The repeating decimal \( 0.0101010101\ldots \) can be expressed as a fraction. Let's solve it step by step.

Let:
\[
x = 0.0101010101\ldots
\]

Since the decimal repeats every two digits (01), multiply both sides of the equation by 100 to shift the decimal point two places:
\[
100x = 1.0101010101\ldots
\]

Now subtract the original equation from this one:
\[
100x - x = (1.0101010101\ldots) - (0.0101010101\ldots)
\]
\[
99x = 1
\]

Now, solve for \( x \):
\[
x = \frac{1}{99}
\]

Thus, \( 0.0101010101\ldots = \frac{1}{99} \).

Do you want further details or have any questions?

Here are 5 related questions to expand this concept:
1. How do you convert a repeating decimal like \( 0.\overline{3} \) to a fraction?
2. What is the fraction form of \( 0.001001001\ldots \)?
3. How do you convert a repeating decimal with more than two repeating digits into a fraction?
4. Can any repeating decimal be expressed as a fraction?
5. How would you convert a mixed repeating decimal like \( 0.12\overline{3} \)?

**Tip:** Repeating decimals can always be expressed as fractions of integers!

The query submitted by the user: .0101010101 . . . . as a fraction

Learn how to convert the repeating decimal 0.0101010101... into the fraction 1/99. This detailed guide covers the algebraic method for shifting decimals, subtracting equations, and solving for x. Suitable for students in Grades 6-8.

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