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FRACTIONS
A fraction of a whole
When you talk about a fraction, you are talking about a part of a whole, namely a fraction of
a whole.
1
3
1
4
In the above two figures, the sections shaded-in represent a fraction of the whole. In the
first one the shaded area is a quarter of the whole and in the second one it is a third.
Numerators other than one
Fractions can be made up of more than one piece of the whole.
2
3
The above fraction is two-thirds, where two pieces of the same size have been taken
together.
When you place the third piece in position you complete the circle and have:
3
3
Three thirds or in other words, one whole circle.
1
Updated March 2013
 The number on the bottom tells you the size (name) of the fraction you have, third,
quarter etc. and is called the denominator.
 The number on the top tells you how many of that fraction you have and is called the
numerator.
NUMERATOR
DENOMINATOR
3
- the numerator is 3 and the denominator is 5, telling you that you have three,
5
one fifth fractions.
e.g.
Equivalent fractions
How many different ways can you express the fraction
1
?
2
1
2
3
4
6




etc.
2
4
6
8 12
To find equivalent fractions you multiply the numerator and the denominator by the same
number.

1
1 2
2


3
3 2
6

1
1 4
4


5
5 4
20
 You can also do this with fractions that have numerators other than 1.

2
2 2
4


3
3 2
6

3
3 5
15


4
45
20
2
Updated March 2013
 Sometimes you need to work the other way round, you have a fraction that needs to be
simplified

Given
12
you need to express it in its simplest form.
24
You need to find a number that will divide into both the numerator and the denominator.
In this case 12 will divide into both.
12  12
1

24  12
2

15
15  5
3


25
25  5
5

12
12  6

18
18  6

2
3
Relative size of fractions
It is important to be able to compare fractions and decide which is the larger and which the
smaller.
 Compare
1
1
and
3
4
1
1
and the darker area is . As you can see, the darker area is
3
4
smaller than the shaded area.
The total shaded area is
3
Updated March 2013
 You represent greater than by using the symbol >
and less than by using the symbol <
So we can write:
1 1

4 3
 The rule is: The larger the number on the bottom, the smaller the actual
size of the piece.
1 1

2 8
e.g.
1 1

15 5
Mixed numbers and improper fractions
Both Mixed Numbers and Improper Fractions are greater than one. They are just different
ways of writing a fractional expression that is greater than one.
A Mixed Number consists of a number of whole parts together with a fractional part.
e.g.
2
3
4
1
4
5
13
1
2
An Improper Fraction is a fraction where the numerator is greater than the denominator
e.g.
11
4
9
5
27
2
It is important to be able to convert from one to the other.
Mixed and whole numbers to improper fractions
 You have 5 wholes and you want to find how many halves there are
There are 2 halves in a whole, so there are 2  5 halves = 10 halves
10
Which can be written as
2
 How many thirds are there in 4 wholes
i.e. 4 
?
12

3
3
4
Updated March 2013
 How many thirds are there in 3
1
?
3
There are 3 thirds in a whole so you have:
(3  3) + 1 thirds = 9 + 1 = 10 thirds
Which can be written as
10
3
 How many quarters are there in 4
(4  4) + 3 = 19
i.e.
3
4
19
4
 The rule is: Multiply the whole number by the denominator, add the numerator and
write above denominator

5
3
(5  5)  3
25  3
28



5
5
5
5

2
4
(2  7)  4
14  4
18



7
7
7
7
Improper fractions to mixed numbers
You need to be able to work the other way. Given an improper fraction, you need to be able
to write it as a mixed number.
 Write
5
as a mixed number.
3
The denominator tells you that the type of fraction is a third.
The numerator tells you that you have 5 thirds.
If you divide 5 by 3, you get 1 with 2 thirds left over.
So you have 1 whole plus 2 thirds: 1
2
3
5
Updated March 2013
 Write
19
as a mixed number
4
Divide 19 by 4 to find how many wholes you have:
19  4 = 4 and 3 left over 
 Write
4
3
4
2
4
5
14
as a mixed number
5
14  5 = 2 and 4 left over 
Fraction multiplication
 What is
1
1
?
of
2
4
The above diagram shows the quarter shaded in. If you halve the shaded part each
1
of the now two shaded parts will be of the whole.
8
 Find
1
1
of
3
2
The shaded part is
1
2
This has been divided into 3 parts.
Each of the shaded parts is
Each shaded part is
1
1
of
3
2
1
of the whole.
6
6
Updated March 2013

