Number: Integers and Powers Worksheet with Answers

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KS3 · Number

KS3.M-N1 Number: Integers and Powers

AQA AQA KS3 Maths progression framework - Number · Calculator allowed · about 75 minutes

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

1

Write these integers in order, starting with the smallest.
-8, 3, -1, 0, -5, 6

(2)

2

Work out the value of each calculation.

(a)-3 + 7(1)

(b)-6 - 5(1)

(c)8 - 12(1)

(d)-9 + (-4)(1)

3

Work out the value of each calculation.

(a)-4 x 6(1)

(b)-3 x -7(1)

(c)-36 / 4(1)

(d)-45 / -9(1)

4

Work out the value of each power.

(a)5²(1)

(b)3³(1)

(c)2⁵(1)

5

Index notation links a power to repeated multiplication.

(a)Write 7 x 7 x 7 as a single power of 7, and work out its value.(2)

(b)Write 6³ as repeated multiplication, and work out its value.(2)

6

Use the correct order of operations (BIDMAS) to work out each expression.

(a)3 + 2³ x 4(2)

(b)(7 - 2)² - 10(2)

(c)40 / 2³ + 1(2)

7

On a winter morning in Sheffield, the temperature at 6am was -4C. By midday it had risen by 9C. By 9pm it had fallen by 15C from the midday temperature.

(a)What was the temperature at midday?(1)

(b)What was the temperature at 9pm?(2)

(c)Work out the difference between the temperature at 6am and the temperature at 9pm.(2)

8

Priya's bank balance is -45 pounds (she is overdrawn by 45 pounds). She deposits 120 pounds and then pays a bill of 85 pounds.

(a)Work out her balance after the deposit.(2)

(b)Work out her balance after paying the bill.(2)

(c)A second bill of 30 pounds arrives. Priya's bank will not let her go more than 50 pounds overdrawn. Show working to determine whether she can pay this bill.(2)

9

From the list 21, 23, 27, 29, 33, 37, write down all the prime numbers.

(2)

10

Factors of a number divide into it exactly.

(a)List all the factors of 84.(2)

(b)Using your list, or otherwise, find the highest common factor of 84 and 60.(2)

11

Multiples of a number are found by multiplying it by 1, 2, 3, and so on.

(a)List the first six multiples of 9.(2)

(b)Find the lowest common multiple of 6 and 8 by listing multiples of each number.(2)

12

Write 360 as a product of its prime factors, giving your answer in index form.

(3)

13

Prime factors can be used to find the highest common factor (HCF) of two numbers.

(a)Write 84 as a product of its prime factors, giving your answer in index form.(2)

(b)Write 126 as a product of its prime factors, giving your answer in index form.(2)

(c)Use your answers to parts a and b to find the highest common factor of 84 and 126.(2)

14

Prime factors can also be used to find the lowest common multiple (LCM) of two numbers.

(a)Write 45 and 60 each as a product of prime factors, in index form.(2)

(b)Hence find the lowest common multiple of 45 and 60.(2)

15

Simplify each expression, writing your answer as a single power. Do not evaluate.

(a)3² x 3⁴(1)

(b)5³ x 5¹(1)

(c)2⁵ x 2²(1)

16

Simplify or work out the value of each expression.

(a)Simplify 7⁸ / 7³, giving your answer as a single power of 7.(2)

(b)Simplify (4²)³, giving your answer as a single power of 4.(2)

(c)Work out the value of 6⁵ / 6³.(2)

17

Work out the value of each root.

(a)√81(1)

(b)√144(1)

(c)cube root of 27(1)

(d)cube root of 125(1)

18

In a quiz of 10 questions, each correct answer scores +4 points and each incorrect answer scores -2 points.

(a)Tom gets 7 questions correct and 3 incorrect. Work out his total score.(2)

(b)Aisha's total score on the same 10 questions is -2. Work out how many questions Aisha answered correctly.(3)

19

Two lighthouses flash together at exactly 9:00pm. Lighthouse A flashes every 8 seconds and lighthouse B flashes every 12 seconds.

