To convert the repeating decimal \( 0.8181818181\ldots \) into a rational number, we can follow these steps:
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Let \( x = 0.8181818181\ldots \).
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Since the decimal repeats every two digits, we can multiply \( x \) by \( 100 \) to move the decimal point two places to the right: \[ 100x = 81.8181818181\ldots \]
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Now, we can set up the equation: \[ 100x = 81 + x \]
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To isolate \( x \), subtract \( x \) from both sides: \[ 100x - x = 81 \] \[ 99x = 81 \]
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Now, divide both sides by \( 99 \): \[ x = \frac{81}{99} \]
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We can simplify \( \frac{81}{99} \) by finding the greatest common divisor (GCD) of 81 and 99. The GCD is 9: \[ x = \frac{81 \div 9}{99 \div 9} = \frac{9}{11} \]
Thus, the repeating decimal \( 0.8181818181\ldots \) can be expressed as the rational number \( \frac{9}{11} \).
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