Let's convert the repeating decimal [tex]\(0.21212121 \ldots\)[/tex] into a fraction step-by-step.
1. Define [tex]\(x\)[/tex] as the repeating decimal:
[tex]\[
x = 0.21212121 \ldots
\][/tex]
2. Multiply [tex]\(x\)[/tex] by [tex]\(100\)[/tex] to shift the decimal point two places to the right:
[tex]\[
100x = 21.21212121 \ldots
\][/tex]
3. Set up the equation by subtracting the original [tex]\(x\)[/tex] from [tex]\(100x\)[/tex]:
[tex]\[
100x - x = 21.21212121 \ldots - 0.21212121 \ldots
\][/tex]
4. Simplify the left side of the equation:
[tex]\[
99x = 21
\][/tex]
5. Solve for [tex]\(x\)[/tex] by dividing both sides by [tex]\(99\)[/tex]:
[tex]\[
x = \frac{21}{99}
\][/tex]
6. Simplify the fraction [tex]\(\frac{21}{99}\)[/tex]:
The greatest common divisor (GCD) of 21 and 99 is 3.
[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]\[
0.2121 \ldots = \frac{7}{33}
\][/tex]
So, let's fill in the blanks:
Let [tex]\( x = 0.21212121 \ldots \)[/tex] \\
[tex]\( 100x = 21.21212121 \ldots \)[/tex] \\
[tex]\( 100x - x = 21.21212121 \ldots - 0.21212121 \ldots \)[/tex] \\
[tex]\( 99x = 21 \)[/tex] \\
[tex]\( x = \frac{21}{99} \)[/tex] \\
simplified, [tex]\( x = \frac{7}{33} \)[/tex]
So [tex]\(0.2121 \ldots\)[/tex] is equal to [tex]\(\frac{7}{33}\)[/tex].
