Find the fraction that creates a repeating decimal that repeats certain digits

Question

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 $z$?

Answer 1

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 $$\frac{10^{l(z)}z}{10^{l(z)}-1}$$

Answer 2

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*}