Question

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

Show your work.

(1 point)

Answer 1

Let \( x = 0.09\overline{9} \), where the overline indicates that the 9 repeats indefinitely.

We can express \( x \) as follows:

\[
x = 0.09999999\ldots
\]

To eliminate the repeating decimal, multiply \( x \) by 10:

\[
10x = 0.99999999\ldots
\]

Next, we subtract the original \( x \) from this equation:

\[
10x - x = 0.99999999\ldots - 0.09999999\ldots
\]

This simplifies to:

\[
9x = 0.9
\]

Now we can solve for \( x \):

\[
x = \frac{0.9}{9}
\]

To express \( 0.9 \) as a fraction:

\[
0.9 = \frac{9}{10}
\]

Substituting it back into our equation gives:

\[
x = \frac{\frac{9}{10}}{9} = \frac{9}{10} \times \frac{1}{9} = \frac{1}{10}
\]

Therefore, the simplest form of \( 0.09\overline{9} \) as a rational number is:

\[
\boxed{\frac{1}{10}}
\]