Arrange the following rational numbers in ascending order :
(i) `3/4, 5/8, 11/16, 21/32`, (ii) `-2/5, 7/(-10), -8/15, 17/(-30)`
(iii) `5/(-12), -2/3, -7/9, 11/(-18)`, (iv) `-4/7, 13/(-28), 9/14, 23/42`

Answer

To arrange the given rational numbers in ascending order, we will follow a systematic approach for each set of numbers.

### (i) Arrange `3/4, 5/8, 11/16, 21/32` in ascending order:

1. **Find the LCM of the denominators**: The denominators are 4, 8, 16, and 32. The LCM is 32.

2. **Convert each fraction to have a common denominator**:
- \( \frac{3}{4} = \frac{3 \times 8}{4 \times 8} = \frac{24}{32} \)
- \( \frac{5}{8} = \frac{5 \times 4}{8 \times 4} = \frac{20}{32} \)
- \( \frac{11}{16} = \frac{11 \times 2}{16 \times 2} = \frac{22}{32} \)
- \( \frac{21}{32} = \frac{21}{32} \)

3. **List the converted fractions**:
- \( \frac{20}{32}, \frac{21}{32}, \frac{22}{32}, \frac{24}{32} \)

4. **Arrange in ascending order**:
- The order is \( \frac{20}{32}, \frac{21}{32}, \frac{22}{32}, \frac{24}{32} \).

5. **Replace with original fractions**:
- \( \frac{5}{8}, \frac{21}{32}, \frac{11}{16}, \frac{3}{4} \)

**Final Ascending Order**:
\[ \frac{5}{8} < \frac{21}{32} < \frac{11}{16} < \frac{3}{4} \]

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### (ii) Arrange `-2/5, 7/(-10), -8/15, 17/(-30)` in ascending order:

1. **Find the LCM of the denominators**: The denominators are 5, -10, 15, and -30. The LCM is 30.

2. **Convert each fraction to have a common denominator**:
- \( \frac{-2}{5} = \frac{-2 \times 6}{5 \times 6} = \frac{-12}{30} \)
- \( \frac{7}{-10} = \frac{7 \times -3}{-10 \times -3} = \frac{-21}{30} \)
- \( \frac{-8}{15} = \frac{-8 \times 2}{15 \times 2} = \frac{-16}{30} \)
- \( \frac{17}{-30} = \frac{17}{-30} \)

3. **List the converted fractions**:
- \( \frac{-21}{30}, \frac{-16}{30}, \frac{-12}{30}, \frac{17}{-30} \)

4. **Arrange in ascending order**:
- The order is \( \frac{-21}{30}, \frac{-16}{30}, \frac{-12}{30}, \frac{17}{-30} \).

5. **Replace with original fractions**:
- \( \frac{7}{-10}, \frac{-8}{15}, \frac{-2}{5}, \frac{17}{-30} \)

**Final Ascending Order**:
\[ \frac{7}{-10} < \frac{-8}{15} < \frac{-2}{5} < \frac{17}{-30} \]

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### (iii) Arrange `5/(-12), -2/3, -7/9, 11/(-18)` in ascending order:

1. **Find the LCM of the denominators**: The denominators are -12, 3, 9, and -18. The LCM is 36.

2. **Convert each fraction to have a common denominator**:
- \( \frac{5}{-12} = \frac{5 \times -3}{-12 \times -3} = \frac{-15}{36} \)
- \( \frac{-2}{3} = \frac{-2 \times 12}{3 \times 12} = \frac{-24}{36} \)
- \( \frac{-7}{9} = \frac{-7 \times 4}{9 \times 4} = \frac{-28}{36} \)
- \( \frac{11}{-18} = \frac{11 \times -2}{-18 \times -2} = \frac{-22}{36} \)

3. **List the converted fractions**:
- \( \frac{-28}{36}, \frac{-24}{36}, \frac{-22}{36}, \frac{-15}{36} \)

4. **Arrange in ascending order**:
- The order is \( \frac{-28}{36}, \frac{-24}{36}, \frac{-22}{36}, \frac{-15}{36} \).

5. **Replace with original fractions**:
- \( \frac{-7}{9}, \frac{-2}{3}, \frac{11}{-18}, \frac{5}{-12} \)

**Final Ascending Order**:
\[ \frac{-7}{9} < \frac{-2}{3} < \frac{11}{-18} < \frac{5}{-12} \]

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### (iv) Arrange `-4/7, 13/(-28), 9/14, 23/42` in ascending order:

1. **Find the LCM of the denominators**: The denominators are -7, -28, 14, and 42. The LCM is 84.

2. **Convert each fraction to have a common denominator**:
- \( \frac{-4}{7} = \frac{-4 \times 12}{7 \times 12} = \frac{-48}{84} \)
- \( \frac{13}{-28} = \frac{13 \times -3}{-28 \times -3} = \frac{-39}{84} \)
- \( \frac{9}{14} = \frac{9 \times 6}{14 \times 6} = \frac{54}{84} \)
- \( \frac{23}{42} = \frac{23 \times 2}{42 \times 2} = \frac{46}{84} \)

3. **List the converted fractions**:
- \( \frac{-48}{84}, \frac{-39}{84}, \frac{46}{84}, \frac{54}{84} \)

4. **Arrange in ascending order**:
- The order is \( \frac{-48}{84}, \frac{-39}{84}, \frac{46}{84}, \frac{54}{84} \).

5. **Replace with original fractions**:
- \( \frac{-4}{7}, \frac{13}{-28}, \frac{23}{42}, \frac{9}{14} \)

**Final Ascending Order**:
\[ \frac{-4}{7} < \frac{13}{-28} < \frac{23}{42} < \frac{9}{14} \]

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