fractions - rose park primary...
TRANSCRIPT
5Chapter 5Fractions
Contents:
A FractionsB Fractions as divisionC Proper and improper fractions
D Fractions of quantities
E Fractions on a number lineF Equal fractions
G Comparing fractions
H Adding and subtracting fractions
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\093AUS06_05.cdr Wednesday, 30 January 2013 9:15:22 AM BRIAN
Opening problem
94 FRACTIONS (Chapter 5)
The students in Amelia’s class have all been given
a week to do a project. So far, Amelia has done3
8,
Charlie has done5
8, and Matilda has done
1
2.
Things to think about:
a Has Amelia completed more of the project than
Charlie?
b Has Amelia completed more of the project than
Matilda?
c Which of the problems a or b was easier to solve? What made the other one harder?
d The next night, Amelia completed another1
2of her project. What total fraction has she
completed now?
Every day we see quantities which can be expressed as fractions. It is therefore important that we
can understand, compare, add, and subtract fractions.
16_ Qw hands
Qt_ open
Qe full
Qr remaining
size 8_ Qw
\\Qw_ apple
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\094AUS06_05.cdr Thursday, 7 February 2013 12:30:29 PM GR8GREG
FRACTIONS (Chapter 5) 95
A fraction is a part of any quantity.
For example, a chocolate bar is divided into 5 equal parts.
If George takes 2 of the parts, we say that George has taken
2
5of the chocolate bar.
2
5is a fraction which shows that we had a whole, we divided it into
5 equal parts, and we are looking at 2 of them.
numerator
bar
denominator
2
5
The numerator shows how many parts we are looking at.
The denominator shows how many equal parts there are altogether.
The fraction wall below shows some of the different ways of dividing a whole into equal parts.
FRACTIONSA
Activity 1 Fraction walls
In words,2
5is
“two fifths”.
| {z }taken byGeorge
Aq_p
1
Qw Qw
Qe
Qt
Qo
Qi
Qu
Qy
Qr
QeQe
QrQrQr
QtQtQtQt
QyQyQyQyQy
QuQuQuQuQuQu
QiQiQiQiQiQiQi
QoQoQoQoQoQoQoQo
Aq_pAq_pAq_pAq_pAq_pAq_pAq_pAq_pAq_p
whole
halves
thirds
quarters
fifths
sixths
sevenths
eighths
ninths
tenths
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\095AUS06_05.cdr Wednesday, 13 February 2013 4:06:37 PM BRIAN
96 FRACTIONS (Chapter 5)
What to do:
Use the fraction wall to complete this table:
Number of
equal parts
One part as
a fractionFraction
in words
All parts form
the fraction
11
1one whole
1
1
a 2 one half
b1
3
c one quarter
d one fifth
e1
6
6
6
f 7
g one eighth
h1
9
9
9
i 10
EXERCISE 5A
1 Write a fraction to show:
a three quarters b two thirds c two fifths
d four fifths e three eighths f five eighths
g two sevenths h three tenths i one hundredth
2 Write these fractions in words:
a1
3b
3
4c
3
5d
7
8
e4
9f
5
7g
5
12h
17
20
i11
30j
4
25k
3
100l
97
100
3 For each of the following fractions, state the numerator:
a2
3b
4
5c
3
7d
1
8
4 For each of the following fractions, state the denominator:
a2
3b
4
5c
3
7d
1
8
PRINTABLETABLE
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\096AUS06_05.cdr Wednesday, 6 February 2013 9:31:44 AM GR8GREG
FRACTIONS (Chapter 5) 97
5 What fraction of the square is shaded?
a b c d
e f g h
i j k l
6 Copy the circles and shade the fractions given:
a1
6b
1
3c
2
3d
5
6
7 Draw a diagram to represent the following fractions:
a2
4b
5
8c
1
9d
4
12
8 Is3
8of this triangle shaded? Explain your answer.
9 What fraction of the dots are red?
a b c
10
What fraction of the cats are: a black b white?
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\097AUS06_05.cdr Wednesday, 30 January 2013 11:02:42 AM GR8GREG
TASTYCHEESE 1 kg
BUTTER
98 FRACTIONS (Chapter 5)
11 a What fraction of the flowers are:
i in the vase ii lying on the table?
b What fraction of the flowers are:
i tulips ii daisies?
