Question

What is the fraction for the decimal expansion above?
0.09 line on the top of 09

Answer 1

The decimal \( 0.\overline{09} \) represents a repeating decimal where "09" repeats indefinitely. To convert this to a fraction, we can use an algebraic method:

Let \( x = 0.\overline{09} \).

Then, we can express \( x \) as:
\[
x = 0.09090909\ldots
\]

To eliminate the repeating part, we can multiply both sides of the equation by 100 (since the repeating part has 2 digits):
\[
100x = 9.09090909\ldots
\]

Now, we have two equations:
1. \( x = 0.09090909\ldots \)
2. \( 100x = 9.09090909\ldots \)

Next, we subtract the first equation from the second:
\[
100x - x = 9.09090909\ldots - 0.09090909\ldots
\]
\[
99x = 9
\]

Now, we can solve for \( x \):
\[
x = \frac{9}{99}
\]

To simplify \( \frac{9}{99} \):
\[
x = \frac{1}{11}
\]

Thus, the fraction for the decimal expansion \( 0.\overline{09} \) is:
\[
\boxed{\frac{1}{11}}
\]