Question

what fraction is equivalent to 0.\overline{1}? type a number in each box. \frac{\boxed{ }}{\boxed{ }}

Explanation:

Step1: Recall decimal to fraction

A repeating decimal \(0.\overline{1}\) (where 1 repeats) can be converted. Let \(x = 0.\overline{1}\), so \(x = 0.111\ldots\)

Step2: Multiply by 10

Multiply both sides by 10: \(10x = 1.111\ldots\)

Step3: Subtract equations

Subtract \(x\) from \(10x\): \(10x - x = 1.111\ldots - 0.111\ldots\)
Which simplifies to \(9x = 1\)

Step4: Solve for x

Divide both sides by 9: \(x=\frac{1}{9}\)

Answer:

\(\frac{1}{9}\)