Question

Find fractional notation for the infinite sum (these are geometric series) 0.1919191919

Answer 1

Let x = 0.191919...
Now, we want to find some sort of relationship with x and 0.191919...

If we multiply x by 100, we get:
$100x = 19.19191919...$

Now, it's clear that the decimal places will cancel when we subtract it:
$100x - x = 19.19191919... - 0.19191919...$
$99x = 19$

Solving for x, we get:
$x = \frac{19}{99}$