Write the number 0.21212121..... as a fraction

The repeating decimal \( 0.21212121\ldots \) can be expressed as a fraction. Let's solve it step by step.

Let:
\[
x = 0.21212121\ldots
\]
This is a repeating decimal with the block "21" repeating. We can multiply both sides of the equation by 100 to shift the decimal point two places to the right:
\[
100x = 21.21212121\ldots
\]
Now, subtract the original equation from this new equation to eliminate the repeating decimal:
\[
100x - x = 21.21212121\ldots - 0.21212121\ldots
\]
\[
99x = 21
\]
Solve for \(x\) by dividing both sides by 99:
\[
x = \frac{21}{99}
\]
Now, simplify the fraction:
\[
\frac{21}{99} = \frac{7}{33}
\]
Thus, the decimal \( 0.21212121\ldots \) is equivalent to the fraction \( \frac{7}{33} \).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you convert other repeating decimals into fractions?
2. How do you simplify fractions?
3. What is the decimal representation of \( \frac{7}{33} \)?
4. Can repeating decimals always be written as fractions?
5. How would you convert a repeating decimal like \( 0.131313\ldots \) to a fraction?

**Tip:** Always simplify fractions by dividing both the numerator and denominator by their greatest common divisor (GCD).

Learn how to convert the repeating decimal 0.21212121... to a fraction with a clear step-by-step solution. This guide includes key mathematical concepts like simplifying fractions and solving repeating decimals. Suitable for Grades 6-8.

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