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Math Help - Recurring decimals

  1. #1
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    Recurring decimals

    Could someone please remind me how to convert a recurring decimal into a fraction

    1) 0.8888888888888....
    2) 0.1888888888888....
    3) 0.656565656565....
    4) 0.8333333333333...
    5) 0.369369369369...
    6) 0.416666666666...
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  2. #2
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    Let N=0.416\overline{6}

    100N=41.6\overline{6}

    1000N=416.6\overline{6}

    900N=375
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  3. #3
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    Oh I understand but how would it work with number 3?

    Let N = 0.65656565
    100N = 65.656565
    1000N = 656.565
    900N = ?
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  4. #4
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    Hello, Natasha1!

    I'll do a few of them . . . It should trigger a memory.


    Please remind me how to convert a recurring decimal into a fraction.

    1)\;0.8888\hdots

    \begin{array}{cccccc}\text{We have:} & N &=& 0.8888\hdots & [1] \\<br />
\text{Multiply by 10:} & 10N &=& 8.8888\hdots & [2] \end{array}


    \text{Subtract [2] - [1]: }\;9N \:=\:8 \quad\Rightarrow\quad N \:=\:\dfrac{8}{9}




    3)\;0.656565\hdots

    \begin{array}{ccccc}\text{We have:} & N &=& \;\;0.656565\hdots & [1] \\<br />
\text{Multiply 100:} & 100N &=& 65.656565\hdots & [2] \end{array}


    \text{Subtract [2] - [1]: }\;99N \:=\:65 \quad\Rightarrow\quad N \;=\;\dfrac{65}{99}




    6)\;0.41666\hdots

    \begin{array}{cccccc}<br />
\text{We have:} & N &=& \qquad 0.41666\hdots & [1] \\ \\[-3mm]<br />
\text{Multiply by 100:} & 100N &=& \;\;41.666\hdots & [2] \\<br />
\text{Multiply by 1000:} & 1000N &=& 416.666\hdots & [3] \end{array}


    \text{Subtract [3] - [2]: }\;900N \:=\:375 \quad\Rightarrow\quad N \:=\:\dfrac{375}{900} \:=\:\dfrac{5}{24}

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  5. #5
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    Notice that 99N=65.
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  6. #6
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    ah yes brilliant!! Thank you
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