Download Number skills

1
eTHINKING
Number skills
1
Jemma and Michael are
playing a game of darts. It is
Jemma’s turn and she has
thrown her darts as shown in
the photograph. Can you
calculate the number of points
scored?
Jemma and Michael
continue to play the game. The
results of their throws of 3
darts in each round are shown
in the table below.
Jemma
Michael
double 13, 20,
triple 9
18, 2,
double 16
8, double 18,
triple 1
triple 20, 25, 7
50, double 17,
12
double 19, 20,
triple 5
double 4, 12, 9
17, double 9, 6
This chapter will revise
your skills in working with
whole numbers, fractions and
decimals.
areyou
2
Maths Quest 8 for Victoria
READY?
Are you ready?
Try the questions below. If you have difficulty with any of them, extra help can be
obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon
next to the question on the Maths Quest 8 CD-ROM or ask your teacher for a copy.
1.1
Can you complete the following questions without a calculator?
Adding the subtracting whole numbers
1.2
1 Calculate each of the following.
a 52 + 39
c 39 – 21
b 4507 + 3243
d 45 386 − 9094
Multiplying whole numbers
1.3
2 Calculate each of the following.
a 9×7
b 41 × 19
Rounding decimals to 2 decimal places
1.4
3 Calculate each of the following, rounding the answer to 2 decimal places where appropriate.
a 162 ÷ 7
b 4785 ÷ 4
Order of operations I
4 Evaluate each of the following.
a 4 + 1--- of 12
1.5
2
Factors
5 Find all the factors of:
a 20
1.6
1.10
6 List the first 5 multiples of:
a 6
1.15
b 8.
Adding and subtracting fractions I
7 Calculate each of the following, writing your answer as a mixed number if appropriate.
3
--7
+
6
--7
b
3
--8
−
1
--4
b
5
--8
÷
3
--4
Multiplying and dividing fractions
8 Calculate each of the following.
a
1.14
b 48.
Multiples
a
1.13
b 5 × (13 − 7)
2
--9
×
7
--6
Adding and subtracting decimals
9 Calculate each of the following.
a 7.6 + 15.1
b 126.35 − 83.49
Multiplying and dividing decimals
10 Calculate each of the following.
a 3.7 × 1.2
b 182.72 ÷ 5
Rounding to the first (leading) digit
1.17
11 Round each number.
a 87 to the nearest ten
b 539 to the nearest 100
Chapter 1 Number skills
3
Whole number operations
Early humans developed a system of numbers for
counting. The Babylonians imprinted cuneiform
(wedge-shaped) numbers into wet clay with a stylus.
They recorded information such as wages due and
ages of animals on baked clay tablets over 3000
years ago. For numbers greater than 60, a type of
arithmetic was used.
As young children we learn to count using our
fingers and toes. Usually there are ten of each, and a
system of ten is used for metric measurement. In
1966, Australia converted to a decimal currency.
Numbers are all around us. We use numbers in all
sorts of ways — counting, shopping, telephone
numbers, measuring, for references, in day-to-day
conversation and for basic calculations. We use numbers to count our age. Can you
think of any other ways that people use numbers in their lives?
It is useful to be able to do some quick mental calculations or use pen and paper,
without having to rely on a calculator.
There are four basic mathematical operations:
+ Addition
− Subtraction
× Multiplication
÷ Division
The following exercise revises each of these operations using whole numbers.
WORKED Example 1
Calculate 96 + 24.
THINK
1
2
Write the numbers in columns with the tens
and units lined up.
Add the units first and then the tens.
WRITE
916
+ 214
11210
WORKED Example 2
Calculate 186 − 38.
THINK
1
Write the numbers in columns with the larger
one above the smaller one and hundreds, tens
and units lined up.
WRITE
17816
– 3181
114181
Continued over page
4
Maths Quest 8 for Victoria
THINK
2
3
4
5
WRITE
Since 8 cannot be subtracted from 6, take one
ten from the tens column of the larger
number and add it to the units column of the
same number. So the 6 becomes 16, and the 8
tens become 7 tens.
Subtract 8 units from 16.
Subtract 3 tens from the 7 remaining tens.
Subtract 0 hundreds from the 1 hundred.
Remember that 2 could be added to both parts of the problem in worked example 2 so
that you could work it out in your head.
186 − 38 becomes 188 – 40 = 148
WORKED Example 3
Calculate 78 × 34.
THINK
1
WRITE
23 3
Write the numbers one under the other.
×
78
314
3
Multiply by the units digit (× 4).
Write a zero and multiply by the tens digit (× 3).
131112
2131410
4
Add the two answers.
2161512
2
A good way to check the answer is to round each number to the nearest 10 or 100.
In this example 78 × 34 ≈ 80 × 30 = 2400, which is close to the answer found.
If multiplying by a three digit number, multiply first by the units, then the tens and
finally the hundreds.
WORKED Example 4
Calculate 6308 × 265.
THINK
1
2
3
4
5
6
Write the numbers one under the other.
Multiply by the units digit (× 5).
Write a zero and multiply by the tens
digit (× 6).
Write two zeros and multiply by the hundreds
digit (× 2).
Add the three answers.
Check the answer by rounding.
Note: The symbol ≈ means approximately.
WRITE
11
63
×
1
44 4
08
2
6
11111 1 15
1 1311151410
131718141810
1121611161010
1161711161210
6308 × 265 ≈ 6000 × 300 = 1 800 000,
which is close to the worked answer.
Chapter 1 Number skills
5
WORKED Example 5
Calculate 687 ÷ 9, rounding the answer to 2 decimal places.
THINK
1
2
3
WRITE
Write the question as shown and divide,
adding zeros until one more than the
required number of decimal places has been
worked.
Write the question and answer,
rounding the answer to 2 decimal
places.
Check the answer by rounding.
9)6756.131313
9)6857.303030
687 ÷ 9 ≈ 76.33 (2 decimal places)
687 ÷ 9 ≈ 700 ÷ 10 = 70, which is
close to the worked answer.
Remember that ≈ means ‘is approximately equal to’. In worked example 5, 76.33 is an
approximate answer because .003 3333 . . . is being left off. An exact answer to this
question would be 76.3̇, using the dot above the three to indicate that it is repeated infinitely.
These four operations are often used to solve worded problems.
WORKED Example 6
At a party 48 people each have 4 glasses of soft drink.
How many glasses of soft drink are consumed
altogether?
THINK
1
2
Read the question carefully and determine the
appropriate operation. Then write a mathematical
expression that will help to solve the problem.
Use the appropriate method to solve the problem.
WRITE
48 × 4
3 3
1 4 8
3
×
3
Write the answer in a sentence.
4
11912
192 glasses of soft drink were drunk
at the party.
6
Maths Quest 8 for Victoria
Order of operations
In the previous examples, the four basic operations of addition, subtraction, multiplication and division were revised. Mathematics, like music, is an international
language, so a few rules are needed to ensure that mathematicians find the same answer
to a given question. Mathematicians have agreed on some rules about the order in which
to do the four operations. An easy way to help remember the order is written below.
This means that any part in brackets must be calculated first, followed by any of
parts of the question. After that, division and multiplication must be done in the order
that they appear in the question from left to right, and finally addition and subtraction
must be worked from left to right.
BODMAS reminds us of the order to work operations.
WORKED Example 7
Evaluate 36 + 6 × 3.
THINK
1
2
3
WRITE
Write the question.
Use BODMAS to decide which
operation to perform first and then
calculate. (Multiplication)
Complete the question. (Add)
36 + 6 × 3
= 36 + 18
= 54
WORKED Example 8
Evaluate 21 + (16 − 5) × (12 ÷ 3) − 4.
THINK
1
2
3
4
WRITE
Write the question.
Use BODMAS to decide which
operation to perform first then
calculate. (Brackets)
Work the next part. (Multiplication)
Work the last part. (Addition and
Subtraction)
21 + (16 − 5) × (12 ÷ 3) − 4
= 21 + 11 × 4 − 4
= 21 + 44 − 4
= 61
remember
1. There are four basic operations:
+ addition
× multiplication
− subtraction
÷ division.
2. Use rounding to check answers to the nearest 10 or 100.
3. The symbol ≈ means ‘is approximately equal to’.
4. BODMAS reminds us that the order to work operations is:
Brackets, Of, Division and Multiplication from left to right, Addition and
Subtraction from left to right.
Chapter 1 Number skills
Example
Example
2
Example
3
Example
4
WORKED
Example
3 Calculate each of the following.
a 11 × 6
b 12 × 7
d 41 × 19
e 698 × 32
g 34 278 × 63
h 3732 × 89
c 25 × 18
f 7891 × 56
i 2745 × 47
4 Calculate each of the following.
a 123 × 245
b 546 × 172
d 3708 × 251
e 3254 × 393
Adding and
subtracting
whole numbers
1.2
SkillS
56 − 18
835 − 57
45 386 − 9034
11 274 − 4187
Multiplying
whole
numbers
1.3
SkillS
Rounding
decimals to
2 decimal
places
Math
c 6401 × 164
f 5137 × 416
L Spre
XCE ad
Tangle
tables
L Spre
XCE ad
sheet
5 Calculate each of the following, rounding the answer to 2 decimal places where
appropriate.
a 64 ÷ 4
b 357 ÷ 6
c 162 ÷ 7
d 890 ÷ 4
e 4785 ÷ 4
f 3692 ÷ 6
g 34 765 ÷ 5
h 325 ÷ 12
i 36 833 ÷ 16
j 87 906 ÷ 23
Basic
operations
sheet
5
c
f
i
l
SkillS
cad
WORKED
2 Calculate each of the following.
a 39 − 21
b 74 − 32
d 43 − 27
e 678 − 89
g 980 − 643
h 6790 − 548
j 76 943 − 31 397
k 10 657 − 3732
1.1
HEET
WORKED
31 + 28
46 + 75
21 567 + 3897
789 + 9086 + 67
6904 + 5789 + 32 027
HEET
WORKED
b
d
f
h
j
HEET
1
1 Calculate each of the following.
a 12 + 17
c 52 + 39
e 4507 + 3243
g 54 + 67 + 908
i 43 890 + 2143 + 78 + 8906
E
WORKED
Whole number operations
E
1A
7
E
Example
8 Stephen was organising a ‘Guess the number of jelly beans’ competition for his
school fete. He put 2347 jelly beans in the jar, but when he wasn’t looking, his little
brother and his friends ate 343 of the jelly beans. How many were left?
10 If each member of a class of 24 students brings in $5 as a donation to a charity, how
much money would the class have raised?
program
GC
Tables
am
progr –C
Tables
asio
9 For their outstanding results in a recent Science project, a group of six students was
given a bag containing 102 Smarties. If they were divided up evenly, how many would
each student receive?
(DIY)
–TI
6
7 The classrooms at Straight Line Secondary College have 13 rows of desks with 5
desks in each row. How many desks are there in each classroom?
