Question

Convert 0.09¯¯¯¯¯
to a rational number in simplest form.

Show your work.

Answer 1

Let \( x = 0.09\overline{09} \) (the bar indicates that the "09" repeats indefinitely). We can express \( x \) mathematically as follows:

\[
x = 0.09090909\ldots
\]

To eliminate the repeating decimal, we can multiply \( x \) by \( 100 \) (since the repeating part is 2 digits long):

\[
100x = 9.09090909\ldots
\]

Now, we have two equations:

1. \( x = 0.09090909\ldots \) (1)
2. \( 100x = 9.09090909\ldots \) (2)

Next, we will subtract equation (1) from equation (2):

\[
100x - x = 9.09090909\ldots - 0.09090909\ldots
\]

This simplifies to:

\[
99x = 9
\]

Now, we can solve for \( x \):

\[
x = \frac{9}{99}
\]

Now we simplify \(\frac{9}{99}\):

The greatest common divisor (GCD) of 9 and 99 is 9. Thus, we divide the numerator and denominator by 9:

\[
x = \frac{9 \div 9}{99 \div 9} = \frac{1}{11}
\]

Therefore, the rational number in simplest form that corresponds to \( 0.09\overline{09} \) is:

\[
\boxed{\frac{1}{11}}
\]