To convert the repeating decimal \(0.264\overline{6}\) into a rational number, you can follow these steps:
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Set \(x\) equal to \(0.264666...\): \[ x = 0.264666... \]
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Multiply both sides by a power of 10 that moves the decimal point to the right just before the repeating part. Since "6" is the repeating digit, you can multiply by 10: \[ 10x = 2.64666... \]
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Now, we can eliminate the repeating part by multiplying by another power of 10. In this case, we multiply by 1000 to shift three decimal places to the right: \[ 1000x = 264.666... \]
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You now have two equations: \[ 1000x = 264.666... \] \[ 10x = 2.64666... \]
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Now, subtract the first equation from the second: \[ 1000x - 10x = 264.666... - 2.64666... \] \[ 990x = 262.02 \]
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Solve for \(x\): \[ x = \frac{262.02}{990} \]
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To simplify this fraction, note that \(262.02\) can be rewritten in fractional form. Here, \(262.02 = \frac{26202}{100}\), so you can substitute this in: \[ x = \frac{26202}{100} \cdot \frac{1}{990} = \frac{26202}{99000} \]
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Now, reduce the fraction \(\frac{26202}{99000}\). First, find the GCD (greatest common divisor) of \(26202\) and \(99000\). After simplifying, you will arrive at the simplest form (if possible).
In this case, the value of \(x\) represents the rational number equivalent of the repeating decimal \(0.264\overline{6}\).