KS3 Mathematics. N9 Mental methods. N9 Mental methods. Contents. N9.2 Addition and subtraction. N9.1 Order of operations. N9.3 Multiplication and division. N9.4 Numbers between 0 and 1. N9.5 Problems and puzzles. Using the correct order of operations. What is 7 – 3 – 2?.
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KS3 Mathematics
N9 Mental methods
N9 Mental methods
Contents
N9.2 Addition and subtraction
N9.1 Order of operations
N9.3 Multiplication and division
N9.4 Numbers between 0 and 1
N9.5 Problems and puzzles
What is 7 – 3 – 2?
When a calculation contains more than one operation it is important that we use the correct order of operations.
The first rule is we work from left to right so,
7 – 3 – 2 = 4 – 2
NOT
7 – 3 – 2 = 7 – 1
= 2
= 6
What is 8 + 2 × 4?
The second rule is that we multiply or divide before we add or subtract.
8 + 2 × 4 = 8 + 8
NOT
8 + 2 × 4 = 10 × 4
= 16
= 40
What is (15 – 9) ÷ 3?
When a calculation contains brackets we always work out the contents of any brackets first.
(15 – 9) ÷ 3 = 6 ÷ 3
= 2
Sometimes we have to use brackets within brackets.
For example,
10 ÷ {5 – (6 – 3)}
These are called nested brackets.
We evaluate the innermost brackets first and then work outwards.
10 ÷ {5 – (6 – 3)}
= 10 ÷ {5 – 3}
= 10 ÷ 2
= 5
13 + 8
7
13 + 8
7
= (13 + 8) ÷ 7
What is ?
When we use a horizontal line for division the dividing line acts as a bracket.
= 21 ÷ 7
= 3
24 + 8
24 + 8
24 – 8
24 – 8
= (24 + 8) ÷ (24 – 8)
What is ?
Again, the dividing line acts as a bracket.
= 32 ÷ 16
= 2
When we multiply by a bracket it is not always necessary to use the symbol for multiplication, ×.
For example,
8 + 3(7 – 3)
is equivalent to 8 + 3 × (7 – 3)
= 8 + 3 × 4
= 8 + 12
= 20
Compare this to the use of brackets in algebraic expressions such as 3(a + 2).
What is 100 – 2(3 + 4)2
When indices appear in a calculation, these are worked out after brackets, but before multiplication and division.
100 – 2(3 + 4)2
Brackets first,
= 100 – 2 × 72
then Indices,
= 100 – 2 × 49
thenDivision and Multiplication,
= 100 – 98
and thenAddition and Subtraction
= 2
B
Remember BIDMAS:
RACKETS
I
NDICES (OR POWERS)
D
IVISION
M
ULTIPLICATION
A
DDITION
S
UBTRACTION
82
64
=
– 8 × 0.5
– 8 × 0.5
=
16
4
(3.4 + 4.6)2
What is
– 8 × 0.5 ?
(6 + 5 × 2)
Brackets first,
then Indices,
thenDivision and Multiplication,
= 16 – 4
and thenAddition and Subtraction
= 12
(52 + 72)
7  5
(
5
5
7
7
(
)
x2
+
x2
)
÷

We can use a calculator to evaluate more difficult calculations.
For example,
This can be entered as:
= 4.3 (to 1 d.p.)
Always use an approximation to check answers given by a calculator.
N9 Mental methods
Contents
N9.1 Order of operations
N9.2 Addition and subtraction
N9.3 Multiplication and division
N9.4 Numbers between 0 and 1
N9.5 Problems and puzzles
What is 276 + 68?
276 + 68 =
200 + 70 + 6 + 60 + 8
= 6 + 8 + 70 + 60 + 200
= 14 + 130 + 200
= 344
What is 63.8 + 4.7?
63.8 + 4.7
= 60 + 3 + 0.8 + 4 + 0.7
= 0.8 + 0.7 + 3 + 4 +60
= 1.5 + 7 + 60
= 68.5
+ 60
+ 8
276
336
344
+ 4
+ 0.7
63.8
67.8
68.5
What is 276 + 68?
276
+ 60
+ 8
= 344
What is 63.8 + 4.7?
63.8
+ 4
+ 0.7
= 68.5
+ 70
– 2
276
344
346
+ 5
– 0.3
63.8
68.5
68.8
What is 276 + 68?
276
+ 70
– 2
= 344
What is 63.8 + 4.7?
63.8
+ 5
– 0.3
= 68.5
– 7
– 30
– 400
127
134
164
564
– 0.4
– 4
– 2
16.1
16.5
20.5
22.5
What is 564 – 437?
564
– 400
– 30
– 7
= 127
What is 22.5 – 6.4?
22.5
– 2
– 4
– 0.4
= 16.1
+ 100
+ 4
+ 20
+ 3
437
537
540
564
560
+ 10
+ 6
+ 0.1
6.4
16.4
22.5
22.4
What is 564 – 437?
100
+ 3
+ 20
+ 4
= 127
What is 22.5 – 6.4?
10
+ 6
+ 0.1
= 16.1
+ 63
– 500
64
127
564
+ 0.1
– 6.5
16
16.1
22.5
What is 564 – 437?
564
– 500
+ 63
= 127
What is 22.5 – 6.4?
