# Thread: Converting a Repeating Decimal to a Fraction

1. ## 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.

$\displaystyle 1.1{\overline{9}}$ 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.

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

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?

2. ## 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 \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*}

3. ## Re: Converting a Repeating Decimal to a Fraction

You could also sum an infinite geometric series.

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

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

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

$\displaystyle =1.1+\frac{\left(\frac{9}{100}\right)}{\left(\frac {9}{10}\right)}=1.1+0.1=1.2=\frac{12}{10}$

4. ## Re: Converting a Repeating Decimal to a Fraction

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