What is
2
3
of ?
3
4
All the shaded area is
3
of the whole.
4
The darker shaded area is
2
of the total shaded
3
area.
The darker shaded area is
So:
1
of the whole
2
2
3
1
of

3
4
2
 Look at the answers:
1
1
1
of

2
4
8
1
1
1
of

3
2
6
2
3
1
of

3
4
2
To find the answer you multiply the numerators together and multiply the denominators
together.
1 1
1

2 4
8
1 1
1

3 2
6
23
6
1


3  4 12
2
 Since the solution can be arrived at by multiplying the numerators together and the
denominators together, the of can be replaced by the x sign.
This corresponds with whole numbers situations where 2 lots of 3 is the same as 2 x 3.
e.g.
1
1
of
2
4
=
1 1

2 4
=
1 1
2 4
=
1
8
 There are 2 other situations where the word of occurs with fractions:
3 lots of
1
4
and
1
of 4
2
7
Updated March 2013
 The same rule can be applied here, remember 3 lots of
Using the above rule, replace lots of with x :
3
1
3
is
4
4
1
4
The rule calls for numerators and denominators in both numbers
What is the denominator for 3?
Think of 3 wholes as 3 whole ones then it is logical to use
3
1
Remember the denominator indicates the number of pieces that the whole is cut into, in
this case it is only 1.
 3 lots of
1
becomes
4
 4 lots of
3
=
5
 Now look at
3 1

1 4
=
3 1
=
1 4
43
12
2

or 2
1 5
5
5
1
of 4
2
The of can be replaced with x again
It becomes
1
of 4 =
2
 What is
1
of $10?
5
1 4
=
2 1
4
2
= 2
8
Updated March 2013
3
4
Or
If I had to share my $10 amongst 5 people, how much would each get?
1
1
1 0 2
10 

 2
5
5 1
1

i.e.
1
of $10 is $2
5
3
of a class of 20 passed the exam, how many passed?
4
3
3
2 0 5
 20 

 15
4
4 1
1
i.e. 15 passed the exam
Solving fraction problems using pictures
Diagrams can be useful when solving problems involving fractions.
 Three-fifths of a sum of money is $60. What is the sum of money?
 Draw rectangle A to represent the whole amount
TOTAL?
A
 Divide it into 5 equal parts, each part representing one-fifth
B
 Shade in three-fifths and label it $60
$60
 The $60 can then be divided into 3 lots of $20 - so $20 is now seen to be onefifth of the total.
$60
$20
Updated March 2013
$20
$20
9
$20
$20
 The rest of the sections can be filled in with the value $20, demonstrating
that the total is $100

3
of a sum of money is $4.50. What is the sum of money?
4
$4.50
$1.50
$1.50
$1.50
$1.50
Each quarter is $1.50 ($4.50  3)
There are 4 quarters, so the whole sum of money is: $1.50  4 = $6
 So far there have only been two fractions multiplied together, but you can multiply any
number of fractions together.

3 7 5
 
10 9 14
You could do as before and multiply the numerators together and the denominators
together
3 7  5
105

10  9 14
1260
Which you now have to simplify by finding numbers that will divide into both the
top and the bottom.
It is better to simplify before multiplying out. Look for a number on the top and a
number on the bottom that have a common factor ( i.e. that will divide by the same
number).
3 and 9 will both divide by 3
3  7  5
10  9 3  14
5 and 10 will both divide by 5
3 1  7  5 1
1 0 2  9 3  14
1
10
Updated March 2013
7 and 14 will both divide by 7
Leaving
3 1  7 1  5 1
1 0 2  9 3  1 4 2
1 1 1
1

2  3 2
12
This method is called cancelling.
The important part is to cancel or simplify before multiplying out.

10 5 22
 
11 8 25
10 5 2 2 2
 
1 11 8 25
11 and 22 both divide by 11
5 and 25 both divide by 5
10 and 8 both divide by 2
10 5 1 2 2 2
 
1 11 8 2 5 5
1 0 5 5 1 2 2 2
 
1 11 8 4 2 5 5
5  1 2
There are still numbers that will cancel out
1 4  5
5 and 5 both divide by 5
5 1 1 2
1 4  5 1
5 1  1  2
2 and 4 both divide by 2
1  4  5 1
Finally leaving
1 1 1
1

1 2  1
2
11
Updated March 2013