Numbers and operations · Ages 13–14

Rational and irrational numbers worksheet

Part of: Surds & irrational numbers

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Numbers and operations · Ages 13–14

Numsprite
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Write each answer in the space. For word problems, show your working and write a sentence answer.

13.Tsz-hin says a square with an area of 64 cm² has sides of rational length. Tsz-hin is right.
True or false?
Working:
Answer:
14.Which of these four numbers is irrational?
- A. 0.7̇
- B. 196
- C. 0.1010010001…
- D. 37
Choose the right answer
Working:
Answer:
15.A square flower bed has an area of 368 m². Each side is between m and m (two whole numbers in a row).
Fill in each blank
Working:
Answer:
16.Someone estimates 248 as about 124. What is wrong? 248 lies between two whole numbers in a row: what is the smaller?
1. (1)248 is about half of 248
2. (2)248 ÷ 2 = 124
Find the wrong step, then write the right answer
Working:
Answer:
17.How many whole numbers k make k lie between 3 and 5 (not counting 3 and 5 themselves)?
Working:
Answer:
18.A square has an area of 147 cm². To the nearest whole number, about how long is each side?
Working:
Answer:

Show the answers and working

Rational and irrational numbers worksheet: Answers and working

1. 16, 17
Find the perfect squares either side of 285: 16² < 285 < 17².
2. 29
Let x = 0.222…; then 10x = 2.22…, and subtracting, 9x = 2, x = 29 = 29.
3. 13
169 < 171 < 196, that is 13² < 171 < 14², so 13 < 171 < 14.
4. 5899
Let x = 0.5858…; then 100x = 58.58…, and subtracting, 99x = 58, x = 5899 = 5899.

5. False
If 35 is a perfect square, 35 is a whole number, so rational; otherwise 35 is irrational.
6. B. 49
49 = 7, which is 71: rational.
7. C. 56
56 isn't a perfect square, so 56's decimal never ends or repeats: irrational.
8. False
When top and bottom are both perfect squares, the root is a fraction, rational; when 83 isn't a perfect square, it isn't.
9. True
3 × 12 = 3 × 12: rational only when 3 × 12 is a perfect square.
10. A. 65
8² < 65 < 9², so 8 < 65 < 9.
11. D. 27
5² < 27 < 6², so 5 < 27 < 6.
12. False
24 × 45 = 24 × 45: rational only when 24 × 45 is a perfect square.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. True
The side is 64: a whole number if 64 is a perfect square, irrational otherwise.
14. C. 0.1010010001…
0.1010010001… has more and more 0s between its 1s: it never ends or repeats, so it is irrational.
15. 19, 20
The side is 368; 19² < 368 < 20², so 19 < 368 < 20.
16. 15
Find the perfect squares either side of 248: 15² < 248 < (15 + 1)², so the smaller is 15.
17. 15
3 < k < 5 means 3² < k < 5²: k runs from 3² + 1 to 5² − 1, 15 numbers.
18. 12 cm
The side is 147, between 12 and 12 + 1; compare 147 with 12.5 squared (12² + 12 + 0.25): 12.

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