This worksheet helps students practice distinguishing between rational and irrational numbers through various examples including fractions, square roots, and decimals.

Let’s go step by step to figure out which numbers are rational and which are irrational.

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First, what’s the difference?

- Rational numbers can be written as a fraction (like 1/2 or -3/4) — even if they look like decimals or square roots, as long as they simplify to a fraction.
- Irrational numbers cannot be written as a simple fraction. They go on forever without repeating (like √2 or π).

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Let’s check each number in the first table:

1. 27/72 → This is already a fraction → Rational

2. 0 → Zero is a whole number → can be written as 0/1 → Rational

3. √10 → 10 is not a perfect square → √10 ≈ 3.162… doesn’t repeat or end → Irrational

4. 2/7 → Already a fraction → Rational

5. -√100 → √100 = 10 → so -√100 = -10 → which is an integer → Rational

6. 12 → Whole number → can be written as 12/1 → Rational

7. √(144/36) → First, 144 ÷ 36 = 4 → √4 = 2 → which is an integer → Rational

✔ So for the first table:
- Rational: 27/72, 0, 2/7, -√100, 12, √(144/36)
- Irrational: √10

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Now the second table:

1. 7/2 → Fraction → Rational

2. 10.3 → Decimal that ends → can be written as 103/10 → Rational

3. √16.45 → 16.45 is not a perfect square → √16.45 ≈ 4.056… doesn’t repeat or end → Irrational

4. 0/8 → That’s 0 → which we already know is rational → Rational

5. -√87 → 87 is not a perfect square → √87 ≈ 9.327… non-repeating, non-ending → Irrational

6. √9 → √9 = 3 → integer → Rational

7. √(81/676) → √81 = 9, √676 = 26 → so this is 9/26 → fraction → Rational

✔ So for the second table:
- Rational: 7/2, 10.3, 0/8, √9, √(81/676)
- Irrational: √16.45, -√87

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Final Answer:

First Table:
- Rational: 27/72, 0, 2/7, -√100, 12, √(144/36)
- Irrational: √10

Second Table:
- Rational: 7/2, 10.3, 0/8, √9, √(81/676)
- Irrational: √16.45, -√87