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Let's determine whether each of the numbers in the table is rational or irrational:

1. **\(\frac{1}{11}\)**:  
   - **Rational**: Any fraction where both the numerator and denominator are integers, and the denominator is not zero, is a rational number.

2. **\(\sqrt{11}\)**:  
   - **Irrational**: The square root of a non-perfect square (like 11) is always irrational because it cannot be expressed as a fraction of two integers.

3. **\(\sqrt{111}\)**:  
   - **Irrational**: Similar to \(\sqrt{11}\), the square root of a non-perfect square (111) is also irrational.

4. **\(0.\overline{1}\)** (repeating decimal 0.111...):  
   - **Rational**: Any repeating decimal is a rational number, as it can be expressed as a fraction (in this case, \( \frac{1}{9} \)).

### Summary:
- \(\frac{1}{11}\): Rational
- \(\sqrt{11}\): Irrational
- \(\sqrt{111}\): Irrational
- \(0.\overline{1}\): Rational

Would you like more details or have any other questions?

Here are 5 related questions for further exploration:
1. What is the difference between rational and irrational numbers?
2. Can the square root of any number be rational?
3. How do you convert repeating decimals into fractions?
4. Is \( \pi \) rational or irrational, and why?
5. Can a number be both rational and irrational?

**Tip**: Any number that can be written as a fraction of two integers is always rational!

Determine whether each number is rational or irrational: 1/11, √11, √111, 0.1̅ (repeating decimal 0.111...).

Learn how to identify whether numbers like 1/11, √11, √111, and repeating decimals like 0.1̅ are rational or irrational. Understand the key concepts of rational and irrational numbers, square roots, and repeating decimals, with detailed explanations suitable for Grades 7-9.

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