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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]
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]
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