To convert the repeating decimal [tex]\(0.\overline{3}\)[/tex] to a simplified fraction, follow these steps:
1. Let [tex]\(x = 0.\overline{3}\)[/tex]. This means [tex]\(x\)[/tex] is equal to the repeating decimal 0.33333...
2. To eliminate the repeating part, multiply [tex]\(x\)[/tex] by 10. This gives:
[tex]\[
10x = 3.33333...
\][/tex]
3. Now, subtract the original equation [tex]\(x = 0.33333...\)[/tex] from this new equation:
[tex]\[
10x - x = 3.33333... - 0.33333...
\][/tex]
4. Simplifying the left side and the right side of the subtraction, we get:
[tex]\[
9x = 3
\][/tex]
5. To solve for [tex]\(x\)[/tex], divide both sides of the equation by 9:
[tex]\[
x = \frac{3}{9}
\][/tex]
6. Simplify the fraction [tex]\(\frac{3}{9}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
[tex]\[
\frac{3}{9} = \frac{1}{3}
\][/tex]
Therefore, the repeating decimal [tex]\(0.\overline{3}\)[/tex] can be written as the simplified fraction [tex]\(\frac{1}{3}\)[/tex].
