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

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 $z$?

• Please update the tags if necessary. I don't know what the proper ones would be. – GamrCorps Mar 16 '15 at 19:55
• Another hint: $x/y-x/(1000y) = 234$. – TonyK Mar 16 '15 at 20:04

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