To arrange the given rational numbers in descending order, we will follow these steps for each part of the question.
### Part (i): Arrange -2, -13/6, 8/(-3), 1/3
1. **Convert all numbers to a common denominator**:
- The common denominator for -2, -13/6, 8/(-3), and 1/3 can be 6.
- Convert -2:
\[
-2 = \frac{-2 \times 6}{6} = \frac{-12}{6}
\]
- -13/6 remains the same:
\[
-\frac{13}{6}
\]
- Convert 8/(-3):
\[
\frac{8}{-3} = \frac{8 \times 2}{-3 \times 2} = \frac{16}{-6} = \frac{-16}{6}
\]
- Convert 1/3:
\[
\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}
\]
2. **List the converted fractions**:
- \(-12/6\), \(-13/6\), \(-16/6\), \(2/6\)
3. **Order the fractions**:
- The largest value is \(2/6\), followed by \(-12/6\), \(-13/6\), and \(-16/6\).
4. **Final arrangement in descending order**:
\[
\frac{1}{3}, -2, -\frac{13}{6}, -\frac{8}{3}
\]
### Part (ii): Arrange -3/10, 7/(-15), -11/20, 17/(-30)
1. **Convert all numbers to a common denominator**:
- The common denominator for -3/10, 7/(-15), -11/20, and 17/(-30) is 60.
- Convert -3/10:
\[
\frac{-3}{10} = \frac{-3 \times 6}{10 \times 6} = \frac{-18}{60}
\]
- Convert 7/(-15):
\[
\frac{7}{-15} = \frac{7 \times 4}{-15 \times 4} = \frac{28}{-60} = \frac{-28}{60}
\]
- Convert -11/20:
\[
\frac{-11}{20} = \frac{-11 \times 3}{20 \times 3} = \frac{-33}{60}
\]
- Convert 17/(-30):
\[
\frac{17}{-30} = \frac{17 \times 2}{-30 \times 2} = \frac{-34}{60}
\]
2. **List the converted fractions**:
- \(-18/60\), \(-28/60\), \(-33/60\), \(-34/60\)
3. **Order the fractions**:
- The largest value is \(-18/60\), followed by \(-28/60\), \(-33/60\), and \(-34/60\).
4. **Final arrangement in descending order**:
\[
-\frac{3}{10}, -\frac{7}{15}, -\frac{11}{20}, -\frac{17}{30}
\]
### Part (iii): Arrange -5/6, -7/2, -13/18, 23/(-24)
1. **Convert all numbers to a common denominator**:
- The common denominator for -5/6, -7/2, -13/18, and 23/(-24) is 36.
- Convert -5/6:
\[
\frac{-5}{6} = \frac{-5 \times 6}{6 \times 6} = \frac{-30}{36}
\]
- Convert -7/2:
\[
\frac{-7}{2} = \frac{-7 \times 18}{2 \times 18} = \frac{-126}{36}
\]
- Convert -13/18:
\[
\frac{-13}{18} = \frac{-13 \times 2}{18 \times 2} = \frac{-26}{36}
\]
- Convert 23/(-24):
\[
\frac{23}{-24} = \frac{23 \times (-3)}{-24 \times (-3)} = \frac{-69}{72} = \frac{-69}{36}
\]
2. **List the converted fractions**:
- \(-30/36\), \(-126/36\), \(-26/36\), \(-69/36\)
3. **Order the fractions**:
- The largest value is \(-26/36\), followed by \(-30/36\), \(-69/36\), and \(-126/36\).
4. **Final arrangement in descending order**:
\[
-\frac{13}{18}, -\frac{5}{6}, -\frac{23}{-24}, -\frac{7}{2}
\]
### Part (iv): Arrange -10/11, -19/11, -23/33, -39/44
1. **Convert all numbers to a common denominator**:
- The common denominator for -10/11, -19/11, -23/33, and -39/44 is 132.
- Convert -10/11:
\[
\frac{-10}{11} = \frac{-10 \times 12}{11 \times 12} = \frac{-120}{132}
\]
- Convert -19/11:
\[
\frac{-19}{11} = \frac{-19 \times 12}{11 \times 12} = \frac{-228}{132}
\]
- Convert -23/33:
\[
\frac{-23}{33} = \frac{-23 \times 4}{33 \times 4} = \frac{-92}{132}
\]
- Convert -39/44:
\[
\frac{-39}{44} = \frac{-39 \times 3}{44 \times 3} = \frac{-117}{132}
\]
2. **List the converted fractions**:
- \(-120/132\), \(-228/132\), \(-92/132\), \(-117/132\)
3. **Order the fractions**:
- The largest value is \(-92/132\), followed by \(-117/132\), \(-120/132\), and \(-228/132\).
4. **Final arrangement in descending order**:
\[
-\frac{23}{33}, -\frac{39}{44}, -\frac{10}{11}, -\frac{19}{11}
\]
