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Is there any way to find the fraction $x/y$ that, when converted to a decimal, repeats a series of digits $z$? For example: ${x}/{y} = z.zzzzzzzz...$ or with actual numbers, $x/y = 234.234234234...$ (z is 234)

If this is impossible, is there a way that does the same but the value to the left of the decimal is not $z$?

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  • $\begingroup$ Please update the tags if necessary. I don't know what the proper ones would be. $\endgroup$ – GamrCorps Mar 16 '15 at 19:55
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    $\begingroup$ Another hint: $x/y-x/(1000y) = 234$. $\endgroup$ – TonyK Mar 16 '15 at 20:04
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There's a nice fact (derived from the expression demonstrated by TonyK): The repeating decimal $0.zzz\dots$ can be represented by $$\frac{z}{10^{l(z)}-1}$$ where $l(z)$ here denotes the number of digits of $z$.

Now if we want instead $z.zzz\dots$, all we have to do is multiply the above expression by $10^{l(z)}$ (or as Joffan points out - simply add $z$), getting $$\frac{10^{l(z)}z}{10^{l(z)}-1}$$

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Here is a way that does not involve yet another formula for students (or you) to memorize. (I am not against formulas; I just realize that many people are.)

The idea is to multiply the number by a power of ten that gets the same repeating pattern, then subtract. For your example,

\begin{align*} u & =234.234\ldots\\ 1000u & =234234.234\ldots\\ 999u & =234.234\ldots-234234.234\ldots\\ & =234000\\ u & =\frac{234000}{999}\\ & =\frac{26000}{111} \end{align*}

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