To convert the repeating decimal [tex]\(0.\overline{601}\)[/tex] into a fraction, we can follow these steps:
1. Assign a variable to the repeating decimal:
Let [tex]\( x = 0.601601601\ldots \)[/tex]
2. Multiply [tex]\( x \)[/tex] by a power of 10 that shifts the repeating part to the left of the decimal point:
Since the repeating part "601" is three digits long, we multiply [tex]\( x \)[/tex] by [tex]\( 1000 \)[/tex]:
[tex]\[
1000x = 601.601601601\ldots
\][/tex]
3. Set up the equations:
[tex]\[
x = 0.601601601\ldots \quad \text{(1)}
\][/tex]
[tex]\[
1000x = 601.601601601\ldots \quad \text{(2)}
\][/tex]
4. Subtract the first equation from the second to eliminate the repeating part:
[tex]\[
1000x - x = 601.601601601\ldots - 0.601601601\ldots
\][/tex]
[tex]\[
999x = 601
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{601}{999}
\][/tex]
Thus, the infinite decimal [tex]\(0.\overline{601}\)[/tex] can be expressed as the fraction [tex]\(\frac{601}{999}\)[/tex].
