PL
E
M
SA
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M
PL
Ordering books
E
7/21/08
Imagine the number of books that are
produced each year and the number
that are in existence now. In order to
categorise these, a system called the
Dewey decimal system was invented.
For example, 510.12 is a specific code
for books concerned with a particular
aspect of mathematics. Books are
ordered on the shelves in a library
exactly as decimal numbers are
ordered, and so a book numbered
510.882 is found after books numbered
510.12 and will be on the same topic.
A newer form of categorising books
is the ISBN (International Standard
Book Number) system. It is a 13-digit
number that uniquely identifies books
and book-like products published
internationally. Each number identifies
a unique edition of a publication, from
one specific publisher, allowing for
more efficient marketing of products
by booksellers, libraries, universities,
wholesalers and distributors.
SA
Chapter 03.qxd
New Zealand
Curriculum
Level 3 Number strategies
Use a range of additive and simple
multiplicative strategies with whole
numbers, fractions, decimals and
percentages
Level 4 Number strategies
and knowledge
Understand addition and subtraction of
fractions, decimals, and integers
Find fractions, decimals and
percentages of amounts expressed as
whole numbers, simple fractions and
decimals
Know the relative size and place value
structure of positive and negative
integers and decimals to three places
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T
EA
1
R
S
et
llshe
ki
CH E
Do now
What is the place value of 4 (in words) in the following numbers?
a 2345
b
14 231
c
2
Write 1, 132, 15, 2004, 123 in order from smallest to largest.
3
Calculate:
4
(5 3) 7
b
d
789 105
37 212
b
12 000 100
b
d
37 21
1980 12
18 3 4
M
PL
c
Evaluate:
a 36 1000
7
b
Estimate the answer for:
a 34 270
c 2765 5
6
34 46 157
421 374
Find the answers to the following:
a (2 8) 4
5
b
d
E
a 34 38
c 156 134
114
Estimate the answers to:
SA
a 56 4
c 676 4
Prior knowledge
Tens
BEDMAS
Sum
88
Hundreds
Decimal point
Multiply
Ones
Product
Thousands
Mathematics and Statistics Year 9
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3-1
Page 89
Decimals and place value
The decimal system was formed using the base 10 system of whole numbers that we saw in
Chapter 1.
Which is bigger, 0.09 or 0.2? John says 0.09 is bigger because 9 is bigger than 2. Sue
says 0.2 is bigger because hundredths are smaller than tenths.
Then John says, ‘what about 0.29, because that’s got hundredths’. Sue says, ‘0.29 is
bigger than 0.2’. Is she right? Why?
Place value houses were also investigated in Chapter 1.
Thousands
T
O
H
3
2
7
5
T
O
1
8
Decimal point
M
PL
H
E
Chapter 03.qxd
9
0
5
3
8
0
Row 1 reads ‘three hundred and twenty-seven thousand, five hundred and eighteen’.
Row 2 reads ‘nine hundred and five thousand, three hundred and eighty’.
1
If one unit is cut into 10 equal parts each piece is called ‘one-tenth’ 0.1
10
1
0.01
100
If one hundredth is cut into 10 equal parts each piece is called ‘one-thousandth’
1
0.001
1000
If one-tenth is cut into 10 equal parts each piece is called ‘one-hundredth’ The tenth/10th
The hundredth/100th The thousandth/1000th
SA
The ones/units
Chapter 3 — Decimals
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Reading decimals
Thousands
thousandths
T
O
H
T
O
t
h
o
3
5
7
2
4
1
6
8
3
7
2
5
0
5
1
7
t
h
4
3
E
H
M
PL
Row 1 reads ‘three hundred and fifty-seven thousand, two hundred and forty-one point
six, eight’
6
8
327 241.68 357 241 and 6 tenths and 8 hundredths 357 241 and
and
10
100
Row 2 reads ‘three point seven, two, five’
7 2
5
3.725 3 and 7 tenths, 2 hundredths and 5 thousandths 3 and ,
and
10 100
1000
Row 3 reads ‘zero point five, one, seven, four, three or point five, one, seven, four, three’
5 tenths, 1 hundredth, 7 thousandths, 4 ten-thousandths and 3 hundred-thousandths
3
5 1
7
4
,
,
,
and
10 100 1000 10 000
100 000
Key ideas
To write a number with a fractional part we use a decimal point to separate the whole
number and the fractional part.
SA
The number 0.346 means 3 tenths and 4 hundredths and 6 thousandths, which we can
write as:
tenths
3
.3
ⴝ3 ⴛ
90
hundredths
4
4
6
=
thousandths
6
.3
.04
.006
3
1
1
1
4
6
ⴝ
ⴝ .3 ⴙ .04 ⴙ .006
ⴙ4 ⴛ
ⴙ6 ⴛ
ⴙ
ⴙ
10
100
1000
10
100
1000
Mathematics and Statistics Year 9
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7/21/08
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Page 91
Example 1
What is the value of 5 in 34.457?
Solution
Explanation
For 34.457, the 5 represents
5 hundredths.
The 5 is in the hundredths column.
Tens Units
5
0.05
100
4
hundredths
thousandths
5
7
E
3
. tenths
. 4
M
PL
The value of 5 is
Example 2
Write these as decimals:
3
1000
a
b
1
27
100
c
Solution
2
19
1000
Explanation
thousandths
a
b
3
0.003
1000
27
1.27
1
100
19
2
2.019
1000
SA
Chapter 03.qxd
c
O
t
h
o
0
0
3
1
2
7
2
0
1
t
h
9
Example 3
Investigate the set of numbers 1.08, 1.191, 1.092, 1.62, 1.602.
a
b
c
Which is the biggest number?
