To convert the decimal \( 12.5\overline{18} \) to a fraction, we can express it as follows:
Let:
\[ x = 12.5\overline{18} \]
**Step 1:** Break down \( x \):
\[ x = 12.5 + 0.0\overline{18} \]
Next, let's focus on the repeating decimal \( 0.0\overline{18} \).
**Step 2:** Set \( y = 0.0\overline{18} \):
\[ y = 0.0181818... \]
Multiply \( y \) by 100 (to shift the repeating part left of the decimal):
\[ 100y = 1.818181... \]
Now, we have:
\[ y = 0.0181818... \]
\[ 100y = 1.818181... \]
**Step 3:** Subtract the two equations:
\[ 100y - y = 1.818181... - 0.0181818... \]
\[ 99y = 1.8 \]
**Step 4:** Solve for \( y \):
\[ y = \frac{1.8}{99} = \frac{18}{990} = \frac{1}{55} \]
**Step 5:** Combining everything:
We have:
\[ x = 12.5 + \frac{1}{55} \]
**Step 6:** Convert \( 12.5 \) to a fraction:
\[ 12.5 = \frac{125}{10} = \frac{25}{2} \]
Now combine the two fractions:
\[ x = \frac{25}{2} + \frac{1}{55} \]
**Step 7:** Get a common denominator (for 2 and 55, it's 110):
Convert \( \frac{25}{2} \) to a fraction with a denominator of 110:
\[ \frac{25}{2} = \frac{25 \times 55}{2 \times 55} = \frac{1375}{110} \]
Convert \( \frac{1}{55} \) to a fraction with a denominator of 110:
\[ \frac{1}{55} = \frac{1 \times 2}{55 \times 2} = \frac{2}{110} \]
**Step 8:** Add the two fractions:
\[ x = \frac{1375}{110} + \frac{2}{110} = \frac{1377}{110} \]
So, the fraction form of \( 12.5\overline{18} \) is:
\[ \frac{1377}{110} \]
Now, based on the choices provided, you are looking for a mixed number expression, where:
The corresponding labels to use:
- A (500)
- B (57)
- C (259)
- D (9)
- E (110)
- F (12)
- G (50)
In the mixed number form, \( 1377 = 12 \times 110 + 57 \).
Thus, the answer would be:
\[ \frac{1377}{110} \text{ can be represented as } F \text{ (12)}/E \text{ (110)}. \]
So, the final answer is:
**Answer: F/E = 12/110**.