GC
WORKED
sheet
6 Calculate each of the following, rounding the answer to 2 decimal places where The four
operations
appropriate.
a 357 ÷ 6
b 284 + 8764
c 8386 × 11
L Spre
XCE ad
d 647 − 32
e 937 ÷ 12
f 1206 + 257 + 7865
g 365 + 422 + 1849
h 473 × 13
i 978 × 12
The four
j 541 ÷ 12
k 13 861 − 3139
l 86 × 132
operations
8
Maths Quest 8 for Victoria
11 In a particular year level there are 184 students. If 98 are boys, how many are girls?
c (51 + 5) ÷ 7
g 7 × ( 1--- of 20)
2
k 11 + 6 ÷ 2
13 Evaluate the following.
a 64 + ( 1--- of 18) − 5
8
2
c 72 − 8 × 3 ÷ (11 − 5)
e 8 + (5 × 4) − (12 ÷ 6)
g 24 + 11 − 3 + 4 − 16
i 13 × 2 + 19 − ( 1--- of 20)
b
d
f
h
j
Example
Order of
operations I
d (36 − 23) × 2
h 18 + 24 ÷ 2
l 15 − 6 × 2
WORKED
d
Example
Mat
Order of
operations
2
36 ÷ (7 + 2) × 11
72 ÷ 8 × 6 + 11
125 − (6 × 7) + 11
8 + 16 × 3 − 5 + 45 ÷ 9
1
- of 360) ÷ (42 ÷ 7) + 5 × 6
( ----10
14 multiple choice
a 4 + 8 × 3 − 20 ÷ 4 + 6 is equal to:
A 10
B 37
C8
b 55 − 9 × (6 − 3) + 50 ÷ 2 is equal to:
A 68
B 53
C 39
D 29
E 30
D 163
E 94
S
c
1
--2
d
e
f
g
h
36 × 2 + 4
36 − 24 ÷ 4 + 2
7 × 12 ÷ 2 × 9
3×7+4×2
84 − 21 ÷ 3 + 7
of 20 × 5
( 1--- of 20) × 5
2
36 × (2 + 4)
(36 − 24) ÷ 4 + 2
7 × (12 ÷ 2) × 9
3 × (7 + 4) × 2
(84 − 21) ÷ 3 + 7
QU EST
E
M AT H
15 State whether the use of brackets makes a difference to the answer in each of the pairs
of questions below. (Remember to use BODMAS.)
a 4+8×3
(4 + 8) × 3
b 27 − 6 ÷ 2
27 − (6 ÷ 2)
E
NG
hca
12 Evaluate the following.
a 56 + 13 − 6
b 3+8×6
7
e 28 ÷ 4 + 9
f 25 × (3 + 7)
i 75 − 100 ÷ 20 j 7 + 22 − 3
WORKED
1.4
SkillS
HEET
CH LL
A
1 Find the number which is as much less than 86 as it is more than 20.
2 Using five 3s and no other numbers, combine them using addition,
subtraction, multiplication, division and brackets as needed, to produce
each of the following values: 1, 2, 3, 4, 5, 6, 7, 8, 9.
(Hint: 1 = 3 − 3 ÷ 3 − 3 ÷ 3)
3 Show a quick way to find the value of 25 × (1958 + 1958 + 1958 + 1958)
without using a calculator.
4 A snail is climbing the stem of a plant that is 150 cm tall. Each day
from 8 am to 8 pm it climbs 20 cm, and each night from 8 pm to 8 am
it slides down 10 cm. Starting from ground level, how many hours will
it take for the snail to reach the top?
9
The New York Museum of Modern Art
discovered this after studying the Matisse
painting ‘Le Bateau’ for 47 days!
Chapter 1 Number skills
Answer the whole number questions
to find the puzzle answer code.
35
64
87
23
+ 68
S
421 – 307
=
6(16 –2 × 3) + 62
=
17
× 24
N
G
A
1026
– 748
1875
– 1549
7 × 23
=
W
P
I
1048 – 889
=
240 + 8 × 8
6
E
U
O
40 + 70 + 30 + 60
=
D
7
18
29
32
14
+ 36
132 + 8 × 7
2
I
12 1164
T
N
H
D
572 ÷ 4
=
52 + 28 + 17 + 33
=
16 × 8
=
7 882
21
46
32
20
+ 19
N
G
A
14
× 19
3(8 + 7 × 4)
=
582
– 397
774 =
9
W
S
I
278 143 326 277 266 128 408 114 185 104 130 122
136
161
159 200 126
97
138
94
86
108
10
Maths Quest 8 for Victoria
Special groups of numbers
Sometimes in mathematics there are terms (or words) that need to be learned so that
mathematicians all around the world can communicate and be sure of understanding
exactly what they all mean. Here are a few of these terms.
Factors
A factor is a whole number that divides exactly into another whole number, with no
remainder. A pair of numbers can have a highest common factor (HCF) or a lowest
common factor (LCF).
Multiples
A multiple of a whole number is found when that number is multiplied by another
whole number. A pair of numbers can have a lowest common multiple (LCM). Two
numbers can’t have a highest common multiple. Why not?
Prime numbers
A prime number is a number that has two factors only: 1 and the number itself. The
number 1 is not a prime number. Are there any even prime numbers?
Composite numbers
A composite number is any number (other than 1) that is not a prime number. A
composite number has more than two different factors. The number 1 is not a composite number.
WORKED Example 9
Find all the factors of 36.
THINK
1
2
Find the factor pairs of the number.
Remember that 1 and the number itself
are both factors.
List the factors in order from smallest
to largest.
WRITE
1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6
1, 2, 3, 4, 6, 9, 12, 18, 36
WORKED Example 10
Find the highest common factor (HCF) of 12 and 15.
THINK
1
2
3
List the factors of 12.
List the factors of 15.
Compare the lists to find the highest
factor that is in both lists and answer
the question.
WRITE
1, 2, 3, 4, 6, 12
1, 3, 5, 15
The HCF of 12 and 15 is 3.
Chapter 1 Number skills
11
WORKED Example 11
Find the lowest common multiple (LCM) of 6 and 4.
THINK
WRITE
1
List the first few multiples of 6.
6, 12, 18, 24, 30
2
List the first few multiples of 4.
4, 8, 12, 16, 24
3
Find the lowest common multiple or the
lowest multiple which is in both lists.
The LCM is 12.
One way of finding a common multiple is to multiply the two numbers together. but
this does not always give the lowest common multiple.
WORKED Example 12
List the numbers from 11 to 20 inclusive that are prime numbers.
THINK
WRITE
1
List all of the numbers from 11 to 20.
11, 12, 13, 14, 15, 16, 17, 18, 19, 20
2
Cross out all of the even numbers
because they have a factor of 2.
11, 12, 13, 14, 15, 16, 17, 18, 19, 20
3
List the factors of the remaining
numbers.
11
13
15
17
19
4
List the numbers that have only two
factors.
The prime numbers from 11 to 20 are 11, 13,
17 and 19.
1, 11
1, 13
1, 3, 5, 15
1, 17
1, 19
Squares and square roots
Squaring a number means multiplying the number by itself. One way to remember this
is to think about drawing the number as a square shape.
32 =
=9
3 squared can be written as 32.
3 × 3 = 9 can be written as 32 = 9.
52 =
= 25
5 squared can be written as 52.
5 × 5 can be written as 52 = 25.
12
Maths Quest 8 for Victoria
The inverse of squaring a number is to find its square root. The square root of a
number is the number that, when multiplied by itself, gives the original number. The
square root of 9 is 3 because 3 × 3 = 9 or 32 = 9.
If a square number is drawn as a square shape, as shown above, the square root is
the side length of the square. The symbol for finding the square root is
. For larger
square roots, use a calculator because it has a square root key on it.
WORKED Example 13
Evaluate 122.
THINK
1
2
WRITE
A number squared means multiply the number by
itself.
Evaluate, either mentally or using a calculator.
122 = 12 × 12
122 = 144
WORKED Example 14
Evaluate: a
64
b
72 (round to 2 decimal places).
THINK
WRITE
a Either use a calculator or compute in your head to
find what number multiplied by itself equals 64.
a
64 = 8
b
b
72 ≈ 8.485
1
2
Use a calculator to find what number multiplied
by itself equals 72. Write one more decimal
place than required.
Round to the required number of decimal
places.
≈ 8.49 (2 decimal places)
The answer to worked example 14 can be checked. The answer to 72 should be
bigger than 8 because 64 = 8 and it should be less than 9 because 81 = 9 . The
answer 8.49 is between 8 and 9.
remember
1. A factor is a whole number that divides exactly into another whole number,
with no remainder.
2. A multiple of a whole number is found when that number is multiplied by
another whole number.
3. A prime number is a number that has two factors only, 1 and the number itself.
The number 1 is not a prime number.
4. A composite number is any number (other than 1) that is not a prime number.
A composite number has more than two factors.
5. The number 1 is the exception. It is neither composite nor prime.
6. When a number is multiplied by itself, the answer is a square number.
7. The square root of a number is the number that multiplies by itself to give the
original number.
Chapter 1 Number skills
1B
WORKED
Example
Special groups of numbers
1 Find all the factors of 24.
1.5
SkillS
HEET
9
13
2 Complete the following table, listing all of the factors for each of the numbers.
Factors
Example
WORKED
Example
12
65
34
100
5
63
60
27
21
62
23
3
9
17
80
14
46
56
Math
Factors,
multiples,
prime and
composite
numbers
program
GC
HCF
and
LCM
am
progr –C
HCF
and
LCM
3 Find the highest common factor of 12 and 20.
4 Using the table in question 2 or otherwise, find the highest common factor for each of
the following pairs of numbers.
a 21 and 63
b 3 and 60
c 80 and 100
d 46 and 62
e 21 and 56
f 21 and 23
g 46 and 80
h 56 and 62
5 Find the lowest common multiple of 6 and 11.
1.6
SkillS
HEET
11
12
6 List the first six multiples of each of the following.
a 3
b 5
c 7
e 4
f 13
g 21
Multiples
d 10
h 22
7 List the numbers from 20 to 40 inclusive that are prime numbers.
8 Copy the table below into your workbook.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
9 List the numbers in the table from question 2 that are prime numbers.
L Spre
XCE ad
Sieve of
Eratosthenes
sheet
a Circle all of the prime numbers.
b Mark all of the composite numbers green.
c How many prime numbers are there between 1 and 50 inclusively?
E
WORKED
10
asio
10
4
Factors
–TI
Example
Number
cad
WORKED
Factors
GC
Number
14
Maths Quest 8 for Victoria
10 Find the lowest common multiple of 11 and 24.
11 List all the multiples of 3 from 20 to 50 inclusive.
12 Find the highest common factor of 120 and 80.
13 List the first 10 multiples of 5. What pattern do you notice?
14 With the exception of 2, prime numbers can end in only one of five digits. What are
they and why?
