To determine which fraction is closest to 1, we can calculate how far each fraction is from 1 by finding the absolute difference between each fraction and 1. We will evaluate the absolute difference for each fraction one-by-one.
1. For the fraction [tex]\(\frac{8}{7}\)[/tex]:
[tex]\[
\text{Absolute difference} = \left| \frac{8}{7} - 1 \right| = \left| \frac{8}{7} - \frac{7}{7} \right| = \left| \frac{8-7}{7} \right| = \left| \frac{1}{7} \right| = \frac{1}{7}
\][/tex]
So, the absolute difference is [tex]\(\frac{1}{7}\)[/tex].
2. For the fraction [tex]\(\frac{9}{8}\)[/tex]:
[tex]\[
\text{Absolute difference} = \left| \frac{9}{8} - 1 \right| = \left| \frac{9}{8} - \frac{8}{8} \right| = \left| \frac{9-8}{8} \right| = \left| \frac{1}{8} \right| = \frac{1}{8}
\][/tex]
So, the absolute difference is [tex]\(\frac{1}{8}\)[/tex].
3. For the fraction [tex]\(\frac{10}{9}\)[/tex]:
[tex]\[
\text{Absolute difference} = \left| \frac{10}{9} - 1 \right| = \left| \frac{10}{9} - \frac{9}{9} \right| = \left| \frac{10-9}{9} \right| = \left| \frac{1}{9} \right| = \frac{1}{9}
\][/tex]
So, the absolute difference is [tex]\(\frac{1}{9}\)[/tex].
4. For the fraction [tex]\(\frac{5}{6}\)[/tex]:
[tex]\[
\text{Absolute difference} = \left| \frac{5}{6} - 1 \right| = \left| \frac{5}{6} - \frac{6}{6} \right| = \left| \frac{5-6}{6} \right| = \left| \frac{-1}{6} \right| = \frac{1}{6}
\][/tex]
So, the absolute difference is [tex]\(\frac{1}{6}\)[/tex].
5. For the fraction [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
\text{Absolute difference} = \left| \frac{3}{4} - 1 \right| = \left| \frac{3}{4} - \frac{4}{4} \right| = \left| \frac{3-4}{4} \right| = \left| \frac{-1}{4} \right| = \frac{1}{4}
\][/tex]
So, the absolute difference is [tex]\(\frac{1}{4}\)[/tex].
Now, we compare the absolute differences we calculated:
- [tex]\(\frac{1}{7} \approx 0.142857\)[/tex]
- [tex]\(\frac{1}{8} = 0.125\)[/tex]
- [tex]\(\frac{1}{9} \approx 0.111111\)[/tex]
- [tex]\(\frac{1}{6} \approx 0.166667\)[/tex]
- [tex]\(\frac{1}{4} = 0.25\)[/tex]
Among these values, [tex]\(\frac{1}{9}\)[/tex] (approximately 0.111111) is the smallest.
Therefore, the fraction that is closest to 1 is [tex]\(\frac{10}{9}\)[/tex], which is option 3.
