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16 The laws of
arithmetic
16.1 The laws of arithmetic
Worked example 1 associative rule brackets
Put brackets in the calculation to make it commutative rule distributive rule
correct. order of operations
4 + 5 × 3 × 2 = 54
4 + 5 × 3 × 2 = 4 + 30 The order of operations is multiplication
= 34 before addition.
Which is not the required answer.
(4 + 5) × 3 × 2 = 9 × 3 × 2 Brackets are worked out before
= 54 multiplication.
Answer: (4 + 5) × 3 × 2 = 54
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16.1 The laws of arithmetic
Calculation Answer
Exercise 16.1
(12 – 3) × 8 18
Focus
10 × 8 + 1 27
1 Work out the answer to each calculation.
6 × (5 – 2) 36
Join each calculation to the correct answer.
7 × (4 + 5) 45
The first one has been done for you.
6 × (3 + 6) 54
(9 – 4) × 9 63
(8 + 4) × 3 72
(12 – 3) × 3 81
2 Which expression has the same value
as 9 × (6 – 1)?
54 – 5 54 – 1 9×5 9×7
3 Work out the answer to each question to help you find your way
through the maze.
start
3 × (14 – 6) 24 20 ÷ (2 + 3) 13 6 × (13 + 2)
36 4 25
10 ÷ (8 – 3) 15 5 × (3 + 4) 35 24 ÷ (3 + 5)
2 19 3
9 × (18 – 9) 81 20 ÷ (7 – 3) 3 end
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16 The laws of arithmetic
4 Safiya is learning to use brackets.
She writes down four calculations but one of her calculations is wrong.
A: 10 + (2 + 8) × 3 = 40 B: (10 + 2 + 8) × 3 = 60
C: (10 + 2) + 8 × 3 = 58 D: 10 + 2 + 8 × 3 = 36
Which calculation is wrong? What is the correct answer?
Practice
5 Complete the calculation.
36 × 97 = 36 × (100 – )
= (36 × 100) – (36 × )
= –
6 Mandy writes 4 + 9 × 5 × 2 = 130
Is she correct?
Explain your answer.
7 Write the missing number to make the calculation correct.
(10 – ) × 10 = 10
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16.1 The laws of arithmetic
8 Which number sentence is equivalent to 8 × 12?
A: (8 × 10) + (8 × 2) B: (8 × 1) + (8 × 2)
C: (8 × 10) + 2 D: 8 + (10 × 2)
Explain why the other sentences are wrong.
9 Write the sign <, > or = to make each expression correct.
a 2 × (3 + 4) 2×3+4
b (10 + 6) ÷ 2 5+6÷2
c (13 + 5) ÷ 9 (13 + 8) ÷ 7
Challenge
10 Use the numbers 2, 3, 4 and 5 to complete the calculations.
a 20 – ( + 9) = 7 b 9+ –3=8
c 11 – ( – 3) = 9 d 11 – + 2 = 10
11 Draw brackets to make each expression equal to 10.
a 14 – 12 × 5 b 11 – 6 – 5
c 20 – 15 – 5 d 20 ÷ 4 – 2
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16 The laws of arithmetic
12 Write the missing numbers.
a 3×( + 4) = 27 b 4 × (8 – ) = 20
c (6 + 7) × = 39 d 16 ÷ ( – 3) = 16
e 3×( – 6) = 6 f (21 – 12) ÷ =3
13 Here are six cards.
7 8 25 ( ) =
Use all these cards together with any of the operation signs +, –, ×
and ÷ to make a number sentence with an answer of 10.
14 Use these numbers together with brackets and operation signs to make
the target number.
Example: 3, 4, 6 Target 42 Answer (3 + 4) × 6
a 2, 5, 6 Target 40
b 2, 3, 5 Target 4
c 3, 4, 6 Target 12
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