Write $0.2154154\overline{154}$ as a fraction Let $x = 0.2154154\overline{154}$ , I have to prove that it is a rational number just by writing it as a fraction with the proper steps.
I note that the repeating part, $154$, is composed by 3 digits. Thus, using a trick that I have learnt, I can write an equation of this type:
$$1000x = ?$$
I am not sure about what goes on the right side of the equation because of this $0.2$
I mean, if, for instance, the number was $x = 0.154$ periodic, I would write the equation as:
$1000x = 154 + x$    and solve for $x$
I have tried various attempts, re-writing $0$, $2$ as $\frac{1}{5}$, but I don't get the number which equals our $x$.
 A: First, ignore the 2, and find a fraction that represents $y = 0.\overline{154}$. Since 
$$y = 0.\overline{154},$$ 
$$1000y = 154.\overline{154} = 154 + y.$$
Solving for $y$, we get $999y = 154$, so $y = 154/999$. Now $x = y/10 + 1/5$, so
$$x = 154/9990 + 1998/9990 = 2152/9990.$$
A: First Method
What you basically do in this method is find a multiple of $x$ that will get rid of the repeating part when you subtract it from x itself. (HINT: $10x$, $100x$, it all depends on the number of digits in the repeating part.)
$$x=0.2154154154...$$
$$10x=2.154154154...$$
$$10000x = 2154.154154...$$
$$10000x - 10x = 9990x = 2154 - 2 = 2152$$
$$x = \frac{2152}{9990}$$
Note we picked $10000x$ here because in the expression of $10x$, the repeating part has three digits.

Second Method
$$2.154154154 = 2 + 0.154 + 0.000154 + 0.000000154 + ...$$
$$= 2 + 0.154 + 0.154(0.001) + 0.154(0.001)^2 +  0.154(0.001)^3 + ...$$
$$2 + \frac{0.154}{(1-0.001)} = 2 + \frac{0.154}{0.999} = \frac{2.152}{0.999}$$
$$= \frac{2152}{999}$$
Thus $$2.\overline{154} = \frac{2152}{999}$$
Since the original number is $0.2\overline{154}$, the fractional form is:
$$ \frac{2152}{9990}$$

