Let \( x = 0.6666\ldots \) (where the 6s are repeating). To convert this repeating decimal into a fraction, we can follow these steps:
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Multiply \( x \) by 10 to shift the decimal point one place to the right: \[ 10x = 6.6666\ldots \]
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Subtract the original \( x \) from this new equation: \[ 10x - x = 6.6666\ldots - 0.6666\ldots \]
This simplifies to: \[ 9x = 6 \]
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Solve for \( x \): \[ x = \frac{6}{9} \]
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Reduce the fraction: \[ x = \frac{2}{3} \]
Therefore, the fraction that is equivalent to \( 0.6666\ldots \) is \( \frac{2}{3} \).