15 Find the highest common factor of 45 and 63.
16 Rewrite the following using the shorthand method.
a 4×4
b 12 × 12
d 20 × 20
e 34 × 34
d
Mat
hca
Squares
and
square
roots
EXCE
WORKED
Example
Squares
and
square
roots
(DIY)
WORKED
Example
14a
WORKED
Example
14b
Work
T
SHEE
c 112
g 342
d 142
h 2452
et
reads
L Sp he
17 Evaluate the following squares.
a 62
b 52
13
2
e 23
f 672
c 14 × 14
f 65 × 65
1.1
18 Evaluate the following.
a
9
b
36
c
49
d
81
e
121
f
576
g
100
h
900
i
1600
j
2500
k
6400
l
14 400
d
333
19 Evaluate the following, rounding the answer to 2 decimal places.
a
65
b
c
21
140
20 Evaluate the following.
a 32 + 42
b 122 − 72
c 5 2 + 100
d
e 152 − 82
f
64 + 400 – 9
144 + 11 2
1
1 Calculate 753 + 2073 + 32 + 3.
2 The result of 3023 × 13 is
.
3 Is the following statement true or false? 9328 ÷ 5 = 1865.6
4 List all the prime numbers from 40 to 50 inclusive.
5 List all the factors of 42.
6 Use BODMAS to calculate 40 + 3 × 18 − 15.
2
7 Evaluate --- of 99 × 5 − (27 ÷ 3).
3
8 Write the highest common factor of 8 and 10.
9 Write the lowest common multiple of 4 and 6.
1
10 Evaluate 132 + 0.032 − --- of 144 , rounding the answer to 2 decimal places.
3
Chapter 1 Number skills
15
Darts competition
THINKING
Have you played darts before? Different regions
20
on the dartboard score a different number of
points. The diagram shows the regions where you
double points (40)
can score double points or triple points.
20
There are a number of different games you can
triple points (60)
play with various rules for scoring. Jemma and
Michael are playing a game where you must throw
20
a double before you can start scoring. Each player
25
takes turns to throw 3 darts in each round.
50 (bullseye)
A player starts with a score of 301 and subtracts
their score obtained in each round until they reach
0. The winner is the person who reaches 0 first. The only condition is that the last
throw must land on a double score.
1 What is the highest score that can be obtained on the throw of one dart?
2 If Jemma’s first throw hit a double 13, the second hit 20 and the third hit triple
9, what is her overall score at the end of round 1? (Remember to start from
301.)
3 Michael’s 3 darts hit 18, 2 and double 16. What is his overall score at the end
of round 1?
4 Use the table on page 1 to calculate each person’s overall score at the end of
each round.
5 For Jemma to win in the next round, she needs to finish with a double. List
three different sets of positions on the board that her darts must hit for her to
win in this round.
6 Repeat question 5 for Michael.
7 If you were playing and your overall score was 45, list 5 possible scenarios for
how you could win in the next round.
Addition and subtraction of fractions
A fraction has two parts: the top part, which is called the
numerator, and the bottom part, which is called the
denominator.
A proper fraction has a numerator that is less than the
denominator, such as 3--- .
5
An improper fraction has a numerator greater than the
2
3
Numerator
Denominator
This line — the vinculum —
means divide.
denominator, such as 7--- .
3
A mixed number contains a whole number part and a proper fraction part, such as
7 5--- .
8
16
Maths Quest 8 for Victoria
Equivalent fractions are fractions that are equal in value, for example,
1
--2
= 2--- .
4
When adding and subtracting fractions, the denominators must be the same.
WORKED Example 15
4 3
Simplify --- – --- .
5 4
THINK
WRITE
1
Write the question.
2
Find the lowest common denominator, that is, the
lowest multiple common to both.
3
Write both fractions with the same denominator,
that is, the lowest common denominator.
Subtract the fractions.
Write the answer.
4
5
4 3
--- – --5 4
4×4 3×5
= ------------ – -----------5×4 4×5
=
16
-----20
=
1
-----20
−
15
-----20
When adding and subtracting mixed numbers, they can be changed to improper
fractions first and then worked as shown in worked example 15.
WORKED Example 16
Calculate 2 2--- + 3 1--- .
3
2
THINK
WRITE
2 2--- + 3 1---
1
Write the question.
2
Change each mixed number to an improper fraction.
=
3
Write both fractions with the same denominator using
equivalent fractions.
8×2 7×3
= ------------ + -----------3×2 2×3
21
------ + -----= 16
3
2
8
--3
+
6
4
Add the fractions.
=
5
Write the answer as a mixed number if appropriate.
=
7
--2
6
37
-----6
6 1--6
A rough estimate can be found by adding or subtracting the whole number. For
example, 2 + 3 = 5, so 2 2--- + 3 1--- > 5.
3
2
17
Chapter 1 Number skills
WORKED Example 17
Calculate 4 1--- − 1 1--- .
5
2
THINK
WRITE
4 1--- − 1 1---
1
Write the question.
2
Change each mixed number to an improper fraction.
=
3
Write both fractions with the same denominator using
equivalent fractions.
21 × 2 3 × 5
= --------------- – -----------5×2
2×5
15
------ – -----= 42
4
Subtract the second fraction from the first.
=
5
Write the answer as a mixed number if appropriate.
=
5
2
21
-----5
–
10
27
-----10
7
2 ----10
3
--2
10
CASI
O
Graphics Calculator tip! Fractions
Fractions
▼
If using a graphics calculator, use the ÷ key to enter
fractions. Remember to end each calculation by
pressing MATH , selecting 1: Frac and pressing
ENTER . This gives an answer expressed as a fraction.
(Note: Mixed numbers will be shown as improper fractions.)
remember
1. To add or subtract fractions with the same denominator, add or subtract the
numerators.
2. To add or subtract fractions with different denominators, make the
denominators the same by using equivalent fractions and then add or subtract.
3. To add or subtract mixed numbers, change the mixed numbers to improper
fractions and then add or subtract.
1.7
1C
+
6
-----17
b
21
-----27
−
16
-----27
c
6
-----17
+
2
-----17
+
4
-----17
d
3
-----15
+
11
-----15
−
2
-----15
Converting
an improper
2 Simplify the following fractions, writing the answer as a mixed number if appropriate. fraction
into a
24
6
mixed
------ − ------ + -----a 3--- + 4--b 7--- + 3--c 7--- + 3--- + 6--d 41
5
5
8
8
8
8
8
50
50
50
number
SkillS
HEET
3
-----17
Simplifying
fractions
1.8
1 Simplify the following fractions, working from left to right.
a
HEET
Addition and subtraction of
fractions
SkillS
18
1.9
WORKED
Example
15
SkillS
HEET
Maths Quest 8 for Victoria
Finding and
converting
to the
lowest
common
denominator
EXCE
et
Adding and
subtracting
fractions
Mat
d
hca
Adding and
subtracting WORKED
fractions Example
16
SkillS
1.10
Adding and
subtracting
fractions I
WORKED
Example
17
SkillS
1.11
HEET
Adding and
subtracting
fractions II
a
2
--5
+
1
--4
b
3
--4
e
3
--4
+
5
--6
f
9
-----10
5
--8
+
−
2
--7
c
6
-----10
−
2
--5
g
8
-----12
+
5
--7
1
--2
−
d
8
-----25
+
34
-----50
h
21
-----30
+
5
--6
−
+
7
-----25
9
-----10
4 Simplify and evaluate. (Remember to write your answers as mixed numbers.)
reads
L Sp he
HEET
3 Simplify the following.
a
7
--3
d
28
-----36
g
4
--7
j
20
-----8
−
6
--4
m
17
-----10
−
6
--5
+
6
--3
11
-----36
−
+
16
-----7
3
--7
−
3
-----15
+
b
16
-----5
−
8
--5
c
54
-----23
+
21
-----23
e
7
-----12
+
14
-----12
f
21
-----24
+
6
-----24
h
125
--------60
i
5
--4
k
15
-----9
+
7
--6
l
14
-----3
n
35
-----24
+
7
--6
−
12
-----60
+
−
34
-----60
9
-----12
−
+
9
-----24
3
--2
+
+
8
--5
5
--3
5 Calculate the following.
a 2 --3- + 4 --15
5
e
6 7--8
+
4 3--8
i
4 3--4
+
5 1--6
b 6 --7- − 3 --5f
9
9
1 4--9
5 5--9
+
c 8 --4- − 4 --1-
2
3 ----12
+
5
5
g
6 1--4
+
3 2--8
j
1 2--5
+
3 1--3
d 8 --5- − 4 --16
h
−
12 2--5
6
+ 8 7--9
4
2 ----15
6 Calculate the following.
3
a 5 3--- − 2 ----5
b 6 1--- − 3 5---
10
7 If Mary eats
2
5
--8
c 10 1--- − 5 2---
6
4
d 4 1--- − 2 3---
5
of a block of chocolate for afternoon tea and
8
3
--8
5
after dinner, how much
of the block has she eaten altogether?
8 Seven bottles of soft drink were put out onto the table at a birthday party. How much
soft drink was left over after 5 --2- bottles were consumed?
9
9 Frances has a part-time job delivering
newspapers. If she spends
magazines and
2
--5
1
--3
of her pay on
on CDs, what fraction
of her pay does she have left over?
10 In my class,
1
--3
bikes to school,
of the students ride their
1
--4
catch the bus and the
rest get a lift. What fraction of my class
get a lift to school?
11 A Year 8 class organised a cake stall to
raise some money. If they had 10 whole
cakes to start with, sold 2 3--- cakes at recess
4
and
5 7--8
at lunchtime, how much cake was
left over?
Chapter 1 Number skills
19
Multiplication and division of fractions
To multiply fractions, multiply the numerators and multiply the denominators. It does
not matter if the denominators are different. Change the mixed numbers to improper
fractions before multiplying.
WORKED Example 18
Simplify 2 1--- × 1 5--- .
4
7
THINK
1
2
3
4
WRITE
Write the question.
2 1--- × 1 5--4
7
9 12 3
= ----1- × -------7
4
Change the mixed numbers to improper fractions and
cancel if possible.
Multiply the numerators and then multiply the
denominators.
Change to a mixed number and simplify if appropriate.
=
27
-----7
= 3 6--7
Division is the same as multiplying by the second fraction turned upside down.
WORKED Example 19
Find 2 1--- ÷ 3--- .
4
8
THINK
WRITE
1
Write the question.
2
Change mixed numbers to improper fractions.
3
Change ÷ to × and tip the second fraction, (× and tip), and
cancel if appropriate.
4
Multiply the numerators and then multiply the
denominators.
Simplify if appropriate.
5
2 --1- ÷
4
3
--8
9 3
= --- ÷ --4 8
93 82
= ----1- × ----14 3
3 2
= --- × --1 1
=6
remember
1. To multiply fractions:
(a) change mixed numbers to improper fractions
(b) cancel if appropriate
(c) multiply numerators and multiply denominators
(d) change the answer to a mixed number if appropriate and simplify.
