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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. 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

Using the correct order of operations

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

Using the correct order of operations

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

Brackets

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

Nested brackets

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

Using a division line

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

Using a division line

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

Multiplying by a bracket

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).

Indices

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

BIDMAS

B

Remember BIDMAS:

RACKETS

I

NDICES (OR POWERS)

D

IVISION

M

ULTIPLICATION

A

DDITION

S

UBTRACTION

Using BIDMAS

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

Using a calculator

(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

Using partitioning to add

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

Adding by counting up

+ 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

Using compensation to add

+ 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

Using partitioning to subtract

– 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

Subtracting by counting up

+ 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

Using compensation to subtract

+ 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

Using partitioning to multiply whole numbers

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

Using partitioning to multiply decimals

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

Using the distributive law to multiply

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

Using factors to multiply whole numbers

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

Using factors to multiply decimals

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

Using doubling and halving

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

Using factors to divide whole numbers

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

Using factors to divide decimals

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

Using partitioning to divide

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

Using fractions to divide whole numbers

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

Using fractions to divide decimals

× 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

Multiplying by multiples of 10, 100 and 1000

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

Dividing by multiples of 10, 100 and 1000

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 multiples of 0.1 and 0.01

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 multiples of 0.1 and 0.01

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

Multiplying by decimals between 1 and 0

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