12
a What fraction of the children are:
i wearing hats ii not wearing hats?
b What fraction of the children are:
i boys ii girls?
13 Give the fraction shown in each diagram:
a b c
14 Copy and complete the following sketches to show:
a
a glass which is half
full of water
b
a petrol gauge
showing the tank is
three quarters full
c
a pizza with one third
missing
d
the container is3
5full
e
five ninths of the balls
are blue.
PRINTABLEDIAGRAMS
?
BUTTER 1 kgBUTTER
BUTTER 1 kg
?
MILK
1 L
MIL
K
1L
?
E
F
FUEL
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\098AUS06_05.cdr Tuesday, 12 February 2013 9:03:50 AM GR8GREG
FRACTIONS (Chapter 5) 99
When we write a fraction such as1
4bar, the bar indicates division.
So,1
4can be written as 1 ¥ 4, and 1 ¥ 4 can be written as
1
4.
We can see this by dividing a pizza into four equal portions.
Each person will get one quarter of the pizza.
1
pizza
¥ 4
people
=1
4of a pizza each
Self Tutor
Write the following divisions as fractions:
a 2 ¥ 7 b 3 ¥ 8
a 2 ¥ 7 =2
7b 3 ¥ 8 =
3
8
EXERCISE 5B
1 Write the following divisions as fractions:
a 4 ¥ 5 b 1 ¥ 7 c 3 ¥ 10
d 8 ¥ 9 e 2 ¥ 11 f 12 ¥ 13
2 Suppose 2 pizzas are shared equally
between 3 people.
a Look at what Kim gets. What
fraction of a pizza is this?
b Check that the other two people each
get the same amount as Kim.
c Copy and complete:
...... pizzas ¥ ...... people = ...... of a pizza each.
Self Tutor
Write the following fractions as divisions:
a1
6b
4
9
a1
6= 1 ¥ 6 b
4
9= 4 ¥ 9
FRACTIONS AS DIVISIONB
Example 2
Example 1
Bill Jane
Emma Tony
Janice
TeeganKim
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\099AUS06_05.cdr Tuesday, 5 February 2013 10:49:03 AM GR8GREG
100 FRACTIONS (Chapter 5)
3 Write the following fractions as divisions:
a1
3b
2
5c
7
8
d3
4e
8
13f
11
20
Self Tutor
Write the following fractions as divisions, and hence as whole numbers:
a12
4b
42
6
a12
4= 12 ¥ 4
= 3
b42
6= 42 ¥ 6
= 7
4 Write the following fractions as divisions, and hence as whole numbers:
a20
5b
27
3c
55
11d
7
7
e24
12f
19
19g
0
8h
108
9
A fraction which has numerator less than its denominator is called a proper fraction.
A fraction which has numerator greater than its denominator is called an improper fraction.
For example:2
3is a proper fraction.
4
3is an improper fraction.
4
3=
3
3+
1
3= 1 +
1
3
To see how improper fractions occur, suppose you
and a friend share three cookies. Each person
receives three halves of a cookie, which is3
2cookies.
We can also see that3
2=
2
2+
1
2= 1 +
1
2.
So each person receives one and a half cookies. We
can write this as 11
2cookies.
PROPER AND IMPROPER FRACTIONSC
Example 3
you friend
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\100AUS06_05.cdr Thursday, 31 January 2013 10:33:15 AM GR8GREG
FRACTIONS (Chapter 5) 101
When an improper fraction is written as a whole number and a proper fraction, it is called a
mixed number.
For example, 21
2is a mixed number. It means two
wholes and one half.
We can write mixed numbers as improper fractions, and vice versa.
For example, at a class picnic there were
3 apple pies, each cut into quarters.
Sam ate one quarter of a pie.
We see there are now 23
4pies remaining.
Each whole pie has 4 quarters, and we have 3 quarters of the third pie, so we have 2 lots of 4 plus
3 quarters = 11 quarters.
So, 23
4=
11
4.
Self Tutor
Write 21
3as an improper fraction.
21
3is 2 wholes and one third.
Each whole has 3 thirds, so there are
2 £ 3 + 1 = 7 thirds in total.