(a)Work out how many seconds it will be until the two lighthouses next flash together.(3)

(b)State the time when the two lighthouses will next flash together.(1)

20

A market trader has 90 apples and 126 oranges. She wants to make identical fruit bags, each with the same number of apples and the same number of oranges, using all the fruit with none left over.

(a)Write 90 and 126 as products of their prime factors.(2)

(b)Work out the greatest number of identical bags she can make.(2)

(c)For this greatest number of bags, work out how many apples and how many oranges go in each bag.(2)

21

Powers of 10 are used to write very large numbers in a compact form.

(a)Write 4,500,000 in the form A x 10ⁿ, where 1 ≤ A < 10.(2)

(b)Work out the value of 3 x 10⁴ x 2 x 10², giving your answer as an ordinary number.(2)

Mark scheme · KS3.M-N1 Number: Integers and Powers

Question 1

B1 at least four of the six values placed in correct relative order (oe)
B1 fully correct order -8, -5, -1, 0, 3, 6 cao
Answer: -8, -5, -1, 0, 3, 6

Question 2

(a) B1 4 cao
(a) Answer: 4
(b) B1 -11 cao
(b) Answer: -11
(c) B1 -4 cao
(c) Answer: -4
(d) B1 -13 cao
(d) Answer: -13

Question 3

(a) B1 -24 cao
(a) Answer: -24
(b) B1 21 cao
(b) Answer: 21
(c) B1 -9 cao
(c) Answer: -9
(d) B1 5 cao
(d) Answer: 5

Question 4

(a) B1 25 cao
(a) Answer: 25
(b) B1 27 cao
(b) Answer: 27
(c) B1 32 cao
(c) Answer: 32

Question 5

(a) B1 7³ cao
(a) B1 343 cao
(a) Answer: 7³ = 343
(b) B1 6 x 6 x 6 cao
(b) B1 216 cao
(b) Answer: 6 x 6 x 6 = 216

Question 6

(a) M1 evaluate 2³ = 8 and multiply by 4 to get 32, before adding (oe)
(a) A1 35 cao
(a) Answer: 35
(b) M1 evaluate the bracket first (7-2=5) then square to get 25 (oe)
(b) A1 15 cao
(b) Answer: 15
(c) M1 evaluate 2³ = 8, then divide 40 by 8 to get 5 (oe)
(c) A1 6 cao
(c) Answer: 6

Question 7

(a) B1 5C cao
(a) Answer: 5C
(b) M1 subtract 15 from their midday temperature (ft from part a)
(b) A1 -10C cao (ft)
(b) Answer: -10C
(c) M1 subtract the two temperatures, e.g. -4 - (-10) (ft)
(c) A1 6C cao (ft)
(c) Answer: 6C

Question 8

(a) M1 -45 + 120 (oe)
(a) A1 75 pounds cao
(a) Answer: GBP 75
(b) M1 subtract 85 from their part a balance (ft)
(b) A1 -10 pounds cao (ft), i.e. 10 pounds overdrawn
(b) Answer: -GBP 10 (10 pounds overdrawn)
(c) M1 subtract 30 from their part b balance, e.g. -10-30 (ft)
(c) A1 correct conclusion: yes, since -40 is not more than 50 pounds overdrawn (oe, ft)
(c) Answer: Yes; balance becomes -GBP 40, within the GBP 50 limit

Question 9

B1 at least two of 23, 29, 37 identified correctly with no non-primes included
B1 all three of 23, 29, 37 identified, and no others, cao
Answer: 23, 29, 37

Question 10

(a) B1 at least 8 of the 12 correct factors listed
(a) B1 fully correct list 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 cao
(a) Answer: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
(b) M1 identify the common factors of 84 and 60, e.g. 1, 2, 3, 4, 6, 12 (ft)
(b) A1 12 cao (ft)
(b) Answer: 12

