To determine which numbers from the provided list are equivalent to the mixed number [tex]\(3 \frac{2}{9}\)[/tex], let's convert [tex]\(3 \frac{2}{9}\)[/tex] into an improper fraction and then into a decimal.
1. Convert [tex]\(3 \frac{2}{9}\)[/tex] to an improper fraction:
- The whole number part is 3.
- The fraction part is [tex]\(\frac{2}{9}\)[/tex].
To convert the mixed number to an improper fraction:
[tex]\[
3 \frac{2}{9} = \frac{3 \times 9 + 2}{9} = \frac{27 + 2}{9} = \frac{29}{9}
\][/tex]
2. Convert the improper fraction [tex]\(\frac{29}{9}\)[/tex] to a decimal:
[tex]\[
\frac{29}{9} \approx 3.2222\ldots
\][/tex]
This is a repeating decimal, so it can be written as [tex]\(3.2\overline{2}\)[/tex].
Now we look at the provided numbers and see which ones are equivalent to 3.2\overline{2} (or [tex]\(\frac{29}{9}\)[/tex]):
- [tex]\(\frac{29}{9}\)[/tex]: This fraction calculates as follows:
[tex]\[
\frac{29}{9} = 3.2222\overline{2}
\][/tex]
This matches exactly with our calculated decimal value.
- [tex]\(32\)[/tex]: As a decimal value, [tex]\(32\)[/tex] is clearly not equivalent to [tex]\(3.2222\overline{2}\)[/tex].
- [tex]\(322\)[/tex]: As a decimal value, [tex]\(322\)[/tex] is clearly not equivalent to [tex]\(3.2222\overline{2}\)[/tex].
- [tex]\(0.2\%\)[/tex]: This is [tex]\(0.2\)[/tex] percent, which can be converted to a decimal:
[tex]\[
0.2\% = \frac{0.2}{100} = 0.002
\][/tex]
This is not equivalent to [tex]\(3.2222\overline{2}\)[/tex].
- [tex]\(329\)[/tex]: As a decimal value, [tex]\(329\)[/tex] is clearly not equivalent to [tex]\(3.2222\overline{2}\)[/tex].
- [tex]\(32978\)[/tex]: As a decimal value, [tex]\(32978\)[/tex] is clearly not equivalent to [tex]\(3.2222\overline{2}\)[/tex].
- [tex]\(\frac{6}{9}\)[/tex]: As a decimal:
[tex]\[
\frac{6}{9} = 0.666\overline{6}
\][/tex]
This is not equivalent to [tex]\(3.2222\overline{2}\)[/tex].
- [tex]\(32.322\)[/tex]: As a decimal value, [tex]\(32.322\)[/tex] is clearly not equivalent to [tex]\(3.2222\overline{2}\)[/tex].
- [tex]\(32970\)[/tex]: As a decimal value, [tex]\(32970\)[/tex] is clearly not equivalent to [tex]\(3.2222\overline{2}\)[/tex].
Given the analysis above, the number from the provided list that is equivalent to [tex]\(3 \frac{2}{9}\)[/tex] or [tex]\(3.2222\overline{2}\)[/tex] is:
[tex]\[ \boxed{\frac{29}{9}} \][/tex]
