# 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 ...

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

and ...

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

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

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

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

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

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

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

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

However, sometimes its not obvious at first and you need to be creative, in cases like $.024\overline{562}$, 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: $10.\overline{34}\%=.01\overline{34}$.

Does this clarify things?

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

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

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

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

However, sometimes its not obvious at first and you need to be creative, in cases like $.024\overline{562}$, 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: $10.\overline{34}\%=.01\overline{34}$.

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

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

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

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

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

$999000x = 24538$

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

Thanks . How did you find this method?