0.09 (the 0 after the decimal and the 9 have lines over them)

Question

What is the fraction of the decimal expansion above?

respoces:
1 over 11
1 over 9
1 over 90
9 over 10

Answer 1

The decimal \(0.\overline{09}\) is a repeating decimal. To convert this into a fraction, we can set \(x = 0.\overline{09}\).

1. First, multiply both sides of the equation by 100 (since the repeating part has 2 digits): \[ 100x = 9.\overline{09} \]

2. Now, subtract the original equation from this equation: \[ 100x - x = 9.\overline{09} - 0.\overline{09} \] This gives: \[ 99x = 9 \]

3. Solve for \(x\): \[ x = \frac{9}{99} \]

4. Simplifying \(\frac{9}{99}\) results in: \[ x = \frac{1}{11} \]

Therefore, the fraction representation of \(0.\overline{09}\) is \(\frac{1}{11}\).

So the correct choice is: 1 over 11.