) 21
3=
7
3
EXERCISE 5C
1 Determine whether each of the following is a proper fraction, an improper fraction, or a mixed
number:
a3
5b
7
6c
1
9d 3
1
3
e11
8f
8
11g 4
2
5h
40
7
2 What mixed number do these diagrams show?
a b c
Example 4
1 2
3 4
5 6
7 8
9 10
11
1 2
3
4 5
6
7
TASTY CHEESE 1 kg
TAST
TASTY CHEESE 1 kg
TASTY CHEESE 1 kg
TASTY CHEESE 1 kg
50 cm 1 m
2L
1L
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\101AUS06_05.cdr Monday, 11 February 2013 3:38:51 PM GR8GREG
102 FRACTIONS (Chapter 5)
3 This diagram shows 31
2pizzas.
a How many halves are there in
31
2pizzas?
b Copy and complete: 31
2=
::::
2
4 a What mixed number is represented
by this diagram?
b How many quarters are shaded?
c Copy and complete: ...... =::::
4
5 Write as an improper fraction:
a 11
4b 2
1
2c 3
2
3d 2
5
6e 1
3
5
f 51
3g 6
1
2h 2
3
8i 4
1
6j 3
7
10
Self Tutor
Write5
3as a mixed number.
5
3is 5 thirds.
This is 1 whole, and 2 thirds left over.
So,5
3= 1
2
3
6 After the school picnic there were
17 quarter sandwiches left over.
a How many whole sandwiches can
be formed from the quarters?
b Once the whole sandwiches have
been formed, how many quarters
are left over?
c Copy and complete:17
4= ::::::
7 Write as a mixed number:
a4
3b
9
4c
11
6d
16
5e
19
4
f15
2g
14
3h
17
7i
33
10j
35
8
Example 5
1 2
3
4 5
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\102AUS06_05.cdr Thursday, 31 January 2013 10:13:54 AM GR8GREG
FRACTIONS (Chapter 5) 103
8 19 carrots are shared equally between 5 horses.
How many carrots does each horse receive?
Give your answer as a mixed number.
Chris has 18 golf balls. He gives one third of them to his
brother Josh. How many golf balls does Josh receive?
To find out, we can divide the golf balls into 3 equal groups.
We see that1
3of 18 balls is 6 balls.
We also notice that 18 ¥ 3 = 6.
Each group contains1
3of the golf balls.
So, to find1
3of a number, we divide the number by 3.
Self Tutor
Find1
4of 24.
1
4of 24 = 24 ¥ 4
= 6
EXERCISE 5D
1 Find:
a1
2of 10 b
1
2of 36 c
1
3of 12 d
1
3of 45
e1
4of 20 f
1
4of 44 g
1
5of 30 h
1
5of 120
i1
6of 30 j
1
10of 70 k
1
8of 48 l
1
12of 600
FRACTIONS OF QUANTITIESD
Example 6
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\103AUS06_05.cdr Thursday, 7 February 2013 12:32:30 PM GR8GREG
104 FRACTIONS (Chapter 5)
2 Find:
a1
3of 30 people b
1
4of 20 lollies c
1
5of 35 drinks
d1
10of 650 g e
1
2of $38 f
1
4of 60 minutes
Self Tutor
On the first day of school this year, there were 27 Year 6
students in a class.1
3of the students were aged 12 years or
older. How many students were 12 years or older?
The full class is 27 students.
So,1
3of 27 is 27 ¥ 3 = 9 students.
There were 9 students aged 12 years or older.
3 Viktor played 15 games of tennis for his school team. He
won one third of them. How many games did Viktor win?
4 Of the 250 students at a school, one fifth were absent with
chicken pox. How many students were absent?
5 One sixth of the cars from an assembly line were painted
white. If 480 cars came from the assembly line, how many
were painted white?
6 Ling had $900 in her bank account. She spent one fifth of
her money on a new badminton racquet. How much did the
racquet cost?
7 While Evan was on holidays, one eighth of the tomato plants
in his greenhouse died. If he had 96 plants alive when he
went away, how many plants:
a died b were still alive?
Self Tutor
Find2
5of 30.
1
5of 30 is 30 ¥ 5 = 6
)2
5of 30 is 6 £ 2 = 12
Example 8
Example 7To find
13
of 27,
we need to divide 27into 3 equal parts.
