To arrange the given rational numbers in ascending order, we will follow these steps for each part of the question.
### Part (i): Arrange `(4)/(-9), (-5)/(12), (7)/(-18), (-2)/(3)`
1. **Convert to Standard Form**:
- Rewrite the numbers with a positive denominator:
- \( \frac{4}{-9} = -\frac{4}{9} \)
- \( \frac{-5}{12} = -\frac{5}{12} \)
- \( \frac{7}{-18} = -\frac{7}{18} \)
- \( \frac{-2}{3} = -\frac{2}{3} \)
2. **Find a Common Denominator**:
- The denominators are \( 9, 12, 18, 3 \). The LCM of these numbers is \( 36 \).
3. **Convert Each Fraction**:
- Convert each fraction to have a denominator of \( 36 \):
- \( -\frac{4}{9} = -\frac{4 \times 4}{9 \times 4} = -\frac{16}{36} \)
- \( -\frac{5}{12} = -\frac{5 \times 3}{12 \times 3} = -\frac{15}{36} \)
- \( -\frac{7}{18} = -\frac{7 \times 2}{18 \times 2} = -\frac{14}{36} \)
- \( -\frac{2}{3} = -\frac{2 \times 12}{3 \times 12} = -\frac{24}{36} \)
4. **Order the Fractions**:
- Now we have:
- \( -\frac{24}{36}, -\frac{16}{36}, -\frac{15}{36}, -\frac{14}{36} \)
- In ascending order:
- \( -\frac{24}{36}, -\frac{16}{36}, -\frac{15}{36}, -\frac{14}{36} \)
5. **Convert Back to Original Form**:
- The ascending order of the original fractions is:
- \( -\frac{2}{3}, -\frac{4}{9}, -\frac{5}{12}, -\frac{7}{18} \)
### Part (ii): Arrange `(-3)/(4), (5)/(-12), (-7)/(16), (9)/(-24)`
1. **Convert to Standard Form**:
- Rewrite the numbers:
- \( -\frac{3}{4} \)
- \( \frac{5}{-12} = -\frac{5}{12} \)
- \( -\frac{7}{16} \)
- \( \frac{9}{-24} = -\frac{9}{24} \)
2. **Find a Common Denominator**:
- The denominators are \( 4, 12, 16, 24 \). The LCM is \( 48 \).
3. **Convert Each Fraction**:
- Convert to \( 48 \):
- \( -\frac{3}{4} = -\frac{3 \times 12}{4 \times 12} = -\frac{36}{48} \)
- \( -\frac{5}{12} = -\frac{5 \times 4}{12 \times 4} = -\frac{20}{48} \)
- \( -\frac{7}{16} = -\frac{7 \times 3}{16 \times 3} = -\frac{21}{48} \)
- \( -\frac{9}{24} = -\frac{9 \times 2}{24 \times 2} = -\frac{18}{48} \)
4. **Order the Fractions**:
- Now we have:
- \( -\frac{36}{48}, -\frac{20}{48}, -\frac{21}{48}, -\frac{18}{48} \)
- In ascending order:
- \( -\frac{36}{48}, -\frac{21}{48}, -\frac{20}{48}, -\frac{18}{48} \)
5. **Convert Back to Original Form**:
- The ascending order of the original fractions is:
- \( -\frac{3}{4}, -\frac{7}{16}, -\frac{5}{12}, -\frac{9}{24} \)
### Part (iii): Arrange `(3)/(-5), (-7)/(10), (-11)/(15), (-13)/(20)`
1. **Convert to Standard Form**:
- Rewrite the numbers:
- \( \frac{3}{-5} = -\frac{3}{5} \)
- \( -\frac{7}{10} \)
- \( -\frac{11}{15} \)
- \( -\frac{13}{20} \)
2. **Find a Common Denominator**:
- The denominators are \( 5, 10, 15, 20 \). The LCM is \( 60 \).
3. **Convert Each Fraction**:
- Convert to \( 60 \):
- \( -\frac{3}{5} = -\frac{3 \times 12}{5 \times 12} = -\frac{36}{60} \)
- \( -\frac{7}{10} = -\frac{7 \times 6}{10 \times 6} = -\frac{42}{60} \)
- \( -\frac{11}{15} = -\frac{11 \times 4}{15 \times 4} = -\frac{44}{60} \)
- \( -\frac{13}{20} = -\frac{13 \times 3}{20 \times 3} = -\frac{39}{60} \)
4. **Order the Fractions**:
- Now we have:
- \( -\frac{36}{60}, -\frac{42}{60}, -\frac{44}{60}, -\frac{39}{60} \)
- In ascending order:
- \( -\frac{44}{60}, -\frac{42}{60}, -\frac{39}{60}, -\frac{36}{60} \)
5. **Convert Back to Original Form**:
- The ascending order of the original fractions is:
- \( -\frac{11}{15}, -\frac{7}{10}, -\frac{13}{20}, -\frac{3}{5} \)
### Part (iv): Arrange `(-4)/(7), (-9)/(14), (13)/(-28), (-23)/(42)`
1. **Convert to Standard Form**:
- Rewrite the numbers:
- \( -\frac{4}{7} \)
- \( -\frac{9}{14} \)
- \( \frac{13}{-28} = -\frac{13}{28} \)
- \( -\frac{23}{42} \)
2. **Find a Common Denominator**:
- The denominators are \( 7, 14, 28, 42 \). The LCM is \( 84 \).
3. **Convert Each Fraction**:
- Convert to \( 84 \):
- \( -\frac{4}{7} = -\frac{4 \times 12}{7 \times 12} = -\frac{48}{84} \)
- \( -\frac{9}{14} = -\frac{9 \times 6}{14 \times 6} = -\frac{54}{84} \)
- \( -\frac{13}{28} = -\frac{13 \times 3}{28 \times 3} = -\frac{39}{84} \)
- \( -\frac{23}{42} = -\frac{23 \times 2}{42 \times 2} = -\frac{46}{84} \)
4. **Order the Fractions**:
- Now we have:
- \( -\frac{48}{84}, -\frac{54}{84}, -\frac{39}{84}, -\frac{46}{84} \)
- In ascending order:
- \( -\frac{54}{84}, -\frac{48}{84}, -\frac{46}{84}, -\frac{39}{84} \)
5. **Convert Back to Original Form**:
- The ascending order of the original fractions is:
- \( -\frac{9}{14}, -\frac{4}{7}, -\frac{23}{42}, -\frac{13}{28} \)
### Final Answers:
- Part (i): \( -\frac{2}{3}, -\frac{4}{9}, -\frac{5}{12}, -\frac{7}{18} \)
- Part (ii): \( -\frac{3}{4}, -\frac{7}{16}, -\frac{5}{12}, -\frac{9}{24} \)
- Part (iii): \( -\frac{11}{15}, -\frac{7}{10}, -\frac{13}{20}, -\frac{3}{5} \)
- Part (iv): \( -\frac{9}{14}, -\frac{4}{7}, -\frac{23}{42}, -\frac{13}{28} \)
