To arrange the given rational numbers in ascending order, we will follow these steps for each part:
### (i) Arrange `frac(2)(5), frac(7)(10), frac(8)(15), frac(13)(30)` in ascending order:
1. **Identify the denominators**: The denominators are 5, 10, 15, and 30. The least common multiple (LCM) of these denominators is 30.
2. **Convert each fraction to have a denominator of 30**:
- `frac(2)(5) = frac(2 * 6)(5 * 6) = frac(12)(30)`
- `frac(7)(10) = frac(7 * 3)(10 * 3) = frac(21)(30)`
- `frac(8)(15) = frac(8 * 2)(15 * 2) = frac(16)(30)`
- `frac(13)(30) = frac(13)(30)` (remains the same)
3. **List the fractions with the common denominator**:
- `frac(12)(30), frac(21)(30), frac(16)(30), frac(13)(30)`
4. **Compare the numerators**:
- The numerators are 12, 21, 16, and 13.
5. **Arrange the numerators in ascending order**:
- 12, 13, 16, 21
6. **Write the fractions in ascending order**:
- `frac(2)(5), frac(13)(30), frac(8)(15), frac(7)(10)`
### Final Answer for (i):
`frac(2)(5), frac(13)(30), frac(8)(15), frac(7)(10)`
---
### (ii) Arrange `frac(-3)(4), frac(5)(-12), frac(-7)(16), frac(9)(-24)` in ascending order:
1. **Identify the denominators**: The denominators are 4, -12, 16, and -24. The LCM of the absolute values is 48.
2. **Convert each fraction to have a denominator of 48**:
- `frac(-3)(4) = frac(-3 * 12)(4 * 12) = frac(-36)(48)`
- `frac(5)(-12) = frac(5 * -4)(-12 * -4) = frac(-20)(48)`
- `frac(-7)(16) = frac(-7 * 3)(16 * 3) = frac(-21)(48)`
- `frac(9)(-24) = frac(9 * -2)(-24 * -2) = frac(-18)(48)`
3. **List the fractions with the common denominator**:
- `frac(-36)(48), frac(-20)(48), frac(-21)(48), frac(-18)(48)`
4. **Compare the numerators**:
- The numerators are -36, -20, -21, and -18.
5. **Arrange the numerators in ascending order**:
- -36, -21, -20, -18
6. **Write the fractions in ascending order**:
- `frac(-3)(4), frac(-7)(16), frac(5)(-12), frac(9)(-24)`
### Final Answer for (ii):
`frac(-3)(4), frac(-7)(16), frac(5)(-12), frac(9)(-24)`
---
### (iii) Arrange `frac(-3)(10), frac(7)(-15), frac(-11)(20), frac(17)(-30)` in ascending order:
1. **Identify the denominators**: The denominators are 10, -15, 20, and -30. The LCM of the absolute values is 60.
2. **Convert each fraction to have a denominator of 60**:
- `frac(-3)(10) = frac(-3 * 6)(10 * 6) = frac(-18)(60)`
- `frac(7)(-15) = frac(7 * -4)(-15 * -4) = frac(-28)(60)`
- `frac(-11)(20) = frac(-11 * 3)(20 * 3) = frac(-33)(60)`
- `frac(17)(-30) = frac(17 * -2)(-30 * -2) = frac(-34)(60)`
3. **List the fractions with the common denominator**:
- `frac(-18)(60), frac(-28)(60), frac(-33)(60), frac(-34)(60)`
4. **Compare the numerators**:
- The numerators are -18, -28, -33, and -34.
5. **Arrange the numerators in ascending order**:
- -34, -33, -28, -18
6. **Write the fractions in ascending order**:
- `frac(17)(-30), frac(-11)(20), frac(7)(-15), frac(-3)(10)`
### Final Answer for (iii):
`frac(17)(-30), frac(-11)(20), frac(7)(-15), frac(-3)(10)`
---
### (iv) Arrange `frac(2)(3), frac(3)(4), frac(5)(-6), frac(-7)(12)` in ascending order:
1. **Identify the denominators**: The denominators are 3, 4, -6, and 12. The LCM of the absolute values is 12.
2. **Convert each fraction to have a denominator of 12**:
- `frac(2)(3) = frac(2 * 4)(3 * 4) = frac(8)(12)`
- `frac(3)(4) = frac(3 * 3)(4 * 3) = frac(9)(12)`
- `frac(5)(-6) = frac(5 * -2)(-6 * -2) = frac(-10)(12)`
- `frac(-7)(12) = frac(-7)(12)` (remains the same)
3. **List the fractions with the common denominator**:
- `frac(8)(12), frac(9)(12), frac(-10)(12), frac(-7)(12)`
4. **Compare the numerators**:
- The numerators are 8, 9, -10, and -7.
5. **Arrange the numerators in ascending order**:
- -10, -7, 8, 9
6. **Write the fractions in ascending order**:
- `frac(5)(-6), frac(-7)(12), frac(2)(3), frac(3)(4)`
### Final Answer for (iv):
`frac(5)(-6), frac(-7)(12), frac(2)(3), frac(3)(4)`
---
