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Math Help - How to convert decimals to fractions

  1. #1
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    Question How to convert decimals to fractions



    How would we solve problems such as 16 and 17? These problems do not repeat in numbers such as 0.235235235.

    Like for 16, how would we put that in geometric form first?
    I tried 0.122 + 0.0012 + 0.000012 ect but cannot find out how to get a fraction that will give that answer.

    Please help me on this. Thank you.
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  2. #2
    Junior Member slovakiamaths's Avatar
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    Solution13: Let x=0.232323............. (1)
    multiply (1) with 100,we get
    100x=23.232323............ (2)
    subtract (1) from (2),we get
    99x=23
    x=23/99
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  3. #3
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    16.

    0.1222222\dots = \frac{1}{10} + \frac{2}{100} + \frac{2}{1000} + \frac{2}{10\,000} + \dots

     = \frac{1}{10} + 2\sum_{k = 2}^{\infty}\frac{1}{10^k}.


    In this case, the geometric series has a = \frac{1}{100} and r = \frac{1}{10}.

    Since S_{\infty} = \frac{a}{1 - r}

    S_{\infty} = \frac{\frac{1}{100}}{1 - \frac{1}{10}}

     = \frac{\frac{1}{100}}{\frac{9}{10}}

     = \frac{10}{900}

     = \frac{1}{90}.



    So \frac{1}{10} + 2\sum_{k = 2}^{\infty}\frac{1}{10^k} = \frac{1}{10} + 2\left(\frac{1}{90}\right)

     = \frac{9}{90} + \frac{2}{90}

     = \frac{11}{90}.


    Therefore 0.122222\dots = \frac{11}{90}.
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  4. #4
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    17.

    0.478888\dots = \frac{47}{100} + \frac{8}{1000} + \frac{8}{10\,000} + \frac{8}{100\,000} + \dots

     = \frac{47}{100} + 8\sum_{k = 3}^{\infty}\frac{1}{10^k}


    In this case, the geometric series has a = \frac{1}{1000} and r = \frac{1}{10}.

    Therefore S_{\infty} = \frac{\frac{1}{1000}}{1 - \frac{1}{10}}

     = \frac{\frac{1}{1000}}{\frac{9}{10}}

     = \frac{10}{9000}

     = \frac{1}{900}.


    Thus \frac{47}{100} + 8\sum_{k = 3}^{\infty}\frac{1}{10^k} = \frac{47}{100} + 8\left(\frac{1}{900}\right)

     = \frac{423}{900} + \frac{8}{900}

     = \frac{431}{900}.


    Therefore 0.478888\dots = \frac{431}{900}.
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