Which is the smallest number?
Arrange the numbers in order from smallest to largest.
Chapter 3 — Decimals
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Solution
a
Explanation
1.62
Write numbers in place value houses.
thousandths
h
1
0
8
1
1
9
1
1
0
9
2
6
2
6
0
1
1
t
h
2
thousandths
1.08
M
PL
b
o
E
t
t
hs
o
1
0
8
1
1
9
1
1
0
9
2
1
6
2
1
6
0
t
h
2
thousandths
1.08, 1.092, 1.191,
1.602, 1.62
SA
c
t
hs
o
1
0
8
1
0
9
2
1
1
9
1
1
6
0
2
1
6
2
t
h
Exercise 3A
Example
1
1
What is the value of the digit 5?
a
f
2
b
g
5.132
30.523
c
h
0.357
65.347
d
i
3.615
0.2357
e
j
56.46
0.354
Give the value of the digit in red:
a
d
92
47.5
347.54
26.543
42.34
b
e
37.264
27.3
c
f
389.2
345.267
Mathematics and Statistics Year 9
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7/21/08
5:52 PM
g
j
Example
2
3
g
m
i
l
20.35
0.3567
3
10
48
1000
5
6
100
b
h
n
5
c
10
466
i
1000
64
o
24
1000
3
10
5
1000
71
2
100
d
5
j
p
three units and five tenths
fourteen thousandths
b
d
35.2
b
0.56
c
6
100
56
4
100
2
16
100
e
k
q
13
100
9
23
10
3
2
10
5.3507
d
4.954003
0.5 and .5
0.05 and .05
c
10.5 and 0.501
0.5
1.405
b
g
10.5
1.070
c
h
1.05
1.700
d
i
1.50
1.003
e
j
1.450
1.3000
0.23, 0.32, 0.63, 0.26, 0.36
0.122, 0.145, 0.169, 0.174
0.00456, 0.00684, 0.00945, 0.00571
Find the smallest number in each set of numbers:
0.68, 0.82, 0.12, 0.32
0.783, 0.258, 0.463, 0.872
0.0075, 0.00695, 00659, 0.0045
SA
a
b
c
10
b
Find the biggest number in each set of numbers:
a
b
c
9
r
In each number, which zero (or zeros) can you leave out without changing the value of
the number?
a
f
8
l
27
100
456
14
1000
24
17
1000
Explore the similarities and differences between:
a
7
f
thirty-four units and three hundredths
twenty-two units and fifteen hundredths
M
PL
6
3
1.341
54.678
Write in words:
a
Example
h
k
Write as decimals:
a
c
5
38.94
0.2896
Write the following as decimals:
a
4
Page 93
E
Chapter 03.qxd
Arrange the numbers in each set in order from smallest to largest:
a
c
e
1.6, 1.06, 10.6, 0.6
0.004, 0.142, 0.0123, 0.222
6.002, 5.24, 60.20, 53.4, 60.020
b
d
f
2.03, 3.74, 0.366, 1.6
0.1211, 0.2111, 0.1121, 0.1112
2.779, 2.007, 27.002, 7.202, 7.002
11
At Mount Hutt ski resort 0.98 m of snow fell. At
Coronet Peak 0.897 m fell. Which resort had more
snow?
12
The hire of skis costs $24.20, boots $24.00,
waterproof pants $22.40 and jacket $20.40.
Arrange these costs in order from lowest to
highest.
Chapter 3 — Decimals
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Matthew went to England as an exchange student. The hours of sunshine for the first
six days were as shown in the table.
a
b
c
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
17.26 h
18.26 h
17.05 h
18.09 h
15.34 h
17.62 h
Which day had the most sunshine?
Which was the dullest day?
Arrange the hours of sunshine in order from smallest to largest.
E
Chapter 03.qxd
Enrichment: Numbers and value
14
a
Using these cards make as many different numbers as you can.
i
How many different numbers can you make?
2
M
PL
1
b
i
How many different numbers can you make?
ii What is the biggest number?
iii What is the smallest number?
Now add a card with the number zero on it and make as many different numbers
as possible.
1
2
3
0
i
How many different numbers can you make?
ii What is the biggest number?
iii What is the smallest number?
Now add a second card with the number zero on it and make as many different
numbers as possible.
1
2
3
0
0
SA
c
ii What is the biggest number?
iii What is the smallest number?
Now add a card with the number three on it and make as many different
numbers as possible.
1
2
3
d
i
ii
iii
94
How many different numbers can you make?
What is the biggest number?
What is the smallest number?
Mathematics and Statistics Year 9
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3-2
Page 95
Adding and subtracting decimals
Three students in a Year 9 class all have different methods of working out the same quiz
questions that their maths teacher gave them.
The students’ working is as follows:
Charlie
0.78 ? 3.6
2.6
0.22
Maddy
−0.22
−2.6
E
Marie
3.6 0.78 3.6 1 2.6
2.6 0.22 2.82
0.78 1
3.6
2.6 0.22 2.82
0.78 1
2
3 3.6
2.6 0.22 2.82
M
PL
With a partner, discuss each method. Can you think of another way of doing the question?
Share with your partner. Which method do you prefer?
Key ideas
When adding and subtracting decimals you can use similar strategies to those used when
adding and subtracting whole numbers.
For adding and subtracting decimal numbers we can use reversibility, rounding and
compensating, and partitioning.
