To convert the repeating decimal \( x = 0.454545454545... \) into a rational number, we can use the following method:
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Set up the equation: \[ x = 0.454545454545... \]
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Multiply both sides by 100 (shifting the decimal point two places to the right): \[ 100x = 45.454545454545... \]
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Now we have two equations: \[ x = 0.454545454545... \quad (1) \] \[ 100x = 45.454545454545... \quad (2) \]
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Subtract equation (1) from equation (2): \[ 100x - x = 45.454545454545... - 0.454545454545... \] This simplifies to: \[ 99x = 45 \]
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Solve for \( x \): \[ x = \frac{45}{99} \]
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Simplify the fraction: To simplify \( \frac{45}{99} \), we can find the greatest common divisor (GCD) of 45 and 99. The GCD is 9. Now we divide both the numerator and the denominator by 9: \[ x = \frac{45 \div 9}{99 \div 9} = \frac{5}{11} \]
Therefore, the repeating decimal \( 0.454545454545... \) can be expressed as the rational number in simplest form: \[ \frac{5}{11} \]
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