To convert the repeating decimal \( 0.78787878\ldots \) to a fraction, we can use the following steps.
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Let \( x \) be the repeating decimal: \[ x = 0.78787878\ldots \]
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Multiply \( x \) by 100 (since the repeating part has 2 digits): \[ 100x = 78.78787878\ldots \]
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Now we have two equations: \[ x = 0.78787878\ldots \quad (1) \] \[ 100x = 78.78787878\ldots \quad (2) \]
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Subtract equation (1) from equation (2): \[ 100x - x = 78.78787878\ldots - 0.78787878\ldots \] \[ 99x = 78 \]
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Solve for \( x \): \[ x = \frac{78}{99} \]
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Simplify the fraction: To simplify \( \frac{78}{99} \), we find the greatest common divisor (GCD) of 78 and 99. The prime factorizations are:
- \( 78 = 2 \times 3 \times 13 \)
- \( 99 = 3^2 \times 11 \)
The GCD is \( 3 \).
Now divide both the numerator and the denominator by \( 3 \): \[ \frac{78 \div 3}{99 \div 3} = \frac{26}{33} \]
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Check if \( \frac{26}{33} \) can be simplified further: The factors of \( 26 \) are \( 2 \times 13 \), and the factors of \( 33 \) are \( 3 \times 11 \). Since they have no common factors, \( \frac{26}{33} \) is in its simplest form.
Thus, the repeating decimal \( 0.78787878\ldots \) can be expressed as the fraction: \[ \boxed{\frac{26}{33}} \]