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5
ES
CH APTER
N
AL
PA
G
Number and Algebra
Fractions – part 2
FI
We have already learned about multiplying and dividing whole numbers, and
we turn now to multiplying and dividing fractions. This will complete our
understanding of the four operations applied to fractions and we can use these
to solve problems.
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125
5 A
Multiplication of fractions
2 3
× ?
3 4
We will use the area model for multiplication that we used in Chapter 1. Begin
with a square of side length 1. Its area will also be 1.
How do we multiply two fractions such as
Now divide the square into rectangles by drawing vertical lines and horizontal
lines, as shown.
1
3
_
4
1
_
2
1
_
4
1_
3
2_
3
1
ES
Thus, we divide the height into quarters and the base into thirds. Each
3
1
2
rectangle has area . Next, shade the rectangle with side lengths and , as
12
4
3
shown in the diagram.
2_
3
PA
G
We can see that the shaded region is made up of 2 × 3 = 6 rectangles, each
1
2 3
of area , so the area of the shaded region is the product of the sides, × .
12
3 4
6
1
= . We conclude that:
But this is also equal to 6 lots of
12 12
2 3 6
× =
3 4 12
3
_
4
From this, we see that the rule for multiplying two fractions is multiply the numerators, multiply the
denominators. This can be written as:
a c a×c
× =
b d b×d
N
AL
Notice that when we find a fraction ‘of’ a whole number, we get the same result as if we were multiplying
1
1
the fraction by that whole number. For example, finding of 6 is the same as evaluating × 6. The same
2
2
2 3
2 3
applies when dealing with two fractions. For example, when we say of , we can write × .
3 4
3 4
Multiplication of fractions
FI
• To multiply two fractions together, multiply the two numerators together and the two
denominators together to form the new fraction and simplify if possible.
a c a×c
In symbols: × =
b d b×d
For example:
2 7 14 7
× =
=
3 8 24 12
n
3
3 3 9
• The whole number n should be written as . For example:
×3=
× =
1
10
10 1 10
• Multiplication of two fractions and the corresponding ‘of’ statement give the same
answer.
For example:
126
3 1
2
2 3 6 1
of = is the same as × =
=
4 2
3
3 4 12 2
I C E - E M M at h em at ic s y e a r 7
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5 A M u lt i p l i cat i o n o f f r ac t i o ns
Example 1
Evaluate:
a 3 × 10
4 7
b
9
×4
8
Solution
3 10 3 × 10
×
=
4 7
4×7
30
=
28
15
=
14
1
= 114
ES
G
Cancelling
9
9 4
b 8 × 4 = 8 × 1
9×4
=
8×1
36
=
8
9
=
2
= 4 12
PA
a
We saw earlier that we can sometimes cancel before multiplying.
3
2
Let us look at the multiplication × .
4
3
N
AL
2 3 6
× =
3 4 12
1
=
2
There is a shorthand way of writing this, which often simplifies the process of multiplication.
FI
2 3 2 3
× = ×
3 4 3 4
2
=
4
21
= 2
4
1
=
2
This process is called cancelling. We can cancel because we are dividing the numerator and the
denominator by the same whole number. We know that this gives an equivalent fraction.
Cancelling
• Cancelling is used to simplify fractions.
• To simplify a product of fractions, find a common factor of a numerator and a
denominator and cancel. Repeat if possible.
C h a p t e r 5 F racti o n s – part 2
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127
5 A M u lt i p l i cat i o n o f f r ac t i o ns
Example 2
a Evaluate
5 2
× .
6 3
b Evaluate
16 18
× .
