Identifying Rational and Irrational Numbers Worksheet | Free ... - Free Printable

Let's solve each problem step by step to determine whether the given number is rational (R) or irrational (I).

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🔹 Definitions:

- Rational Number (R): A number that can be expressed as a fraction $ \frac{p}{q} $, where $ p $ and $ q $ are integers and $ q \neq 0 $. Rational numbers include:
- Integers
- Terminating decimals
- Repeating decimals
- Square roots of perfect squares

- Irrational Number (I): A number that cannot be written as a simple fraction. These have non-repeating, non-terminating decimal expansions. Examples include:
- $ \pi $
- $ \sqrt{2} $
- Non-repeating, non-terminating decimals

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Now let’s go through each item:

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1) 61%

- Percent means "per hundred": $ 61\% = \frac{61}{100} $
- This is a fraction of two integers → Rational
✔ Answer: R

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2) 42

- Integer → can be written as $ \frac{42}{1} $
✔ Answer: R

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3) 75.082106

- This is a terminating decimal → can be written as a fraction
- Example: $ 75.082106 = \frac{75082106}{1000000} $
✔ Answer: R

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4) $ \sqrt{101} $

- 101 is not a perfect square (since $ 10^2 = 100 $, $ 11^2 = 121 $)
- So $ \sqrt{101} $ is irrational
✘ Answer: I

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5) 65.4279

- Terminating decimal → rational
✔ Answer: R

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6) $ \frac{20}{6} $

- This is a fraction of two integers → rational
- Even though it simplifies to $ \frac{10}{3} $, it's still rational
✔ Answer: R

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7) $ \pi $

- Pi is a well-known irrational number — non-repeating, non-terminating
✘ Answer: I

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8) 5.6213

- Terminating decimal → rational
✔ Answer: R

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9) $ \frac{89}{16} $

- Fraction of two integers → rational
✔ Answer: R

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10) 39

- Integer → rational
✔ Answer: R

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11) 89.396668...

- The ellipsis (...) suggests it may not repeat or terminate.
- But here, we need to check: does it repeat?
- The digits are: 89.396668...
- No clear repeating pattern, but if it's non-repeating and non-terminating, then it's irrational.
- However, if this number was meant to be repeating, it should have a bar over digits (e.g., 89.396̅).
- As written, it's ambiguous, but since it's just shown with dots and no indication of repetition, and it doesn't terminate, we assume it's non-repeating → Irrational
⚠️ But in many such worksheets, if a number is written with a few decimal places and an ellipsis, it might be intended as terminating or repeating.

Wait — look closely: 89.396668...

- It has a sequence: 396668...
- Not obviously repeating.
- Since it's not specified as repeating and appears non-terminating, likely irrational.

But actually, in standard math problems like this, unless explicitly stated as repeating or terminating, and especially if the decimal is infinite without a pattern, it's considered irrational.

So assuming it's non-repeating and non-terminating → Irrational

✘ Answer: I

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12) $ \sqrt{17} $

- 17 is not a perfect square → irrational
✘ Answer: I

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13) 67.714813...

- Again, ellipsis, no repeating pattern shown → assumed non-repeating, non-terminating → Irrational
✘ Answer: I

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14) $ \sqrt{64} $

- $ \sqrt{64} = 8 $ → integer → rational
✔ Answer: R

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15) $ \frac{1}{4} $

- Fraction → rational
✔ Answer: R

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16) $ \sqrt{25} $

- $ \sqrt{25} = 5 $ → integer → rational
✔ Answer: R

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17) 71.5186

- Terminating decimal → rational
✔ Answer: R

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18) $ \frac{7}{54} $

- Fraction of integers → rational
✔ Answer: R

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19) 20.455566...

- Digits: 20.455566...
- Looks like it might be trying to show a pattern? But no clear repetition.
- Without a bar or indication of repeating, and since it continues indefinitely without pattern → Irrational
✘ Answer: I

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20) 97.33997

- Terminating decimal → rational
✔ Answer: R

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✔ Final Answers:

| # | Answer |
|---|--------|
| 1 | R |
| 2 | R |
| 3 | R |
| 4 | I |
| 5 | R |
| 6 | R |
| 7 | I |
| 8 | R |
| 9 | R |
|10 | R |
|11 | I |
|12 | I |
|13 | I |
|14 | R |
|15 | R |
|16 | R |
|17 | R |
|18 | R |
|19 | I |
|20 | R |

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📝 Summary of Key Points:

- Integers, fractions, terminating decimals, repeating decimals → Rational
- Non-repeating, non-terminating decimals, square roots of non-perfect squares, $ \pi $ → Irrational
- Always check if the decimal repeats or terminates — if not, and it's infinite, it's likely irrational.

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