Add the following rational numbers :
(i) `(-2)/(5)`and `(3)/(4)`
(ii) `(-5)/(9)` and `(2)/(3)`
(iii) `(-4)` and `(1)/(2)`
(iv) `(-7)/(27)`and `(5)/(18)`
(v) ` (-5)/(36)` and `(-7)/(12)`
(vi) `(1)/(-9)`and `(4)/(-27)`
(vii) `(-9)/(24)` and `(-1)/(18)`
(viii) `(27)/(-4)`and `(-15)/(8)`

Answer

Let's solve the given problems step by step:

### (i) Add `(-2)/(5)` and `(3)/(4)`

1. **Find the LCM of the denominators (5 and 4)**:
- LCM(5, 4) = 20

2. **Convert each fraction to have the common denominator**:
- For `(-2)/(5)`:
\[ \frac{-2}{5} = \frac{-2 \times 4}{5 \times 4} = \frac{-8}{20} \]
- For `(3)/(4)`:
\[ \frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20} \]

3. **Add the two fractions**:
\[ \frac{-8}{20} + \frac{15}{20} = \frac{-8 + 15}{20} = \frac{7}{20} \]

### (ii) Add `(-5)/(9)` and `(2)/(3)`

1. **Find the LCM of the denominators (9 and 3)**:
- LCM(9, 3) = 9

2. **Convert each fraction to have the common denominator**:
- For `(-5)/(9)`:
\[ \frac{-5}{9} = \frac{-5}{9} \quad \text{(no change needed)} \]
- For `(2)/(3)`:
\[ \frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9} \]

3. **Add the two fractions**:
\[ \frac{-5}{9} + \frac{6}{9} = \frac{-5 + 6}{9} = \frac{1}{9} \]

### (iii) Add `(-4)` and `(1)/(2)`

1. **Convert `(-4)` to a fraction**:
\[ -4 = \frac{-4}{1} \]

2. **Find the LCM of the denominators (1 and 2)**:
- LCM(1, 2) = 2

3. **Convert each fraction to have the common denominator**:
- For `(-4)`:
\[ \frac{-4}{1} = \frac{-4 \times 2}{1 \times 2} = \frac{-8}{2} \]
- For `(1)/(2)`:
\[ \frac{1}{2} = \frac{1}{2} \quad \text{(no change needed)} \]

4. **Add the two fractions**:
\[ \frac{-8}{2} + \frac{1}{2} = \frac{-8 + 1}{2} = \frac{-7}{2} \]

### (iv) Add `(-7)/(27)` and `(5)/(18)`

1. **Find the LCM of the denominators (27 and 18)**:
- LCM(27, 18) = 54

2. **Convert each fraction to have the common denominator**:
- For `(-7)/(27)`:
\[ \frac{-7}{27} = \frac{-7 \times 2}{27 \times 2} = \frac{-14}{54} \]
- For `(5)/(18)`:
\[ \frac{5}{18} = \frac{5 \times 3}{18 \times 3} = \frac{15}{54} \]

3. **Add the two fractions**:
\[ \frac{-14}{54} + \frac{15}{54} = \frac{-14 + 15}{54} = \frac{1}{54} \]

### (v) Add `(-5)/(36)` and `(-7)/(12)`

1. **Find the LCM of the denominators (36 and 12)**:
- LCM(36, 12) = 36

2. **Convert each fraction to have the common denominator**:
- For `(-5)/(36)`:
\[ \frac{-5}{36} = \frac{-5}{36} \quad \text{(no change needed)} \]
- For `(-7)/(12)`:
\[ \frac{-7}{12} = \frac{-7 \times 3}{12 \times 3} = \frac{-21}{36} \]

3. **Add the two fractions**:
\[ \frac{-5}{36} + \frac{-21}{36} = \frac{-5 - 21}{36} = \frac{-26}{36} = \frac{-13}{18} \]

### (vi) Add `(1)/(-9)` and `(4)/(-27)`

1. **Find the LCM of the denominators (9 and 27)**:
- LCM(9, 27) = 27

2. **Convert each fraction to have the common denominator**:
- For `(1)/(-9)`:
\[ \frac{1}{-9} = \frac{1 \times 3}{-9 \times 3} = \frac{3}{-27} \]
- For `(4)/(-27)`:
\[ \frac{4}{-27} = \frac{4}{-27} \quad \text{(no change needed)} \]

3. **Add the two fractions**:
\[ \frac{3}{-27} + \frac{4}{-27} = \frac{3 + 4}{-27} = \frac{7}{-27} = \frac{-7}{27} \]

### (vii) Add `(-9)/(24)` and `(-1)/(18)`

1. **Find the LCM of the denominators (24 and 18)**:
- LCM(24, 18) = 72

2. **Convert each fraction to have the common denominator**:
- For `(-9)/(24)`:
\[ \frac{-9}{24} = \frac{-9 \times 3}{24 \times 3} = \frac{-27}{72} \]
- For `(-1)/(18)`:
\[ \frac{-1}{18} = \frac{-1 \times 4}{18 \times 4} = \frac{-4}{72} \]

3. **Add the two fractions**:
\[ \frac{-27}{72} + \frac{-4}{72} = \frac{-27 - 4}{72} = \frac{-31}{72} \]

### (viii) Add `(27)/(-4)` and `(-15)/(8)`

1. **Find the LCM of the denominators (4 and 8)**:
- LCM(4, 8) = 8

2. **Convert each fraction to have the common denominator**:
- For `(27)/(-4)`:
\[ \frac{27}{-4} = \frac{27 \times 2}{-4 \times 2} = \frac{54}{-8} \]
- For `(-15)/(8)`:
\[ \frac{-15}{8} = \frac{-15}{8} \quad \text{(no change needed)} \]

3. **Add the two fractions**:
\[ \frac{54}{-8} + \frac{-15}{8} = \frac{54 - 15}{-8} = \frac{39}{-8} = \frac{-39}{8} \]

### Summary of the Answers:
1. \( \frac{7}{20} \)
2. \( \frac{1}{9} \)
3. \( \frac{-7}{2} \)
4. \( \frac{1}{54} \)
5. \( \frac{-13}{18} \)
6. \( \frac{-7}{27} \)
7. \( \frac{-31}{72} \)
8. \( \frac{-39}{8} \)