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KS4 Mathematics
N3 Fractions
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Contents
N3 Fractions
N3.1 Equivalent fractions
N3.2 Finding fractions of quantities
N3.3 Comparing and ordering fractions
N3.4 Adding and subtracting fractions
N3.5 Multiplying and dividing fractions
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Equivalent fractions
Look at this diagram:
×2
3
4
=
×2
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×3
6
8
=
18
24
×3
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Equivalent fractions
Look at this diagram:
×3
2
3
=
×3
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×4
6
9
=
24
36
×4
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Equivalent fractions
Look at this diagram:
÷3
18
30
=
÷3
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÷2
6
10
=
3
5
÷2
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Equivalent fractions spider diagram
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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
7
8
20
27
3
13
15
21
5
7
14
35
2
5
32
15
Fractions which are not shown in their lowest terms can be
simplified by cancelling.
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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
is an improper fraction.
4
We can write improper fractions as mixed numbers.
15
4
can be shown as
15
=
4
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3
3
4
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Improper fraction to mixed numbers
37
Convert
to a mixed number.
8
37
8
8
8
8
+
+
+
=
8
8
8
8
8
+
1+1+1+1+
5
= 4
8
=
37 ÷ 8 = 4 remainder 5
5
8
5
8
37
=
8
This number is the remainder.
4
5
8
This is the number of times 8 divides into 37.
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Mixed numbers to improper fractions
2
to a mixed number.
7
3
2
37 =1 + 1 + 1 +
Convert
2
7
7
7
7
2
=
+
+
+
7
7
7
7
23
=
7
To do this in one step,
… and add this number …
3
2
23
=
7
7
… to get the numerator.
Multiply these numbers together …
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Find the missing number
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Contents
N3 Fractions
N3.1 Equivalent fractions
N3.2 Finding fractions of quantities
N3.3 Comparing and ordering fractions
N3.4 Adding and subtracting fractions
N3.5 Multiplying and dividing fractions
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Finding a fraction of an amount
2
What is
of £18?
3
We can see this in a diagram:
2
of £18 = £18 ÷ 3 × 2 = £12
3
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Finding a fraction of an amount
7
What is
of £20?
10
Let’s look at this in a diagram again:
7
of £20 = £20 ÷ 10 × 7 = £14
10
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Finding a fraction of an amount
5
What is
of £24?
6
5
1
of £24 =
of £24 × 5
6
6
= £24 ÷ 6 × 5
= £4 × 5
= £20
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Finding a fraction of an amount
4
What is
of 9 kg?
7
4
To find
of an amount we can multiply by 4 and divide by 7.
7
We could also divide by 7 and then multiply by 4.
4 × 9 kg = 36 kg
36 kg ÷ 7 =
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36
7
kg =
5
1
kg
7
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Finding a fraction of an amount
When we work out a fraction of an amount we
multiply by the numerator
and
divide by the denominator
For example,
2
of 18 litres = 18 litres ÷ 3 × 2
3
= 6 litres × 2
= 12 litres
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Finding a fraction of an amount
What is
2
of 3.5m?
5
1
To find 1 25 of an amount we need to add 1 times the
amount to two fifths of the amount.
1 × 3.5 m = 3.5 m
so,
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1
and
2
of 3.5 m = 1.4 m
5
2
of 3.5 m = 3.5 m + 1.4 m = 4.9 m
5
7
We could also multiply by
5
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MathsBlox
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Contents
N3 Fractions
N3.1 Equivalent fractions
N3.2 Finding fractions of quantities
N3.3 Comparing and ordering fractions
N3.4 Adding and subtracting fractions
N3.5 Multiplying and dividing fractions
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Using decimals to compare fractions
3
7
Which is bigger
or
?
8
20
We can compare two fractions by converting them to
decimals.
3
8
= 3 ÷ 8 = 0.375
7 = 7 ÷ 20 = 0.35
20
0.375 > 0.35
so
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3
8
>
7
20
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Using equivalent fractions
3
5
Which is bigger
or
?
8
12
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.
3
5
Now, write
and
as equivalent fractions over 24.
8
12
×3
3
8
9
=
24
×3
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×2
and
5
10
=
12
24
so,
3
8
<
5
12
×2
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Ordering fractions
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Mid-points
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Contents
N3 Fractions
N3.1 Equivalent fractions
N3.2 Finding fractions of quantities
N3.3 Comparing and ordering fractions
N3.4 Adding and subtracting fractions
N3.5 Multiplying and dividing fractions
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Adding and subtracting 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:
+
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=
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Adding and subtracting fractions
7
8
–
3
8
=
7–3
8
=
4
1
8
2
=
1
2
Fractions should always be cancelled down to their lowest
terms.
We can show this calculation in a diagram:
–
27 of 54
=
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Adding and subtracting fractions
1
7
4
1+7+4
12
+
+
=
=
=
9
9
9
9
9
1
3
9
1
3
=
1
1
3
Top-heavy or improper fractions should be written as mixed
numbers.
Again, we can show this calculation in a diagram:
+
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+
=
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Fractions with common denominators
Fractions are said to have a common denominator if
they have the same denominator.
For example,
11
4
5
,
and
12 12
12
all have a common denominator of 12.
