Which of the following rational numbers will have terminating decimal expansion or a non-terminating repeating (recurring) decimal expansion
(i) `(135)/(50)` (ii) `(4)/(11)` (iii) `(8)/(7)` (iv) `6 (3)/(8)`
(v) `(55)/(9)` (vi) `(5^(2) xx 3^(3))/(2 xx 5^(3) xx 27)` (vii) `(51)/(60)`

Answer

To determine whether the given rational numbers have terminating decimal expansions or non-terminating repeating (recurring) decimal expansions, we will analyze each number based on its denominator after simplification. A rational number has a terminating decimal expansion if, after simplification, its denominator has only the prime factors 2 and/or 5. If it has other prime factors, it will have a non-terminating repeating decimal expansion.

Let's analyze each option step by step:

### Step 1: Analyze each rational number

**(i) \( \frac{135}{50} \)**
- Simplify: \( \frac{135}{50} = \frac{27}{10} \) (after dividing both numerator and denominator by 5).
- Denominator: \( 10 = 2 \times 5 \) (only has prime factors 2 and 5).
- **Conclusion**: Terminating decimal.

**(ii) \( \frac{4}{11} \)**
- Denominator: \( 11 \) (a prime number).
- **Conclusion**: Non-terminating repeating decimal.

**(iii) \( \frac{8}{7} \)**
- Denominator: \( 7 \) (a prime number).
- **Conclusion**: Non-terminating repeating decimal.

**(iv) \( 6 \frac{3}{8} \)**
- Convert to improper fraction: \( 6 \frac{3}{8} = \frac{51}{8} \).
- Denominator: \( 8 = 2^3 \) (only has prime factor 2).
- **Conclusion**: Terminating decimal.

**(v) \( \frac{55}{9} \)**
- Denominator: \( 9 = 3^2 \) (a prime factor other than 2 or 5).
- **Conclusion**: Non-terminating repeating decimal.

**(vi) \( \frac{5^2 \times 3^3}{2 \times 5^3 \times 27} \)**
- Simplify: \( 27 = 3^3 \), thus \( \frac{5^2 \times 3^3}{2 \times 5^3 \times 3^3} = \frac{5^2}{2 \times 5^3} = \frac{1}{2 \times 5} = \frac{1}{10} \).
- Denominator: \( 10 = 2 \times 5 \) (only has prime factors 2 and 5).
- **Conclusion**: Terminating decimal.

**(vii) \( \frac{51}{60} \)**
- Simplify: \( \frac{51}{60} = \frac{17}{20} \) (after dividing both numerator and denominator by 3).
- Denominator: \( 20 = 2^2 \times 5 \) (only has prime factors 2 and 5).
- **Conclusion**: Terminating decimal.

### Final Summary of Results:
1. \( \frac{135}{50} \) - Terminating
2. \( \frac{4}{11} \) - Non-terminating repeating
3. \( \frac{8}{7} \) - Non-terminating repeating
4. \( 6 \frac{3}{8} \) - Terminating
5. \( \frac{55}{9} \) - Non-terminating repeating
6. \( \frac{5^2 \times 3^3}{2 \times 5^3 \times 27} \) - Terminating
7. \( \frac{51}{60} \) - Terminating