© boardworks ltd 2005 1 of 54 n3 fractions ks4 mathematics

© Boardworks Ltd 2005 1 of 54

N3 Fractions

KS4 Mathematics


N3.1 Equivalent fractions

N3 Fractions

Contents

N3.2 Finding fractions of quantities

N3.3 Comparing and ordering fractions

N3.4 Adding and subtracting fractions

N3.5 Multiplying and dividing fractions


Look at this diagram:

3

4=

6

8

×2

×2

=18

24

×3

×3

Equivalent fractions



2

3=

6

9

×3

×3

=24

36

×4

×4




18

30=

6

10

÷3

÷3

=3

5

÷2

÷2



Equivalent fractions spider diagram


Cancelling fractions to their lowest terms

A fraction is said to be expressed in its lowest terms if the numerator and the denominator have no common factors.

Which of these fractions are expressed in their lowest terms?

14

16

20

27

3

13

15

21

14

35

32

15

Fractions which are not shown in their lowest terms can be simplified by cancelling.

7

8

5

7

2

5


Mixed numbers and improper fractions

When the numerator of a fraction is larger than the denominator it is called an improper fraction.

For example,

15

4is an improper fraction.

We can write improper fractions as mixed numbers.

15

4can be shown as

15

4= 3

3

4


Improper fraction to mixed numbers

Convert to a mixed number.378

378

=88

+ + +88

88

88

+58

581 + 1 + 1 += 1 +

= 4 5

8437 ÷ 8 = 4 remainder 5 37

8= 4

5

84This is the number of times 8 divides into 37.

4

This number is the remainder.

5


227733

Mixed numbers to improper fractions

Convert to a mixed number.273

273 =

271 + 1 + 1 +

=77

+ + +77

77

27

=237

To do this in one step,

=

Multiply these numbers together …

… and add this number …

… to get the numerator.237


Find the missing number



Contents

N3 Fractions






Finding a fraction of an amount

We can see this in a diagram:

23

of £18 = £18 ÷ 3 × 2 = £12

23

of £18?What is


Let’s look at this in a diagram again:

710

of £20 = £20 ÷ 10 × 7 = £14

710

of £20?What is



56

of £24 =16

of £24 × 5

= £24 ÷ 6 × 5

= £4 × 5

= £20

56

of £24?What is



What is of 9 kg?4

7

To find of an amount we can multiply by 4 and divide by 7.4

7

We could also divide by 7 and then multiply by 4.

4 × 9 kg = 36 kg

36 kg ÷ 7 = =36

7kg 5

1

7kg



When we work out a fraction of an amount we

multiply by the numerator

and

divide by the denominator

For example,

23

of 18 litres = 18 litres ÷ 3 × 2

= 6 litres × 2

= 12 litres



To find of an amount we need to add 1 times the amount to two fifths of the amount.

251

1 × 3.5 m =3.5 m and2

5of 3.5 m = 1.4 m

so, of 3.5 m =2

51 3.5 m + 1.4 m =4.9 m

What is of 3.5m?2

51


We could also multiply by 7

5


MathsBlox



Contents

N3 Fractions






Using decimals to compare fractions

Which is bigger or ?38

720

We can compare two fractions by converting them to decimals.

38

= 3 ÷ 8 = 0.375

= 7 ÷ 20 = 0.35720

0.375 > 0.35

so 38

>720


Which is bigger or ?38

512

Another way to compare two fractions is to convert them to equivalent fractions.

First we need to find the lowest common multiple of 8 and 12.

The lowest common multiple of 8 and 12 is 24.

Now, write and as equivalent fractions over 24. 38

512

38

=24

×3

×3

9and

512

=24

×2

×2

10so,

38

512

<

Using equivalent fractions


Ordering fractions


Mid-points



Contents

N3 Fractions






When fractions have the same denominator it is quite easy to add them together and to subtract them.

For example,

3

5+

1

5=

3 + 1

5=

4

5

We can show this calculation in a diagram:

+ =

Adding and subtracting fractions


7

8–

3

8=

7 – 3

8=

4

8

Fractions should always be cancelled down to their lowest terms.

1

2 =1

2

We can show this calculation in a diagram:

– =



1

9+

7

9+

4

9=

1 + 7 + 4

9=

12

9

Top-heavy or improper fractions should be written as mixed numbers.

= 1 3

9

1

3 = 1 1

3

Again, we can show this calculation in a diagram:

+ + =



Fractions with common denominators

Fractions are said to have a common denominator if they have the same denominator.