Example 4
SA
Find the sum of:
a
c
10.2 and 11.34 using place value
3.41 11.2 0.098 using place value
Solution
a
b
10.2 11.34
10.2 11 0.3 0.04
21.2 0.3 0.04
21.5 0.04
21.54
2.6 3.87
3 3.87
6.87
6.87 0.4 6.47
b
2.6 3.87 by rounding and
compensating
Explanation
By partitioning, break up 11.34 into
11 0.3 0.04.
Add the whole number 11.
Then add 0.3 and 0.04.
By rounding and compensating, add 0.4
to 2.6 to make 3 and add to 3.87 to make
6.87, and compensate by subtracting the
0.4 which we previously added.
Chapter 3 — Decimals
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Using place value, add the first two
numbers and then add 0.098, but rewrite
as 0.09 and 0.008 and add separately
to 14.61.
3.41 11.2 0.098
3.41 11.2 14.61
14.61 0.098
14.61 0.09 0.008
14.70 0.008
14.708
Example 5
a
Subtract 3.4 from 8.6.
b
5.243 2.67
Explanation
M
PL
Solution:
E
Chapter 03.qxd
a
Using reversibility,
3.4 ? 8.6
8.6 3.4 5.2
4.6
0.6
3.4
4
8.6
0.6 4.6 5.2
Using rounding and compensating, add
0.33 to 2.67 to make 3 and subtract the
3 from 5.243. To compensate add 0.33
to 2.243 (we add the 0.33 because we
subtracted 3, which is a larger number
than 2.67).
5.243 2.67
5.243 (2.67 0.33)
5.243 3 2.423
2.243 0.33
2.573
SA
b
Exercise 3B
Example
4a
1
Work out the following by choosing a strategy. Show all your working.
a
d
Example
4b
2
5a
3
16.8 2.7
5.76 8.92
c
f
18.74 6.7
6.98 7.45
3.42 1.37 0.5
7.4 4.6 444.44
43.8 8.25 0.43
b
d
f
0.04 2.35 34.8
1.243 7.2 2.7
3.423 1.85 2.461
Work out the following by choosing a strategy. Show all your working.
a
d
96
b
e
Work out each sum, showing all your working:
a
c
e
Example
12.30 6.04 3.40
0.8 21.91
25.34 15.23
13.68 2.89
b
e
324.46 21.25
31.85 6.47
c
f
7.873 6.24
4.826 3.475
Mathematics and Statistics Year 9
Cambridge University Press • Uncorrected Sample Pages • 2008 © Brookie, Halford, Lawrence, Tiffen, Wallace
5b
4
5:52 PM
Work out these subtractions, showing all your working:
a
d
5
b
e
4.81 1.72
12.26 10.35
c
f
12.046 7.33
11.1 4.289
2.91 32.5 2.05
14.62 6.372 13.5
15 5.475 2.22
b
d
f
3.64 2.26 12.45
2024.5 1876.436
345.6 12.76 1.547
The following times in seconds were recorded in a semi-final of a world series
100 m race:
a
b
10.72, 10.31, 10.97, 10.68, 10.76, 10.17, 10.87, 10.35
What is the difference in time between the fastest and slowest athlete?
What is the difference in time between first and second place?
Sam biked 1.85 km to Tim’s house. Tim and Sam then biked 2.76 km to the movies
in town.
M
PL
7
5.3 2.8
3 0.55
Work out the following:
a
c
e
6
Page 97
a
b
How far did Sam bike to get to the movies?
How much further than Tim did Sam bike to get to the movies?
8
John has $10.50 in his pocket. If he buys a hamburger with cheese that costs $2.85,
how much money will he have left?
9
A family meal of four Kiwi burgers, four small fruit juices, and four small serves of fries is
on sale for $37.50. Look at the menu to find how much is saved by buying a family meal.
10
11
Blade works at the local restaurant after
school. In one week he earned $35.79 and
payed $6.84 in tax. What was his takehome pay?
Today’s Special
KIWI BURGER
$5.55
SMALL FRUIT JUICE
$2.40
SMALL FRIES
$1.90
Samantha was given a piano for Christmas. She then purchased a stool for $107.95,
some sheet music for $16.60, a music stand for $26 and a candelabra for $7.90.
SA
Example
7/21/08
E
Chapter 03.qxd
a
b
What was the total cost?
How much money did she have left from $200?
Enrichment: Do you get the point?
12
John performs the following calculation on his calculator:
3.14 27.23 3.054
He compares his answer to the answers of three of his friends and they are all
different. The answers are 2.809, 26.83, 27.316 and 55.576.
a
b
Determine the correct answer.
Discuss the key-stroke errors that were made in the other calculations.
Chapter 3 — Decimals
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Page 98
Converting fractions to decimals
John says, ‘All decimals are recurring.’
1
Sue says, ‘What about 0.2?’
5
John replies, ‘It is 0.2000000000 . . .’
Do you agree? What does John mean by recurring decimals?
E
Key ideas
numerator
denominator
To change a fraction to a decimal, divide the numerator by the denominator. In changing
from a fraction to a decimal we often create recurring decimals.
M
PL
Fraction ⴝ
A decimal that repeats is called a recurring decimal and we show the repeating pattern
using dots or bars over the numbers.
#
#
#
0.33333 p ⴝ 0.3 or 0.3 and 0.28571428571428 ⴝ 0.285714 or 0.285714
Throughout this section you are permitted to use a calculator for any division.
Some common fractions that you would have seen already and their decimal equivalents are
shown below.