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\104AUS06_05.cdr Wednesday, 13 February 2013 4:08:26 PM BRIAN
FRACTIONS (Chapter 5) 105
8 Find:
a2
3of 9 b
3
4of 24 c
2
5of 45
d3
5of 35 e
4
7of 21 f
5
6of 54
g7
10of 120 h
8
9of 72 i
13
20of 400
9 55 passengers were on the bus one morning. Two fifths of the passengers were school children.
How many school children were on the bus?
10 Richard spent three quarters of his working day
installing computers, and the remainder of the time
travelling between jobs. If his working day was
8 hours, how much time did Richard spend installing
computers?
11 Sasha shot for goal 16 times during a netball match.
She scored a goal with seven eighths of her shots.
How many goals did Sasha score?
12 A business hired a truck to transport boxes of
equipment. The total weight of the equipment was
3000 kg, but the truck could only carry5
8of the
boxes in one load.
a What weight did the truck carry in the first
load?
b If there were 80 boxes of equal weight, how
many did the truck carry in the first load?
In Chapter 1, we placed natural numbers on a number
line.
We can do the same thing with fractions.
For example, to place the fraction3
4on a number line,
we divide the interval between 0 and 1 into 4 equal parts.
Each of the small intervals has length1
4.
We count 3 intervals from 0, and mark3
4with a dot.
FRACTIONS ON A NUMBER LINEE
0 5 10
3 7
0 1
Er
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\105AUS06_05.cdr Wednesday, 6 February 2013 9:36:38 AM GR8GREG
0 1 2
1EuTu
0 1 0 1
0 1 0 1 2
0 1 2 0 1 32
106 FRACTIONS (Chapter 5)
Self Tutor
Place the fractions5
7and 1
3
7on a number line.
Since these fractions both involve sevenths, we divide the number line into
intervals of length1
7.
EXERCISE 5E
1 Place the following fractions on a number line:
a2
5and
4
5b
3
6and
5
6c
1
4and
7
4
d2
3and 2
1
3e
6
5and
13
5f
5
8and 1
7
8
g2
10,
5
10, and
9
10h
1
6,
11
6, and 1
1
6i
6
7,
12
7, and 2
2
7
2 Identify the value indicated by the red dot:
a b
c d
e f
g h
3 a Place the fraction1
4on a number line.
b On the same number line, place the fraction3
8.
c Which value is larger,1
4or
3
8? Explain your answer.
4 a Place the fractions2
3and
4
6on the same number line.
b What do you notice about these fractions?
Example 9
2 4 3 4 5
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\106AUS06_05.cdr Monday, 11 February 2013 3:01:20 PM GR8GREG
Investigation Equal fractions
FRACTIONS (Chapter 5) 107
Two fractions are equal if they describe the same amount.
They lie at the same place on the number line.
For example, we can represent the
fractions2
3and
4
6by shading
diagrams.
We see the same amount is shaded
in each case, so2
3=
4
6.
2
3is shaded
4
6is shaded
What to do:
1 a Use grid paper to construct three identical squares with sides 6 cm
long, or click on the icon to obtain a template.
b Divide the first square into two equal halves.
Shade one of the halves, so that1
2of the square is shaded.
c Divide the second square into four equal quarters.
Shade two of the quarters, so that2
4of the square is shaded.
d Do you think that the fractions1
2and
2
4are equal? Check your answer by placing
them on a number line.
e Divide the third square into six equal sixths. How many sixths do you need to shade,
to make a fraction equal to1
2and
2
4?
2 a Draw two circles with radius 3 cm, or print them using the template.
b From the centre of the first circle, measure and rule 3 lines, 120±
apart. Since 3 £ 120± = 360±, you have divided the circle into
thirds. Shade two of the sectors, which is2
3of the circle.
c From the centre of the second circle, measure and rule 9 lines,
40± apart. Since 9 £ 40± = 360±, you have divided the circle
into ninths. Shade six of the sectors, which is6
9of the circle.
d Do you think that the fractions2
3and
6
9are equal? Check your answer by placing
them on a number line.
EQUAL FRACTIONSF
DEMO
TEMPLATE
We
Yo
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\107AUS06_05.cdr Friday, 1 February 2013 12:38:02 PM BRIAN
108 FRACTIONS (Chapter 5)
In the Investigation, you should have found that1
2=
2
4, and
2
3=
6
9.
