To convert the repeating decimal \(0.27\overline{27}\) to a rational number, we can perform the following steps:
Let \(x = 0.272727...\)
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Multiply both sides by 100 (since the repeat is 2 digits): \[ 100x = 27.272727... \]
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Now, we can write the first equation and the second equation: \[ x = 0.272727... \] \[ 100x = 27.272727... \]
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Now subtract the first equation from the second: \[ 100x - x = 27.272727... - 0.272727... \] \[ 99x = 27 \]
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Now, solve for \(x\): \[ x = \frac{27}{99} \]
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Next, simplify \(\frac{27}{99}\):
- The greatest common divisor of 27 and 99 is 9. \[ x = \frac{27 \div 9}{99 \div 9} = \frac{3}{11} \]
Thus, \(0.27\overline{27} = \frac{3}{11}\).
The correct response is: \[ \text{Start Fraction 3 over 11 end fraction} \]