Add the following rational numbers:
(i) `(3)/(4) and (-3)/(5)`
(ii) `(5)/(8) and (-7)/(12)`
,(iii) `(-8)/(9) and (11)/(6)`
,((iv) `(-5)/(16) and (7)/(24)`
,(v) `(7)/(-18) and (8)/(27)`
,(vi) `(1)/(-12) and (2)/(-15)`,
(vii) `-1 and (3)/(4)`
,(viii) `2 and (-5)/(4)`
,(ix) `0 and (-2)/(5) `

Answer

To solve the given problem of adding rational numbers step by step, we will follow the procedure of finding the least common multiple (LCM) of the denominators, adjusting the numerators accordingly, and then performing the addition. Here’s the detailed solution for each part of the question:

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

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

2. **Adjust the fractions to have the same denominator**:
- For `(3)/(4)`:
\[ \frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20} \]
- For `(-3)/(5)`:
\[ \frac{-3}{5} = \frac{-3 \times 4}{5 \times 4} = \frac{-12}{20} \]

3. **Add the adjusted fractions**:
\[ \frac{15}{20} + \frac{-12}{20} = \frac{15 - 12}{20} = \frac{3}{20} \]

### (ii) Add `(5)/(8)` and `(-7)/(12)`

1. **Find the LCM of the denominators (8 and 12)**:
- The LCM of 8 and 12 is 24.

2. **Adjust the fractions**:
- For `(5)/(8)`:
\[ \frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24} \]
- For `(-7)/(12)`:
\[ \frac{-7}{12} = \frac{-7 \times 2}{12 \times 2} = \frac{-14}{24} \]

3. **Add the adjusted fractions**:
\[ \frac{15}{24} + \frac{-14}{24} = \frac{15 - 14}{24} = \frac{1}{24} \]

### (iii) Add `(-8)/(9)` and `(11)/(6)`

1. **Find the LCM of the denominators (9 and 6)**:
- The LCM of 9 and 6 is 18.

2. **Adjust the fractions**:
- For `(-8)/(9)`:
\[ \frac{-8}{9} = \frac{-8 \times 2}{9 \times 2} = \frac{-16}{18} \]
- For `(11)/(6)`:
\[ \frac{11}{6} = \frac{11 \times 3}{6 \times 3} = \frac{33}{18} \]

3. **Add the adjusted fractions**:
\[ \frac{-16}{18} + \frac{33}{18} = \frac{-16 + 33}{18} = \frac{17}{18} \]

### (iv) Add `(-5)/(16)` and `(7)/(24)`

1. **Find the LCM of the denominators (16 and 24)**:
- The LCM of 16 and 24 is 48.

2. **Adjust the fractions**:
- For `(-5)/(16)`:
\[ \frac{-5}{16} = \frac{-5 \times 3}{16 \times 3} = \frac{-15}{48} \]
- For `(7)/(24)`:
\[ \frac{7}{24} = \frac{7 \times 2}{24 \times 2} = \frac{14}{48} \]

3. **Add the adjusted fractions**:
\[ \frac{-15}{48} + \frac{14}{48} = \frac{-15 + 14}{48} = \frac{-1}{48} \]

### (v) Add `(7)/(-18)` and `(8)/(27)`

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

2. **Adjust the fractions**:
- For `(7)/(-18)`:
\[ \frac{7}{-18} = \frac{7 \times 3}{-18 \times 3} = \frac{21}{-54} \]
- For `(8)/(27)`:
\[ \frac{8}{27} = \frac{8 \times 2}{27 \times 2} = \frac{16}{54} \]

3. **Add the adjusted fractions**:
\[ \frac{21}{-54} + \frac{16}{54} = \frac{21 + 16}{-54} = \frac{37}{-54} = \frac{-37}{54} \]

### (vi) Add `(1)/(-12)` and `(2)/(-15)`

1. **Find the LCM of the denominators (12 and 15)**:
- The LCM of 12 and 15 is 60.

2. **Adjust the fractions**:
- For `(1)/(-12)`:
\[ \frac{1}{-12} = \frac{1 \times 5}{-12 \times 5} = \frac{5}{-60} \]
- For `(2)/(-15)`:
\[ \frac{2}{-15} = \frac{2 \times 4}{-15 \times 4} = \frac{8}{-60} \]

3. **Add the adjusted fractions**:
\[ \frac{5}{-60} + \frac{8}{-60} = \frac{5 + 8}{-60} = \frac{13}{-60} = \frac{-13}{60} \]

### (vii) Add `-1` and `(3)/(4)`

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

2. **Add the fractions**:
\[ \frac{-4}{4} + \frac{3}{4} = \frac{-4 + 3}{4} = \frac{-1}{4} \]

### (viii) Add `2` and `(-5)/(4)`

1. **Convert 2 to a fraction**:
\[ 2 = \frac{8}{4} \]

2. **Add the fractions**:
\[ \frac{8}{4} + \frac{-5}{4} = \frac{8 - 5}{4} = \frac{3}{4} \]

### (ix) Add `0` and `(-2)/(5)`

1. **Add the fractions**:
\[ 0 + \frac{-2}{5} = \frac{-2}{5} \]

### Summary of Answers:
1. \( \frac{3}{20} \)
2. \( \frac{1}{24} \)
3. \( \frac{17}{18} \)
4. \( \frac{-1}{48} \)
5. \( \frac{-37}{54} \)
6. \( \frac{-13}{60} \)
7. \( \frac{-1}{4} \)
8. \( \frac{3}{4} \)
9. \( \frac{-2}{5} \)