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

  1. #1
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    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?
    Last edited by Chris L T521; Jan 19th 2010 at 07:02 PM. Reason: fixed tex error
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Masterthief1324 View Post
    [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?
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  3. #3
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    Quote Originally Posted by Chris L T521 View Post
    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?
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