Certainly! Let's convert the repeating decimal [tex]\( 0.585858\ldots \)[/tex] (which can be written as [tex]\( 0.5\overline{85} \)[/tex]) into a fraction in the form [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex].
To do this, we will follow these steps:
1. Let [tex]\( x = 0.585858\ldots \)[/tex]:
[tex]\[
x = 0.585858\ldots
\][/tex]
2. Multiply both sides by 100 (since the repeating block "85" is two digits long):
[tex]\[
100x = 58.585858\ldots
\][/tex]
3. Now, we have the two equations:
[tex]\[
x = 0.585858\ldots \quad \text{(Equation 1)}
\][/tex]
[tex]\[
100x = 58.585858\ldots \quad \text{(Equation 2)}
\][/tex]
4. Subtract Equation 1 from Equation 2 to eliminate the repeating part:
[tex]\[
100x - x = 58.585858\ldots - 0.585858\ldots
\][/tex]
[tex]\[
99x = 58
\][/tex]
5. Solve for [tex]\( x \)[/tex] by dividing both sides by 99:
[tex]\[
x = \frac{58}{99}
\][/tex]
So, the repeating decimal [tex]\( 0.585858\ldots \)[/tex] can be expressed in fraction form as [tex]\( \frac{58}{99} \)[/tex].
Thus, [tex]\( \frac{58}{99} \)[/tex] is the fraction representation for [tex]\( 0.585858\ldots \)[/tex], where [tex]\( p = 58 \)[/tex] and [tex]\( q = 99 \)[/tex].