27 20
Solution
5 2 5 21
×
× =
6 3 63 3
5 1
= ×
3 3
5×1
=
3×3
5
=
9
b
16 18 164 182
=
×
×
27 20 273 205
4 2
= ×
3 5
4×2
=
3×5
8
=
15
ES
a
Calculate:
a
2 3 5
× ×
3 4 8
PA
Example 3
G
Cancelling should be done before multiplication. Cancelling is very helpful when three or more
fractions are involved. Note that, as for multiplication of whole numbers, order does not matter.
b
11 10 7
×
×
5 14 11
N
AL
Solution
111 102 71
b 11 × 10 × 7 =
×
×
5 14 11 51 142 111
2
=
2
=1
FI
21 31
a 2 × 3 × 5 = × × 5
3 4 8 31 42 8
5
=
16
Exercise 5A
Example 1
128
1 Evaluate each product, simplifying where possible.
3
a 1 ×
4 5
b
3 5
×
4 8
c 1 × 1
2 8
d
1 2
×
3 3
e
5 3
×
4 7
f
5 3
×
11 8
g
7 7
×
2 5
h
3 3
×
2 7
i
7 5
×
6 7
I C E - E M M at h em at ic s y e a r 7
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Example 1
2 Find:
a 4 ×
5
7
d 11 ×
b 7 ×
2
3
e
3
11
3
×5
4
c
5
×7
12
f
4
×3
11
3 Find:
Example 3
b
1
2
of
3
6
c
5
3
of
8
4
d
3
2
of
3
2
e
5
3
of
4
5
f
2
4
of
9
3
g
2
of 4
3
h
4
of 13
5
4 Find the value of each of:
ES
1
1
of
3
2
a
4 6
×
3 8
b
5 3
×
9 7
c
6 7
×
14 8
d
8
3
×
21 40
e
17 10
×
15 34
f
12 49
×
7 60
g
44 21
×
28 66
h
30 27
×
54 75
G
Example 2
a
5 Complete each multiplication.
3 5 2
× ×
4 6 5
g
3 5 8
× ×
8 9 15
3 5 3
× ×
4 6 8
3 7
c 1 × ×
2 4 12
e
7 6 5
× ×
12 7 9
5
f 2 × × 4
3 6 15
h
24 14 5
×
×
7 15 16
i
9
8 5
×
×
9 18 10
l
8 22 15
×
×
11 25 32
N
AL
d
b
PA
a 1 × 1 × 1
2 3 6
j
3 5 4
× ×
10 9 5
14 16 5
×
×
15 21 8
FI
5 B
k
Division of fractions
Dividing by whole numbers
We saw in Section 4C how to divide two whole numbers.
17
4
= 3 25.
For example, 4 ÷ 3 = = 1 13 and 17 ÷ 5 =
3
5
1 4
Thus, dividing 4 by 3 gives the same answer as finding 4 × = , and dividing 17 by 5 is the same
3 3
1 17
as 17 × = .
5 5
C h a p t e r 5 F racti o n s – part 2
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129
5 B D i v i s i o n o f f r ac t i o ns
1
In general, for whole numbers m and n dividing m by n is the same as multiplying m by .
n
1
m÷n=m×
n
1
The fraction is called the reciprocal of n.
n
1
1
Thus, the reciprocal of 3 is and the reciprocal of 5 is .
3
5
Division of a fraction by a whole number
This means that we take
3
and divide it into 6 equal parts.
5
ES
The idea of performing a division by multiplying by the reciprocal can also be used to divide a
fraction by a whole number. For example:
3
÷6
5
N
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PA
G
This can be illustrated by drawing a unit square divided into fifths.
3
Shade , as shown in the first diagram, and further divide the square into 6 equal horizontal strips, as
5
shown in the second diagram.
The orange shaded area represents both
3 3 1
= × , so the rule of multiplying by the reciprocal also holds.
30 5 6
FI
Notice that
3
3
÷ 6 and .
30
5
Dividing by a fraction
1
Suppose we wish to divide 8 by . We can express this question as ‘How many halves in 8 wholes?’
2
1
Since there are 2 halves in a whole, there will be 16 halves in 8 wholes. Thus, 8 ÷ = 16. Once
2
again, the idea of multiplying by the reciprocal gives us the correct answer, since:
8÷
1
2
= 8 × = 16
2
1
Similarly, 6 divided by
130
2
means that we divide 6 into 18 thirds, and then divide this by 2.
