To determine which fractions are equivalent to [tex]\(\frac{2}{6}\)[/tex], we need to examine each provided fraction and simplify them to see if any of them reduce to the same value as [tex]\(\frac{2}{6}\)[/tex].
Let's first simplify [tex]\(\frac{2}{6}\)[/tex]:
[tex]\[
\frac{2}{6} = \frac{2 \div 2}{6 \div 2} = \frac{1}{3}
\][/tex]
Now, we will simplify each fraction provided:
1. [tex]\(\frac{4}{8}\)[/tex]:
[tex]\[
\frac{4}{8} = \frac{4 \div 4}{8 \div 4} = \frac{1}{2}
\][/tex]
So, [tex]\(\frac{4}{8} \neq \frac{1}{3}\)[/tex].
2. [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\frac{1}{3}
\][/tex]
This fraction is already simplified and matches [tex]\(\frac{1}{3}\)[/tex].
3. [tex]\(\frac{5}{9}\)[/tex]:
[tex]\[
\frac{5}{9}
\][/tex]
This fraction is already simplified, and [tex]\(\frac{5}{9} \neq \frac{1}{3}\)[/tex].
4. [tex]\(\frac{4}{12}\)[/tex]:
[tex]\[
\frac{4}{12} = \frac{4 \div 4}{12 \div 4} = \frac{1}{3}
\][/tex]
So, [tex]\(\frac{4}{12} = \frac{1}{3}\)[/tex].
5. [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\frac{1}{4}
\][/tex]
This fraction is already simplified, and [tex]\(\frac{1}{4} \neq \frac{1}{3}\)[/tex].
Thus, the fractions that are equivalent to [tex]\(\frac{2}{6}\)[/tex] (or [tex]\(\frac{1}{3}\)[/tex]) are:
[tex]\[
\boxed{\frac{1}{3} \text{ and } \frac{4}{12}}
\][/tex]
