Arrange the following rational numbers in descending order :
(i) `11/12, 13/18, 5/6, 7/9`, (ii) `-11/20, 3/(-10), 17/(-30), -7/15`
(iii) `9/(-24), -1, 2/(-3), -7/(-6)`, (iv) `7/(-10), 11/15, -17/(-30), -2/5`

Answer

To arrange the given rational numbers in descending order, we will follow these steps for each set of numbers:

### (i) Arrange `11/12, 13/18, 5/6, 7/9`

1. **Find the LCM of the denominators**: The denominators are 12, 18, 6, and 9. The LCM is 36.
2. **Convert each fraction to have the same denominator**:
- \( \frac{11}{12} = \frac{11 \times 3}{12 \times 3} = \frac{33}{36} \)
- \( \frac{13}{18} = \frac{13 \times 2}{18 \times 2} = \frac{26}{36} \)
- \( \frac{5}{6} = \frac{5 \times 6}{6 \times 6} = \frac{30}{36} \)
- \( \frac{7}{9} = \frac{7 \times 4}{9 \times 4} = \frac{28}{36} \)
3. **Compare the numerators**: Now we have \( \frac{33}{36}, \frac{26}{36}, \frac{30}{36}, \frac{28}{36} \).
4. **Arrange in descending order**:
- The largest is \( \frac{33}{36} \) (which is \( \frac{11}{12} \)),
- Next is \( \frac{30}{36} \) (which is \( \frac{5}{6} \)),
- Then \( \frac{28}{36} \) (which is \( \frac{7}{9} \)),
- Finally, \( \frac{26}{36} \) (which is \( \frac{13}{18} \)).
5. **Final answer**: \( \frac{11}{12}, \frac{5}{6}, \frac{7}{9}, \frac{13}{18} \)

### (ii) Arrange `-11/20, 3/(-10), 17/(-30), -7/15`

1. **Find the LCM of the denominators**: The denominators are 20, -10, -30, and -15. The LCM is 60.
2. **Convert each fraction to have the same denominator**:
- \( -\frac{11}{20} = -\frac{11 \times 3}{20 \times 3} = -\frac{33}{60} \)
- \( \frac{3}{-10} = -\frac{3 \times 6}{10 \times 6} = -\frac{18}{60} \)
- \( \frac{17}{-30} = -\frac{17 \times 2}{30 \times 2} = -\frac{34}{60} \)
- \( -\frac{7}{15} = -\frac{7 \times 4}{15 \times 4} = -\frac{28}{60} \)
3. **Compare the numerators**: Now we have \( -\frac{33}{60}, -\frac{18}{60}, -\frac{34}{60}, -\frac{28}{60} \).
4. **Arrange in descending order**:
- The least negative (greatest value) is \( -\frac{18}{60} \) (which is \( \frac{3}{-10} \)),
- Next is \( -\frac{28}{60} \) (which is \( -\frac{7}{15} \)),
- Then \( -\frac{33}{60} \) (which is \( -\frac{11}{20} \)),
- Finally, \( -\frac{34}{60} \) (which is \( \frac{17}{-30} \)).
5. **Final answer**: \( \frac{3}{-10}, -\frac{7}{15}, -\frac{11}{20}, \frac{17}{-30} \)

### (iii) Arrange `9/(-24), -1, 2/(-3), -7/(-6)`

1. **Find the LCM of the denominators**: The denominators are -24, 1, -3, and -6. The LCM is 24.
2. **Convert each fraction to have the same denominator**:
- \( \frac{9}{-24} = -\frac{9}{24} \)
- \( -1 = -\frac{24}{24} \)
- \( \frac{2}{-3} = -\frac{2 \times 8}{3 \times 8} = -\frac{16}{24} \)
- \( -\frac{7}{-6} = \frac{7 \times 4}{6 \times 4} = \frac{28}{24} \)
3. **Compare the numerators**: Now we have \( -\frac{9}{24}, -\frac{24}{24}, -\frac{16}{24}, \frac{28}{24} \).
4. **Arrange in descending order**:
- The largest is \( \frac{28}{24} \) (which is \( -\frac{7}{-6} \)),
- Next is \( -\frac{16}{24} \) (which is \( \frac{2}{-3} \)),
- Then \( -\frac{9}{24} \) (which is \( \frac{9}{-24} \)),
- Finally, \( -\frac{24}{24} \) (which is \( -1 \)).
5. **Final answer**: \( -\frac{7}{-6}, \frac{2}{-3}, \frac{9}{-24}, -1 \)

### (iv) Arrange `7/(-10), 11/15, -17/(-30), -2/5`

1. **Find the LCM of the denominators**: The denominators are -10, 15, -30, and -5. The LCM is 30.
2. **Convert each fraction to have the same denominator**:
- \( \frac{7}{-10} = -\frac{7 \times 3}{10 \times 3} = -\frac{21}{30} \)
- \( \frac{11}{15} = \frac{11 \times 2}{15 \times 2} = \frac{22}{30} \)
- \( \frac{-17}{-30} = \frac{17}{30} \)
- \( \frac{-2}{5} = -\frac{2 \times 6}{5 \times 6} = -\frac{12}{30} \)
3. **Compare the numerators**: Now we have \( -\frac{21}{30}, \frac{22}{30}, \frac{17}{30}, -\frac{12}{30} \).
4. **Arrange in descending order**:
- The largest is \( \frac{22}{30} \) (which is \( \frac{11}{15} \)),
- Next is \( \frac{17}{30} \) (which is \( -\frac{17}{-30} \)),
- Then \( -\frac{12}{30} \) (which is \( -\frac{2}{5} \)),
- Finally, \( -\frac{21}{30} \) (which is \( \frac{7}{-10} \)).
5. **Final answer**: \( \frac{11}{15}, \frac{-17}{-30}, -\frac{2}{5}, \frac{7}{-10} \)