Sure, let's go through the steps to convert the repeating decimal [tex]\(0.21212121 \ldots\)[/tex] into a fraction.
1. Let [tex]\( x = 0.21212121 \ldots \)[/tex].
2. Multiply both sides by 100 (since the repeating part has 2 digits) to shift the decimal places:
[tex]\[ 100x = 21.21212121 \ldots \][/tex]
3. Subtract the original [tex]\( x \)[/tex] from this new equation to eliminate the repeating part:
[tex]\[ 100x - x = 21.21212121 \ldots - 0.21212121 \ldots \][/tex]
4. This simplifies to:
[tex]\[ 99x = 21 \][/tex]
5. Solve for [tex]\( x \)[/tex] by dividing both sides by 99:
[tex]\[ x = \frac{21}{99} \][/tex]
6. The fraction [tex]\(\frac{21}{99}\)[/tex] can be simplified. Find the greatest common divisor (GCD) of 21 and 99. The GCD is 3.
7. Simplify the fraction by dividing the numerator and the denominator by their GCD:
[tex]\[ \frac{21 \div 3}{99 \div 3} = \frac{7}{33} \][/tex]
Therefore, the repeating decimal [tex]\(0.21212121 \ldots\)[/tex] as a fraction is [tex]\(\frac{7}{33}\)[/tex].
Here are the answers filled into the blanks for the practice problem:
1. Let [tex]\( x = 0.21212121 \ldots \)[/tex]
2. [tex]\( 100x = 21.21212121 \ldots \)[/tex]
3. [tex]\( 100x - x = 21.21212121 \ldots - 0.21212121 \ldots \)[/tex]
4. [tex]\( 100x - x = \)[/tex]
5. [tex]\( 99x = 21 \)[/tex]
6. [tex]\( x = \)[/tex]
7. [tex]\( \frac{7}{33} \)[/tex]
So when filled in:
```
7. Write the number [tex]\(0.21212121 \ldots\)[/tex] as a fraction.
Let [tex]\(x = 0.21212121 \ldots\)[/tex]
[tex]\(100x = 21.21212121 \ldots\)[/tex]
8.
[tex]\[
100x - x = 21.21212121 \ldots - 0.21212121 \ldots \][/tex]
[tex]\[
\begin{array}{r}
100x - x = \\
99x = 21 \\
x = \frac{21}{99} = \frac{7}{33}
\end{array}
\][/tex]
```