3
I C E - E M M at h em at ic s y e a r 7
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5 B D i v i s i o n o f f r ac t i o ns
Thus:
6÷
3 18
2
=6× =
=9
2 2
3
In both examples, we turn the second fraction upside down and multiply.
The reciprocal of a fraction is the fraction obtained by swapping the numerator and denominator.
To divide a number by a fraction, we multiply the number by the reciprocal.
Dividing a fraction by a fraction
ES
One way of dividing a fraction by a fraction is to express each fraction using a common denominator
so that we are dividing two fractions of the same type.
3 2
For example, to find ÷ , we write:
5 3
3 2 9 10
÷ =
÷
5 3 15 15
Since the denominators are the same, we divide the numerators to obtain
3 3 9
× = .
5 2 10
G
Notice that
PA
Thus:
3 2 3 3 9
÷ = × =
5 3 5 2 10
9
.
10
3
3
3
2
2
by is the same as multiplying it by . We say that is the reciprocal of . It is the
2
2
3
3
5
fraction obtained when the numerator and denominator are swapped.
So dividing
N
AL
Thus, the rule ‘To divide by a fraction, multiply by the reciprocal’ holds in all situations.
Division of fractions
• The reciprocal of a fraction is the fraction we get by swapping the numerator and
denominator. The product of a fraction and its reciprocal is 1.
FI
• Division and multiplication of fractions are inverse operations.
2 1
1
2
For example: ÷ means the fraction that, when multiplied by , gives .
9 8
8
9
• Dividing a fraction by a fraction is the same as multiplying the first fraction by the
reciprocal of the second. So:
3 4 3 5
2 1 2
÷ = ×8
÷ = ×
9 8 9
and 7 5 7 4
16
15
=
=
9
28
In general:
a c a d
÷ = ×
b d b c
C h a p t e r 5 F racti o n s – part 2
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131
5 B D i v i s i o n o f f r ac t i o ns
Example 4
2 3
÷ by multiplying by the reciprocal. Check that the answer you get is the
3 5
3
2
number that, when multiplied by the divisor , gives .
3
5
Work out
Solution
(
3
5
is the reciprocal of
3
5
)
Checking:
3 10 30
×
=
5 9 45
2
= (common factor 15)
3
ES
2 3 2 5
÷ = ×
3 5 3 3
2×5
=
3×3
10
=
9
G
Example 5
a 6 ÷
5
7
Solution
5
7
=6×
7
5
6 7
= ×
1 5
42
=
5
= 8 25
N
AL
a 6 ÷
b
3 4
÷
5 7
b
3 4 3 7
÷ = ×
5 7 5 4
3×7
=
5×4
21
=
20
1
= 120
PA
Find:
Example 6
FI
Evaluate:
a 3 ÷ 1
4 4
b 7 ÷ 3
10 5
c 3 ÷ 9
7 14
1
b 7 ÷ 3 = 7 × 5
10 5 102 3
7
=
6
= 116
1
2
c 3 ÷ 9 = 3 × 14
7 14 71 93
1 2
= ×
1 3
2
=
3
Solution
41
a 3 ÷ 1 = 3 ×
4 4 41 1
3
=
1