Fraction
Decimal
1
8
0.125
1
5
0.2
1
4
0.25
1
3
#
0.3
1
2
0.5
2
3
#
0.6
3
4
0.75
4
5
0.8
7
8
0.875
Example 6
SA
Convert the following to decimals:
3
2
a
b 3
8
5
Solution
a
b
98
Essential Mathematics 9 for VELS
c
d
98
3
0.375
8
2
3 3.4
5
#
1
0.33333 0.3 or 0.3
3
#
#
2
0.285714 or 0.285714
7
c
1
3
d
2
7
Explanation
Divide 3 by 8 0.375
Look at the fractional part of the number:
Divide 2 by 5 0.4, so the decimal is 3.4
Divide 1 by 3 0.3333333333p
2
5
Divide 2 by 7 0.285714285714285 p
Mathematics and Statistics Year 9
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Chapter 03.qxd
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Page 99
Example 7
Convert the following to fractions:
a
0.2
b
0.25
c
2.437
Solution
b
25 represents 25 hundredths.
437 represents 437 thousandths.
M
PL
c
2 represents 2 tenths.
1
2
10
5
1
25
0.25 100
4
437
2.437 2
1000
0.2 E
a
Explanation
Exercise 3C
Example
6a
1
Convert to decimals:
a
f
b
g
3
1
100
1
4
c
h
1
1000
1
5
d
i
7
10
1
8
7
100
1
16
e
j
Convert to decimals:
a
2
5
b
f
3
200
g
SA
2
1
10
1
2
We know that
3
4
3
16
c
h
3
5
4
5
d
i
5
8
3
4
1
100
1
20
e
j
2
1
0.125 and 0.25. Use that information to convert the following
8
8
to decimals:
a
4
6b
5
b
4
8
c
5
8
d
6
8
e
7
8
d
2
8
8
f
1
0.2 so how many fifths are there in:
5
a
Example
3
8
0.4?
b
0.6?
Convert to a decimal:
a
f
1
10
1
6
2
2
b
g
1
100
1
10
4
4
c
h
1
100
1
12
5
5
i
7
10
1
5
8
e
j
7
100
1
2
16
4
Chapter 3 — Decimals
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5:52 PM
Convert to decimals:
a
e
i
7
e
8
9
Example
6d
b
f
j
1
4
2
3
5
2
1
c
g
3
100
k
3
4
3
4
8
5
1
8
3
d
h
l
2
,
5
2
,
5
3
8
1 3
,
9 4
b
f
3
,
5
3
,
7
5
9
5 2
,
9 3
c
2 1
,
7 8
2 9 6
, ,
11 20 13
g
d
h
10
a
1
6
e
3
5
,
10
16
,
31
6
11
12 1
,
50 2
2
3
b
1
9
f
1
4
15
c
g
7
12
1
3
3
#
1
0.16, so what is the decimal value of:
6
2
3
5
a
b
c
6
6
6
6
What do you know about ?
6
d
5
9
h
2
d
6
6
5
12
We know that
Convert to a decimal (calculators may be used):
a
1
11
3
2
11
SA
e
11
3
10
2
5
3
3
3
4
4
Convert to a decimal, rounding your answers to 3 decimal places as necessary.
(Calculators may be used).
M
PL
6c
1
5
5
2
8
3
2
5
1
Convert each set of fractions to decimals and then write the biggest:
a
Example
Page 100
E
Chapter 03.qxd
b
f
1
22
3
3
22
c
1
7
g
2
3
7
d
h
3
11
5
5
7
1 2 3
Write , , to 9 decimal places. What patterns do you notice?
7 7 7
4 5 6
Can you now predict , , to 6 decimal places?
7 7 7
Example
12
7
Convert to fractions:
a
e
i
m
100
0.3
0.03
0.23
0.004
b
f
j
n
0.5
0.05
0.35
0.235
c
g
k
o
0.6
0.06
0.46
3.271
d
h
l
p
0.8
0.08
0.58
4.333
Mathematics and Statistics Year 9
Cambridge University Press • Uncorrected Sample Pages • 2008 © Brookie, Halford, Lawrence, Tiffen, Wallace
7/21/08
Convert to fractions:
a 0.300
e 2.47
i 0.237
14
Page 101
b
f
j
0.50
2.40
1.35
c
g
k
0.601
2.44
4.6
d
h
l
0.57
3.08
0.0021
In an archery contest, the best performance is determined by the points scored divided
by the total number of points attempted.
a
M
PL
b
In the first round Anna attempted
75 points and scored 48 points. In
the next round, she scored 54
points and attempted 82 points.
Which was the better performance?
Jean scored 69 out of 95 possible
points. Joseph scored 54 out of 80
possible points. Who had the better
performance?
From 170 points attempted, Peter
scored 147 points, and John scored
200 from 240 points attempted.
Who performed better?
E
13
5:52 PM
c
15
Joseph and Alicia played chess on their computers. Alicia said, ‘I have played 38
games and beaten the computer 25 times’. Joseph said, ‘I have played 52 games and
beaten it 35 times so I am a better player than you’. Was Joseph correct in saying this?
Enrichment: Decimal patterns
16
a
3
1 2
Express , and as decimals and use the pattern to predict the decimals for
9 9
9
4 5
6
, and . Use a calculator to check your predictions.
9 9
9
SA
Chapter 03.qxd
b
1 4 12
15 24 98
, ,
and then predict the decimals for , , .
99 99 99
99 99 99
Use a calculator to check your predictions.
Write the decimals for
Chapter 3 — Decimals
101
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Page 102
Multiplying and dividing by
multiples of 10
By using place value houses we can see what happens when we multiply or divide a
decimal by 10, or 100, or 1000 and so on.
