To convert the repeating decimal [tex]\(0.1\overline{6}\)[/tex] into a fraction, follow these steps:
1. Let [tex]\( x \)[/tex] be the repeating decimal:
[tex]\[
x = 0.1666666\ldots
\][/tex]
2. Multiply both sides of the equation by 10 to move the decimal point one place to the right:
[tex]\[
10x = 1.666666\ldots
\][/tex]
3. Subtract the original [tex]\( x \)[/tex] from this new equation:
[tex]\[
10x - x = 1.666666\ldots - 0.1666666\ldots
\][/tex]
4. Perform the subtraction on the right side:
[tex]\[
9x = 1.5
\][/tex]
5. Now, solve for [tex]\( x \)[/tex] by dividing both sides by 9:
[tex]\[
x = \frac{1.5}{9}
\][/tex]
6. Simplify the fraction:
[tex]\[
x = \frac{1.5}{9} = \frac{3/2}{9} = \frac{3}{2} \times \frac{1}{9} = \frac{3}{18} = \frac{1}{6}
\][/tex]
Therefore,
[tex]\[
0.1\overline{6} = \frac{1}{6} = 0.16666666666666666\ldots
\][/tex]
The repeating decimal [tex]\(0.1\overline{6}\)[/tex] can be expressed as the fraction [tex]\(\frac{1}{6}\)[/tex].