=3
132
I C E - E M M at h em at ic s y e a r 7
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5 B D i v i s i o n o f f r ac t i o ns
Example 7
Evaluate (work from left to right):
a
3 11 11
×
÷
4 3
2
b 7 ÷ 1 ÷ 2
8 2 7
Solution
b 7 ÷ 1 ÷ 2 = 7 ×
8 2 7 8
7
= ×
8
49
=
8
= 6 18
2 7
×
1 2
21 7
×
1 21
G
ES
1
1
1
a 3 × 11 ÷ 11 = 3 × 11 × 2
4 3
2 42 31 111
1
=
2
Example 5a
1 Calculate:
3
5
b 2 ÷
8
9
f 5 ÷
e 3 ÷
c 2 ÷
5
6
d 5 ÷
6
11
g 7 ÷
2
3
h 11 ÷
4
7
5
6
2 Find:
a
3 5
÷
4 7
b
2 5
÷
3 11
c
3 5
÷
11 8
d
3 2
÷
11 3
e
5 2
÷
8 3
f
1 3
÷
2 7
g
5
7
÷
12 11
h
5 11
÷
7 13
FI
Example 6
3
7
N
AL
a 4 ÷
Example 5b
PA
Exercise 5B
3 Evaluate:
a
3 3
÷
4 8
b
3 3
÷
8 4
c
5 5
÷
6 9
d
2 5
÷
3 6
e
3 5
÷
4 8
f
2 4
÷
3 9
g
5
7
÷
12 16
h
9
3
÷
16 20
i
8 16
÷
9 27
j
7 14
÷
18 27
k
5 15
÷
24 16
l
72 16
÷
100 25
C h a p t e r 5 F racti o n s – part 2
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133
4 Evaluate:
a
4 3 2
× ÷
9 4 3
b
5 2 5
× ÷
6 3 9
c
3 8
2
of ÷
3
4 9
d
7 3 14
÷ ×
9 4 27
e
2 3 2
÷ ×
9 4 3
f
5 2 5
÷ ×
12 3 8
g
3 9 1
÷
×
4 16 6
h
5 2 3
÷ ×
9 3 5
i
3 4 3
× ÷
12 5 25
j 4 ÷
k
2 5
÷
÷4
3 12
l
4 3 7
× ÷
9 5 15
n
1 1 1 1 1
× ÷ × ÷
2 2 2 2 2
p
24 96 132 55
÷
×
÷
81 18 72 24
2 3
×
3 5
1 1 1 1
m ÷ ÷ ÷
2 2 2 2
o
34 11
4 14
×
÷
÷
7 100 110 17
ES
Example 7
2
of a chocolate bar if I have 4 chocolate bars?
3
3
6 I have 12 oranges. To how many people can I give of an orange?
4
5
7 A ribbon that is 20 m long is to be divided into lengths of of a metre. How many such
8
lengths are there?
Multiplication and division of
mixed numerals
PA
5 C
G
5 To how many people can I give
N
AL
When multiplying and dividing mixed numerals, convert to improper fractions and then proceed as
you have already learned.
Example 8
a Evaluate 1 23 × 4 12.
b Evaluate 7 34 ÷ 2 13.
Solution
FI
5 93
×
31 2
15
=
2
= 7 12
a 1 23 × 4 12 =
b 7 3 ÷ 2 1 = 31 ÷
4
3
4
31
=
×
4
93
=
28
9
= 3 28
134
7
3
3
7
(Change to improper fractions.)
(Convert to a mixed numeral.)
(Change to improper fractions.)
7
(Multiply by the reciprocal of .)
3
(Convert to a mixed numeral.)
I C E - E M M at h em at ic s y e a r 7
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5 C M u lt i p l i cat i o n and d i v i s i o n o f m i x ed n u me r a l s
Operations with mixed numbers
When dealing with multiplication and division and mixed numerals, the following strategy is
normally used.
Convert each mixed numeral to an improper fraction and proceed as usual.