Consider: 34 10 340 34 100 3400
and
3.4 10 34
3.4 100 340
E
A pattern develops:
When multipying by 10, the digits glide 1 place value to the left.
When multipying by 100, the digits glide 2 place values to the left.
When multilpying by 1000, the digits glide 3 place values to the left.
M
PL
Multiplication produces a larger value number.
Th
H
× 10
=
5
× 100
=
1
7
T
O
t
h
5
4
3
2
4
3
2
1
7
0
0
4
3
× 1000
0
=
7
th
4
3
7
6
6
SA
Conversely, when we divide, the digits glide to the right. Division produces a smaller
value number.
Th
÷ 10
H
T
O
t
4
5
3
1
4
5
3
0
2
6
3
0
2
6
0
7
1
=
÷ 100
3
=
÷ 1000
7
=
102
h
th
1
1
Mathematics and Statistics Year 9
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7/21/08
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Page 103
Key ideas
When multiplying a decimal by 10 or 100 or 1000 . . . we glide the digits to the left as
many places as the number of zeros.
When dividing a decimal by 10 or 100 or 1000 . . . we glide the decimal point to the right
as many places as the number of zeros.
E
When multiplying or dividing, an empty ‘cell’ either side of the decimal point is filled with
a zero. This shows that there are none of that particular place value.
Example 8
M
PL
Complete a place value table to help you carry out the following calculations:
0.245 10
4.2 1000
a
c
3934 1000
8.6 100
b
d
Solution
Explanation
0.245 10 2.45
a
H
T
O
t
h
th
0
2
4
5
2
4
5
3934 1000
b
TH
H
T
O
3
9
3
4
÷ 1000
3
t
h
th
9
3
4
4.2 1000
c
TH
H
T
4
2
0
4.2 × 1000
t
4
2
h
TH
H
b
Glide digits 3 place value to the
right when dividing by 1000.
c
Glide digits 3 place values to the
left. Because there are no tens or
ones, zeros are placed in these cells.
d
Glide digits 2 place values to the
right. Because there are no tenths, a
zero is placed in the tenths place
value cell.
th
8.6 100
d
Glide digits one place value to the
left when multiplying by 10.
0
SA
= 4200
O
a
T
8.6 ÷ 100
O
t
8
6
= 0.086
0
h
th
8
6
Example 9
Complete these calculations:
a
0.723 400
b
89.4 6000
Chapter 3 — Decimals
103
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Page 104
Solution
Explanation
0.723 400 0.723 4 100
1.446 2 100
2.892 100
289.2
a
b
89.4 6000 89.4 6 1000
14.9 1000
0.0149
b
8a
M
PL
Exercise 3D
Example
1
Use place value charts to help find the answers to these calculations:
a
d
g
j
Example
8b
2
8c
3
0.345 10
245.45 10
34.4567 10
7.34 100 000
37.54 10
37.54 100
37.54 1000
4
5
9
0.345 1000
245.45 1000
34.4567 1000
345.6 1 000 000
4.38 10
4.38 100
4.38 1000
c
f
i
0.345 10
0.345 100
0.345 1000
b
e
h
k
36.456 10
567.7 100
17.24 1000
0.0035 100
c
f
i
l
2.347 10
2.56 100
456.7 1000
0.0579 1000
380 100
1203 1000
81.23 100000
0.056 10
b
e
h
k
45 10
1347 10
345.98 10 000
0.003 1000
c
f
i
l
17.3 10
23 1000
2.456 100
0.347 10000
Work out the answers to the following:
a
e
i
m
104
c
f
i
l
Work out the answers to the following:
a
d
g
j
Example
b
e
h
1.65 10
47.467 100
3.7 1000
0.24 10
SA
8d
0.345 100
245.45 100
34.4567 100
0.7854 10 000
Work out the answers to the following:
a
d
g
j
Example
b
e
h
k
Use place value charts to help find the answers to these calculations:
a
d
g
Example
Rewrite 400 as 4 100 and, because
4 2 2, we can use the doubling
strategy twice.
So to multiply by 100, we glide digits 2
place values to the left.
Rewrite 6000 as 6 1000.
Divide 89.4 by 6 (you may use a
calculator). So to divide by 1000, we
glide digits 3 place values to the right.
E
a
2.6 30
46.4 20
8.94 200
154.8 600
b
f
j
n
3.6 200
56.7 300
7.94 1100
12.6 1200
c
g
k
o
27.2 50
0.16 400
62.4 120
0.068 11000
d
h
l
p
5.5 50
0.008 70
18.6 5000
98.4 6000
6
On average 15.6 mm of rain fell every day for 30 days. What is the total rainfall for
the 30 days?
7
A builder requires 300 m of timber at $4.78 per metre. What is the overall cost?
Mathematics and Statistics Year 9
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7/21/08
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Page 105
8
Paul paid $156.00 for 400 plastic soldiers. How much was each soldier?
9
A house cost $145 000. If the house is 200 m2, what is the average price per m2?
10
The overall cost of a reception for 70 people was $1071. What was the cost for each
couple?
11
Tanya buys 3000 sequins for her new dress. If they cost $0.35 per 20 how much do the
sequins cost all together?
12
E
Enrichment: Standard form and the calculator
Large numbers and small numbers are often written in standard form. This is useful
if numbers are too large for the display.
