To arrange the given rational numbers in increasing (ascending) order, we'll start by converting each fraction to its decimal equivalent. This will help us compare and sort them more easily.
Here are the given fractions and their decimal equivalents:
- [tex]\(\frac{4}{5} = 0.8\)[/tex]
- [tex]\(\frac{-2}{5} = -0.4\)[/tex]
- [tex]\(\frac{2}{1} = 2.0\)[/tex]
- [tex]\(\frac{-1}{5} = -0.2\)[/tex]
- [tex]\(\frac{2}{10} = 0.2\)[/tex]
- [tex]\(\frac{4}{6} = \frac{2}{3} = 0.6666666666666666\)[/tex]
- [tex]\(\frac{5}{2} = 2.5\)[/tex]
- [tex]\(\frac{3}{3} = 1.0\)[/tex]
- [tex]\(\frac{-2}{6} = \frac{-1}{3} = -0.3333333333333333\)[/tex]
- [tex]\(\frac{1}{6} = 0.16666666666666666\)[/tex]
- [tex]\(0 = 0\)[/tex]
- [tex]\(\frac{3}{6} = \frac{1}{2} = 0.5\)[/tex]
Next, we compare and order these decimals from the smallest to the largest:
1. [tex]\(\frac{-2}{5} = -0.4\)[/tex]
2. [tex]\(\frac{-2}{6} = -0.3333333333333333\)[/tex]
3. [tex]\(\frac{-1}{5} = -0.2\)[/tex]
4. [tex]\(0 = 0\)[/tex]
5. [tex]\(\frac{1}{6} = 0.16666666666666666\)[/tex]
6. [tex]\(\frac{2}{10} = 0.2\)[/tex]
7. [tex]\(\frac{3}{6} = 0.5\)[/tex]
8. [tex]\(\frac{4}{6} = 0.6666666666666666\)[/tex]
9. [tex]\(\frac{4}{5} = 0.8\)[/tex]
10. [tex]\(\frac{3}{3} = 1.0\)[/tex]
11. [tex]\(\frac{2}{1} = 2.0\)[/tex]
12. [tex]\(\frac{5}{2} = 2.5\)[/tex]
So here is the list of fractions in ascending order:
[tex]\[
\frac{-2}{5}, \frac{-2}{6}, \frac{-1}{5}, 0, \frac{1}{6}, \frac{2}{10}, \frac{3}{6}, \frac{4}{6}, \frac{4}{5}, \frac{3}{3}, \frac{2}{1}, \frac{5}{2}
\][/tex]
For completeness, the corresponding decimals in ascending order are:
[tex]\[
-0.4, -0.3333333333333333, -0.2, 0, 0.16666666666666666, 0.2, 0.5, 0.6666666666666666, 0.8, 1.0, 2.0, 2.5
\][/tex]