2. To divide fractions:
(a) change mixed numbers to improper fractions
(b) change the ÷ to × and tip the second fraction (× and tip)
(c) cancel if appropriate
(d) multiply numerators and multiply denominators
(e) change the answer to a mixed number and simplify if appropriate.
20
Maths Quest 8 for Victoria
Multiplication and division of
fractions
1D
1 Simplify the following.
1.12
SkillS
HEET
Converting
a mixed
number
into an
improper
fraction
1.13
WORKED
Example
18
SkillS
HEET
Multiplying
and dividing
fractions
3
--4
×
1
--2
b
1
--8
×
1
--7
c
2
--5
e
1
--2
×
5
--6
f
3
--7
×
7
--9
g
11
-----20
i
5
--8
×
11
-----20
j
2
--3
×
9
-----10
k
6
--7
×
d
5
--7
×
1
--3
2
--3
h
1
--3
×
3
--5
14
-----15
l
5
--6
×
3
-----10
2
1
- × 1 --b 1 -----
c 2 2--- × 1 1---
d 3 2--- × 2 1---
9
- ×
e 8 -----
f 5 3--- × 2 2---
2
5
2
2 1--6
et
EXCE
×
3
--5
a 3 1--- × 1 3---
g 6×
Multiplying
and dividing
fractions
×
2 Simplify the following.
4
reads
L Sp he
10
5
7
-----10
10
h
1 3--5
5
--8
×
3
2
4
5
i
4 3--4
× 2 1---
÷
2
3 Simplify the following.
d
Mat
hca
a
Multiplying
and dividing
fractions
WORKED
Example
19
a
1
--3
÷
1
--2
b
7
--8
÷
3
--2
c
4
-----14
d
2
--5
÷
1
--4
e
3
--4
÷
7
--8
f
5
--6
÷
8
--9
g
12
-----15
h
1
--5
÷
10
-----12
i
3
--4
÷
3
--8
4
--3
÷
1
--3
4 Find the following.
6
3
- ÷ 1 --a 1 -----
b 3 5--- ÷ 2 1---
c 1 1--- ÷
d 1 5--- ÷
1
--3
e 1 1--- ÷
f
7
--9
3 1--2
1 3--5
i
7 8--9
b 1 2--- ÷ 3 1---
c
8
--9
e 5 5--- × 3 2---
f 22 2--- ÷ 2 6---
10
5
7
g
÷
7
6
3
h
10 4--5
5
--6
÷
2 1--2
6
2
--1
7
÷ 1 ----18
÷ 7 1--2
5 Simplify the following.
a
3
--4
×
8
--9
5
2
d 2 1--- ×
7
--8
g
2 1--6
÷
2 3--5
h 12 ÷
j
7
-----12
÷
5
-----18
k 3 1--- × 5 2---
m
9
-----11
4
6 Find
÷3
3
--4
6
n
3
1 1--5
3
3
2 2--3
4 2--3
×
÷ 1 1--3
3
i
4 5--6
9
×7
l 4 1--- ÷ 2 3--8
o 4÷
4
1
--3
of 16. (‘of’ has the same meaning as multiplying.)
7 An assortment of 75 lollies is to be divided evenly among 5 children.
a What fraction of the total number of lollies will each child receive?
b How many lollies will each child receive?
Chapter 1 Number skills
21
8 Sam has been collecting caps from all around the world. If he has a total of 160 caps
and 1--- of them are from the USA, how many non-US caps does he have?
5
9 In the staffroom there is 7--- of a cake left over from a meeting. If 14 members of staff
8
would all like a piece, what fraction will they each receive?
GAME
time
Number
skills
— 001
10 The Year 8 cake stall raised $120. If they plan to give 1--- to a children’s charity and 2--4
3
to a charity for the prevention of cruelty to animals, how much will each group
receive and how much is left over?
2
1 Calculate 6790 – 54 + 283.
2 Is the following statement true or false? 12 + (5 × 9) − (108 ÷ 2) = 3
3 Write all the common factors of 32 and 40.
4 Write down the lowest common multiple of 5 and 15.
5 Write the highest common factor of 15 and 60.
6 Evaluate 0.22 +
1
--2
× ( 50 − 1) × 3, correct to 2 decimal places.
7 A shop sold 1372 newspapers at $1.30 each. How much money was taken?
8 Simplify 1 2--- +
5
3
--4
−
6
------ .
10
9 Write 2 fractions that, when they are multiplied together, give a result of 6 2--- .
3
10 What are the missing numbers?
4
--9
÷
?
--3
=
1
--?
22
Maths Quest 8 for Victoria
THINKING
Nude and cute numbers
A ‘nude’ number is a number whose digits are factors of the number. The number
1 is a factor of all numbers and so is not considered in this definition. An example
of a nude number is 24 as both 2 and 4 are factors of 24.
A number is called ‘cute’ if it has exactly 4 factors including the number itself.
For example, 10 is a cute number as it has the factors 1, 2, 5 and 10.
1 What is the smallest 2-digit nude number?
2 List all the 2-digit nude numbers.
3 What is the smallest cute number?
4 What is the smallest number that is both cute and nude?
5 How many cute numbers are less than 100?
6 Give an example of a square number that is nude.
7 Are there any square numbers that are cute? Can you explain your answer?
8 List 4 cubes that are cute. Cubes are numbers like 1 (= 13 = 1 × 1 × 1),
8 (= 23 = 2 × 2 × 2), 27 (= 33 = 3 × 3 × 3) and so on.
Fractions to decimals, decimals to
fractions
It is useful to be able to convert fractions to decimals and vice versa. Divide the
numerator of the fraction by the denominator and round the answer to 2 decimal places
if it is not otherwise specified.
WORKED Example 20
Convert the following fractions to decimals, giving answers correct to 2 decimal places
where appropriate.
a
1
--5
5
b 4 ----12
THINK
WRITE
a
a
1
Write the question.
2
Rewrite the question using division.
Divide, adding zeros as required.
3
4
Write the answer.
1
--5
=1÷5
0.2
5)1.0
1
--- = 0.2
5
Chapter 1 Number skills
THINK
WRITE
b
5
b 4 -----
1
Write the question.
2
Convert mixed numbers to improper fractions.
=
3
Rewrite the question using division.
= 53 ÷ 12
4
Divide, adding three zeros so that the answer is
initially to 3 decimal places.
4. 4 1 6
12)53.502080
5
Round the answer to 2 decimal places.
5
- ≈ 4.42
4 -----
23
12
53
-----12
12
A calculator can also be used to convert fractions to decimals by entering 53 ÷ 12 and
rounding the answer correctly.
When changing a decimal to a fraction, rewrite the decimal as a fraction with the
same number of zeros in the denominator as there are decimal places in the question.
Simplify the fraction by cancelling. If the decimal has a whole number part, it is easier
to write it in expanded form.
2.365 = 2 + 0.365
WORKED Example 21
Convert the following decimals to fractions in simplest form.
a 0.25
b 1.342
THINK
WRITE
a
a 0.25
b
1
Write the question.
2
Rewrite as a fraction with the same
number of zeros in the denominator
as there are decimal places in the
question. Simplify the fraction by
cancelling.
3
Write the answer.
1
Write the question.
2
Rewrite the decimal in expanded
form.
3
Write as a mixed number with the
same number of zeros in the
denominator as there are decimal
places in the question and cancel.
4
Write the answer.
25 1
= ----------4100
=
1
--4
b 1.342
= 1 + 0.342
342 171
= 1 + -----------------1000 500
171
= 1 -------500
24
Maths Quest 8 for Victoria
remember
1. When changing fractions to decimals, divide the numerator of the fraction by
the denominator and round the answer to 2 decimal places if it is not otherwise
specified.
2. When changing a mixed number to a decimal, write it as an improper fraction
before dividing.
3. When changing a decimal to a fraction, rewrite the decimal as a fraction with
the same number of zeros in the denominator as there are decimal places in the
question.
4. Simplify the fraction by cancelling.
1E
Mat
d
hca
Fractions
to decimals,
decimals to
fractions
WORKED
Example
20a
EXCE
et
reads
L Sp he
Converting
fractions to
decimals
WORKED
Example
reads
L Sp he
EXCE
et
20b
1 Convert the following fractions to decimals, giving exact answers or correct
to 2 decimal places where appropriate.
a
4
--5
b
1
--4
c
3
--4
d
5
-----12
e
9
-----11
f
21
-----25
g
7
--4
h
13
-----6
i
7
-----15
j
2
--3
2 Convert the following mixed numbers to decimals, correct to 2 decimal places.
a 1 --5-
b 1 --3-
6 3--4
5 4--7
6
f
Converting
decimals to
fractions
Fractions to decimals,
decimals to fractions
c 3 --2-
4
g
5
h
-----11 11
15
d 8 --45
i
6 1--2
3 Convert the following decimals to fractions in simplest form.
a 0.4
b 0.8
c 1.2
d 3.2
21
f 0.75
g 1.30
h 7.14
i 4.21
k 1.333
l 8.05
m 7.312
n 9.940
p 84.126
q 73.90
r 0.0042
9
e 12 -----
10
j
4 1--3
WORKED
Example
GC p
am –
rogr TI
Converting
fractions to
decimals
GC p
sio
am–
rogr Ca
Converting
fractions to
decimals
4 Of the people at a school social,
3
--4
e 5.6
j 10.04
o 12.045
were boys. Write this fraction as a decimal.
5 Alfonzo ordered a pizza to share with three friends, but he ate 0.6 of it. What
fraction was left for his friends?
6 Alison sold the greatest number of chocolates in her Scout troop. She sold 5--- of
9
all chocolates sold by the troop. Write this as a decimal, correct to 2 decimal places.
7 On a recent Science test, Katarina worked the bonus question correctly as well
--------- . What is this as a decimal value?
as everything else, and her score was 110
100
Work
T
SHEE
1.2
8 The opposition leader’s approval rating was 0.35. Write this decimal as a fraction.
9 Stephanie decided to place
4
--7
of her weekly pay into her savings account. Write this
fraction as a decimal, correct to 2 decimal places.
Chapter 1 Number skills
25
Addition and subtraction of decimals
Adding and subtracting decimals is a very useful skill, particularly when working with
money. When adding and subtracting decimals, be sure that the decimal points are lined
up one underneath the other.
WORKED Example 22
Find 4.622 + 38 + 210.07 + 21.309.
THINK
WRITE
Write the numbers one underneath the
other with the decimal points lined up and
fill the spaces with zeros. Then add as for
whole numbers putting the decimal point in
the answer directly under the decimal
points in the question.
4 .6 2 2
3 8 .0 0 0
2 1 0 .0 7 0
+ 2111.31019
2 7 4 .0 0 1
WORKED Example 23
Find 37.6 − 12.043.
