Let’s go through each problem one by one. We’ll decide if each number is
rational or
irrational.
Quick reminder:
- A
rational number can be written as a fraction (like 1/2, -3/4) or as a decimal that either stops (terminates) or repeats.
- An
irrational number cannot be written as a fraction — its decimal goes on forever without repeating (like π or √2).
---
Problem 1: 7
7 is an integer. All integers are rational because you can write them as a fraction: 7 = 7/1.
→ Rational
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Problem 2: –4/8
This is already a fraction! Even though it simplifies to –0.5, it’s still rational.
→ Rational
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Problem 3: √49
√49 = 7 (because 7 × 7 = 49). And 7 is rational.
→ Rational
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Problem 4: 3√25
First, √25 = 5 → so 3 × 5 = 15. 15 is an integer → rational.
→ Rational
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Problem 5: √24 / 7
√24 is not a perfect square. Let’s check:
√24 ≈ 4.898979... and it doesn’t repeat or stop. So 24 is irrational.
Dividing an irrational number by 7 (a rational number) still gives an irrational number.
→ Irrational
*(Note: You might think “but it’s over 7!” — but unless the numerator becomes rational after simplifying, the whole thing stays irrational.)*
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Problem 6: –6√8
√8 = (4×2) = 2√2 → so –6 × 2√2 = –12√2
√2 is irrational → multiplying by –12 keeps it irrational.
→ Irrational
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Problem 7: 5.8̅5̅38̅
The bar means the digits “8538” repeat forever: 5.853885388538...
Repeating decimals are always rational.
→ Rational
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Problem 8: 8.7̅08̅
Digits “708” repeat: 8.708708708...
Again, repeating decimal → rational.
→ Rational
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Problem 9: –9π
π is irrational. Multiplying by –9 (a rational number) doesn’t make it rational.
→ Irrational
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Problem 10: 9.76904...
The “...” means it continues forever — and there’s no indication it repeats. If it doesn’t repeat and doesn’t stop, it’s irrational.
*(Important: If it were something like 9.7690476904..., with a repeating pattern, we’d say rational. But here, no pattern is shown — just dots meaning “goes on randomly.” In math problems like this, if they don’t show a repeat and use “...”, it usually means non-repeating → irrational.)*
→ Irrational
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✔ Final Answers:
1. Rational
2. Rational
3. Rational
4. Rational
5. Irrational
6. Irrational
7. Rational
8. Rational
9. Irrational
10. Irrational
Parent Tip: Review the logic above to help your child master the concept of sets of real numbers worksheet.
