Question

Convert 0.09¯¯¯¯¯(09 repeated) to a rational number in simplest form.

Show your work.

Answer 1

To convert the repeating decimal \( 0.09\overline{09} \) to a rational number, follow these steps:

Let:
\[
x = 0.090909\ldots
\]

This indicates that the digits "09" repeat indefinitely. To eliminate the repeating part, we can multiply \( x \) by a power of 10 that shifts the decimal point two places to the right.

Multiply \( x \) by 100:
\[
100x = 9.090909\ldots
\]

Now we have two equations:
1. \( x = 0.090909\ldots \)
2. \( 100x = 9.090909\ldots \)

Next, we can subtract the first equation from the second:
\[
100x - x = 9.090909\ldots - 0.090909\ldots
\]
This simplifies to:
\[
99x = 9
\]

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

To simplify \( \frac{9}{99} \), divide both the numerator and the denominator by their greatest common divisor, which is 9:
\[
x = \frac{9 \div 9}{99 \div 9} = \frac{1}{11}
\]

Thus, the repeating decimal \( 0.09\overline{09} \) can be expressed as the simplified rational number:
\[
\boxed{\frac{1}{11}}
\]