22.5
– 6.5
+ 0.1
= 16.1
N9 Mental methods
Contents
N9.1 Order of operations
N9.2 Addition and subtraction
N9.3 Multiplication and division
N9.4 Numbers between 0 and 1
N9.5 Problems and puzzles
What is 7 × 43?
We can work out 7 × 43 mentally using partitioning.
43 = 40 + 3
So,
7 × 43 = (7 × 40) + (7 × 3)
= 280 + 21
= 301
What is 3.2 × 40?
We can work out 3.2 × 40 by partitioning 3.2
3.2 = 3 + 0.2
So,
3.2 × 40 = (3 × 40) + (0.2 × 40)
= 120 + 8
= 128
What is 0.6 × 29?
We can work out 0.6 × 29 using the distributive law.
29 = 30 – 1
So,
0.6 × 29 = (0.6 × 30) – (0.6 × 1)
= 18 – 0.6
= 17.4
What is 26 × 12?
We can work out 26 × 12 by dividing 12 into factors.
12 = 4 × 3 = 2 × 2 × 3
So we can multiply 26 by 2, by 2 again and then by 3:
52 × 2 × 3
26 × 2 × 2 × 3 =
= 104 × 3
= 312
What is 0.7 × 18?
We can work out 0.7 × 18 by dividing 18 into factors.
18 = 9 × 2
So we can multiply 0.7 by 9 and then by 2:
0.7 × 18 =
= 0.7 × 9 × 2
= 6.3 × 2
= 12.6
What is 7.5 × 8?
Two numbers can be multiplied together mentally by doubling one number and halving the other.
We can repeat this until the numbers are easy to work out mentally.
7.5 × 8 =
15 × 4
= 30 × 2
= 60
What is 68 ÷ 20?
We can work out 68 ÷ 20 by dividing 20 into factors.
20 = 2 × 10
So we can divide 68 by 2 and then by 10:
68 ÷ 20 =
68 ÷ 2 ÷ 10
= 34 ÷ 10
= 3.4
What is 12.4 ÷ 8?
We can work out 12.4 ÷ 8 by dividing 8 into factors.
8 = 2 × 2 × 2
So we can divide 12.4 by 2, by 2 again and then by 2 a third time:
12.4 ÷ 8 =
12.4 ÷ 2 ÷ 2 ÷ 2
31 ÷ 2 = 15.5 so
3.1 ÷ 2 =1.55
= 6.2 ÷ 2 ÷ 2
= 3.1 ÷ 2
= 1.55
What is 486 ÷ 6?
We can work out 486 ÷ 6 by partitioning 486.
486 = 480 + 6
So,
486 ÷ 6 = (480 ÷ 6) + (6 ÷ 6)
= 80 + 1
= 81
420
420
40
40
21
2
What is 420 ÷ 40?
We can simplify 420 ÷ 40 by writing the division as a fraction and then cancelling.
420 ÷ 40 =
21
=
2
= 101/2
= 10.5
× 10
2.6
26
26
13
0.8
8
8
4
× 10
What is 2.6 ÷ 0.8?
We can simplify 2.6 ÷ 0.8 by writing the division as a fraction.
2.6 ÷ 0.8 =
=
13
=
4
= 31/4
= 3.25
We can use our knowledge of place value to multiply by multiples of 10, 100 and 1000.
What is 7 × 600?
What is 2.3 × 4000?
2.3 × 4000 =
2.3 × 4 × 1000
7 × 600 =
7 × 6 × 100
= 42 × 100
= 9.2 × 1000
= 4200
= 9200
We can use our knowledge of place value to divide by multiples of 10, 100 and 1000.
What is 24 ÷ 80?
What is 4.5 ÷ 500?
24 ÷ 80 =
24 ÷ 8 ÷ 10
4.5 ÷ 500 =
4.5 ÷ 5 ÷ 100
= 3 ÷ 10
= 0.9 ÷ 100
= 0.3
= 0.009
N9 Mental methods
Contents
N9.1 Order of operations
N9.2 Addition and subtraction
N9.4 Numbers between 0 and 1
N9.3 Multiplication and division
N9.5 Problems and puzzles
Multiplying by 0.1
is the same as
Dividing by 10
Multiplying by 0.01
is the same as
Dividing by 100
What is 4 × 0.8?
What is 15 × 0.03?
4 × 0.8 =
4 × 8 × 0.1
15 × 0.03 =
15 × 3 × 0.01
= 32 × 0.1
= 45 × 0.01
= 32 ÷ 10
= 45 ÷ 100
= 3.2
= 0.45
Dividing by 0.1
is the same as
Multiplying by 10
Dividing by 0.01
is the same as
Multiplying by 100
What is 36 ÷ 0.4?
What is 3 ÷ 0.02?
36 ÷ 0.4 =
36 ÷ 4 ÷ 0.1
3 ÷ 0.02 =
3 ÷ 2 ÷ 0.01
= 9 ÷ 0.1
= 1.5 ÷ 0.01
= 9 × 10
= 1.5 × 100
= 90
= 150
When we multiply a number n by a number greater than 1 the answer will be bigger than n.
When we multiply a number n by a number between 0 and 1 the answer will be smaller than n.
When we divide a number n by a number greater than 1 the answer will be smaller than n.
When we divide a number n by a number between 0 and 1 the answer will be bigger than n.
N9 Mental methods
Contents
N9.1 Order of operations
N9.2 Addition and subtraction
N9.5 Problems and puzzles
N9.3 Multiplication and division
N9.4 Numbers between 0 and 1