THINK
WRITE
Write the numbers one under the other with
the larger number on top and the decimal
points lined up. Add in the required zeros
and subtract using the method shown.
37.569010
− 12. 0 4 3
25. 5 5 7
remember
When adding and subtracting decimals, be sure that the decimal points are lined
up one underneath the other.
1F
Addition and subtraction of
decimals
HEET
1.14 SkillS
Example
b
d
f
h
j
l
n
7.2 + 5.8
7.9 + 12.4
5.34 + 2.80
5.308 + 33.671 + 3.74
5.67 + 3 + 12.002
306 + 5.2 + 6.032 + 76.9
34.2 + 7076 + 2.056 + 1.3
Adding
and
subtracting
decimals
Math
cad
1 Find the following.
a 8.3 + 4.6
22
c 16.45 + 3.23
e 13.06 + 4.2
g 128.09 + 4.35
i 0.93 + 4.009 + 1.3
k 56.830 + 2.504 + 0.1
m 25.3 + 89 + 4.087 + 7.77
WORKED
Adding
and
subtracting
decimals
26
2 Find the following.
a 4.56 − 2.32
23
d 63.872 − 9.051
g 87.25 − 34.09
j 35 − 8.97
WORKED
Example
et
EXCE
reads
L Sp he
Adding
decimals
EXCE
et
reads
L Sp he
b
e
h
k
c
f
i
l
19.97 − 12.65
43.58 − 1.25
125.006 − 0.04
42.1 − 9.072
124.99 − 3.33
1709.53 − 34.6
24.86 − 1.963
482 − 7.896
3 multiple choice
Adding
decimals
(DIY)
a The difference between 47.09 and 21.962 is:
A 17.253
B 26.93
C 25.932
b The sum of 31.5 and 129.62 is:
A 98.12
B 161.12
C 150.12
et
reads
L Sp he
EXCE
Maths Quest 8 for Victoria
Subtracting
decimals
4 Calculate the following.
a 56.3 + 52.09 + 6.7
c 908.52 − 87.04
e 1495.945 − 2.07
g 7.286 + 5.4 + 2.083 + 1538.82
i 603.9 − 5.882
b
d
f
h
j
D 26.128
E 25.128
D 444.62
E 132.77
7.9 + 3 + 21.053
53.091 + 6 + 1895.2
439.98 − 6
12.784 − 3.9
3965.09 + 3.2 + 256 + 0.006
5 Round to the nearest whole number to find an approximate answer to the following.
a 33.2 + 4.8 − 10.5
b 59.62 − 17.71 + 3.6
c 29.5 − 15.3 + 5.7
d 99.9 + 35.3 − 5.5
6 a On a recent shopping trip, Salmah spent the following amounts:
$45.23, $102.78, $0.56 and $8.65. How much did he spend
altogether?
b If Salmah started with $200.00, how much did he
have left after the trip?
7 Dagmar is in training for the school athletic carnival.
The first time she ran the 400 m it took her 87.04
seconds. After a week of intensive training she had
reduced her time to 75.67 seconds. By how much
had she cut her time?
8 Kathie runs each morning before school. On Monday
she ran 1.23 km, on Tuesday she ran 3.09 km, she rested
on Wednesday, and on both Thursday and Friday she ran
2.78 km. How many kilometres has she run for the week?
Multiplication and division of decimals
Multiplication
The method for multiplying decimals is almost the same as for multiplying whole
numbers. Ignore the decimal point when multiplying, count the number of digits after
the decimal point in each of the multiplying numbers, and then add these numbers
together to find the number of decimal places in the answer.
It is often a good idea to use your estimating skills with decimal multiplication to
check that the answer makes sense.
Chapter 1 Number skills
27
WORKED Example 24
Calculate, giving an exact answer, 125.678 × 0.23.
THINK
1
2
WRITE
1 1 11
1 2 22
Write the numbers with the larger one on top.
Multiply, starting with the last digit and ignoring
the decimal point.
Count the number of digits after the decimal
point in each of the multiplying numbers and use
this total as the number of decimal places in the
answer. There are 3 decimal places in 125.678
and 2 in 0.23 so there will be 5 decimal places in
the answer.
12 5678
× 023
37 7034
2511 3560
289 0594
125.678 × 0.23 = 28.905 94
Division
When dividing decimals, make sure that the divisor (the number being divided by) is a
whole number. If the divisor is not a whole number, either:
1. write the question as a fraction and multiply the numerator and the denominator by
an appropriate multiple of 10 or
2. multiply both parts of the question (dividend and divisor) by an appropriate multiple
of ten.
Then set out the question as for division of whole numbers and divide as for whole
numbers, placing the decimal point in the answer directly in line with the decimal point
in the question.
WORKED Example 25
Calculate: a 54.6 ÷ 8
b 89.356 ÷ 0.06.
Give answers correct to 2 decimal places.
THINK
WRITE
6. 8 2 5
a
1
2
Write the question as shown, adding zeros to
one more decimal place than is required.
Write the decimal point in the answer directly
above the decimal point in the question and
divide as for short division.
Write the question and answer, rounded to the
required number of decimal places.
a
8)54.662040
54.6 ÷ 8 ≈ 6.83 (2 decimal places)
Continued over page
28
Maths Quest 8 for Victoria
THINK
WRITE
b
b 89.356 ÷ 0.06
1
Write the question.
2
Multiply both parts by an appropriate multiple
of 10 so that the divisor is a whole number. (In
this case, 100.)
3
4
Divide, adding zeros to one more decimal
place than required. Write the decimal point in
the answer directly above the decimal point in
the question and divide as for short division.
Write the question and answer, rounded to the
required number of decimal places.
= (89.356 × 100) ÷ (0.06 × 100)
= 8935.6 ÷ 6
1 4 8 9. 2 6 6
6)8295355.164040
89.356 ÷ 0.06 ≈ 1489.27
(2 decimal places)
Again, rounding to the nearest whole number can be useful when finding how much
material is required to complete a task.
remember
1. When multiplying decimals, count the number of digits after the decimal point
in each of the numbers being multiplied and add these together to find the total
number of decimal points in the answer.
2. When dividing, make sure that the divisor is a whole number.
3. When the divisor is a decimal, make it a whole number either by:
(a) writing the question as a fraction and multiplying the numerator and
denominator by a multiple of 10 or
(b) multiplying both dividend and divisor by an appropriate multiple of ten.
4. When dividing decimals by a whole number, place the decimal point in the
answer directly in line with the decimal point in the question.
1G
1.15
Example
Multiplying
and dividing
decimals
EXCE
et
reads
L Sp he
1 Calculate the following giving an exact answer.
a 6.2 × 0.8
b 7.9 × 1.2
24
d 109.5 × 5.6
e 5.09 × 0.4
g 123.97 × 4.7
h 576.98 × 2
j 0.6 × 67.9
k 23.4 × 6.7
m 52.003 × 12
n 22.97 × 0.015
WORKED
SkillS
HEET
Multiplication and division
of decimals
Multiplying
decimals WORKED
c
f
i
l
o
65.7 × 3.2
32.76 × 2.4
3.4 × 642.1
0.006 × 43.6
13.42 × 0.011
2 Calculate the following. Give answers correct to 2 decimal places.
a 43.2 ÷ 7
b 523.9 ÷ 4
c 6321.09 ÷ 8
25a
d 286.634 ÷ 3
e 76.96 ÷ 12
f 27.8403 ÷ 11
Example
Chapter 1 Number skills
29
Example
Math
cad
4 Evaluate the following, giving the answer correct to 1 decimal place.
a 4.6 × 2.1 + 1.2 × 3.5
b 5.9 × 1.8 − 2.4 × 3.8
c 6.2 + 4.5 ÷ 0.5 − 7.6
d 11.4 − 7.6 × 1.5 + 2
Multiplying
and dividing
decimals
5 multiple choice
E
L Spre
XCE ad
1
--- (3.6
2
+ 1.4 × 7.5) is equal to:
A 18.75
B 14.1
b Rounded to 2 decimal places,
A 1.06
B 6.57
C 9.375
3
--- (10.5
4
D 7.05
E 28.2
Dividing
decimals
− 5.8 ÷ 4 × 1.2) is equal to:
C 0.73
D 11.68
E 2.19
6 Round each of the following to the nearest whole number to find an estimate.
a 3.5 × 24.9 + 33.2
b 4.8 × 19.6 − 10.4
c 15.6 + 50.1 × 9.5 − 15.4
d 49.8 − 20.3 ÷ 4.7
GAME
time
Number
7 A group of 21 Year 8 students were going on an excursion to the planetarium. If the skills
— 002
total cost is $111.30, how much would each student have to pay?
Estimation
It is always wise to make an estimate of a calculation to check whether the answer is
appropriate. These estimates generally involve rounding the digits to a particular place
value before conducting the estimate.
Clustering around a common value
Worked example 26 illustrates an estimation technique that can be employed when a
basic calculation involving similar values is required.
WORKED Example 26
Marilyn and Kim disagree about the answer to the following calculation: 7.3 + 7.1 + 6.9 +
6.8 + 7.2 + 7.3 + 7.4 + 6.6. Marilyn says the answer is 56.6, but Kim thinks it is 46.6. Obtain
an estimate for the calculation and determine who is correct.
THINK
1
2
3
WRITE
Carefully analyse the values and devise
a method to estimate the total.
Perform the calculation using the
rounded numbers.
Answer the question.
sheet
a
SkillS
HEET
3 Calculate the following. Give answers correct to 2 decimal places, where appropriate.
1.16
a 53.3 ÷ 0.6
b 960.43 ÷ 0.5
c 21.42 ÷ 0.004
25b
d 3219.09 ÷ 0.006
e 478.94 ÷ 0.016
f 76.327 ÷ 0.000 08
Dividing a
g 25.865 ÷ 0.004
h 26.976 ÷ 0.0003
i 0.0673 ÷ 0.0005
decimal by a
decimal
j 12.000 53 ÷ 0.007
k 35.064 ÷ 0.005
l 0.059 ÷ 0.009
WORKED
Each of the values can be approximated to 7
and there are 8 values.
7 × 8 = 56
Marilyn is correct because the approximate
value is very close to 56.6.
We will now consider the process involving whole numbers, rather than decimals,
although a similar process follows for decimals.
30
Maths Quest 8 for Victoria
Rounding, rounding up, rounding down
It is important to understand the difference between the terms rounding, rounding up
and rounding down. Rounding was discussed in Year 7. When rounding to a given place
value, the procedure applied is as follows:
• If the next lower place value digit is less than 5, leave the given place value digit as
it is and add zeros to all lower place values, if necessary.
• If the next lower place value digit is 5 or greater, increase the given place value digit
by 1 and add zeros to all lower place values, if necessary.
