To determine the rational number equivalent to the repeating decimal [tex]$0.33333\ldots$[/tex], follow these steps:
1. Define the Repeating Decimal:
Let [tex]\( x = 0.33333\ldots \)[/tex].
2. Express as an Equation:
Since [tex]\( 0.33333\ldots \)[/tex] repeats every digit, it can be expressed as [tex]\( x = 0.\overline{3} \)[/tex].
3. Eliminate the Repeating Decimal by Multiplication:
Multiply both sides by 10 (which is the base of the decimal representation), so:
[tex]\[
10x = 3.33333\ldots
\][/tex]
This step shifts the repeating decimal one place to the left.
4. Set Up a System of Equations to Eliminate the Decimal:
Now, subtract the original [tex]\( x = 0.33333\ldots \)[/tex] from this new equation:
[tex]\[
10x - x = 3.33333\ldots - 0.33333\ldots
\][/tex]
Simplifying the left-hand side and the right-hand side, you get:
[tex]\[
9x = 3
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Divide both sides by 9:
[tex]\[
x = \frac{3}{9}
\][/tex]
Simplify this fraction:
[tex]\[
x = \frac{1}{3}
\][/tex]
Therefore, the rational number equivalent to the repeating decimal [tex]\( 0.33333\ldots \)[/tex] is [tex]\( \frac{1}{3} \)[/tex].
Hence, the correct answer is [tex]\( \boxed{1 / 3} \)[/tex].
