Write the number [tex]$0.21212121 \ldots$[/tex] as a fraction.

1. Let [tex]$x = 0.21212121 \ldots$[/tex]
2. [tex][tex]$100x = 21.21212121 \ldots$[/tex][/tex]
3. [tex]$100x - x = 21.21212121 \ldots - 0.21212121 \ldots$[/tex]
4. [tex]$99x = 21$[/tex]
5. [tex]$x = \frac{21}{99}$[/tex]
6. So, [tex][tex]$0.21212121 \ldots$[/tex][/tex] is equal to [tex]$\frac{7}{33}$[/tex] after simplifying.



Answer :

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].