Question 11

(a) B1 at least four correct multiples listed
(a) B1 fully correct list 9, 18, 27, 36, 45, 54 cao
(a) Answer: 9, 18, 27, 36, 45, 54
(b) M1 list multiples of both 6 and 8 until a common value appears (oe)
(b) A1 24 cao
(b) Answer: 24

Question 12

M1 begin correct repeated division/factor tree, e.g. 360=2x180
dM1 continue fully to obtain all prime factors 2, 2, 2, 3, 3, 5
A1 2³ x 3² x 5 cao (index form)
Answer: 2³ x 3² x 5

Question 13

(a) M1 correct prime factors identified: 2, 2, 3, 7
(a) A1 2² x 3 x 7 cao
(a) Answer: 2² x 3 x 7
(b) M1 correct prime factors identified: 2, 3, 3, 7
(b) A1 2 x 3² x 7 cao
(b) Answer: 2 x 3² x 7
(c) M1 select the lowest power of each common prime factor (2¹, 3¹, 7¹) (ft)
(c) A1 42 cao (ft)
(c) Answer: 42

Question 14

(a) B1 45 = 3² x 5 cao
(a) B1 60 = 2² x 3 x 5 cao
(a) Answer: 45 = 3² x 5; 60 = 2² x 3 x 5
(b) M1 select the highest power of each prime present (2², 3², 5¹) (ft)
(b) A1 180 cao (ft)
(b) Answer: 180

Question 15

(a) B1 3⁶ cao
(a) Answer: 3⁶
(b) B1 5⁴ cao
(b) Answer: 5⁴
(c) B1 2⁷ cao
(c) Answer: 2⁷

Question 16

(a) M1 subtract the indices, 8-3 (oe)
(a) A1 7⁵ cao
(a) Answer: 7⁵
(b) M1 multiply the indices, 2x3 (oe)
(b) A1 4⁶ cao
(b) Answer: 4⁶
(c) M1 simplify to 6² (oe)
(c) A1 36 cao
(c) Answer: 36

Question 17

(a) B1 9 cao
(a) Answer: 9
(b) B1 12 cao
(b) Answer: 12
(c) B1 3 cao
(c) Answer: 3
(d) B1 5 cao
(d) Answer: 5

Question 18

(a) M1 7x4=28 and 3x(-2)=-6 found (oe)
(a) A1 22 cao
(a) Answer: 22
(b) M1 form an equation using c correct and (10-c) incorrect, e.g. 4c - 2(10-c) = -2 (oe)
(b) dM1 correctly simplify to a linear equation, e.g. 6c - 20 = -2 or 6c=18 (oe, dependent on M1)
(b) A1 c = 3 cao
(b) Answer: 3 questions correct

Question 19

(a) M1 recognise this requires the lowest common multiple of 8 and 12
(a) M1 correct method to find the LCM, e.g. 8=2³, 12=2²x3 or listing multiples until a match (oe)
(a) A1 24 seconds cao
(a) Answer: 24 seconds
(b) B1 9:00:24pm (ft from part a)
(b) Answer: 9:00:24pm

Question 20

(a) B1 90 = 2 x 3² x 5 cao
(a) B1 126 = 2 x 3² x 7 cao
(a) Answer: 90 = 2 x 3² x 5; 126 = 2 x 3² x 7
(b) M1 identify the common prime factors with lowest powers, 2¹ and 3² (ft)
(b) A1 18 cao (ft)
(b) Answer: 18 bags
(c) M1 divide 90 by their HCF or 126 by their HCF (ft)
(c) A1 5 apples and 7 oranges per bag cao (ft)
(c) Answer: 5 apples and 7 oranges per bag

Question 21

(a) M1 A = 4.5 identified (oe)
(a) A1 4.5 x 10⁶ cao
(a) Answer: 4.5 x 10⁶
(b) M1 multiply the coefficients (3x2=6) and add the indices of 10 (4+2=6) to get 6 x 10⁶ (oe)
(b) A1 6,000,000 cao
(b) Answer: 6,000,000

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