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 â€“ 3 â€“ 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 â€“ 3 â€“ 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 â€“ 3 â€“ 2 = 4 â€“ 2
NOT
7 â€“ 3 â€“ 2 = 7 â€“ 1
= 2
= 6
What is 8 + 2 Ã— 4?
The second rule is that we multiply or divide before we add or subtract.
8 + 2 Ã— 4 = 8 + 8
NOT
8 + 2 Ã— 4 = 10 Ã— 4
= 16
= 40
What is (15 â€“ 9) Ã· 3?
When a calculation contains brackets we always work out the contents of any brackets first.
(15 â€“ 9) Ã· 3 = 6 Ã· 3
= 2
Sometimes we have to use brackets within brackets.
For example,
10 Ã· {5 â€“ (6 â€“ 3)}
These are called nested brackets.
We evaluate the innermost brackets first and then work outwards.
10 Ã· {5 â€“ (6 â€“ 3)}
= 10 Ã· {5 â€“ 3}
= 10 Ã· 2
= 5
13 + 8
7
13 + 8
7
= (13 + 8) Ã· 7
What is ?
When we use a horizontal line for division the dividing line acts as a bracket.
= 21 Ã· 7
= 3
24 + 8
24 + 8
24 â€“ 8
24 â€“ 8
= (24 + 8) Ã· (24 â€“ 8)
What is ?
Again, the dividing line acts as a bracket.
= 32 Ã· 16
= 2
When we multiply by a bracket it is not always necessary to use the symbol for multiplication, Ã—.
For example,
8 + 3(7 â€“ 3)
is equivalent to 8 + 3 Ã— (7 â€“ 3)
= 8 + 3 Ã— 4
= 8 + 12
= 20
Compare this to the use of brackets in algebraic expressions such as 3(a + 2).
What is 100 â€“ 2(3 + 4)2
When indices appear in a calculation, these are worked out after brackets, but before multiplication and division.
100 â€“ 2(3 + 4)2
Brackets first,
= 100 â€“ 2 Ã— 72
then Indices,
= 100 â€“ 2 Ã— 49
thenDivision and Multiplication,
= 100 â€“ 98
and thenAddition and Subtraction
= 2
B
Remember BIDMAS:
RACKETS
I
NDICES (OR POWERS)
D
IVISION
M
ULTIPLICATION
A
DDITION
S
UBTRACTION
82
64
=
â€“ 8 Ã— 0.5
â€“ 8 Ã— 0.5
=
ïƒ–16
4
(3.4 + 4.6)2
What is
â€“ 8 Ã— 0.5 ?
ïƒ–(6 + 5 Ã— 2)
Brackets first,
then Indices,
thenDivision and Multiplication,
= 16 â€“ 4
and thenAddition and Subtraction
= 12
ïƒ–(52 + 72)
7  5
ïƒ–
(
5
5
7
7
(
)
x2
+
x2
)
Ã·

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
â€“ 2
276
344
346
+ 5
â€“ 0.3
63.8
68.5
68.8
What is 276 + 68?
276
+ 70
â€“ 2
= 344
What is 63.8 + 4.7?
63.8
+ 5
â€“ 0.3
= 68.5
â€“ 7
â€“ 30
â€“ 400
127
134
164
564
â€“ 0.4
â€“ 4
â€“ 2
16.1
16.5
20.5
22.5
What is 564 â€“ 437?
564
â€“ 400
â€“ 30
â€“ 7
= 127
What is 22.5 â€“ 6.4?
22.5
â€“ 2
â€“ 4
â€“ 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 â€“ 437?
100
+ 3
+ 20
+ 4
= 127
What is 22.5 â€“ 6.4?
10
+ 6
+ 0.1
= 16.1
+ 63
â€“ 500
64
127
564
+ 0.1
â€“ 6.5
16
16.1
22.5
What is 564 â€“ 437?
564
â€“ 500
+ 63
= 127
What is 22.5 â€“ 6.4?
22.5
â€“ 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 Ã— 43?
We can work out 7 Ã— 43 mentally using partitioning.
43 = 40 + 3
So,
7 Ã— 43 = (7 Ã— 40) + (7 Ã— 3)
= 280 + 21
= 301
What is 3.2 Ã— 40?