Exercise 5C
e
1
× 4 12
7
i 3 13 × 5
Example 8b
f
1
3
k 6 × 2 37
l 5 × 3 35
d 6 ÷ 1 13
7
× 3 35
8
g 5 19 ×
j 8 12 × 4
2 Find:
a 1 13 ÷
2
3
1
× 3 23
3
1
h 2 14 ×
5
c 2 29 × 1 13
b 4 45 × 3 34
b 2 23 ÷
5
6
d
ES
a 5 12 × 6 14
G
1 Find:
5
c 1 12
÷
3
4
PA
Example 8a
5
8
f 2 15 ÷ 1 13
g 5 13 ÷ 4 14
h 3 25 ÷
a 2 12 ÷ 1 14
b 4 13 ÷ 1 12
c 6 35 ÷ 2 13
d 10 12 ÷ 5 78
e 5 13 ÷ 6 14
f 4 15 ÷ 1 14
g 2 18 ÷ 7 15
h 5 12 ÷ 7 13
i 2 19 ÷ 2 14
j 1 18 ÷ 7 12
1
k 5 25
÷ 4 15
l 21 12 ÷ 14 14
e 7 14 ÷ 1 15
N
AL
3 Find:
4 Five boys each have 1 14 litres of lemonade. How many litres of lemonade do the boys have
in total?
5 Find:
2
of 3 14
3
FI
a
b
3
1
of 5 12
4
c
7
of 6 15
8
6 Jake shares his liquorice strap equally between himself and five friends. If the strap is 1 23
metres long, how much will each person get?
7 Alexa cuts a yellow ribbon, of length 2 56 metres, into four equal pieces. How long is each
piece?
8 Sarah’s pension is $105 per fortnight. How much is this per day?
9 Fran’s family want to make their 689 Easter eggs last for one year.
a How many Easter eggs can they eat per week?
b I f there are 7 people in Fran’s family, how many Easter eggs will one person eat per week
if the eggs are shared equally?
C h a p t e r 5 F racti o n s – part 2
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135
5 D
Word problems
involving fractions
There are a number of things to consider when tackling word problems involving fractions. We need
to think about which operations are required, and to think carefully about what is ‘the whole’.
Example 9
Two-thirds of the chocolates in a box contain nuts. There are 54 chocolates in the box.
ES
a How many chocolates in the box contain nuts?
b What fraction of the chocolates in the box do not contain nuts?
2
2 5418
of 54 = 1 ×
1
3
3
= 2 × 18
= 36
36 of the chocolates in the box contain nuts.
b 1 −
PA
a
G
Solution
2 1
=
3 3
N
AL
1
of the chocolates in the box do not contain nuts.
3
Example 10
At Toocastle College,
3
of the students are boys.
5
FI
a What fraction of students are girls?
b If there are 360 boys, how many girls are there?
Solution
136
a
3
3 2
of the students are boys. Therefore, 1 − = of the students are girls.
5
5 5
b
3
1
is 360. This means that of the number of students is 360 ÷ 3 = 120. Therefore, the
5
5
number of girls is 2 × 120 = 240.
I C E - E M M at h em at ic s y e a r 7
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5 D W o r d p r o b l ems i n v o lv i n g f r ac t i o ns
Exercise 5D
Example 9
1 Two-thirds of the jelly beans in a jar are black. There are 96 jelly beans in the jar. How
many of them are black?
2 Five-twelfths of the students in a particular school are girls. There are 1344 students in the
school.
a How many girls are there in the school?
b What fraction of the students in the school are boys?
a How many eucalypts are there in the forest?
b What fraction of the trees are not eucalypts?
3
4 In a box of chocolates, of the chocolates have soft centres.
5
a What fraction of the chocolates do not have soft centres?
G
Example 10
ES
3 Seven-eighths of the trees in a forest are eucalypts. It is known that there are 5000 trees in
the forest.
PA
b If there are 15 chocolates with soft centres, how many chocolates are there altogether?
7
5 A piece of land is divided among Jan, David and Greg. Jan is allocated
of the land and
12
1
David receives of the land.
6
a What fraction of the land does Greg receive?
N
AL
b The land is worth $96 000. What is the value of each person’s share?
5
1
of the seats are in the back stalls, are in the front stalls and the remainder
6 In a theatre,
12
4
are in the balcony.
a What fraction of the seating capacity of the theatre is in the balcony?
b If the theatre can hold 1080 people, how many people can sit in each of the sections?