Notice how these numbers are related:
1
2=
2
4or
1
2=
2
4
2
3=
6
9or
2
3=
6
9
This suggests that:
Multiplying or dividing both the numerator and the denominator by the same non-zero number
produces an equal fraction.
This rule allows us to write a given fraction with a different numerator or with a different
denominator, without changing the fraction’s value.
Self Tutor
Express with denominator 18:
a7
9b
5
6c
22
36
a7
9
=7£ 2
9£ 2f9 £ 2 = 18g
=14
18
b5
6
=5£ 3
6£ 3f6 £ 3 = 18g
=15
18
c22
36
=22¥ 2
36¥ 2f36 ¥ 2 = 18g
=11
18
EXERCISE 5F.1
1 Write3
4with denominator:
a 8 b 12 c 16 d 20
2 Write4
10with denominator:
a 20 b 30 c 50 d 5
3 Express with denominator 8:
a1
4b
1
2c
3
4d 1 e
10
16
4 Express with denominator 30:
a1
2b
4
5c
5
6d
3
10e
1
5
f2
3g 1 h
3
5i
14
60j
18
90
Example 10
*2
*2
/2
/2
*3
*3
/3
/3
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\108AUS06_05.cdr Monday, 11 February 2013 3:14:01 PM GR8GREG
FRACTIONS (Chapter 5) 109
5 Express in hundredths:
a1
2b
1
4c
4
5d
9
10e
7
25
f13
50g 1 h
17
20i
34
200j
44
400
6 Which two of the following fractions are equal?
A8
20B
4
12C
3
8D
1
5
E6
9F
9
24G
3
5H
10
20
SIMPLEST FORM
We say that a fraction is written in simplest form if it is written with the
smallest possible whole number numerator and denominator.
For example, the fraction9
12is not written in simplest form,
because we can write9
12as
3
4.
To write a fraction in simplest form, we must find the largest number that is a factor of both the
numerator and the denominator. We then divide the numerator and denominator by this value.
Self Tutor
Write in simplest form:
a5
20b
8
12
a5
20
=5¥ 5
20¥ 5
f5 is a factor of
both 5 and 20g=
1
4
b8
12
=8¥ 4
12¥ 4
f4 is the largest
factor of both
8 and 12g=
2
3
Example 11
Game Equal fractions
Click on the icon to play a game where you must find equal fractions. GAME
9
12=
3
4
/3
/3
A fraction is in simplest form
when the numerator and
denominator do not have any
factors in common, except 1.
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\109AUS06_05.cdr Friday, 15 March 2013 11:29:49 AM BRIAN
110 FRACTIONS (Chapter 5)
EXERCISE 5F.2
1 Write in simplest form:
a2
4b
4
8c
3
9d
2
10e
5
15
f4
24g
6
10h
20
30i
18
21j
24
32
2 Which of these fractions is written in simplest form?
A6
8B
3
12C
10
21D
14
20E
7
28
We often wish to compare the size of two fractions.
For example, if you were offered3
5or
7
10of a block of chocolate, which would you choose?
If two fractions are written with the same denominator, we can simply compare the sizes of the
numerators.
Self Tutor
Which is larger: a4
7or
6
7b
13
5or 2
1
5?
a 6 is larger than 4, so6
7is larger than
4
7.
b 21
5as an improper fraction is
11
5.
13 is larger than 11, so13
5is larger than 2
1
5.
If two fractions do not have the same denominator, we write one of them as an equal fraction
which has the same denominator as the fraction we are comparing with. We can then compare the
numerators.
Self Tutor
Which is larger:3
5or
7
10?
We multiply the numerator and denominator of3
5by 2, so that both
fractions have denominator 10.
3
5=
3£ 2
5£ 2=
6
10
7
10is greater than
6
10, so
7
10is greater than
3
5.
COMPARING FRACTIONSG
Example 13
Example 12 Convert mixed numbers
to improper fractions
before comparing them.
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\110AUS06_05.cdr Wednesday, 13 February 2013 4:09:46 PM BRIAN
FRACTIONS
Discussion
(Chapter 5) 111
EXERCISE 5G
1 Which is larger:
a5
12or
7
12b
4
5or
3
5c
8
9or
13
9
d11
7or 1
3
7e
19
4or 5
1
4f
28
6or 4
5
6?
