To convert the repeating decimal \(0.27\overline{27}\) (which can also be written as \(0.272727...\)) into a rational number, we can use the following steps:
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Let \(x = 0.27\overline{27}\).
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Multiply both sides by 100 (since the repeating part is two decimal places): \[ 100x = 27.27\overline{27} \]
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Now subtract the original equation from this new equation: \[ 100x - x = 27.27\overline{27} - 0.27\overline{27} \] This simplifies to: \[ 99x = 27 \]
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Solve for \(x\): \[ x = \frac{27}{99} \]
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Simplify \( \frac{27}{99} \):
- The greatest common divisor (GCD) of 27 and 99 is 9.
- Divide both the numerator and the denominator by 9: \[ \frac{27 \div 9}{99 \div 9} = \frac{3}{11} \]
Thus, the repeating decimal \(0.27\overline{27}\) is converted to the rational number \( \frac{3}{11} \).
The correct response is: \[ \frac{3}{11} \]