Question

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

Show your work.

(1 point)

Answer 1

To convert the repeating decimal \(0.09\ldots\) into a rational number in simplest form, we'll follow these steps:

1. Let \(x = 0.09\ldots\).
\[
x = 0.090909\ldots
\]

2. Since the repeating part "09" has a length of 2, we can multiply by \(100\) to shift the decimal point two places to the right:
\[
100x = 9.090909\ldots
\]

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

4. Now we'll subtract the first equation from the second:
\[
100x - x = 9.090909\ldots - 0.090909\ldots
\]
\[
99x = 9
\]

5. Now, divide both sides by \(99\):
\[
x = \frac{9}{99}
\]

6. Simplify the fraction \(\frac{9}{99}\):
\[
x = \frac{9 \div 9}{99 \div 9} = \frac{1}{11}
\]

Thus, the repeating decimal \(0.09\ldots\) can be expressed as the rational number \(\frac{1}{11}\) in simplest form.