2 Keith and Caroline ate sushi for dinner. Keith had
31
3pieces of sushi. Caroline cut her sushi pieces into
thirds, and ate 8 of the thirds.
Who had more sushi for dinner?
3 Which is larger:
a1
2or
3
4b
1
3or
3
6c
3
4or
7
8
d5
8or
1
2e
2
3or
5
9f
1
4or
5
20
g4
3or
5
6h
13
15or
6
5i
15
4or 3
1
2?
4 Arnold spends1
3of his income on rent, and
2
9of his income on groceries. Does he spend
more on rent or on groceries?
5 Trent and Meredith each own a cage of birds.
a What fraction of Trent’s birds are yellow?
b What fraction of Meredith’s birds are yellow?
c In which cage is there a greater fraction of yellow birds?
Are improper fractions always larger in size than proper fractions?
Trent’s cage Meredith’s cage
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\111AUS06_05.cdr Monday, 11 February 2013 3:33:05 PM GR8GREG
112 FRACTIONS (Chapter 5)
A pizza is divided into 8 equal pieces. Sam takes 3 pieces,
and Pam takes 2 pieces. This means that together they have
taken a total of 5 pieces.
Notice that Sam has taken3
8of the pizza, Pam has taken
2
8,
and together they have taken5
8.
So,
3
8+
2
8=
5
8
Sam eats 1 of his pieces of pizza, so he has 2 pieces remaining. We can also say that Sam took3
8
of the pizza, he ate1
8of the pizza, and he has
2
8of the pizza remaining.
So,
3
8¡ 1
8=
2
8
To add or subtract fractions with the same denominator, we add or subtract the numerators.
The denominator stays the same.
Self Tutor
Find: a2
5¡ 1
5b
4
9+
7
9
a2
5¡ 1
5
=2¡ 1
5
=1
5
b4
9+
7
9
=4 + 7
9
=11
9
= 12
9
ADDING AND SUBTRACTING FRACTIONSH
Example 14
+ =
¡ =
Sam
Pam
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\112AUS06_05.cdr Friday, 1 February 2013 12:39:12 PM BRIAN
FRACTIONS (Chapter 5) 113
EXERCISE 5H.1
1 Find:
a1
4+
2
4b
2
3¡ 1
3c
5
4¡ 2
4
d3
8+
4
8e
1
5+
3
5f
5
7¡ 3
7
g6
11¡ 2
11h
9
20+
8
20i
21
25¡ 13
25+
1
25
2 Find:
a4
5+
2
5b
7
10+
6
10c
6
7+
5
7
d11
15+
8
15e
10
13+
8
13+
11
13f
11
14+
13
14¡ 1
14
Self Tutor
Find: 2 +4
7+
6
72 +
4
7+
6
7
= 2 +4 + 6
7
= 2 +10
7
= 2 + 13
7
= 33
7
3 Find:
a 3 +1
9+
4
9b 2 +
3
10+
4
10c 5 +
6
7¡ 4
7
d 1 +5
6+
2
6e 4 +
13
15+
4
15f 7 +
12
17+
10
17
Self Tutor
Find3
8¡ 2
8+
5
8, giving your
answer in simplest form.
3
8¡ 2
8+
5
8
=3¡ 2 + 5
8
=6
8
=6¥ 2
8¥ 2f2 is a factor of both 6 and 8g
=3
4
Example 16
Example 15
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\113AUS06_05.cdr Wednesday, 30 January 2013 4:14:11 PM GR8GREG
114 FRACTIONS (Chapter 5)
4 Find, giving your answer in simplest form:
a3
4¡ 1
4b
2
9+
1
9c
7
6¡ 3
6
d1
8+
2
8+
3
8e
8
4¡ 3
4¡ 3
4f
3
10+
7
10¡ 2
10
Self Tutor
Find 23
5+ 8
4
5.
23
5+ 8
4
5
=13
5+
44
5
=13 + 44
5
=57
5
= 112
5
5 Find:
a 22
3+ 1
2
3b 4
3
5¡ 2
1
5c 3
2
7+ 5
3
7
d 62
8¡ 3
5
8e 1
5
9+ 3
2
9+ 4
7
9f 8
1
10¡ 5
7
10+ 6
3
10
6 Simon and Shane went hiking. On the first day
they walked5
9of the total distance. They had a
steep climb on the second day and only walked
2
9of the total distance.
