Express each of the following rational numbers as the sum of an integer and a rational number :
(i) `(12)/(5)`
(ii) `(-11)/(7)`
(iii) `(-25)/(9)`
(iv) `(-103)/(20)`

Answer

To express each of the given rational numbers as the sum of an integer and a rational number, we will follow these steps:

### Step-by-Step Solution:

#### (i) \( \frac{12}{5} \)

1. **Divide 12 by 5**:
- \( 12 \div 5 = 2 \) (integer part)
- Remainder: \( 12 - (5 \times 2) = 2 \)

2. **Express as a sum**:
- The integer part is 2 and the remaining fraction is \( \frac{2}{5} \).
- Therefore, \( \frac{12}{5} = 2 + \frac{2}{5} \).

#### (ii) \( \frac{-11}{7} \)

1. **Divide -11 by 7**:
- \( -11 \div 7 = -2 \) (integer part)
- Remainder: \( -11 - (7 \times -2) = -11 + 14 = 3 \)

2. **Express as a sum**:
- The integer part is -2 and the remaining fraction is \( \frac{3}{7} \).
- Therefore, \( \frac{-11}{7} = -2 + \frac{3}{7} \).

#### (iii) \( \frac{-25}{9} \)

1. **Divide -25 by 9**:
- \( -25 \div 9 = -3 \) (integer part)
- Remainder: \( -25 - (9 \times -3) = -25 + 27 = 2 \)

2. **Express as a sum**:
- The integer part is -3 and the remaining fraction is \( \frac{2}{9} \).
- Therefore, \( \frac{-25}{9} = -3 + \frac{2}{9} \).

#### (iv) \( \frac{-103}{20} \)

1. **Divide -103 by 20**:
- \( -103 \div 20 = -5 \) (integer part)
- Remainder: \( -103 - (20 \times -5) = -103 + 100 = -3 \)

2. **Express as a sum**:
- The integer part is -5 and the remaining fraction is \( \frac{-3}{20} \).
- Therefore, \( \frac{-103}{20} = -5 + \frac{-3}{20} \).

### Final Answers:
- \( \frac{12}{5} = 2 + \frac{2}{5} \)
- \( \frac{-11}{7} = -2 + \frac{3}{7} \)
- \( \frac{-25}{9} = -3 + \frac{2}{9} \)
- \( \frac{-103}{20} = -5 + \frac{-3}{20} \)