We can work out 3.2 Ã— 40 by partitioning 3.2
3.2 = 3 + 0.2
So,
3.2 Ã— 40 = (3 Ã— 40) + (0.2 Ã— 40)
= 120 + 8
= 128
What is 0.6 Ã— 29?
We can work out 0.6 Ã— 29 using the distributive law.
29 = 30 â€“ 1
So,
0.6 Ã— 29 = (0.6 Ã— 30) â€“ (0.6 Ã— 1)
= 18 â€“ 0.6
= 17.4
What is 26 Ã— 12?
We can work out 26 Ã— 12 by dividing 12 into factors.
12 = 4 Ã— 3 = 2 Ã— 2 Ã— 3
So we can multiply 26 by 2, by 2 again and then by 3:
52 Ã— 2 Ã— 3
26 Ã— 2 Ã— 2 Ã— 3 =
= 104 Ã— 3
= 312
What is 0.7 Ã— 18?
We can work out 0.7 Ã— 18 by dividing 18 into factors.
18 = 9 Ã— 2
So we can multiply 0.7 by 9 and then by 2:
0.7 Ã— 18 =
= 0.7 Ã— 9 Ã— 2
= 6.3 Ã— 2
= 12.6
What is 7.5 Ã— 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 Ã— 8 =
15 Ã— 4
= 30 Ã— 2
= 60
What is 68 Ã· 20?
We can work out 68 Ã· 20 by dividing 20 into factors.
20 = 2 Ã— 10
So we can divide 68 by 2 and then by 10:
68 Ã· 20 =
68 Ã· 2 Ã· 10
= 34 Ã· 10
= 3.4
What is 12.4 Ã· 8?
We can work out 12.4 Ã· 8 by dividing 8 into factors.
8 = 2 Ã— 2 Ã— 2
So we can divide 12.4 by 2, by 2 again and then by 2 a third time:
12.4 Ã· 8 =
12.4 Ã· 2 Ã· 2 Ã· 2
31 Ã· 2 = 15.5 so
3.1 Ã· 2 =1.55
= 6.2 Ã· 2 Ã· 2
= 3.1 Ã· 2
= 1.55
What is 486 Ã· 6?
We can work out 486 Ã· 6 by partitioning 486.
486 = 480 + 6
So,
486 Ã· 6 = (480 Ã· 6) + (6 Ã· 6)
= 80 + 1
= 81
420
420
40
40
21
2
What is 420 Ã· 40?
We can simplify 420 Ã· 40 by writing the division as a fraction and then cancelling.
420 Ã· 40 =
21
=
2
= 101/2
= 10.5
Ã— 10
2.6
26
26
13
0.8
8
8
4
Ã— 10
What is 2.6 Ã· 0.8?
We can simplify 2.6 Ã· 0.8 by writing the division as a fraction.
2.6 Ã· 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 Ã— 600?
What is 2.3 Ã— 4000?
2.3 Ã— 4000 =
2.3 Ã— 4 Ã— 1000
7 Ã— 600 =
7 Ã— 6 Ã— 100
= 42 Ã— 100
= 9.2 Ã— 1000
= 4200
= 9200
We can use our knowledge of place value to divide by multiples of 10, 100 and 1000.
What is 24 Ã· 80?
What is 4.5 Ã· 500?
24 Ã· 80 =
24 Ã· 8 Ã· 10
4.5 Ã· 500 =
4.5 Ã· 5 Ã· 100
= 3 Ã· 10
= 0.9 Ã· 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 Ã— 0.8?
What is 15 Ã— 0.03?
4 Ã— 0.8 =
4 Ã— 8 Ã— 0.1
15 Ã— 0.03 =
15 Ã— 3 Ã— 0.01
= 32 Ã— 0.1
= 45 Ã— 0.01
= 32 Ã· 10
= 45 Ã· 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 Ã· 0.4?
What is 3 Ã· 0.02?
36 Ã· 0.4 =
36 Ã· 4 Ã· 0.1
3 Ã· 0.02 =
3 Ã· 2 Ã· 0.01
= 9 Ã· 0.1
= 1.5 Ã· 0.01
= 9 Ã— 10
= 1.5 Ã— 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