To determine which fractions are equivalent to [tex]\(\frac{4}{10}\)[/tex], we'll simplify and compare each fraction one by one.
First, we simplify the fraction [tex]\(\frac{4}{10}\)[/tex]:
[tex]\[
\frac{4}{10} = \frac{4 \div 2}{10 \div 2} = \frac{2}{5}
\][/tex]
So, [tex]\(\frac{4}{10} = \frac{2}{5}\)[/tex].
Now, let's check each given fraction:
1. [tex]\(\frac{2}{5}\)[/tex]:
[tex]\[
\frac{2}{5} = \frac{2}{5} \quad \text{(already simplified)}
\][/tex]
[tex]\(\frac{2}{5}\)[/tex] is equivalent to [tex]\(\frac{4}{10}\)[/tex].
2. [tex]\(\frac{40}{100}\)[/tex]:
[tex]\[
\frac{40}{100} = \frac{40 \div 10}{100 \div 10} = \frac{4}{10}
\][/tex]
[tex]\(\frac{40}{100}\)[/tex] simplifies to [tex]\(\frac{4}{10}\)[/tex], so it is also equivalent to [tex]\(\frac{4}{10}\)[/tex].
3. [tex]\(\frac{1}{6}\)[/tex]:
[tex]\[
\frac{1}{6} = \frac{1}{6} \quad \text{(already simplified)}
\][/tex]
[tex]\(\frac{1}{6}\)[/tex] is not equivalent to [tex]\(\frac{4}{10}\)[/tex].
4. [tex]\(\frac{8}{10}\)[/tex]:
[tex]\[
\frac{8}{10} = \frac{8 \div 2}{10 \div 2} = \frac{4}{5}
\][/tex]
[tex]\(\frac{8}{10}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex], which is not equivalent to [tex]\(\frac{4}{10}\)[/tex].
5. [tex]\(\frac{2}{8}\)[/tex]:
[tex]\[
\frac{2}{8} = \frac{2 \div 2}{8 \div 2} = \frac{1}{4}
\][/tex]
[tex]\(\frac{2}{8}\)[/tex] simplifies to [tex]\(\frac{1}{4}\)[/tex], which is not equivalent to [tex]\(\frac{4}{10}\)[/tex].
Therefore, the fractions that are equivalent to [tex]\(\frac{4}{10}\)[/tex] are:
[tex]\[
\boxed{\frac{2}{5}, \frac{40}{100}}
\][/tex]
