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Which fraction represents the decimal [tex]$0 . \overline{12}$[/tex]?

A. [tex]\frac{1}{12}[/tex]
B. [tex]\frac{3}{25}[/tex]
C. [tex]\frac{4}{33}[/tex]
D. [tex]\frac{33}{4}[/tex]


Sagot :

To determine which fraction represents the repeating decimal [tex]\(0.\overline{12}\)[/tex], we'll follow a systematic method to convert it to a fraction.

1. Let [tex]\( x \)[/tex] be the repeating decimal:
[tex]\[ x = 0.121212\ldots \][/tex]

2. Multiply both sides of this equation by 100 to shift the decimal point two places to the right (since the repeating block is two digits long):
[tex]\[ 100x = 12.121212\ldots \][/tex]

3. Subtract the original [tex]\( x = 0.121212\ldots \)[/tex] from this new equation:
[tex]\[ 100x - x = 12.121212\ldots - 0.121212\ldots \][/tex]
[tex]\[ 99x = 12 \][/tex]

4. Solve for [tex]\( x \)[/tex] by dividing both sides of the equation by 99:
[tex]\[ x = \frac{12}{99} \][/tex]

5. Simplify the fraction [tex]\( \frac{12}{99} \)[/tex]:
To do this, find the greatest common divisor (GCD) of 12 and 99, which is 3:
[tex]\[ \frac{12 \div 3}{99 \div 3} = \frac{4}{33} \][/tex]

So, the fraction that represents the repeating decimal [tex]\(0.\overline{12}\)[/tex] is [tex]\( \frac{4}{33} \)[/tex].

Thus, the correct answer is:
[tex]\[ \boxed{\frac{4}{33}} \][/tex]