Question

what fraction is equivalent to 0.1111111111

Answer 1

The decimal \(0.1111111111\) can be expressed as a fraction. This decimal is equivalent to \( \frac{1}{9} \). This is because the repeating decimal \(0.1111...\) (where the 1 repeats indefinitely) represents one-ninth of 1.

To confirm:
1. Let \(x = 0.1111...\).
2. Multiply both sides by 10: \(10x = 1.1111...\).
3. Now, subtract the first equation from the second:
\[
10x - x = 1.1111... - 0.1111...
\]
\[
9x = 1
\]
4. Divide by 9:
\[
x = \frac{1}{9}
\]

Therefore, \(0.1111111111\) is equivalent to \( \frac{1}{9} \).