Let \( x = 0.755555\ldots \), where the digit \( 5 \) is repeating.
To convert this repeating decimal to a fraction, we can follow these steps:
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Multiply by a power of 10 to shift the decimal point so that the repeating part is aligned. Since the repeating portion (5) has one digit, we multiply by 10: \[ 10x = 7.55555\ldots \]
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Multiply by another power of 10 to shift the decimal so that both \( x \) and \( 10x \) have the same repeating digits. Here, we multiply by 10 again: \[ 100x = 75.55555\ldots \]
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Set up the equations: \[ 10x = 7.55555\ldots \quad \text{(1)} \] \[ 100x = 75.55555\ldots \quad \text{(2)} \]
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Subtract equation (1) from equation (2): \[ 100x - 10x = 75.55555\ldots - 7.55555\ldots \] \[ 90x = 68 \]
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Solve for \( x \): \[ x = \frac{68}{90} \]
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Simplify the fraction: To simplify \( \frac{68}{90} \), find the greatest common divisor (GCD) of 68 and 90. The GCD is 2. \[ \frac{68 \div 2}{90 \div 2} = \frac{34}{45} \]
Thus, the repeating decimal \( 0.755555\ldots \) as a fraction in simplest form is: \[ \boxed{\frac{34}{45}} \]
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