Add the following rational numbers:
(i) `-2/3` and `3/4`, (ii) `-4/9` and `5/6`, (iii) `-5/18` and `11/27`
(iv) `-7/12` and `-5/24`, (v) `-1/18` and `-7/27`, (vi) `21/(-4)` and `-11/8`

Answer

Let's solve the given rational number addition step by step.

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

1. **Write the expression**:
\[ -\frac{2}{3} + \frac{3}{4} \]

2. **Find the LCM of the denominators (3 and 4)**:
The LCM of 3 and 4 is 12.

3. **Convert each fraction to have the same denominator**:
- For `-2/3`:
\[ -\frac{2}{3} = -\frac{2 \times 4}{3 \times 4} = -\frac{8}{12} \]
- For `3/4`:
\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \]

4. **Add the fractions**:
\[ -\frac{8}{12} + \frac{9}{12} = \frac{-8 + 9}{12} = \frac{1}{12} \]

**Final Answer**:
\[ \frac{1}{12} \]

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### (ii) Add `-4/9` and `5/6`

1. **Write the expression**:
\[ -\frac{4}{9} + \frac{5}{6} \]

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

3. **Convert each fraction to have the same denominator**:
- For `-4/9`:
\[ -\frac{4}{9} = -\frac{4 \times 2}{9 \times 2} = -\frac{8}{18} \]
- For `5/6`:
\[ \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18} \]

4. **Add the fractions**:
\[ -\frac{8}{18} + \frac{15}{18} = \frac{-8 + 15}{18} = \frac{7}{18} \]

**Final Answer**:
\[ \frac{7}{18} \]

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### (iii) Add `-5/18` and `11/27`

1. **Write the expression**:
\[ -\frac{5}{18} + \frac{11}{27} \]

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

3. **Convert each fraction to have the same denominator**:
- For `-5/18`:
\[ -\frac{5}{18} = -\frac{5 \times 3}{18 \times 3} = -\frac{15}{54} \]
- For `11/27`:
\[ \frac{11}{27} = \frac{11 \times 2}{27 \times 2} = \frac{22}{54} \]

4. **Add the fractions**:
\[ -\frac{15}{54} + \frac{22}{54} = \frac{-15 + 22}{54} = \frac{7}{54} \]

**Final Answer**:
\[ \frac{7}{54} \]

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### (iv) Add `-7/12` and `-5/24`

1. **Write the expression**:
\[ -\frac{7}{12} + -\frac{5}{24} \]

2. **Find the LCM of the denominators (12 and 24)**:
The LCM of 12 and 24 is 24.

3. **Convert each fraction to have the same denominator**:
- For `-7/12`:
\[ -\frac{7}{12} = -\frac{7 \times 2}{12 \times 2} = -\frac{14}{24} \]
- For `-5/24`:
\[ -\frac{5}{24} = -\frac{5}{24} \]

4. **Add the fractions**:
\[ -\frac{14}{24} + -\frac{5}{24} = \frac{-14 - 5}{24} = -\frac{19}{24} \]

**Final Answer**:
\[ -\frac{19}{24} \]

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### (v) Add `-1/18` and `-7/27`

1. **Write the expression**:
\[ -\frac{1}{18} + -\frac{7}{27} \]

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

3. **Convert each fraction to have the same denominator**:
- For `-1/18`:
\[ -\frac{1}{18} = -\frac{1 \times 3}{18 \times 3} = -\frac{3}{54} \]
- For `-7/27`:
\[ -\frac{7}{27} = -\frac{7 \times 2}{27 \times 2} = -\frac{14}{54} \]

4. **Add the fractions**:
\[ -\frac{3}{54} + -\frac{14}{54} = \frac{-3 - 14}{54} = -\frac{17}{54} \]

**Final Answer**:
\[ -\frac{17}{54} \]

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### (vi) Add `21/(-4)` and `-11/8`

1. **Write the expression**:
\[ \frac{21}{-4} + -\frac{11}{8} \]

2. **Find the LCM of the denominators (4 and 8)**:
The LCM of 4 and 8 is 8.

3. **Convert each fraction to have the same denominator**:
- For `21/(-4)`:
\[ \frac{21}{-4} = -\frac{21 \times 2}{4 \times 2} = -\frac{42}{8} \]
- For `-11/8`:
\[ -\frac{11}{8} = -\frac{11}{8} \]

4. **Add the fractions**:
\[ -\frac{42}{8} + -\frac{11}{8} = \frac{-42 - 11}{8} = -\frac{53}{8} \]

**Final Answer**:
\[ -\frac{53}{8} \]

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