This means that if you are rounding 25 354 to the nearest thousand, the answer would
be 25 000 (3 in the hundreds place is less than 5). Rounding to the nearest hundred
would give the number 25 400 (5 in the tens position is in the category 5 or greater),
while rounding to the nearest ten would be 25 350 (4 in the units place is less than 5).
When we round up to a given place value, the digit in the desired place value is
increased by 1 regardless of the digits in the lower place positions (as long as they are
not all zeros). Zeros are added to the lower place positions to retain the place value. So,
rounding up the number 3176 to the nearest hundred would produce the number 3200.
When we round down to a given place value, all digits following the desired place
value are replaced by zeros, leaving the digit in the given place position unchanged. So,
rounding down the number 632 to the nearest ten would give the number 630.
WORKED Example 27
Consider the number 39 461 and perform the following.
a Round to the nearest thousand.
b Round up to the nearest hundred.
c Round down to the nearest ten.
THINK
WRITE
a
Consider the digit in the thousands
place position and the digit in the
next lower place position.
Write the answer, adding the
required number of zeros.
a The digit 9 lies in the thousands position.
The digit 4, which is less than 5, lies in the
hundreds position.
The number 39 461 rounded to the nearest
thousand is 39 000.
1
Consider the digit in the hundreds
place position and the digit in the
next lower place position.
2
Write the answer, adding the
required number of zeros.
b The digit 4 lies in the hundreds position. The
digit 6 lies in the tens position. When
rounding up to the nearest hundred, the 4
will increase to 5.
The number 39 461 rounded up to the
nearest hundred is 39 500.
1
Consider the digit in the tens place
position and the digit in the next
lower place position.
2
Write the answer, adding the
required number of zeros.
1
2
b
c
c The digit 6 lies in the tens position. The digit
1 lies in the units position. When rounding
down to the nearest ten, the 1 will be converted to 0.
The number 39 461 rounded down to the
nearest ten is 39 460.
Chapter 1 Number skills
31
Rounding to the first digit
In estimating answers to calculations, sometimes it is simplest to round all numbers in
the calculation to the first digit and then perform the operation.
WORKED Example 28
Provide an estimate to the following calculations by first rounding each number to its first
digit. Check your estimate with a calculator. Comment on the accuracy of your estimate.
692 × 32
a 394 + 76 – 121
b --------------------19 × 87
THINK
WRITE
a
Round each of the numbers to the
first digit.
Perform the calculation using the
rounded numbers.
Check using a calculator. Comment
on how the rounded result compares
with the actual answer.
a Rounded to the first digit, 394 becomes 400,
76 becomes 80 and 121 becomes 100.
394 + 76 – 121≈ 400 + 80 – 100
394 + 76 – 121≈ 380
Using a calculator, the result is 349. The
estimate compares well to the actual
(calculator) value.
1
Round each of the numbers to the
first digit.
2
Perform the calculation using the
rounded numbers.
b Rounded to the first digit, 692 becomes 700,
32 becomes 30, 19 becomes 20 and 87
becomes 90 35
1
692 × 32
700 × 30
--------------------- ≈ --------------------19 × 87
201× 90 3
35
≈ ------
1
2
3
b
3
3
Check using a calculator. Comment
on how rounded result compares
with actual answer.
≈ 12
Using a calculator, the result is 13.4
(rounded to 1 decimal place). The estimate
is very close to the actual (calculator) value.
The technique employed in worked example 28 may also be referred to as ‘leading
digit’ or ‘front-end estimation’.
Rounding the dividend to a multiple of the divisor
When performing division, an estimation technique that can be used is to round the
dividend to a multiple of the divisor. For example, in estimating the answer to
20 532 ÷ 7 we could round 20 532 (the dividend) to 21 000 (knowing that 21 is a
multiple of 7); then the division could be performed mentally to give an answer of
3000. This is a sound estimate for the calculated answer of 2933 (to the nearest whole
number).
Each of the estimation techniques may provide a slightly different answer to calculations. Let us illustrate this with a simple example.
32
Maths Quest 8 for Victoria
WORKED Example 29
Provide estimates for the calculation
537
--------40
by:
a rounding the dividend up to the nearest hundred
b rounding the dividend to the nearest ten
c rounding the dividend to a multiple of the divisor.
THINK
WRITE
a
a 537 rounded up to the nearest hundred is
600.
537
600 15
--------- ≈ --------40
401
≈ 15
b
c
1
Round the dividend up to the nearest
hundred.
2
Perform the division. Write the
estimation.
1
Round the dividend to the nearest
ten.
2
Perform the division. Write the
estimation.
1
Round the dividend to a multiple of
the divisor.
2
Perform the division. Write the
estimation.
b 537 rounded up to the nearest ten is 540.
537
540 27
--------- ≈ --------- 2
40
40
≈ 13.5
c 52 is a multiple of 4.
537 520 13
--------- ≈ --------40
401
≈ 13
Although each of these techniques gives a slightly different answer, they are all good
estimates of the division, which has an exact value of 13.425. Care must be taken to
ensure that the calculated estimate has the correct place value.
WORKED Example 30
132 × 77
The exact answer to --------------------- has the digits 1848. Use any estimation technique to locate
55
the position of the decimal point.
THINK
1
Round each of the numbers to the first
digit.
2
Perform the calculation using the
rounded numbers and write the estimate
ignoring the decimal.
3
Use the estimate obtained to locate the
position of the decimal point. Write the
correct answer.
WRITE
Rounded to the first digit, 132 becomes 100, 77
becomes 80 and 55 becomes 60.
132 × 77 100 × 80 4
--------------------- ≈ --------------------55
60 3
400
≈ --------3
≈ 133
The estimate gives an answer between 100 and
200. This indicates that the decimal point
should be between the last two digits. The
correct answer is 184.8.
Chapter 1 Number skills
33
It should be noted that any of the rounding techniques could have been used in worked
example 30. A different estimated value may have been obtained, but interpretation of
this estimated value will provide the same answer.
remember
1. Estimation is a method of checking the reasonableness of an answer or a
calculator computation.
2. Clustering around a common value can be employed when a basic calculation
involving similar values is required.
3. When estimating, numbers can be rounded, rounded up or rounded down.
4. Rounding involves increasing the value of the desired digit if the following
digit is 5 or greater. If the following digit is less than 5, the value of the desired
digit remains the same. Zeros are added to maintain the place value of the
number, if necessary.
5. When rounding up, the desired digit is increased by 1 irrespective of the digits
in the lower place value positions (as long as they are not all zeros). Zeros are
added to maintain the place value of the number, if necessary.
6. If rounding down, the desired digit remains unchanged, irrespective of the
digits in the lower place value positions. Zeros are added to maintain the place
value of the number, if necessary.
7. When rounding to the first digit, apply the process of rounding to the first digit
of the number.
8. Making the dividend a multiple of the divisor is another useful technique for
estimating an answer involving division.
1H
Example
WORKED
Example
27
to the first
(leading)
digit
Math
cad
2 For each of the following numbers:
iii round to the first digit
iii round up to the first digit
iii round down to the first digit.
a 239
b 4522
d 53 624
e 592
c 21
f 1044
3 Round each of the numbers in question 2 down to the nearest ten.
Example
28
5 Find an estimate for each of the following.
a 78 ÷ 21
b 297 + 36
d 235 + 67 + 903
e 1256 − 678
g 56 × 891
h 1108 ÷ 53
j 907 ÷ 88
k 326 × 89 × 4
m (426 + 1076) × 21
n 7 × 211 − 832
p (12 384 − 6910) × (214 + 67)
c
f
i
l
o
587 − 78
789 × 34
345 + 8906 − 23 + 427
2378 ÷ 109
977 ÷ 10 × 37
Estimation
L Spre
XCE ad
The four
operations
sheet
4 Round each of the numbers in question 1 up to the nearest hundred.
WORKED
SkillS
HEET
26
1 Marilyn and Kim disagree about the answer to the following calculation: 8.6 + 9.2 + 1.17
8.7 + 8.8 + 8.9 + 9.3 + 9.4 + 8.6. Marilyn says the answer is 81.5, but Kim thinks it is
71.5. Obtain an estimate for the calculation and determine who is correct.
Rounding
E
WORKED
Estimation
34
Maths Quest 8 for Victoria
6 Estimate the whole numbers between which each of the following will lie.
a 20
b 120
c 180
d 240
7 Complete the table below with the rounded question, the estimated answer and the
exact answer. The first one has been worked.
Question
WORKED
Example
29
WORKED
Example
30
a
789 × 56
b
124 ÷ 5
c
678 + 98 + 46
d
235 × 209
e
7863 − 908
f
63 × 726
g
39 654 ÷ 227
h
1809 − 786 + 467
i
21 × 78 × 234
j
942 ÷ 89
k
492 × 94
--------------------38 × 49
l
54 296
--------------------97 × 184
Rounded
question
Estimated
answer
Exact
answer
800 × 60
48 000
44 184
8 Provide estimates for each of the following by first rounding the dividend to a
multiple of the divisor.
a 35 249 ÷ 9
b 2396 ÷ 5
c 526 352 ÷ 7
d 145 923 ÷ 12
e 92 487 ÷ 11
f 5249 ÷ 13
9 Use any of the estimation techniques to locate the position of the decimal point in
each of the following calculations. The correct digits for each one are shown in
brackets.
369 × 16
42 049
a --------------------(205)
b -----------------(150175)
288
14 × 20
99 × 270
c --------------------1320
(2025)
285 × 36
d --------------------16 × 125
(513)
256 × 680
e -----------------------32 × 100
(544)
7290 × 84
f -----------------------27 × 350
(648)
10 If 127 people came to a school social and each paid $5 admission, find an estimate for
the amount of money collected.
35
11 Find an approximate answer to each of the worded problems below. Remember to write
your answer in a sentence.
a A company predicted that it would sell 13 cars in a month at $28 999 each. About
how much money would they take in sales?
b A tap was leaking 8 mL of water each hour. Approximately how many millilitres of
water would be lost if the tap was allowed to leak for 78 hours?
c The Year 8 cake stall sold 176 pieces of cake for 95 cents each. How much money
did they make?
d Steven swam 124 laps of a 50 m pool and, on average, each lap took him 47 seconds.
If he swam non-stop, for approximately how many seconds was he swimming?
e An audience of 11 784 people attended a recent Kylie concert at Rod Laver Arena
and paid $89 each for their tickets. How much money was taken at the door?
f A shop sold 4289 articles at $4.20 each. How much money was paid altogether?
g On Clean Up Australia Day, 19 863 people volunteered to help. If they each picked
up 196 pieces of rubbish, how many pieces of litter were collected altogether?
Binary numbers
Our number system is a decimal system based
on counting in ‘lots’ of 10s. It uses the digits 0,
1, 2, 3, 4, 5, 6, 7, 8 and 9. The binary number
system is based on counting in lots of 2s.