)
M
PL
For example, 2 000 000 000 000 can be written as 2 1012, meaning the 2 is
followed by 12 zeros. The calculator shows it as 2 E12 ( Keys: 2 EXP 12
0.00000000002 can be written as 2 1011, meaning the 2 is 11 value places after
the decimal point. The calculator shows it as 2 E11. (Keys: 2 EXP 11 )
a
b
c
What does the E mean?
How many zeros before your calculator changes
to standard form?
Start with 1, multiply by 10 and keep
multiplying the answer by 10 until your answer
becomes standard form.
How many decimal places before your calculator
changes to standard form?
Start with 1, divide by 10 and keep dividing the
answer by 10 until your answer becomes standard
form.
While in this form we can perform normal
calculations. Here we will consider
multiplication and division.
For example, 30 000 000 000 0.005 is the same as 3 E10 5 E3 or 150 000 000.
To write 29, press 2 and X y 9.
To write 2 12, press 2 and X y 12.
Use your calculator to evaluate the following:
i
5 000 000 000 000 0.000 000 000 04
ii 600 000 000 000 000 0.000 000 000 000 03
iii 70 000 000 000 000 900 000 000 000
iv 0.000 000 000 000 4 0.000 000 000 000 05
v 5 000 000 000 000 8 000 000 000 000 000
vi 170 000 000 000 000 000 14 000 000 000 000 000
vii 0.00 000 000 000 000 12 0.000 000 000 03
viii 13 000 000 000 000 000 0.000 000 000 000 007
SA
Chapter 03.qxd
d
Chapter 3 — Decimals
105
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Page 106
Multiplying a whole number by a
number less than one
A carpenter needs three pieces of timber 0.4 m long.
E
Will he need more than 3 m of timber? Discuss this with your partner.
What does this tells us about multiplying a whole number by a number less than one?
Discuss.
Key idea
M
PL
When multiplying by a number less than one, the answer is smaller than the whole number.
Example 10
Work out the answer: 4 0.3
Solution
Explanation
4 0.3 2 2 0.3
Multiplying by 2 and
then multiplying 2 again
is the same as
multiplying by 4.
2 0.6
1.2
0.3 × 4
double double
0.3 0.6 1.2
×4
SA
Example 11
Evaluate: 5 0.6.
Solution
Explanation
5 0.6
10 0.3
3
Using the double/halve strategy:
double 5 and halve 0.6
glide one place to the left
Exercise 3E
Example
10
1
Evaluate:
a
e
106
5 0.5
12 0.8
b
f
7 0.6
21 0.5
c
g
3 0.8
16 0.7
d
h
9 0.4
34 0.3
Mathematics and Statistics Year 9
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Chapter 03.qxd
11
2
5:53 PM
Page 107
Evaluate:
a
e
3 0.86
4 0.07
b
f
8 0.35
7 0.19
c
g
6 0.7 9
9 0.004
d
h
4 0.92
5 0.108
Pere requires 7 pieces of decking timber to make steps. Each piece must be 0.75 m
long. What length of timber will he need?
4
Wiremu has been asked to tie the flowers in the garden to stakes. He uses 0.48 m lengths
of twine. How much twine will he have to buy if there are 23 flowers to be tied?
5
A 10-cent coin is 0.013 m in diameter. A coin trail for 10-cent pieces is used to help
raise funds for a class trip. How long will the trail be if there are 250 coins?
6
Tilly drinks 0.33 L of milk each morning. How much milk does she drink in a week?
7
Each of Sam’s cows drinks 23 L of water every day. He adds 0.17 L of dissolved
minerals for every litre of water that they drink. If he has 10 cows, what quantity of
dissolved minerals must he add every day?
M
PL
E
3
8
Pani decides to buy her friends some
chew bars. She buys 4 coconut, 8 caramel,
3 chocolate and 7 peppermint bars. How
much will she spend?
Tempting Times
Coconut delights
Chewy caramels
Chocolate puffs
Peppermints
$0.55
$0.72
$0.38
$0.25
Enrichment
9
Evaluate:
a
e
10
11 0.78
21 0.24
b
f
15 0.42
26 0.103
c
g
18 0.05
42 0.421
Terry built a small ramp to use with his skateboard.
He used 0.58 m of plywood for the slope and 0.32 m
for the rise. His friends were so impressed that he
was asked to make another seven ramps. Each
piece of slope plywood costs $0.82 and each piece
of rise plywood costs $0.55.
SA
Example
7/21/08
a
b
c
d
e
d
h
14 0.86
52 0.007
0.58 m
0.32 m
How much plywood is required for seven slopes?
His friend Parekura has sufficient plywood for three rises and two slopes and will
provide it for no cost. What is the total length of plywood supplied by Parekura?
What length of extra plywood is required for the rises?
What length of extra plywood is required for the slopes?
What is the total cost of the ramps Terry builds?
Chapter 3 — Decimals
107
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Page 108
Multiplying decimals
Most real data is in decimal form and
calculations often arise that involve the use
of these decimal fractions.
From the table, what does the time taken
to travel around the Sun tell you about the
position of the planets from the Sun? Which
planet is closest to the Sun? Which planet is
third closest?
Often, calculations involving decimals
require multiplication; for example, how
long does it take Saturn to orbit the Sun five
times?
Time taken to orbit the Sun
Period of revolution
(years)
M
PL
E
Planet
Key ideas
When multiplying decimals:
Determine how many decimal places there are in each number.
Perform normal multiplication.
Write your answer to the total number of decimal places in the question.
Example 12
Calculate:
3.24 2
a
b
2.42 3.3
SA
Solution
a
b
3.24 2
324 100 2
324 2 100
648 100
6.48
2.42 3.3
242 100 33 10
242 33 100 10
7986 100 10
7.986
Explanation
a
Rewrite the decimal as a whole number.