3
7 A container of cooking oil is full. A further 4 12 L of cooking oil is required to fill it. How
5
much oil does the container hold?
FI
8 Complete each of the following magic squares. Each row, column and diagonal sums to the
same mixed number.
a
b
1
12
1
3
1
4
2
1
4
114
214
2
3
3
are girls.
5
a What fraction of the group is boys?
9 Of a group of 50 students,
b How many of the students are boys?
C h a p t e r 5 F racti o n s – part 2
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137
2
of the students are girls. There are 600 boys.
5
a How many girls are there?
10 In a school,
Order of operations with
fractions
G
5 E
ES
b How many students are there?
5
1
1
of its income on food, on housing and
on transport. If
11 Each month, a family spends
12
3
24
there is $960 left over, calculate the family’s income.
7
1
of his land cultivated. He also has of his land used for buildings. Of the
12 A farmer has
12
6
1
remaining land, is marsh and is unable to be farmed. The rest of the land is forest. What
3
fraction of his land is forest?
Order of operations
PA
The conventions of order of operations that we discussed for whole numbers in Chapter 1 apply also
to the arithmetic of fractions.
• Evaluate expressions inside brackets first.
• In the absence of brackets, carry out operations in the following order:
N
AL
–– powers
–– multiplication and division from left to right, then
–– addition and subtraction from left to right.
Example 11
FI
Evaluate:
a 1 −
1 1
−
4 3
b
( )
7
1 1
−
+
8
4 3
Solution
a 1 − 1
4
138
−
3
1
4
=1−
−
12 12
3
4
12 3
−
−
=
12 12 12
5
=
12
b
( )
( )
7
1 1
3+4
7
−
+
−
=
8
4 3
8
12
21 14
=
−
24 24
7
=
24
I C E - E M M at h em at ic s y e a r 7
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5 E O r de r o f o p e r at i o ns w i t h f r ac t i o ns
Example 12
Evaluate:
a
()
1
2
+
3
6
2
b
4 3 6
− ÷
3 5 11
Solution
2
1 2 2
+ ×
6 3 3
1 4
= +
6 9
8
3
=
+
18 18
11
=
18
b
=
1 Evaluate:
a
b 1 −
( )
1 1
+
5 3
c 1 −
( )
1 1
−
3 5
N
AL
2 Evaluate:
a
3 3 3
+ ×
4 4 4
c
3 1 1
− ×
4 2 4
e
5
2
+
3
6
()
2
FI
Example 12
( )
11
1 1
−
+
8
4 3
PA
Exercise 5E
Example 11
4 3 6 4 31 11
×
− ÷
= −
3 5 11 3 5 62
4 11
= −
3 10
40 33
=
−
30 30
7
=
30
ES
()
1
2
+
3
6
G
a
b
1 4 3
÷ +
2 3 4
d
11 44 7
÷
+
18 3 9
f
() ()
3
1
−
4
2
2
g
1 1 1 1
+ × −
2 2 2 2
h
3 5 1 3
+ × ÷
4 4 4 4
i
16 8 5
÷
+
15 25 4
j
5
7 5 4
+ ×
−
8 4 15 12
k
2 1 5
× ÷
3 4 9
l
3 8 1
÷
×
5 20 4
( )
( ) ( ) ( )
(
)
m
2 5
5
+
×
3 6
9
n
5
1 1 1
÷
+ +
4
2 4 8
o
4 2
2 5
1 2
−
×
+
÷
+
5 3
6 8
2 3
p
2 4 2 5 1 2
− × + ÷ +
3 5 6 8 4 3
C h a p t e r 5 F racti o n s – part 2
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400
139
Review exercise
1 Evaluate:
a
2 3
×
3 4
e 6 ÷
1
11
4 22
×
11 25
b 5 16 × 2
c 3 14 ÷ 13
d
f 5 12 × 3 14
g 3 14 ÷ 1 23
h 5 23 ÷
1
5
2 Evaluate:
b
5 3
× +2
11 5
3 Find:
a 6 23 × 1 14
b 5 23 ÷ 1 14
1 1
+
3 2
d 1 13 × 1 14 + 2
c 6 78 × 5 13
d 6 13 ÷ 5 23
PA
2
4 Find of 63.
3
3
5 of a number is 27. Find the number.
