Which of the following numbers can be represented as non-terminating, repeating decimals?

Options

A. `(39)/(24)`
B. `3/(16)`
C. `3/(11)`
D. `(137)/(25)`

Answer

To determine which of the given numbers can be represented as non-terminating, repeating decimals, we need to analyze each fraction based on the properties of their denominators. A fraction will be a non-terminating, repeating decimal if its denominator, in simplest form, contains prime factors other than 2 and 5.

Let's evaluate each option step by step:

### Step 1: Analyze Number A: \( \frac{39}{24} \)

1. **Simplify the fraction**:
\[ \frac{39}{24} = \frac{3 \times 13}{3 \times 8} = \frac{13}{8} \]
2. **Identify the denominator**:
The denominator \(8\) can be expressed as \(2^3\).
3. **Conclusion**:
Since the denominator is a power of \(2\), \( \frac{39}{24} \) is a terminating decimal.

### Step 2: Analyze Number B: \( \frac{3}{16} \)

1. **Simplify the fraction**:
\[ \frac{3}{16} = \frac{3}{2^4} \]
2. **Identify the denominator**:
The denominator \(16\) can be expressed as \(2^4\).
3. **Conclusion**:
Since the denominator is a power of \(2\), \( \frac{3}{16} \) is a terminating decimal.

### Step 3: Analyze Number C: \( \frac{3}{11} \)

1. **Simplify the fraction**:
The fraction is already in simplest form.
2. **Identify the denominator**:
The denominator \(11\) is a prime number and does not factor into \(2\) or \(5\).
3. **Conclusion**:
Since the denominator contains a prime factor other than \(2\) or \(5\), \( \frac{3}{11} \) is a non-terminating, repeating decimal.

### Step 4: Analyze Number D: \( \frac{137}{25} \)

1. **Simplify the fraction**:
The fraction is already in simplest form.
2. **Identify the denominator**:
The denominator \(25\) can be expressed as \(5^2\).
3. **Conclusion**:
Since the denominator is a power of \(5\), \( \frac{137}{25} \) is a terminating decimal.

### Final Conclusion:

Among the given options, only \( \frac{3}{11} \) can be represented as a non-terminating, repeating decimal.

### Answer:
The number that can be represented as a non-terminating, repeating decimal is \( \frac{3}{11} \).

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