Use multiplication strategy to solve.
3.24 324 100
Divide through by 100.
b
Rewrite decimal as whole numbers:
2.42 242 100 and 3.3 33 10.
Use a multiplication strategy to solve
242 33 7986
Divide through by 100 and 10.
ⴛ
30
3
108
200
6000
600
40
1200
120
2
60
6
Mathematics and Statistics Year 9
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Page 109
Example 13
Calculate:
0.2 0.4
b
3.678 90
Solution
a
0.2 0.4
2 10 4 10
2 4 10 10
8 10 10
0.08
3.678 90
3678 1000 90
331 020 1000
331.020 or 331.02
a
Rewrite decimal as a whole number:
0.2 2 10 and 0.4 4 10
Use multiplication strategy to solve 2 4 8.
Divide through by 10 and 10, or 100.
b
Rewrite numbers as whole numbers and
use a multiplication strategy to solve
3678 90 331 020.
Divide through by 100 and 10, or 1000.
If the last digit is zero it can be removed as is
has no place value.
M
PL
b
Explanation
E
a
Exercise 3F
Example
12a
1
Find the answers to the following:
a
e
Example
12b
2
2.4 2
3.73 8
7.3 2.4
9.3 4.2
26.5 8.3
SA
13a
3
13b
4
5.2 4
9.54 9
d
h
7.1 7
3.42 6
b
e
h
3.6 5.8
4.6 2.7
45.2 9.4
c
f
i
5.3 6.2
7.9 5.2
3.4 47.2
0.2 0.5
2.34 0.6
c
g
b
f
0.4 0.3
4.31 0.5
0.7 0.7
7.93 0.4
d
h
0.2 0.4
6.45 0.7
c
g
3.5 40
1.4 300
d
h
4.2 20
2.67 500
c
f
i
0.2, 0.8
5.4, 7.7
4.13, 2.22
Find the answers to the following:
a
e
5
c
g
Find the answers to the following:
a
e
Example
3.3 3
4.67 6
Multiply the following:
a
d
g
Example
b
f
3.74 70
3.14 100
b
f
2.74 50
2.735 200
Find the product of each pair of numbers:
a
d
g
1.2, 0.02
2.3, 3.6
32.24, 2.3
b
e
h
8.6, 0.01
3.4, 2.7
16.5, 12.04
Chapter 3 — Decimals
109
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5:53 PM
Use your calculator to find the answer and then round your answers to the nearest
dollar:
a
d
g
7
b
e
h
4 $8.50
17 $6.10
5.9 $14
c
f
i
7 $6.25
12 $10.67
34.2 $6.12
312 $23.20
0.45 $22.98
23.568 $23.30
b
e
h
$0.34 26
$116 0.0435
$3.58 401.4
c
f
i
$34.32 22
0.23 $35.24
$100.01 345.45
Paula needs seven pieces of timber, each 6.8 m long.
a
b
What is the total length needed?
Determine the total cost if the price of the timber is $4.20 per metre.
A new water tank can store 750 litres of water. The average water collected in the tank
is 1.75 litres per day. Will the tank fill to capacity over a year if no water is removed?
If so, how much excess water will there be?
M
PL
9
3 $2
5 $5.15
2.6 $3.46
Use your calculator to find the answer and the round your answers to the nearest cent:
a
d
g
8
Page 110
E
Chapter 03.qxd
10
David earns $5.67 per hour as an apprentice. If he works
38.3 hours, how much will he earn?
11
A timber supplier purchases 47 m of timber at $2.75 per
metre and then sells it for $4.36 per metre. How much
profit is made?
12
A plumber requires 18.57 m of drainage pipe. If the pipe
sells at $2.78 per metre, how much will it cost?
13
A fireplace requires 800 bricks, which weigh 0.60 kg each. Can the builder use his
truck to carry them if the truck takes a maximum load of 500 kg? Explain your answer.
Enrichment: Modular kitchens
The cost of a modular kitchen is decided by the number
and type of cabinets required.
Tim and Mary require six normal cupboards costing
$89.70 each, three sets of drawers costing $105.30
a set, one sink unit costing $126 and two corner
units costing $99.95 each.
SA
14
Packages are also available:
Package 1: 4 cupboards, 2 drawers and a sink unit for $680
Package 2: 5 cupboards, 3 drawers and a sink unit for $800
Package 3: 6 cupboards and 2 drawers for $900.
a
b
c
110
Calculate the overall cost if Tim and Mary buy each component separately.
Calculate the cost if they use each package and buy the extra pieces needed.
What is the cheapest way for Tim and Mary to buy their kitchen?
Mathematics and Statistics Year 9
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Page 111
Dividing a whole number by
a number less than one
Division introduces more language. The number you are dividing by is called the divisor.
The answer is called the quotient. When you divide 20 by 4, the divisor is 4 and the
quotient (answer) is 5.
When we divide 10 by 2, we can change the problem to multiplication and say: ‘What do
I multiply 2 by to give me 10?’
2 䊐 10
1
2
3
4
5
6
7
8
9
10
E
Chapter 03.qxd
10 whole
M
PL
There are 5 lots of 2 in 10 whole.
2 5 10
10 2 5
When we divide by a decimal less than one, we can carry out the same operation:
1 0.2 becomes 0.2 䊐 1
.1
.1
.2
.1
.3
.1
.4
.1
.5
.1
.6
.1
.7
.1
Q: How many ‘lots of’ 0.2 are there in one whole?
