# Thread: Writing repeating decimals as fractions (complex repeating decimals)

1. ## Writing repeating decimals as fractions (complex repeating decimals)

Just something I know will come in handy later on.
So ...

$\displaystyle 1/3 = \overline{.3}$
$\displaystyle 2/3 = \overline{.6}$

and ...

$\displaystyle 7/3 = 2\overline{.3}$
$\displaystyle 8/3 = 2\overline{.3}$

but ...
How can I find a fraction that represents a decimal like:
$\displaystyle \overline{.2}, $$\displaystyle \overline{.23},$$\displaystyle \overline{.54}?$

And with regards to percent how would I represent something like
$\displaystyle 10\overline{.34}\%$ as a decimal?

2. Originally Posted by Masterthief1324
[snip]
How can I find a fraction that represents a decimal like:
$\displaystyle \overline{.2}, $$\displaystyle \overline{.23},$$\displaystyle \overline{.54}?$

And with regards to percent how would I represent something like
$\displaystyle 10\overline{.34}\%$ as a decimal?
[/snip]
To answer the first part, let $\displaystyle x$ be the repeating decimal.

For instance, let's say that $\displaystyle x=0.\overline{124}$. We then want to multiply by a power of 10 such that we maintain the repeating decimal. In this case, we see that $\displaystyle 1000x=124.\overline{124}$, preserving the repeating decimal.

Next, we observe that $\displaystyle 1000x-x=124.\overline{124}-0.\overline{124}\implies 999x=124$

Solving for $\displaystyle x$, we now observe that $\displaystyle x=\frac{124}{999}$.

However, sometimes its not obvious at first and you need to be creative, in cases like $\displaystyle .024\overline{562}$, but the same idea applies.

To answer the second question, write it like you would if you were dealing with percentages that don't have repeating decimals: $\displaystyle 10.\overline{34}\%=.01\overline{34}$.

Does this clarify things?

3. Originally Posted by Chris L T521
To answer the first part, let $\displaystyle x$ be the repeating decimal.

For instance, let's say that $\displaystyle x=0.\overline{124}$. We then want to multiply by a power of 10 such that we maintain the repeating decimal. In this case, we see that $\displaystyle 1000x=124.\overline{124}$, preserving the repeating decimal.

Next, we observe that $\displaystyle 1000x-x=124.\overline{124}-0.\overline{124}\implies 999x=124$

Solving for $\displaystyle x$, we now observe that $\displaystyle x=\frac{124}{999}$.

However, sometimes its not obvious at first and you need to be creative, in cases like $\displaystyle .024\overline{562}$, but the same idea applies.

To answer the second question, write it like you would if you were dealing with percentages that don't have repeating decimals: $\displaystyle 10.\overline{34}\%=.01\overline{34}$.

Does this clarify things?
I don't mean to put you on the spotlight but let me show you how well you clarified it!

$\displaystyle .024\overline{562} \implies$fraction

Let x = $\displaystyle .024\overline{562}$
$\displaystyle 1000000x = 24562.\overline{562}$

$\displaystyle 1000x = 24.\overline{562}$

$\displaystyle 1000000x - 1000x = 24562.\overline{562} - 24.\overline{562}$

$\displaystyle 999000x = 24538$

$\displaystyle x = \frac{24562} {999000} = .024\overline{562}$

Thanks . How did you find this method?