Which of these is a RATIONAL number ?

Options

A. `3pi`
B. `5 sqrt()5`
C. `0.3466666......`
D. ` 0.345210651372849...`

Answer

To determine which of the given options is a rational number, we need to understand the definition of rational and irrational numbers.

### Step-by-Step Solution:

1. **Understanding Rational Numbers**:
- A rational number is defined as any number that can be expressed in the form \( \frac{P}{Q} \), where \( P \) and \( Q \) are integers and \( Q \neq 0 \).
- Rational numbers can also be represented as terminating or repeating decimals.

2. **Analyzing the Options**:
- Let's assume the options are:
1. \( \pi \)
2. \( 5\sqrt{5} \)
3. \( 0.346666... \) (where 6 repeats)
4. \( 0.345206513... \) (non-repeating)

3. **Evaluating Each Option**:
- **Option 1: \( \pi \)**
- \( \pi \) is known to be an irrational number. It cannot be expressed as a fraction of integers.
- **Conclusion**: Not a rational number.

- **Option 2: \( 5\sqrt{5} \)**
- The square root of 5 is also an irrational number. Thus, \( 5\sqrt{5} \) is irrational as it cannot be expressed as a fraction.
- **Conclusion**: Not a rational number.

- **Option 3: \( 0.346666... \)**
- This is a non-terminating decimal, but it is repeating (the digit 6 repeats indefinitely). Therefore, it can be expressed as a fraction.
- **Conclusion**: This is a rational number.

- **Option 4: \( 0.345206513... \)**
- This is a non-terminating decimal and does not repeat. Such numbers cannot be expressed as a fraction of integers.
- **Conclusion**: Not a rational number.

4. **Final Conclusion**:
- The only rational number among the options is **Option 3: \( 0.346666... \)**.

### Answer:
**Option 3 is the rational number.**