Comparing Rational and Irrational Numbers | Worksheet - Free Printable

Let's solve each problem on the worksheet "Comparing Rational and Irrational Numbers" step by step. We need to compare each pair of numbers using either `<` or `>`.

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1. √7 ○ -3.28

- √7 ≈ 2.645 (since 2.6² = 6.76, 2.7² = 7.29 → so around 2.645)
- -3.28 is negative
- Positive number > Negative number

✔ √7 > -3.28

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2. 2/π ○ π

- 2/π ≈ 2 / 3.1416 ≈ 0.6366
- π ≈ 3.1416
- 0.6366 < 3.1416

✔ 2/π < π

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3. 2.97 ○ √8

- √8 = √(4×2) = 2√2 ≈ 2 × 1.414 ≈ 2.828
- 2.97 > 2.828

✔ 2.97 > √8

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4. √15 ○ -8.64

- √15 ≈ 3.873 (since 3.8² = 14.44, 3.9² = 15.21 → ~3.87)
- -8.64 is negative
- Positive > Negative

✔ √15 > -8.64

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5. -3π ○ -16/5

- -3π ≈ -3 × 3.1416 ≈ -9.4248
- -16/5 = -3.2
- Compare: -9.4248 vs -3.2
- More negative means smaller

So: -9.4248 < -3.2

✔ -3π < -16/5

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6. -0.39 ○ √15

- √15 ≈ 3.873 (positive)
- -0.39 is negative
- Negative < Positive

✔ -0.39 < √15

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7. -0.02 ○ 6π

- -0.02 is negative
- 6π ≈ 6 × 3.1416 ≈ 18.8496 (positive)
- Negative < Positive

✔ -0.02 < 6π

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8. 25/4 ○ √16

- 25/4 = 6.25
- √16 = 4
- 6.25 > 4

✔ 25/4 > √16

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9. 16.025 ○ 3π

- 3π ≈ 3 × 3.1416 ≈ 9.4248
- 16.025 > 9.4248

✔ 16.025 > 3π

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10. -0.86 ○ -25/3

- -25/3 ≈ -8.333...
- -0.86 vs -8.333
- -0.86 is greater than -8.333 (closer to zero)

✔ -0.86 > -25/3

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11. 7π ○ √100

- 7π ≈ 7 × 3.1416 ≈ 21.991
- √100 = 10
- 21.991 > 10

✔ 7π > √100

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12. 4 + π ○ √30

- 4 + π ≈ 4 + 3.1416 ≈ 7.1416
- √30 ≈ 5.477 (since 5.4² = 29.16, 5.5² = 30.25 → ~5.477)
- 7.1416 > 5.477

✔ 4 + π > √30

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13. √3 - 1 ○ -π

- √3 ≈ 1.732 → √3 - 1 ≈ 0.732
- -π ≈ -3.1416
- 0.732 > -3.1416

✔ √3 - 1 > -π

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14. 3π ○ √9 + 6

- 3π ≈ 9.4248
- √9 + 6 = 3 + 6 = 9
- 9.4248 > 9

✔ 3π > √9 + 6

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15. √82 + 10 ○ √85 + 16

- √82 ≈ 9.055 → √82 + 10 ≈ 19.055
- √85 ≈ 9.219 → √85 + 16 ≈ 25.219
- 19.055 < 25.219

✔ √82 + 10 < √85 + 16

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16. 46 - 3 ○ √75

- 46 - 3 = 43
- √75 = √(25×3) = 5√3 ≈ 5 × 1.732 ≈ 8.66
- 43 > 8.66

✔ 46 - 3 > √75

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17. √18 + 8 ○ √100 - 4

- √18 = √(9×2) = 3√2 ≈ 3 × 1.414 ≈ 4.242 → +8 = 12.242
- √100 - 4 = 10 - 4 = 6
- 12.242 > 6

✔ √18 + 8 > √100 - 4

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18. 9π - 2 ○ 40 - √50

- 9π ≈ 28.274 → 9π - 2 ≈ 26.274
- √50 = √(25×2) = 5√2 ≈ 7.071 → 40 - 7.071 ≈ 32.929
- 26.274 < 32.929

✔ 9π - 2 < 40 - √50

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19. 12 - √24 ○ √17 + 4

- √24 = √(4×6) = 2√6 ≈ 2 × 2.449 ≈ 4.898 → 12 - 4.898 ≈ 7.102
- √17 ≈ 4.123 → +4 = 8.123
- 7.102 < 8.123

✔ 12 - √24 < √17 + 4

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20. √50 ○ 10π - 27

- √50 ≈ 7.071
- 10π ≈ 31.416 → 31.416 - 27 = 4.416
- 7.071 > 4.416

✔ √50 > 10π - 27

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21. 6π ○ √25 + 20

- 6π ≈ 18.8496
- √25 + 20 = 5 + 20 = 25
- 18.8496 < 25

✔ 6π < √25 + 20

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | √7 > -3.28 |
| 2 | 2/π < π |
| 3 | 2.97 > √8 |
| 4 | √15 > -8.64 |
| 5 | -3π < -16/5 |
| 6 | -0.39 < √15 |
| 7 | -0.02 < 6π |
| 8 | 25/4 > √16 |
| 9 | 16.025 > 3π |
| 10 | -0.86 > -25/3 |
| 11 | 7π > √100 |
| 12 | 4 + π > √30 |
| 13 | √3 - 1 > -π |
| 14 | 3π > √9 + 6 |
| 15 | √82 + 10 < √85 + 16 |
| 16 | 46 - 3 > √75 |
| 17 | √18 + 8 > √100 - 4 |
| 18 | 9π - 2 < 40 - √50 |
| 19 | 12 - √24 < √17 + 4 |
| 20 | √50 > 10π - 27 |
| 21 | 6π < √25 + 20 |

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📌 Key Concepts Used:

- Approximate irrational numbers like √n and π.
- Compare rational numbers (fractions, decimals).
- Remember: negative numbers are less than positive numbers.
- When comparing two negatives, the one closer to zero is greater.
- Use estimation for square roots and multiples of π.

Let me know if you'd like a printable version or explanation for any specific problem!