What fraction of the total distance was
completed after 2 days?
7 Leah wrote 11
4pages of a story before tea, and another 2
1
4pages after tea. How many pages
had she completed?
8 Spiros had9
10of a bag of fertiliser. He used
6
10of a bag for his tomatoes.
a What fraction of a bag of fertiliser was left?
b Suppose a full bag contains 20 kg of fertiliser. How many kilograms of fertiliser does
Spiros have left?
Example 17
them to improper fractions.
To add or subtract mixed
numbers, we first convert
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\114AUS06_05.cdr Wednesday, 13 February 2013 4:10:14 PM BRIAN
FRACTIONS (Chapter 5) 115
9 Sarah and Jane went apple picking.
Sarah picked 13
5bags and Jane picked 2
4
5bags.
a How many bags of apples did they pick
altogether?
b How many more bags did Jane pick than
Sarah?
ADDING AND SUBTRACTING FRACTIONS WITH UNEQUAL
DENOMINATORS
Sometimes the fractions we want to add or subtract do not have the same denominator.
For example, suppose Anita drinks1
2of a can of soft drink, and
Melissa drinks3
8of the can. What fraction of the can did they
drink?
In the same way that it is easier to compare two fractions if they have the same denominator, it is
also easier to add or subtract fractions if they have the same denominator.
In the situation above, we need to find1
2+
3
8.
We can write1
2with denominator 8 by multiplying the numerator and denominator by 4.
So, we have1
2+
3
8
=1£ 4
2£ 4+
3
8
=4
8+
3
8
=7
8
So, Anita and Melissa drank7
8of the can of soft drink.
Qw
Ei
Multiplying the numerator
an equal fraction.
and denominator by the
same number produces
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\115AUS06_05.cdr Wednesday, 13 February 2013 4:10:41 PM BRIAN
116 FRACTIONS (Chapter 5)
Self Tutor
Find:
a5
9¡ 1
3b
3
5+
7
10
a5
9¡ 1
3
=5
9¡ 1£ 3
3£ 3fconverting to 9thsg
=5
9¡ 3
9
=2
9
b3
5+
7
10
=3£ 2
5£ 2+
7
10fconverting to 10thsg
=6
10+
7
10
=13
10
= 13
10
EXERCISE 5H.2
1 Find:
a1
2¡ 1
4b
1
6+
2
3c
5
8¡ 1
4
d1
3+
1
12e
19
30¡ 2
5f
2
7+
30
49
2 Find:
a4
5+
3
10b
10
12+
3
4c
7
9+
21
45d
23
25+
51
100
Self Tutor
Find:
a 11
4+ 2
1
2b 4 ¡ 1
2
3
a 11
4+ 2
1
2
=5
4+
5
2
fconverting to
improper fractionsg=
5
4+
5£ 2
2£ 2fconverting to quartersg
=5
4+
10
4
=15
4
= 33
4
b 4 ¡ 12
3
=4
1¡ 5
3
fconverting to
improper fractionsg=
4£ 3
1£ 3¡ 5
3fconverting to thirdsg
=12
3¡ 5
3
=7
3
= 21
3
Example 19
Example 18
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\116AUS06_05.cdr Wednesday, 30 January 2013 4:53:07 PM GR8GREG
FRACTIONS (Chapter 5) 117
3 Find:
a 11
2+ 2
3
8b 3
1
3¡ 2
1
6c 1
2
5+ 1
9
10
d 71
2¡ 5
3
4e 2
2
7+ 1
10
21f 6
2
3¡ 3
2
15
4 Find:
a 1 ¡ 4
9b 3 ¡ 1
5
8c 7 ¡ 4
2
7d 9 ¡ 6
5
12
5 Joshua baked a cake to share with friends. Lisa ate2
9, and Rebecca ate
1
3of the cake. What
fraction of the cake did the girls eat between them?
6 Every day, Angus feeds his chickens1
5of a large
tub of feed. If Angus’ tub is9
10full at the start
of the day, how much is left after he has fed his
chickens?
7 Samantha is an artist. She spends 31
2hours on
Saturday painting a portrait, and a further 21
4hours
finishing it off on Sunday.
