To express the decimal \( 12.5\overline{18} \) in fraction form, we can break it down into its non-repeating and repeating parts.
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Separate the parts: The decimal can be written as: \[ 12.5 + 0.0\overline{18} \]
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Convert \( 12.5 \) to a fraction: \[ 12.5 = \frac{125}{10} = \frac{25}{2} \]
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Convert \( 0.0\overline{18} \) to a fraction: Let \( x = 0.0\overline{18} \).
Multiply both sides by \( 100 \) to shift the decimal point two places: \[ 100x = 1.81818\ldots \]
Now, subtract the original \( x \) from this equation: \[ 100x - x = 1.81818\ldots - 0.01818\ldots
\] \[ 99x = 1.8 \]Now we convert \( 1.8 \) to a fraction: \[ 1.8 = \frac{18}{10} = \frac{9}{5} \]
Thus, we have: \[ 99x = \frac{9}{5} \] \[ x = \frac{9}{5 \cdot 99} = \frac{9}{495} \] Simplifying \( \frac{9}{495} \): \[ \frac{9}{495} = \frac{1}{55} \] Hence, \[ 0.0\overline{18} = \frac{1}{55} \]
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Combine the fractions: Now, we combine \( 12.5 \) and \( 0.0\overline{18} \): \[ 12.5\overline{18} = \frac{25}{2} + \frac{1}{55} \]
To add these fractions, we need a common denominator. The least common multiple of \( 2 \) and \( 55 \) is \( 110 \).
Convert each fraction: \[ \frac{25}{2} = \frac{25 \times 55}{2 \times 55} = \frac{1375}{110} \] \[ \frac{1}{55} = \frac{1 \times 2}{55 \times 2} = \frac{2}{110} \]
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Add the fractions: \[ 12.5\overline{18} = \frac{1375}{110} + \frac{2}{110} = \frac{1377}{110} \]
Therefore, the fraction form of \( 12.5\overline{18} \) is: \[ \boxed{\frac{1377}{110}} \]
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