Under this system we can have only two digits:
0 and 1. This is a two-state system that can be
simulated in many ways — a light can be
switched on or off, a door can be open or
closed, an object can move clockwise or anticlockwise. Computers operate using a binary
system; they use a series of 0s and 1s to store
all numbers, letters and characters. The computer term ‘bit’ is short for binary digit.
T
SHEE
Work
Chapter 1 Number skills
1.3
36
Maths Quest 8 for Victoria
Converting from the decimal system to the binary
system
An easy way to convert a number from the decimal system to its equivalent in the
binary system is to repeatedly divide it by 2. The decimal number 23 is written as 2310
(and read as 23 to base 10) and 10111, its binary equivalent, is written as 101112
(10111 to base 2).
WORKED Example 31
a Convert 1910 into binary form.
b Convert 1210 into binary form.
THINK
WRITE
a
Divide 19 by 2, clearly displaying
the quotient (result) and remainder.
Divide the result obtained in step 1
by 2, again clearly displaying the
quotient and remainder.
Repeat the process outlined in step 2
until the quotient is 0.
Note: Write each remainder obtained
even if the remainder is 0.
Take all the remainders from the
bottom up and write them as the
binary number.
a 2 ) 19
2) 9
2) 4
2) 2
2) 1
2 0
Divide 12 by 2, clearly displaying
the quotient (result) and remainder.
Note: Write the remainder obtained
even if the remainder is 0.
Divide the result obtained in step 1
by 2, again clearly displaying the
quotient and remainder.
Repeat the process outlined in step 2
until the quotient is 0.
Take all the remainders from the
bottom up and write them as the
binary number.
b 2 ) 12
2) 6
2) 3
2) 1
2 0
1
2
3
4
b
1
2
3
4
Remainder 1
Remainder 1
Remainder 0
Remainder 0
Remainder 1
1910 = 100112
Remainder 0
Remainder 0
Remainder 1
Remainder 1
1210 = 11002
Converting from the binary system to the decimal
system
Recall from Year 7 that numbers can be written in expanded notation by breaking them
up into their place values, for example, 28 734 = 20 000 + 8000 + 700 + 30 + 4.
Therefore, the decimal number 60910 = 600 + 9
Therefore, the decimal number 60910 = 6 × 100 + 0 × 10 + 9 × 1.
Another way to express this is 60910 = 6 × 102 + 0 × 101 + 9 × 1.
Similarly, the expanded form of 1012 is 1 × 22 + 0 × 21 + 1 × 1. Note that we are
counting in lots of 2 here instead of lots of 10. We use this method to convert from
binary form to decimal form.
Chapter 1 Number skills
37
WORKED Example 32
Convert 100112 to its decimal equivalent.
THINK
WRITE
2
Write the expanded form of the
binary number.
Simplify the expanded form.
3
Write the answer.
1
100112 = 1 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 1 × 1
100112 = 1 × 16 + 0 × 8 + 0 × 4 + 1 × 2 + 1 × 1
100112 = 16 + 0 + 0 + 2 + 1
100112 = 19
100112 = 1910
Note that the length of the binary number is greater than its decimal equivalent. Using
the binary system for everyday calculations would be very cumbersome and probably
lead to frequent errors. Computers are able to cope with calculations in base 2 with
speed and accuracy.
remember
1. The decimal system uses 10 as its base and uses the digits 0, 1, 2, 3, 4, 5, 6, 7,
8 and 9.
2. The binary system uses 2 as its base and uses only the digits 0 and 1.
3. To convert a decimal number to its binary equivalent, use the process of
repeated division by 2.
4. To convert a binary number to its decimal equivalent, expand the number using
powers of 2.
38
Maths Quest 8 for Victoria
1I
Binary numbers
1 The following questions show the technique for converting a decimal number to its
equivalent binary form. Copy and complete each of them by filling in the blank
spaces.
a 2) 2
2 ) 1 Rem __
0 Rem __
b 2)
2)
2)
2)
d 2 )24
2)
2)
2)
2)
2)
e 2 ) 30
2)
2)
2)
2)
2)
Rem 0
Rem 0
Rem 0
Rem 1
Rem 1
9
4
2
1
0
Rem __
Rem __
Rem __
Rem __
Rem 0
Rem 1
Rem __
Rem __
Rem __
c 2 ) 11
2) 5
2) 2
2) 1
0
Rem __
Rem __
Rem __
Rem __
f 2 ) 40
2)
2)
2)
2)
2)
2)
Rem __
Rem __
Rem __
Rem __
Rem __
Rem __
2 Use your answers to question 1 to convert the following to binary form.
a 210 =
b 910 =
c 1110 =
d 2410 =
e 3010 =
f 4010 =
3 Convert the following decimal numbers to binary form.
a 710
b 1210
31
d 3910
e 4210
WORKED
Example
c 2210
f 5010
4 Copy and complete the expansions of the following binary numbers.
a 112 = 1 × 2K + 1 × K
b 1012 = 1 × 2K + 0 × 2K + 1 × K
c 11002 = 1 × 2K + 1 × 2K + 0 × 2K + 0 × K
d 1102 = K × 2K + K × 2K + K × K
e 11012 = K × 2K + K × 2K + K × 2K + K × K
f 101012 = K × 2K + K × 2K + K × 2K + K × 2K + K × K
5 Complete the calculations of the binary numbers in question 4 to find their equivalent
decimal form.
6 Convert each of the following binary numbers to their decimal equivalent.
a 1002
b 1112
c 10102
32
d 100002
e 111112
f 1001002
WORKED
Example
7 Convert the binary number 1 000 0002 to its equivalent decimal form. Convert your
answer back to binary form to confirm your conversion.
Chapter 1 Number skills
39
Operations on binary numbers
Binary counting
From the previous exercise, we were able to convert a decimal number to its binary
equivalent using the process of repeated division by 2. The first five binary numbers
below were obtained using this process.
2) 1
2) 2
2) 3
2) 4
2) 5
2 0 Rem 1 2 ) 1 Rem 0 2 ) 1 Rem 1 2 ) 2 Rem 0 2 ) 2 Rem 1
2 0 Rem 1 2 0 Rem 1 2 ) 1 Rem 0 2 ) 1 Rem 0
2 0 Rem 1 2 0 Rem 1
110 = 12
210 = 102
310 = 112
410 = 1002
510 = 1012
With practice, the method of counting in a binary base becomes quite easy. To
clearly see the counting pattern, the first 10 binary numbers and their decimal equivalent have been placed in the table below. What do you notice?
Decimal number
1
2
3
4
5
6
7
Binary number
1
10
11
100
101
110
8
9
10
111 1000 1001 1010
Binary addition
Using the same techniques that we use on decimals, we may perform calculations on
binary numbers. The only difference is that we must remember to group the numbers in
lots of 2 rather than in lots of 10. Once the numeral 9 is reached in the decimal system,
we move to the next place value. Applying this principle to binary arithmetic, after we
reach 1, we move to the next higher place value. For example, when adding 1 + 1, the
answer is 2, which represents 1 lot of 2 and 0 remainder. Using binary notation, this
would be expressed as 12 + 12 = 102.
WORKED Example 33
Perform the following binary additions.
a 11012 + 1102
b 10012 + 10102 + 100012
THINK
WRITE
a
a
1
2
3
Write the numbers one under the
other, taking care to line up the place
values.
Add the numbers in the first column.
In this case the total is 1, so write the
remainder of 1.
Add the numbers in the second
column. The total is 1, so write the
remainder of 1.
+
+
+
11012
1102
11012
1102
12
11012
1102
112
Continued over page
40
Maths Quest 8 for Victoria
THINK
4
b
WRITE
Add the numbers in the third
column. This column adds to 2, so
carry over 1 lot of 2 to the fourth
column and write the remainder of 0
in the third column.
+
11012
1 1102
0112
5
Add the numbers in the fourth
column. This column adds to 2, so
carry over 1 lot of 2 to the fifth
column and write the remainder of 0
in the fourth column.
11012
+ 1 1 1102
00112
6
Write the 1 that was carried over into
the fifth column.
+
7
Write the answer.
11012 + 1102 = 100112
1
Write the numbers one under the
other, taking care to line up the place
values.
2
Add the numbers in the first column.
In this case, the total is 2, so carry
over 1 lot of 2 to the second column
and write the remainder of 0 in the
first column.
10012
10102
+ 100012
1
12
02
3
Add the numbers in the second
column. The total is 2, so carry over
1 lot of 2 to the third column and
write the remainder of 0 in the
second column.
10012
10102
+ 100012
11
1
0022
4
Add the numbers in the third
column. This column adds to 1, so
write the remainder of 1.
10012
10102
+ 100012
11
1
10022
5
Add the numbers in the fourth
column. This column adds to 2, so
carry over 1 lot of 2 to the fifth
column and write the remainder of 0
in the fourth column.
10012
10102
+ 100012
111
1
010022
b
11012
1 1 1002
100112
10012
10102
+ 100012
Chapter 1 Number skills
THINK
6
7
8
41
WRITE
Add the numbers in the fifth column.
This column adds to 2, so carry over
1 lot of 2 to the sixth column and
write the remainder of 0 in the fifth
column.
Write the 1 that was carried over into
the sixth column.
10012
10102
+ 100012
11 1 1
1
0010022
10012
10102
+ 100012
11 1 1
1
10010022
10012 + 10102 + 100012 = 1001002
Write the answer.
Binary multiplication
When multiplying binary numbers, follow the given steps:
1. Multiply the digits as usual.
Note: As we are multiplying only 1s and 0s, we do not have to carry over lots of 2.
2. Add the resulting rows, carrying over lots of 2 to the next higher column and writing
the remainders.
WORKED Example 34
Perform the following binary multiplications.
a 10112 × 112
b 1112 × 1012
THINK
WRITE
a
a
1
2
3
4
5
Write the binary numbers to be
multiplied with the larger one on
top.
Multiply by the last digit and write
the first row values.
Write a zero in the second row and
multiply by the first digit. Carefully
enter the second row values in the
correct place positions.
Add the columns resulting from the
multiplication. Remember to carry
over lots of 2 to the next higher
column and write the remainders.
Write the answer.
×
10112
112
10112
112
10112
10112
×
112
10112
101102
10112
×
112
10112
101102
1111
10
10000122
10112 × 112 = 1000012
×
Continued over page
42
Maths Quest 8 for Victoria
THINK
WRITE
b
b
1
2
3
4
5
6
Write the binary numbers to be
multiplied.
Multiply by the last digit and write
the first row values.
Write a zero in the second row and
multiply by the middle digit.
Carefully enter the second row
values in the correct place positions.
Write two zeros in the third row and
multiply by the first digit. Carefully
enter the third row values in the
correct place positions.
Add the columns resulting from the
multiplication. Remember to carry
over lots of 2 to the next higher
column and write the remainders.
Write the answer.
