Topic 4: Multiplying and dividing fractions
To help understand the concept of multiplying fraction, let’s consider a question such as a half of a
third. A diagram that represents this is:
1
3
1
3
1
3
1
2
The double shaded region of the diagram
represents a half of a third which is one sixth.
1
2
The diagram above can be represented in number form as
1 1 1
× =.
2 3 6
Using this example, an observation is possible.
1 1 1
× =
2 3 6
Multiply numerators
→ Numerator of answer
Multiply denominators → Denominator of answer
This observation works with all possible questions, so the rule for multiplying fractions is:
Multiply two fractions by multiplying the numerators and multiplying the denominators.
Examples:
3 1
×
4 5
3 ×1
=
4×5
=
3
20
Multiply numerators, multiply
denominators.
1 8 2
× ×
4 9 7
1× 8 × 2
=
4×9×7
16 ÷4
252÷4
4
=
63
=
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Multiply numerators,
multiply
denominators.
This answer can be
simplified
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In the second question (above), the answer had to be simplified, which meant dividing the
numerator and the denominator by a common factor. There is shortcut that allows this division to
be performed before multiplying the fractions.
For example:
No short cut
3 8
3 8
×
4 11
3 82
=
×
4 1 11
6
=
11
×
4 11
24÷4
= ÷4
44
6
=
11
Simplify
Shortcut method
4 goes into 8, 2 times
4 goes into 4, 1 time
For this method to work, one numerator and one denominator is divided by a common factor.
More examples are below.
4 9
×
3 16
41 9 3
=
×
3 1 16 4
3
=
4
1 4 10
× ×
8 5 11
4 goes into 4, 1 time
4 goes into 16, 4 times
3 goes into 9, 3 times
3 goes into 3, 1 time
=
1
82
1
=
11
1
4 1 10
× 1×
11
5
21
4 goes into 4, 1 time
4 goes into 8, 2 times
5 goes into 10, 2 times
5 goes into 5, 1 time
Looking at the 2 in the numerator and
the 2 in the denominator, divide these
by 2
2 goes into 2, 1 time
2 goes into 2, 1 time
If the question contains mixed numbers, these must be made improper before multiplying using
either method. Some examples containing mixed numbers are below:
2 6
2 ×
3 7
Change mixed numbers to
improper fractions
8 62
×
31 7
16
=
7
2
=2
7
3 goes into 6, 2 times
3 goes into 3, 1 time
=
3
3 Change mixed numbers to improper fractions
1 ×2
4
8
Answer is improper, change to a
mixed number.
Using a calculator:
2A2
3
7 19 There is no common factor.
×
4 8
Answer is improper, change to a mixed
133 number.
=
32
5
=4
32
=
Using a calculator:
O6a7=
16
7
To change to a mixed number, press N gives 2
1A3
2
7
4
O2A3
To change to a mixed number, press N gives 4
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8=
133
32
5
32
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If the question contains an integer, remember any integer can be written over 1.
Write 1400 with a denominator
of 1.
2
× 1400
7
2 1400
×
71
1
=
200
7 goes into 1400, 200 times
7 goes into 7, 1 time
400
= 400
1
Using a calculator:
2a7
O1400=400
Video ‘Multiplying Fractions’
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Dividing Fractions
To introduce this section, consider this scenario: Dividing a pizza into 2 is the same as taking half
of a pizza?


The operation ÷2  or ÷
1
2
is the same as ×

2
1
When the operation changed from ÷ to × , the number changed from
2
1
to .
1
2
When a fraction ‘flips’ like this, it is called taking the reciprocal of a number.
The reciprocal of
2
3
is
.
3
2
The reciprocal of 2
2
5
12 
5

 which as an improper fraction is 5  is 12 .


