Sure! Let's convert the repeating decimal [tex]\( 3.0\overline{1} \)[/tex] into a mixed number.
1. Express the repeating decimal as a fraction:
- Let [tex]\( x = 3.0111111\ldots \)[/tex]
- To eliminate the repeating part, multiply by 10:
[tex]\[
10x = 30.111111\ldots
\][/tex]
- Now, subtract the original [tex]\( x \)[/tex] from [tex]\( 10x \)[/tex]:
[tex]\[
10x - x = 30.111111\ldots - 3.011111\ldots
\][/tex]
[tex]\[
9x = 27.1
\][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{27.1}{9}
\][/tex]
2. Convert the improper fraction to a mixed number:
- Divide [tex]\( 27.1 \)[/tex] by [tex]\( 9 \)[/tex] to find the integer part and the fractional part:
[tex]\[
\frac{27.1}{9} = 3 + \frac{0.1}{9}
\][/tex]
- Here, the integer part is [tex]\( 3 \)[/tex].
3. Express the fractional part:
- The remaining fractional part is [tex]\( \frac{0.1}{9} \)[/tex], which can be simplified as:
[tex]\[
\frac{0.1 \times 10}{9} = \frac{1}{90}
\][/tex]
4. Combine the integer and fractional parts:
- The mixed number is:
[tex]\[
3 + \frac{1}{90}
\][/tex]
Therefore, the repeating decimal [tex]\( 3.0\overline{1} \)[/tex] written as a mixed number is [tex]\( 3 + \frac{1}{90} \)[/tex].
