Which of the following numbers can be written as a non-terminating but recurring decimal?

Options

A. 9
B. `(43)/(8)`
C. `sqrt()6`
D. `(5)/(12)`

Answer

To determine which of the given numbers can be expressed as a non-terminating but recurring decimal, we need to understand the characteristics of such decimals.

### Step-by-Step Solution:

1. **Understanding Non-Terminating Recurring Decimals**:
- Non-terminating recurring decimals are decimals that go on forever but have a repeating pattern. For example, \(0.333...\) (which is \( \frac{1}{3} \)) or \(0.666...\) (which is \( \frac{2}{3} \)).

2. **Identifying the Options**:
- We need to analyze the options provided (though not explicitly listed in the question, we will assume they are common fractions or integers).

3. **Evaluating Each Option**:
- **Option 1: Integer (e.g., 9)**:
- An integer does not have a decimal part, so it cannot be a non-terminating decimal.
- **Conclusion**: Not a non-terminating recurring decimal.

- **Option 2: Fraction (e.g., \( \frac{1}{8} \))**:
- When we divide \(1\) by \(8\), we get \(0.125\), which is a terminating decimal.
- **Conclusion**: Not a non-terminating recurring decimal.

- **Option 3: Irrational Number (e.g., \( \sqrt{6} \))**:
- The decimal representation of an irrational number is non-terminating and non-recurring.
- **Conclusion**: Not a non-terminating recurring decimal.

- **Option 4: Fraction (e.g., \( \frac{5}{12} \))**:
- When we divide \(5\) by \(12\), we perform long division:
- \(5 \div 12 = 0.41666...\) (where \(6\) repeats).
- This results in a non-terminating decimal with a repeating part.
- **Conclusion**: This is a non-terminating recurring decimal.

4. **Final Answer**:
- The only number that can be expressed as a non-terminating but recurring decimal from the options given is \( \frac{5}{12} \).