To convert the repeating decimal [tex]\( n = 0.2121\ldots \)[/tex] to its fractional equivalent, follow these steps:
1. Identify the repeating block: The repeating block here is "21," which has 2 repeating digits.
2. Set up the equation:
[tex]\[
n = 0.2121\ldots
\][/tex]
3. Multiply [tex]\( n \)[/tex] by a power of 10 that matches the length of the repeating block: Since the repeating block is 2 digits long, multiply [tex]\( n \)[/tex] by [tex]\( 100 \)[/tex]:
[tex]\[
100n = 21.2121\ldots
\][/tex]
4. Set up another equation by subtracting the original [tex]\( n \)[/tex] from this new equation:
[tex]\[
100n - n = 21.2121\ldots - 0.2121\ldots
\][/tex]
5. Simplify the left-hand side and the right-hand side of the equation:
[tex]\[
99n = 21
\][/tex]
6. Solve for [tex]\( n \)[/tex] by isolating [tex]\( n \)[/tex]:
[tex]\[
n = \frac{21}{99}
\][/tex]
7. Simplify the fraction: The greatest common divisor (GCD) of 21 and 99 is 3. Divide the numerator and the denominator by their GCD to simplify the fraction:
[tex]\[
\frac{21 \div 3}{99 \div 3} = \frac{7}{33}
\][/tex]
So, the fractional equivalent of the repeating decimal [tex]\( 0.2121\ldots \)[/tex] is [tex]\( \frac{7}{33} \)[/tex].
