Question

What is the fraction form of the following decimal expansion?
12.5 overline18

Answer 1

To express the decimal \( 12.5\overline{18} \) in fraction form, we can break it down into its non-repeating and repeating parts.

1. **Separate the parts**: The decimal can be written as:
\[
12.5 + 0.0\overline{18}
\]

2. **Convert \( 12.5 \) to a fraction**:
\[
12.5 = \frac{125}{10} = \frac{25}{2}
\]

3. **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,
\[
0.0\overline{18} = \frac{1}{55}
\]

4. **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:
\[
\frac{25}{2} = \frac{25 \times 55}{2 \times 55} = \frac{1375}{110}
\]
\[
\frac{1}{55} = \frac{1 \times 2}{55 \times 2} = \frac{2}{110}
\]

5. **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:
\[
\boxed{\frac{1377}{110}}
\]