For example,

1112

,412

and512

all have a common denominator of 12.

We can add them together:

1112

+412

+512

=11 + 4 + 5

12=

2012

= 1 812

= 1 23


Fractions with different denominators

Fractions with different denominators are more difficult to add and subtract.

For example,

We can show this calculation using diagrams:

What is 56

–29

?

–

1518

–418

=

=15 – 4

18=

1118


+

1220

+1520

=

=12 + 15

20=

2720

= 1 720

What is 35

+34

?

Using diagrams


–

2520

–1420

=

=25 – 14

20=

1120

What is –710

?1 14

Using diagrams


Using a common denominator

1) Write any mixed numbers as improper fractions.

1 34

= 74

2) Find the lowest common multiple of 4, 9 and 12.

The multiples of 12 are: 12, 24, 36 . . .

36 is the lowest common denominator.

What is +19

?134

+512


3) Write each fraction over the lowest common denominator.

74

=36

×9

×9

63 19

=36

×4

×4

4 512

=36

×3

×3

15

4) Add the fractions together.

3663

+364

+3615

=36

63 + 4 + 15=

3682

= 2 3610

= 2 185

What is +19

?134

+512

Using a common denominator


Using a calculator

It is also possible to add and subtract fractions using the

key on a calculator.abc

For example, to enter 84

we can key in abc4 8

The calculator displays this as:

Pressing the = key converts this to:


To calculate: 23

+45

using a calculator, we key in:

abc2 3 + abc4 5 =

The calculator will display the answer as:

We write this as 1 157

Using a calculator


Fraction cards



Contents

N3 Fractions






When we multiply a fraction by an integer we:


and


For example,

49

54 × = 54 ÷ 9 × 4

= 6 × 4

= 24

Multiplying fractions by integers

This is equivalent to of 54.4

9


57

12 × ?What is

57

12 × = 12 × 5 ÷ 7

= 60 ÷ 7

=607

= 8 47

Multiplying fractions by integers


Using cancellation to simplify calculations

712

What is 16 × ?

We can write 16 × as:712

161

×712

4

3=

283

=139


825

What is × 40?

We can write × 40 as:825

825

×401

8

5=

645

=4512

Using cancellation to simplify calculations


Multiplying a fraction by a fraction

To multiply two fractions together, multiply the numerators together and multiply the denominators together:

38

What is × ?25

38

45

× =1240

3

10

=310

We could also cancel at

this step.


56

What is × ?12255

Start by writing the calculation with any mixed numbers as improper fractions.

To make the calculation easier, cancel any numerators with any denominators.

1225

356

× =

7

5

2

1

145

= 2 45

Multiplying a fraction by a fraction


Multiplying fractions


Dividing an integer by a fraction

13

What is 4 ÷ ?

13

4 ÷ means, “How many thirds are there in 4?”

Here are 4 rectangles:

Let’s divide them into thirds.

4 ÷ = 1213


25

What is 4 ÷ ?

25

4 ÷ means, “How many two fifths are there in 4?”

Here are 4 rectangles:

Let’s divide them into fifths, and count the number of two fifths.

4 ÷ = 1025



34

6 ÷ means, ‘How many three quarters are there in six?’

6 ÷ = 6 × 414

= 24

So,

6 ÷ = 24 ÷ 334

= 8

We can check this by multiplying.

8 × = 8 ÷ 4 × 334

= 6

34

What is 6 ÷ ?


There are 4 quarters in

each whole.


Dividing a fraction by a fraction

18

What is ÷ ?12

means, ‘How many eighths are there in one half?’18

÷12

Here is of a rectangle:12

Now, let’s divide the shape into eighths.

= 418

÷12


45

What is ÷ ?23

To divide by a fraction we multiply by the denominator and divide by the numerator.

45

23

÷ can be written as54

23

×

Swap the numerator and the denominator and multiply.

54

23

× =1012

=56


This is the reciprocal of .45


67

What is ÷ ?353

Start by writing as an improper fraction. 353

185

353 =

185

÷67

=185

×76

3

1

=215

=154



Dividing fractions


Multiplying and dividing are inverse operations.


and


When we multiply by a fraction we:

When we divide by a fraction we:

divide by the numerator

and

multiply by the denominator

Multiplying and dividing by fractions

© boardworks ltd 2005 1 of 54 n3 fractions ks4 mathematics

Documents