4
c 6 ÷
ES
2 3 1
× ÷
3 5 4
G
a
6 A rope 6 m long is divided into 4 equal pieces. How long is each piece?
7 Evaluate
( )
3 1
2
−
×
3
4 2 .
3
of 20 m.
5
3
b Find of $20.
4
7
c Find of 1600 kg.
8
1
1
9 What is the sum of and half of ?
5
5
3
2
10 Omar reads
of the pages of a book one day, the next day and the remaining 84
11
5
pages the following day. How many pages are in the book?
FI
N
AL
8 a Find
3
4
of a sum of money is $21. How much is of the sum of money?
7
5
12 Five-sixths of a farm covers 325 hectares (ha). What is the area of the whole farm?
11
140
I C E - E M M at h e matic s y e ar 7
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400
Challenge exercise
1 A half is a third of the number x. What is the number x?
2 A quarter is a third of the number x. What is the number x?
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
1
1
1
1
× 1−
× 1−
× 1− ?
2
3
4
5
4 What is the value of 1 +
1
1
1
1
1
× 1+
× 1+
× 1+
× 1+ ?
7
8
9
6
5
ES
3 What is the value of 1 −
5 Evaluate:
1
b 1 +
1
1+
2
1
1+
1
1+
1
2
G
a 1 +
6 Using the numbers 1, 2 and 3, fill in the boxes to make these statements true.
N
AL
2
5
PA
□□ × 23 = □□
□
□
□ × 1 = □□
b □
□
□
a
1
7 How many times do you need to write the fraction to make this statement true?
3
1 1 1 … 1
÷ ÷ ÷
÷ = 81
3 3 3
3
1
1
and are special because when they are multiplied the answer is the
7
8
1 1 1 1 1
same as the difference between them. That is, × = − = . Find five other such
7 8 7 8 56
pairs of numbers.
FI
8 The fractions
9 Johan was hungry late one night. He looked in the freezer and found a 4-litre tub of ice
cream. Johan started eating the ice cream, but so he would not get into trouble for eating
it all, he stopped when he had eaten half of it. The next night he was hungry again, so he
went back to the ice cream and again stopped when he had eaten half of what was there.
This went on for five nights in a row before his mother finally caught him. How much
ice cream was left at the end of each of the five nights? If his mother had not caught
him, would he ever have finished all the ice cream?
C h a p t e r 5 F racti o n s – part 2
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400
141
C H A LL E N G E e x e r c i se
10 The school lap-a-thon circuit is 234 kilometres in length. Benjamin walked 413 laps,
Nathan walked 512 laps and Alicia walked 314 laps.
a Who covered the greatest distance?
b What was the total distance walked by the three students?
c How much farther than Benjamin did Nathan walk?
12 Calculate
1 97 × 95
+
− 95.
96
96
37
= 2+
13
1
a+
.
1
b+
1
c
G
13 Find the whole numbers a, b and c if
ES
1
1
11 In a cricket test match lasting 5 days, of the runs were scored in the first day, in the
6
5
7
2
in the fourth day and remaining 21 runs on the last day.
second day, in the third day,
32
5
How many runs were scored on each day?
FI
N
AL
PA
14 A train from Alston to Brampton stops at two intermediate stations. At the first of these,
1
1
of the train’s passengers leave and 135 new passengers board. At the second station, of
2
3
the passengers who arrived on the train at the first station leave, and 110 new passengers
board. The train arrives at Brampton with 350 passengers. How many passengers were
on the train when it left Alston?
142
I C E - E M M at h e matic s y e ar 7
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400