×
×
×
1112
1012
1112
1012
1112
00002
111002
1112
1012
1112
00002
111002
111
00
10001122
1112 × 1012 = 1000112
We can check the result of binary operations by performing the operation of the equivalent decimal numbers. In part a of worked example 34, 10112 = 1110 and 112 = 310.
The product of these two decimal numbers is 3310, which has a binary equivalent of
1000012.
Similarly, in part b of worked example 34, 1112 = 710 and 1012 = 510. The product of
these two decimals is 3510, which has a binary equivalent of 1000112.
remember
1. Binary operations are based on the same techniques as decimal operations.
2. Counting in binary involves moving to the next highest place value if the place
value in question is occupied by a 1.
3. When adding binary numbers, carry over lots of 2 and write the remainders.
4. When multiplying binary numbers, multiply by each digit and then perform
binary addition on the resulting rows.
Chapter 1 Number skills
1J
43
Operations on binary
numbers
1 Referring to the table on page 39, write the next five binary numbers after the number
10102.
2 What is the third binary number after 110012?
3 What is the binary number before 101102?
4 Count in 2s to give the next the next four binary numbers after 10012.
WORKED
Example
33
5 Perform the following binary additions.
a 1112 + 1002
b 1012 + 1012
d 11002 + 10112
e 11112 + 1112
c 10012 + 1012
f 101112 + 11112
6 Take the binary number 1102.
a What are the next two binary numbers?
b Determine the sum of these three consecutive binary numbers.
7 a
b
c
d
WORKED
Example
34
Convert 2510 to binary form.
Write the binary number before this one.
Add these two binary numbers.
Convert your answer back to a decimal number.
8 Perform the following binary multiplications.
a 11002 × 102
b 1012 × 112
d 1112 × 1112
e 11012 × 1012
c 1102 × 1102
f 11112 × 11112
9 a Convert the decimal number 1310 to binary form.
b Square your binary answer.
c Convert this binary number back to decimal form.
10 In the suburb of Binaryville the streets have house numbers in binary form. Bit
Avenue is a cul-de-sac with ten houses on either side of the street. The numbering
starts from the corner with even numbers on one side and odd numbers on the other.
Sam’s house is number 1001.
a Is he on the odd- or even-numbered side of the street? How can you tell without
working out the decimal equivalent of his house number?
b What is the decimal equivalent of his house number?
c What house numbers are either side of his house?
d What is the number of the house directly opposite his?
e How many houses are there between Sam’s house and the corner?
f What is the house number at the far end of Sam’s side of the street?
g What is the highest-numbered house in the street?
h Sam’s friend Tom lives in house number 10000. What house number is directly
opposite Tom’s house?
44
Maths Quest 8 for Victoria
Funny definitions
Answer the division questions
to find the puzzle code.
Boycott
Each
Y = 1.2 ÷ 6 =
L = 0.87 ÷ 0.6 =
W = 17.6 ÷ 8 =
J = 1.16 ÷ 0.2 =
V = 6.8 ÷ 4 =
I = 0.648 ÷ 0.8 =
U = 15.9 ÷ 3 =
H = 0.92 ÷ 0.4 =
T = 32.4 ÷ 9 =
G = 4.2 ÷ 1.2 =
S = 7.49 ÷ 7 =
F = 0.462 ÷ 0.11 =
R = 9.8 ÷ 2 =
E = 26.1 ÷ 10 =
P = 13.75 ÷ 5 =
D = 56.1 ÷ 30 =
O = 0.35 ÷ 0.7 =
C = 0.304 ÷ 0.08 =
N = 2.79 ÷ 0.9 =
B = 1.4 ÷ 0.25 =
M = 1.32 ÷ 0.3 =
A = 0.564 ÷ 0.12 =
3.8 4.9 4.7 1.87 1.45 2.61 4.2 0.5 4.9 4.4 4.7 1.45 2.61 5.6 4.7 5.6 0.81 2.61 1.07
4.4 0.81 3.1 0.5 4.9 0.81 4.9 4.9 0.81 3.6 4.7 3.6 0.81 0.5 3.1
Flattery
1.45 0.81 1.7 0.81 3.1 3.5 0.81 3.1 4.7 3.1 4.7 2.75 4.7 4.9 3.6 4.4 2.61 3.1 3.6
Inkling
4.7
Kidney
5.8 0.5 0.81 3.1 3.6 0.81 3.1 4.7 3.8 2.3 0.81 1.45 1.87 1.07 1.45 2.61 3.5
Tortoise
2.2 2.3 4.7 3.6 0.5 5.3 4.9 3.6 2.61 4.7 3.8 2.3 2.61 4.9 1.87 0.81 1.87
5.6
4.7 5.6 0.2 2.75 2.61 3.1
Chapter 1 Number skills
45
summary
Copy and complete the sentences below using words from the word list
that follows.
1
There are four basic mathematical operations. These are addition,
subtraction,
and
.
2
BODMAS can be used to help remember the correct order in which
operations should be completed. BODMAS stands for:
B
O
D ivision
M ultiplication
A
S
.
3
A
4
A number that divides exactly into another number is a
5
If a number isn’t a prime, then it must be 1 or a
6
A
of a number is one in which that number has been
multiplied by another whole number.
7
Multiplying a number by itself is known as
8
The inverse of squaring a number is to find its
9
In a fraction, the
the bottom number.
number has only two factors: one and itself.
.
number.
the number.
.
is the top number and the
10
A fraction that also has a whole number part is called a
11
A
denominator.
12
In an improper fraction the numerator is
denominator.
13
Fractions that are equal in value are known as
14
When adding and subtracting decimals, the
underneath one another.
is
.
fraction is one in which the numerator is less than the
than the
fractions.
must be lined up
46
Maths Quest 8 for Victoria
15
rounding
process.
can be used to find an approximate answer. Rounding,
or rounding
can be used in this
16
The decimal system uses
uses
different digits.
17
Repeated division by 2 will convert a
number.
18
Counting in binary involves moving to the next highest place value if the
place value in question is occupied by a
.
19
When adding and multiplying binary numbers the digits must be grouped
in
of 2.
WORD
multiple
greater
addition
mixed number
squaring
up
binary
different digits. The binary system
number to a
LIST
factor
multiplication
composite
of
subtraction
two
one
decimal points
prime
estimation
numerator
square root
ten
lots
denominator
brackets
equivalent
division
proper
down
decimal
Chapter 1 Number skills
47
CHAPTER
review
1 Calculate the following.
a 743 + 2094 + 26 + 14
d 58 246 − 3071
b 9327 ÷ 6
e 2583 + 27 + 156 + 4
1A
c 1258 × 36
f 3061 × 12
2 Use the order of operations (BODMAS) to calculate the following.
a 12 × 7 + 32 − 26 ÷ 2
b ( 1--- of 60) × 4 + 7
c 302 − 74 ÷ (16 + 11 − 25)
d 38 + 2 × 17 − 11
e 210 ÷ 3 + 16 − 48
f 32 × 4 + 6 − 15 ÷ 3 + 11
2
1A
3 The gymnasium at Straight Line Secondary College is to be set up for the end-of-year
exams. The gymnasium will have 18 rows of desks with 8 desks in each row. How many
desks are required to be set up for the end-of-year exams?
1A
4 If the first division prize in the lottery is $12 000 000, how much will each of the 18 winners
receive?
1A
5 a List all prime numbers from 30 to 50 inclusive.
b List all the factors of:
i 26
ii 4
iii 30.
c Find the HCF of the following pairs of numbers:
i 27 and 42
ii 15 and 60
d List the first four multiples of:
i 6
ii 11
1B
iii 18 and 96.
iii 20.
6 Evaluate the following.
a
64
g 82
b
25
h 272
c
i
10 000
3.62
b
3
--5
e
127
--------64
d
169
j 0.062
e
144
k 25.22
f
l
361
6.42
7 Simplify the following.
a
2
--3
d
5
--6
+
6
--7
+
3
-----12
+
4
-----15
+ 4 1--−
5
--8
1C
c 2 3--- − 1 1---
2
4
+
2 3--4
f
2 1--2
1B
8
+ 3 1--- − 1 3--2
5
8 A Year 8 class organised a cake stall to raise money for their school charity. If they had 12
whole cakes to start with, and sold 5 2--- cakes at recess and 5 2--- cakes at lunchtime, how much
5
3
cake was left over?
1C
9 Simplify the following.
1D
a
2
--5
×
7
--8
b
d 4 1--- × 9 1--3
2
3
--4
÷
7
--8
e 7 1--- ÷
5
8
-----20
10 The Year 8 cake stall raised $240. If they plan to give
and
1
--5
over?
2
--3
c
22
-----6
f
9
--4
×
8
-----11
÷ 8 1--2
to the victims of Hurricane Katrina
to the Starlight Foundation, how much will each group receive and how much is left
1D
48
1E
Maths Quest 8 for Victoria
11 Convert the following fractions to decimals (correct to 2 decimal places).
a
3
--4
b
7
--5
c
6 1--4
d
9
--5
e 4 1--7
f
12 3---
f
17.04
8
1E
12 Convert the following decimals to fractions in simplest form.
a 0.7
b 0.45
c 1.23
d 3.08
e 24.365
1F
13 Evaluate the following.
a 2.4 + 3.7
d 5.63 − 0.07
1F
14 Steve runs each morning before school. On Monday he ran 4.42 km, on Tuesday he ran 5.81
km, he rested on Wednesday, and on both Thursday and Friday he ran 4.86 km. How many
kilometres has he run for the week?
1G
15 Evaluate the following, correct to 2 decimal places where appropriate.
a 432.9 × 2
b 78.02 × 3.4
c 543.7 ÷ 0.12
d 9.65 ÷ 1.1
e 923.06 × 0.000 45
f 74.23 ÷ 0.0007
1H
16 For each of the following numbers:
iii round to the first digit
iii round up to the first digit
iii round down to the first digit
a 39 260
b 222
1H
1H
b 11.62 − 4.89
e 34.2 − 4.008
c 12.04 + 2.9
f 34.09 + 1.2 + 3479.3 + 0.0003
c 3001
17 Provide estimates for each of the following by first rounding the dividend to a multiple of
the divisor.
a 809 ÷ 11
b 7143 ÷ 9
c 13 216 ÷ 12
99 × 1560
18 The answer to ------------------------ contains the digits 375, in that order. Use an estimating technique
132 × 312
to determine the position of the decimal point and write the true answer.
1H
19 Use your estimation skills to find approximate answers for the following.
a 306 × 12
b 268 + 3075 + 28 + 98 031
c 4109 ÷ 21
d 19 328 − 4811
1I
1I
1J
1J
1J
20 Change 10010 to binary form.
CHAPTER
test
yourself
1
21 Express the number 100000002 in decimal form.
22 What are the three binary numbers immediately before 101012?
23 Add the three binary numbers from your previous answer.
24 a Convert 1210 to binary form.
b Square your answer to part a
c Convert your answer to part b back to decimal form.