Write $0.2154154\overline{154}$ as a fraction

Question

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 $x$.

Answer 1

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.

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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:

$$ \frac{2152}{9990}$$

Answer 2

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 $$x = 154/9990 + 1998/9990 = 2152/9990.$$