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Math Help - Converting a Repeating Decimal to a Fraction

  1. #1
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    Converting a Repeating Decimal to a Fraction

    Having no guidance but my textbook, which explains basic concepts, but leaves to more in depth stuff to the teacher (of which I have none), I turn yet again to the forums to help me with the following problem.

    1.1{\overline{9}} converted to a fraction.

    Now, knowing how to convert a decimal if there are TWO repeating numbers directly after the decimal point, but not having any explanation of what to do if there is one non-repeating number followed by a repeating number, I figured I'd follow the same line
    of reasoning.

    n = 1.1{\overline{9}}
    100n = 119.\overline{9}}

    etc, but I feel like this is the wrong path to be going down. I know it works for 2 repeating numbers after the decimal point, but what do I do when there's one number after the point followed by a repeating number?
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  2. #2
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    Re: Converting a Repeating Decimal to a Fraction

    You're on the right track. Just get two numbers that have exactly the same repeating digits.

    \displaystyle \begin{align*}n &= 1.1\overline{9} \\ 10n &= 11.\overline{9} \\ 100n &= 119.\overline{9} \\ 100n - 10n &= 119.\overline{9} - 11.\overline{9} \\ 90n &= 108 \\ n &= \frac{108}{90} \\ n &= \frac{6}{5} \end{align*}
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  3. #3
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    Re: Converting a Repeating Decimal to a Fraction

    You could also sum an infinite geometric series.

    1.19999999....=1.1+\frac{9}{100}+\frac{9}{1000}+..  .....

    For the series, a=\frac{9}{100};\;\;\;r=\frac{1}{10}

    n=1.1+\frac{\frac{9}{100}}{1-\frac{1}{10}}

    =1.1+\frac{\left(\frac{9}{100}\right)}{\left(\frac  {9}{10}\right)}=1.1+0.1=1.2=\frac{12}{10}
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  4. #4
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    Re: Converting a Repeating Decimal to a Fraction

    In fact, for this particular number, 1.1999....= 1.2= \frac{12}{10}
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