Question
The number \(0.\overline{01}\) in the form \( \frac{p}{q} \), where \( q \neq 0 \), is:
(a) \( \frac{1}{1000} \)
(b) \( \frac{1}{100} \)
(c) \( \frac{1}{1999} \)
(d) \( \frac{1}{99} \)
Solution
Let \( x = 0.\overline{01} = 0.010101\ldots \)
Multiply both sides by 100:
\[ 100x = 1.010101\ldots \]
Subtract the original equation:
\[ 100x – x = 1.010101\ldots – 0.010101\ldots \]
\[ 99x = 1 \]
\[ x = \frac{1}{99} \]
Final Answer
✔ Correct option: (d) \( \frac{1}{99} \)