Which of the following fractions represent a non terminating decimal?

Options

A. `63/24`
B. `18/15`
C. `14/21`
D. `33/44`

Answer

To determine which of the given fractions represents a non-terminating decimal, we need to analyze each fraction. A fraction will yield a non-terminating decimal if, when simplified, its denominator has prime factors other than 2 and 5.

Let's analyze the given fractions step by step:

### Step 1: Analyze Fraction A (63/24)
1. **Simplify the fraction**:
- The prime factorization of 24 is \(2^3 \times 3\).
- The prime factorization of 63 is \(3^2 \times 7\).
- Therefore, \(63/24\) can be simplified to \(63 \div 3 = 21\) and \(24 \div 3 = 8\), giving us \(21/8\).

2. **Check the denominator**:
- The denominator \(8\) is \(2^3\), which has only the prime factor 2.

**Conclusion**: This fraction is a terminating decimal.

### Step 2: Analyze Fraction B (18/15)
1. **Simplify the fraction**:
- The prime factorization of 15 is \(3 \times 5\).
- The prime factorization of 18 is \(2 \times 3^2\).
- Therefore, \(18/15\) can be simplified to \(18 \div 3 = 6\) and \(15 \div 3 = 5\), giving us \(6/5\).

2. **Check the denominator**:
- The denominator \(5\) has only the prime factor 5.

**Conclusion**: This fraction is a terminating decimal.

### Step 3: Analyze Fraction C (14/21)
1. **Simplify the fraction**:
- The prime factorization of 21 is \(3 \times 7\).
- The prime factorization of 14 is \(2 \times 7\).
- Therefore, \(14/21\) can be simplified to \(14 \div 7 = 2\) and \(21 \div 7 = 3\), giving us \(2/3\).

2. **Check the denominator**:
- The denominator \(3\) has the prime factor 3, which is neither 2 nor 5.

**Conclusion**: This fraction is a non-terminating decimal.

### Step 4: Analyze Fraction D (33/44)
1. **Simplify the fraction**:
- The prime factorization of 44 is \(2^2 \times 11\).
- The prime factorization of 33 is \(3 \times 11\).
- Therefore, \(33/44\) can be simplified to \(33 \div 11 = 3\) and \(44 \div 11 = 4\), giving us \(3/4\).

2. **Check the denominator**:
- The denominator \(4\) is \(2^2\), which has only the prime factor 2.

**Conclusion**: This fraction is a terminating decimal.

### Final Answer:
The fraction that represents a non-terminating decimal is **C: 14/21**.

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