To determine which fraction is equivalent to the decimal [tex]\(0.09\)[/tex], let's evaluate each of the given fractions step-by-step.
1. Check [tex]\(\frac{10}{9}\)[/tex]:
- [tex]\(\frac{10}{9}\)[/tex] is approximately equal to [tex]\(1.111\)[/tex].
- Therefore, [tex]\(\frac{10}{9}\)[/tex] is not equivalent to [tex]\(0.09\)[/tex].
2. Check [tex]\(\frac{9}{10}\)[/tex]:
- [tex]\(\frac{9}{10}\)[/tex] is equal to [tex]\(0.9\)[/tex].
- Therefore, [tex]\(\frac{9}{10}\)[/tex] is not equivalent to [tex]\(0.09\)[/tex].
3. Check [tex]\(\frac{90}{1000}\)[/tex]:
- Simplify [tex]\(\frac{90}{1000}\)[/tex]:
[tex]\[
\frac{90}{1000} = \frac{9}{100} = 0.09
\][/tex]
- We see that after simplification, [tex]\(\frac{90}{1000}\)[/tex] is indeed equal to [tex]\(0.09\)[/tex].
4. Check [tex]\(\frac{9}{100}\)[/tex]:
- [tex]\(\frac{9}{100} = 0.09\)[/tex].
- Therefore, [tex]\(\frac{9}{100}\)[/tex] is equivalent to [tex]\(0.09\)[/tex].
Since both [tex]\(\frac{90}{1000}\)[/tex] and [tex]\(\frac{9}{100}\)[/tex] are correct and equal to [tex]\(0.09\)[/tex], we have verified that these choices match the given decimal.
In conclusion:
- [tex]\(\frac{90}{1000}\)[/tex] and [tex]\(\frac{9}{100}\)[/tex] are both correct answers, but traditionally we use the simplest form, which is [tex]\(\frac{9}{100}\)[/tex].
Therefore, the fractions which match [tex]\(0.09\)[/tex] are [tex]\(\frac{90}{1000}\)[/tex] and [tex]\(\frac{9}{100}\)[/tex]. If we need to choose the simplest form, it is:
[tex]\[
\boxed{\frac{9}{100}}
\][/tex]
