To convert the repeating decimal \(0.27\overline{27}\) to 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 of the equation by \(100\) (to move the decimal point two places to the right): \[ 100x = 27.\overline{27} \]
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Now, subtract the original \(x = 0.27\overline{27}\) from this equation: \[ 100x - x = 27.\overline{27} - 0.27\overline{27} \] This simplifies to: \[ 99x = 27 \]
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Now, solve for \(x\): \[ x = \frac{27}{99} \]
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To simplify \(\frac{27}{99}\), we can find the greatest common divisor (GCD) of 27 and 99, which is 9: \[ \frac{27 \div 9}{99 \div 9} = \frac{3}{11} \]
Thus, \(0.27\overline{27}\) as a rational number in simplest form is \(\frac{3}{11}\).
The answer is \(\frac{3}{11}\).
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