8 1

The reciprocal of 8  which as a fraction is  is .
1 8

To solve division of fraction questions, the idea above can be used. The division can be changed
into a multiplication, multiplying by the reciprocal.
For example:
3 5
÷
4 7
=
=
3 7
×
4 5
12 6
÷
5 7
2
12
7
× 1
=
5
6
14
4
= = 2
5
5
−
5 3
÷
8 4
−
5 41
×
82 3
=
21
1
= 1
20
20
2 6
2 ÷
5 7
=
Change the operation from
÷ to × , multiply by the
reciprocal.
−
=
Change the mixed number to
an improper fraction
Change the operation from
÷ to × , multiply by the
reciprocal.
6 goes into 12, 2 times
6 goes into 6, 1 time
Change the improper fraction
back to a mixed number
−
5
6
5 ÷1
Change the operation from
÷ to × , multiply by the
reciprocal.
4 goes into 4, 1 time
4 goes into 8, 2 times
7
8
=
5 15
÷
1 8
=
51
8
×
1 15 3
8
2
=2
3
3
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× + =−
Change the mixed numbers to
an improper fractions
Change the operation from
÷ to × , multiply by the
reciprocal.
5 goes into 5, 1 time
5 goes into 15, 3 times
Change the improper fraction
back to a mixed number
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4 −3
1 ÷
7 5
Change the mixed number
to an improper fraction
11 − 3
÷
7
5
Change the operation from
÷ to × , multiply by the
reciprocal.
11 − 5
×
7
3
−
55
=
21
=
−
2
13
21
The three fractions below are
equal. Keep the negative sign
either with the numerator or in
front of the entire fraction.
−
3
3
3
or − not −
5
5
5
Never leave the negative sign
with the denominator
Change the improper
fraction back to a mixed
number
Video ‘Dividing Fractions’
Some of these, done on a calculator, are below.
3 5
÷
4 7
−
5 3
÷
8 4
5 ÷1
7
8
3a4
z5a8
P5a7= 21
20
P3a4= − 5
5P1A7
6
8= 8
3
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Mixed operation fraction operations
When doing mixed operation questions, the order of operations rules must be followed.
3 6 4
÷ ×
8 11 5
3 1 11 4 1
×
×
82 62 5
11
=
20
=
Change the operation from
÷ to × , multiply by the
reciprocal.
Use shortcut method
7 2 2
1 + ×1
8 5 3
Change the mixed numbers to an
improper fractions
15 2 5 1
+
×
8 51 3
Do multiplication first
15×3 2×8
+
8×3 3×8
45 16
=
+
24 24
61
13
= = 2
24
24
=
Add using a common denominator
of 24
Change the improper fraction
back to a mixed number
Video ‘Mixed Operations with Fractions’
Some of these examples, done on a calculator, are below.
3 6 4
÷ ×
8 11 5
3a8
P6a11
7 2 2
1 + ×1
8 5 3
1A7
8
+2a5
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O4a5=
O1A2
11
20
3=
61
24
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Activity
1.
(a)
(d)
2.
Perform the operation indicated in the question, check with you calculator.
4 3
×
5 8
2
× 800
5
3 7
×
4 12
2 3
4
× ×1
9 10 11
(b)
(e)
(d)
3.
(f)
3
9
1 × −2
7
20
1 2 3
2 ×1 ×
3 7 4
Perform the operation indicated in the question, check with you calculator.
−
(a)
(c)
1 −1
÷
2 3
3
÷4
4
3 9
÷
4 11
2
8÷
9
(b)
(e)
A recent survey showed that
2
5
(c)
(f)
4 5
÷
7 9
3 1 3
× ÷
4 7 8
3
of students have hearing problems. In a school of 450
students, how many would you expect to have hearing problems?
1
are
8
7
?
9
4.
How many
5.
Joanne has a 1 km circuit that she jogs around. How far will she cover is she jogs 5
there in
1
3
circuits?
6.
1
4
Jim has a 5 Ha block of land, a large dam covers
2
of the property. What area does the
11
dam cover?
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