Sure, let's express [tex]\(0.3333\ldots = 0.\overline{3}\)[/tex] in the form [tex]\(\frac{p}{q}\)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex].
1. Let's represent the repeating decimal [tex]\(0.3333\ldots\)[/tex] as [tex]\(x\)[/tex]. Therefore, we have:
[tex]\[
x = 0.3333\ldots
\][/tex]
2. To eliminate the repeating part, multiply both sides of the equation by 10:
[tex]\[
10x = 3.3333\ldots
\][/tex]
3. Now, let's subtract the original equation from this new equation:
[tex]\[
10x - x = 3.3333\ldots - 0.3333\ldots
\][/tex]
Simplifying both sides, we get:
[tex]\[
9x = 3
\][/tex]
4. Next, solve for [tex]\(x\)[/tex] by dividing both sides by 9:
[tex]\[
x = \frac{3}{9}
\][/tex]
5. Simplify the fraction [tex]\(\frac{3}{9}\)[/tex]:
[tex]\[
x = \frac{1}{3}
\][/tex]
6. Therefore, [tex]\(p = 1\)[/tex] and [tex]\(q = 3\)[/tex], and thus:
[tex]\[
0.3333\ldots = \frac{1}{3}
\][/tex]
Hence, [tex]\(0.3333\ldots\)[/tex] can be expressed as [tex]\(\frac{1}{3}\)[/tex], where [tex]\(p = 1\)[/tex] and [tex]\(q = 3\)[/tex].
