Prove that the decimal 0.303030... (recurring) has the value of 10/33

Let x = 0.303030...
We do not want to deal with a recurring decimal, and so we want to cancel this out. The easiest way to do this is to multiply x by 10 and this means we multiple our decimal by 10 until we find one that matches the original decimal, _.3030... .
10x = 3.030303... (does not work as the decimal starts with _.0303..., which does not match our original decimal).
100x = 30.303030... (this has the same recurring decimal as x, _.3030...).
Now that we have a matching recurring decimal, we can subtract one from another to give us a whole number.
100x - x = 30.303030... - 0.303030...99x = 30
Now this is an equation that is easier to deal with, as we can divide 30 by 99 to give us:
x = 30/99, and when simplified, this makes 10/33 (dividing by 3)