We can add them together:
11
4
5
11 + 4 + 5
20
=
+
+
=
=
12
12
12
12
12
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1
8
=
12
1
2
3
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Fractions with different denominators
Fractions with different denominators are more difficult to add
and subtract.
For example,
What is
5
2
–
?
6
9
We can show this calculation using diagrams:
–
15
18
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–
=
4
18
15 – 4
11
=
=
18
18
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Using diagrams
What is
+
12
20
31 of 54
+
3
3
+
?
5
4
=
15
20
12 + 15
=
=
20
27
20
=
1
7
20
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Using diagrams
What is
1
1
7
–
?
4
10
–
25
20
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–
=
14
20
25 – 14 11
=
=
20
20
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Using a common denominator
What is
1
3
5
1
+
+
?
4
12
9
1) Write any mixed numbers as improper fractions.
1
3
4
=
7
4
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.
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Using a common denominator
What is
1
3
5
1
+
+
?
4
12
9
3) Write each fraction over the lowest common denominator.
×9
7
4
= 63
36
×9
×4
1
9
= 4
36
×4
×3
5 = 15
12
36
×3
4) Add the fractions together.
63
4
15
63 + 4 + 15
82
+
+
=
=
=
36
36
36
36
36
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2
10
=
36
2
5
18
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Adding and subtracting fractions
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Using a calculator
It is also possible to add and subtract fractions using the
abc key on a calculator.
4
For example, to enter 8 we can key in
4
a bc
8
The calculator displays this as:
Pressing the
36 of 54
=
key converts this to:
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Using a calculator
To calculate:
2
4
+
3
5
using a calculator, we key in:
2
a bc
3
+
4
a bc
5
=
The calculator will display the answer as:
We write this as
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1
7
15
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Fraction cards
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Contents
N3 Fractions
N3.1 Equivalent fractions
N3.2 Finding fractions of quantities
N3.3 Comparing and ordering fractions
N3.4 Adding and subtracting fractions
N3.5 Multiplying and dividing fractions
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Multiplying fractions by integers
When we multiply a fraction by an integer we:
multiply by the numerator
and
divide by the denominator
For example,
4
54 ×
= 54 ÷ 9 × 4
9
=6×4
This is
equivalent to
4
of 54.
9
= 24
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Multiplying fractions by integers
5
What is 12 ×
?
7
5
12 ×
= 12 × 5 ÷ 7
7
= 60 ÷ 7
60
=
7
=
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8
4
7
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Using cancellation to simplify calculations
7
What is 16 ×
?
12
We can write 16 × 7 as:
12
4
16
7
28
×
=
1
12 3
3
=
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9
1
3
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Using cancellation to simplify calculations
8
What is
× 40?
25
8
We can write
× 40 as:
25
8
8
40
64
×
=
25 5
1
5
=
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12
4
5
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Multiplying a fraction by a fraction
3
2
What is
×
?
8
5
To multiply two fractions together, multiply the numerators
together and multiply the denominators together:
3
3
4
12
=
×
8
5
40 10
We could also
cancel at this
step.
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3
=
10
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Multiplying a fraction by a fraction
What is
5
5 12
×
?
6 25
Start by writing the calculation with any mixed numbers as
improper fractions.
To make the calculation easier, cancel any numerators with
any denominators.
7
2
35
14
12
×
=
6 1 25 5
5
=
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2
4
5
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Multiplying fractions
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Dividing an integer by a fraction
1
What is 4 ÷ ?
3
1
4÷
means, “How many thirds are there in 4?”
3
Here are 4 rectangles:
Let’s divide them into thirds.
1
4÷
= 12
3
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Dividing an integer by a fraction
2
What is 4 ÷ ?
5
2
4÷
means, “How many two fifths are there in 4?”
5
Here are 4 rectangles:
Let’s divide them into fifths, and count the number of two fifths.
2
4÷
= 10
5
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Dividing an integer by a fraction
3
What is 6 ÷ ?
4
3
6÷
means, ‘How many three quarters are there in six?’
4
There are 4
1
6÷
= 6 × 4 = 24
quarters in
4
each whole.
So,
3
6÷
= 24 ÷ 3 = 8
4
We can check this by multiplying.
3
8×
=8÷4×3=6
4
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Dividing a fraction by a fraction
1
1
What is
÷
?
8
2
1
1
÷
means, ‘How many eighths are there in one half?’
8
2
1
Here is of a rectangle:
2
Now, let’s divide the shape into eighths.
1
1
÷
=4
8
2
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Dividing a fraction by a fraction
4
2
What is
÷
?
5
3
To divide by a fraction we multiply by the denominator and
divide by the numerator.
4
5
2
2
can be written as
÷
×
5
4
3
3
Swap the numerator and
the denominator and
multiply.
This is the reciprocal of
5
10
2
×
=
4
12
3
4
.
5
5
=
6
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Dividing a fraction by a fraction
What is
Start by writing
3
3
6
3
÷
?
7
5
3 as an improper fraction.
5
3
18
=
5
5
3
3
18
6
18
7
÷
×
=
5
7
5
6 1
21
=
5
1
=
5
4
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Dividing fractions
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Multiplying and dividing by fractions
Multiplying and dividing are inverse operations.
When we multiply by a fraction we:
multiply by the numerator
and
divide by the denominator
When we divide by a fraction we:
divide by the numerator
and
multiply by the denominator
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