Operations With Rational And Irrational Numbers Worksheet Pdf ... - Free Printable

Sure! Let’s solve each problem step by step and determine whether the result is Rational or Irrational.

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🔢 Definitions to Remember:

- Rational Number: A number that can be expressed as a fraction `a/b` where `a` and `b` are integers and `b ≠ 0`. This includes terminating decimals, repeating decimals, and integers.
- Irrational Number: A number that cannot be expressed as a simple fraction. Its decimal goes on forever without repeating. Examples: √2, π, √3, etc.

Also:
- The sum or product of two rational numbers is always rational.
- The sum or product of a rational and an irrational number is usually irrational (unless it cancels out, which is rare).
- The sum or product of two irrational numbers can be either rational or irrational — we must check case by case.

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## ✔ Problem-by-Problem Solutions:

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1. `1.10 + 0.01`

→ Both are terminating decimals → Rational

✔ Answer: Rational

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2. `1.2 ÷ 0.5`

→ `1.2 / 0.5 = 12/5 = 2.4` → Terminating decimal → Rational

✔ Answer: Rational

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3. `1.15 + 0.07`

→ `1.22` → Terminating decimal → Rational

✔ Answer: Rational

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4. `1.17 ÷ 0.02`

→ `1.17 / 0.02 = 117 / 2 = 58.5` → Terminating decimal → Rational

✔ Answer: Rational

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5. `√7 + 8`

→ `√7` is irrational (≈2.645...), 8 is rational.

→ Rational + Irrational = Irrational

✔ Answer: Irrational

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6. `√11 + 9`

→ `√11` is irrational, 9 is rational.

→ Rational + Irrational = Irrational

✔ Answer: Irrational

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7. `√17 × 5`

→ `√17` is irrational, 5 is rational.

→ Rational × Irrational = Irrational

✔ Answer: Irrational

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8. `√20 × 7`

→ `√20 = √(4×5) = 2√5` → Still irrational.

→ Multiply by 7: `7 × 2√5 = 14√5` → Still irrational.

✔ Answer: Irrational

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9. `√225 × √825`

→ First, simplify:

- `√225 = 15` → Rational
- `√825 = √(25×33) = 5√33` → Irrational

→ So: `15 × 5√33 = 75√33` → Irrational

✔ Answer: Irrational

> 💡 Note: Even though one part is rational, multiplying it by an irrational gives irrational.

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10. `√36 × √64`

→ `√36 = 6`, `√64 = 8` → both rational

→ `6 × 8 = 48` → Rational

✔ Answer: Rational

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## 📝 Final Answers:

1. Rational
2. Rational
3. Rational
4. Rational
5. Irrational
6. Irrational
7. Irrational
8. Irrational
9. Irrational
10. Rational

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