To arrange the given rational numbers in ascending order, we will follow these steps:
### (i) Rational Numbers: [tex]\(\frac{2}{5}, \frac{1}{3}, \frac{3}{4}, \frac{1}{6}\)[/tex]
1. [tex]\(\frac{2}{5} = 0.4\)[/tex]
2. [tex]\(\frac{1}{3} \approx 0.333\)[/tex]
3. [tex]\(\frac{3}{4} = 0.75\)[/tex]
4. [tex]\(\frac{1}{6} \approx 0.167\)[/tex]
Now sort these decimal values in ascending order:
- [tex]\(\frac{1}{6} \approx 0.167\)[/tex]
- [tex]\(\frac{1}{3} \approx 0.333\)[/tex]
- [tex]\(\frac{2}{5} = 0.4\)[/tex]
- [tex]\(\frac{3}{4} = 0.75\)[/tex]
So, the ascending order of the rational numbers is:
[tex]\[
\boxed{\frac{1}{6}, \frac{1}{3}, \frac{2}{5}, \frac{3}{4}}
\][/tex]
### (ii) Rational Numbers: [tex]\(\frac{5}{6}, \frac{7}{8}, \frac{11}{12}, \frac{3}{10}\)[/tex]
1. [tex]\(\frac{5}{6} \approx 0.833\)[/tex]
2. [tex]\(\frac{7}{8} = 0.875\)[/tex]
3. [tex]\(\frac{11}{12} \approx 0.917\)[/tex]
4. [tex]\(\frac{3}{10} = 0.3\)[/tex]
Now sort these decimal values in ascending order:
- [tex]\(\frac{3}{10} = 0.3\)[/tex]
- [tex]\(\frac{5}{6} \approx 0.833\)[/tex]
- [tex]\(\frac{7}{8} = 0.875\)[/tex]
- [tex]\(\frac{11}{12} \approx 0.917\)[/tex]
So, the ascending order of the rational numbers is:
[tex]\[
\boxed{\frac{3}{10}, \frac{5}{6}, \frac{7}{8}, \frac{11}{12}}
\][/tex]
### (iii) Rational Numbers: [tex]\(\frac{3}{7}, \frac{-5}{9}, \frac{7}{-11}, \frac{6}{1}, \frac{0}{1}, \frac{9}{3}\)[/tex]
1. [tex]\(\frac{3}{7} \approx 0.429\)[/tex]
2. [tex]\(\frac{-5}{9} \approx -0.556\)[/tex]
3. [tex]\(\frac{7}{-11} \approx -0.636\)[/tex]
4. [tex]\(\frac{6}{1} = 6\)[/tex]
5. [tex]\(\frac{0}{1} = 0\)[/tex]
6. [tex]\(\frac{9}{3} = 3\)[/tex]
Now sort these decimal values in ascending order:
- [tex]\(\frac{7}{-11} \approx -0.636\)[/tex]
- [tex]\(\frac{-5}{9} \approx -0.556\)[/tex]
- [tex]\(\frac{0}{1} = 0\)[/tex]
- [tex]\(\frac{3}{7} \approx 0.429\)[/tex]
- [tex]\(\frac{9}{3} = 3\)[/tex]
- [tex]\(\frac{6}{1} = 6\)[/tex]
So, the ascending order of the rational numbers is:
[tex]\[
\boxed{\frac{7}{-11}, \frac{-5}{9}, \frac{0}{1}, \frac{3}{7}, \frac{9}{3}, \frac{6}{1}}
\][/tex]