How long did it take her to paint the portrait?
8 31
8tonnes of earth needs to be removed to level
a housing block. A truck moves 11
2tonnes in the
first load.
How much earth still needs to be moved?
KEY WORDS USED IN THIS CHAPTER
² denominator ² equal fractions ² fraction
² improper fraction ² mixed number ² number line
² numerator ² proper fraction ² simplest form
1 What fraction is represented by the following?
a b c
Review set 5
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\117AUS06_05.cdr Wednesday, 30 January 2013 5:14:59 PM GR8GREG
Practice test 5A Multiple Choice
118 FRACTIONS (Chapter 5)
2 What fraction of the cars in this car park are blue?
3 Write the following divisions as fractions:
a 6 ¥ 11 b 15 ¥ 19
4 Write as a mixed number:
a9
5b
13
3c
35
6
5 Express with denominator 12:
a5
6b
2
3c
10
24
6 Find:
a1
4of $200 b
2
5of 100 g c
3
8of 56 cm
7 Place the following fractions on a number line:
a1
6and
4
6b
4
8and
11
8c
6
7and 1
4
7
8 An athlete runs2
5of a race in the first hour and
3
10in
the second hour.
What fraction of the race has he completed?
9 Which is larger:
a6
10or
3
10b
19
7or 2
3
7c
4
5or
22
25d 5
2
3or
31
6?
10 Find:
a12
7¡ 8
7b
8
11+
9
11c
3
8+
1
4d 5
1
3¡ 1
1
9
Click on the icon to obtain this printable test.PRINTABLE
TEST
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\118AUS06_05.cdr Tuesday, 5 February 2013 11:10:08 AM GR8GREG
FRACTIONS (Chapter 5) 119
Practice test 5B Short response
1 Copy this circle and shade5
8of it.
2 a What mixed number is represented by
this diagram?
b Write the mixed number as an improper
fraction.
3 Write these fractions as divisions, and hence as whole numbers:
a40
8b
72
9c
99
11
4 Write as an improper fraction:
a 35
6b 4
3
7c 5
2
5
5 Write in simplest form:
a2
16b
25
45
6 Write5
6with denominator:
a 12 b 30 c 48
7 Sarah went on a holiday for 20 days. It rained
on one quarter of the days. On how many days
did it rain?
8 Which is greater,3
7or
10
21?
9 Find:
a 3 +3
5+
4
5b 4
1
10¡ 2
4
10c
5
6+
14
18
10 A cyclist completes5
14of her training ride in the first hour and
3
7in the second hour.
What fraction of her ride has she completed?
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\119AUS06_05.cdr Tuesday, 5 February 2013 4:15:38 PM GR8GREG
Practice test 5C Extended response
120 FRACTIONS (Chapter 5)
1 Answer the Opening Problem on page 94.
2 At a barbecue, Adam ate 51
3sausages, and Jill ate
32
3sausages.
a Write each of these values as an improper
fraction.
b How many sausages did they eat in total?
c How many more sausages did Adam eat than
Jill?
3 a Place the fraction5
9on a number line.
b On the same number line, place the fraction2
3.
c Which of these fractions do you think is larger?
d Check your answer to c by writing the fractions with the same denominator.
4 Judy has to write 60 Christmas cards to send to her friends and family. She writes1
3of
them on Monday.
a How many cards did Judy write on Monday?
b How many cards does she still need to write?
c Judy writes remaining cards on Tuesday.
i How many cards did she write on Tuesday?
ii How many cards does she still need to write?
5 Caleb had a mathematics test and a spelling test on the same day. The results of each test
are shown below.
Mathematics test
1 6 X 11 16
2 X 7 X 12 X 17 X
3 X 8 13 X 18
4 9 X 14 19 X
5 X 10 X 15 X 20 X
Spelling test
1 X 6 X
2 7 X
3 X 8 X
4 X 9
5 10 X
a What fraction of the mathematics questions did Caleb answer correctly?
b What fraction of the spelling questions did Caleb answer correctly?
c In which test did Caleb answer a greater fraction of questions correctly?
2
5of the
AUS_11magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\AUS_06\AUS06_05\120AUS06_05.cdr Monday, 11 February 2013 3:22:14 PM GR8GREG