To convert the decimal \( 12.5\overline{18} \) into a fraction, we need to separate the whole number, the decimal part, and the repeating part.
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Separate the components:
- Whole number part: 12
- Non-repeating decimal part: 0.5
- Repeating decimal part: \( 0.\overline{18} \)
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Convert the non-repeating part: The decimal \( 0.5 \) can be expressed as: \[ 0.5 = \frac{5}{10} = \frac{1}{2} \]
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Convert the repeating part: Let \( x = 0.\overline{18} \).
- Thus, \( 100x = 18.181818\ldots \)
- Subtracting the first equation from the second gives: \[ 100x - x = 18.181818\ldots - 0.181818\ldots \] \[ 99x = 18 \implies x = \frac{18}{99} = \frac{2}{11} \quad (\text{after simplifying}) \]
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Combine the parts: Thus, the decimal \( 12.5\overline{18} \) can be written as: \[ 12 + 0.5 + 0.\overline{18} = 12 + \frac{1}{2} + \frac{2}{11} \]
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Finding a common denominator: The common denominator for \( 2 \) and \( 11 \) is \( 22 \). \[ \frac{1}{2} = \frac{11}{22} \quad \text{and} \quad \frac{2}{11} = \frac{4}{22} \]
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Add the fractions: \[ \frac{11}{22} + \frac{4}{22} = \frac{15}{22} \] Thus, \[ 12 + \frac{15}{22} = \frac{12 \times 22 + 15}{22} = \frac{264 + 15}{22} = \frac{279}{22} \]
So the fraction form of \( 12.5\overline{18} \) is \( \frac{279}{22} \).
Placing the numbers in the mixed number: Given the mixed number format || |/___|:
- The whole number part is \( 12 (F) \).
- The numerator of the fraction part is \( 15 (E) \) and the denominator is \( 22 (not listed) \).
Therefore, the completed mixed number should be displayed as: | 12 | 15 / ___ | (where the denominator is implied to be 22, but not listed).
The correct letters to place in the mixed number: \[ | F | E / ___ | \]
Thus, the answer for this problem is: | 12 | 15 / ___ | where \( F = 12 \) and \( E = 15 \).