What is the fractional equivalent of the repeating decimal
n=0.2121...?
This number (represented by n) has
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repeating digit(s),so it should be multiplied by
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The value of 100n is
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The value of 99n is
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The fractional value of nis
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After simplifying, the fractional equivalent of n is
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6 of 13 QUESTIONS



Answer :

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].