 9,999,990/285,714?

 

[Ascendant0]

What's 9,999,990/285,714?

Ah, ok I see. So you flip the numerator and denominator, divide it that way, then after you simplify it that way, then swap the numerator and denominator back (and if it's a mixed number, simply convert the mixed number into a fraction, *then* flip it).

Wow, such an easy method, yet something I have never seen to date! You are awesome, thanks so much for the help!