A: 5
1
2
3
.1
.1
.1
.1
.1
.8
.1
.9 1.0
.1 .1 1 whole
4
.1
.1
5
.1
.1 1 whole
.1
This means: 1 0.2 5
For the equation 3 0.2 䊐, it becomes 0.2 䊐 3
We can use three deci-strips.
1
2
3
4
5
SA
1
2
3 whole
Q: How many ‘lots of’ 0.2 are there in 3 whole?
A: 3 5 15
This means: 3 0.2 15
If we divide by a decimal less than one with 2 decimal places, we could divide 1 whole
into 100 cells, each with a value of 0.01, and carry out the same process:
1 0.02 䊐, which becomes 0.2 䊐 1
Q: How many lots of 0.02 are there in one whole?
A: 50
This means: 1 0.02 50
Chapter 3 — Decimals
111
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Page 112
If we had 3 whole and wished to divide by 0.02, there would be 3 50 lots of 0.02:
3 50 150 or
3 0.02 150
Use your calculator to solve 1 0.002 䊐. It gives 䊐 500.
Q: How many lots of 0.002 are there in 1 whole?
A: 500
Q: How many lots of 0.002 are there in 3 whole?
A: 1500
E
Have a look at all the examples we have calculated. What happens to the quotient
(answer) as the value of the divisor becomes smaller and smaller?
M
PL
Key ideas
A division equation may be changed to a multiplication equation.
The smaller the divisor the larger the quotient (answer to a division equation).
Make the divisor a whole number by gliding the place value to the left.
Do the same for the whole number.
For example: 4 ⴜ 0.03 is the same as 400 ⴜ 3
Estimate the answer to a division equation to check that it is sensible.
Example 14
Calculate:
a
4 0.2
b
9 0.03
SA
Solution
a
4 0.2
40 2
20
8 0.004
c
Explanation
Make the divisor into a whole number by gliding the
digits for both numbers one place value to the left.
H
O
t
h
4
0
0
4
0
0
H
O
t
h
2
2
How many ‘lots of’ 2 are there in 40?
112
Mathematics and Statistics Year 9
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7/21/08
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Page 113
Solution
b
Explanation
9 0.03
900 3
300
The divisor is made into a whole number by gliding the
digits for both numbers two place values to the left.
H
T
O
t
h
0
3
O
t
h
9
0
0
H
T
9
0
0
E
3
So we can now ask: ‘How many ‘lots of’ 3 are there in 900?’
8 0.004
8000 4
2000
The divisor is made into a whole number by gliding the
digits for both numbers three place value to the left.
M
PL
c
Th
H
T
O
t
h
th
0
0
4
O
t
h
th
8
0
0
0
4
Th
H
8
0
T
0
0
So we can now ask: ‘How many ‘lots of’ 4 are there in 8000?’
Example 15
Evaluate: 51 0.17
SA
Chapter 03.qxd
Solution
Explanation
51 0.17 5100 17
300
Make the divisor into a whole number by gliding the
digits of both numbers two place values to the left.
How many lots of 17 are there in 5100?
(Remember: 3 17 51)
Example 16
For these calculations, estimate the quotient and then use your calculator to check your
answer:
a
3 0.4
b
5 0.12
c
11 0.437
Chapter 3 — Decimals
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Solution
Explanation
3 0.4 30 4
4 7 28 and 4 8 32
A sensible estimate would give a
quotient between 7 and 8.
Calculator quotient 7.5
b
5 0.12 500 12
40 12 480
A sensible estimate would give a
quotient a little larger than 40.
Calculator quotient 41.67 (2 d.p.)
c
11 0.437 11 000 437
⬇ 11 000 500
22
A sensible estimate would give a
quotient a little larger than 22.
Calculator quotient 25.17162471
25.17 (2 d.p.)
Glide both numbers one place value to
the left. Q: What number multiplied by 4
gives an answer close to 30? A: 30 is half
way between 28 and 32 so the quotient is
thus half way between 7 and 8.
By calculator: 7.5....
Glide both numbers two place values to
the left. Q: What number multiplied by
12 gives an answer close to 500?
A: 4 12 48 thus 40 12 480,
which is very close to 500.
By calculator: 41.66666 . . .
Glide both numbers three place values to
the left. Q: What number multiplied by
437 gives an answer close to 11 000?
A: 437 is close to 500, and 500 22
11 000 or 11 000 500 22
By calculator: 25.171... Round sensibly
to 2 decimal places.
M
PL
a
E
Chapter 03.qxd
Exercise 3G
Example
14a
1
Find the answers to the following:
3 0.3
12 0.3
346 0.2
SA
a
e
i
Example
14b
2
14c
3
15
4
9 0.3
36 0.9
48 0.3
d
h
l
4 0.2
49 0.7
126 0.9
4 0.04
3 0.12
7 0.07
b
f
j
2 0.05
6 0.04
9 0.06
c
g
k
3 0.06
4 0.04
6 0.03
d
h
l
4 0.08
7 0.05
9 0.09
5 0.005
9 0.001
b
f
2 0.008
4 0.008
c
g
3 0.003
6 0.004
d
h
7 0.002
8 0.005
36 0.12
56 0.28
38 0.19
c
f
i
42 0.07
144 0.24
18 0.09
Work out the answers:
a
d
g
114
c
g
k
Find the answer:
a
e
Example
8 0.4
24 0.6
45 0.5
Find the answers to the following:
a
e
i
Example
b
f
j
26 0.13
48 0.48
35 0.25
b
e
h
Mathematics and Statistics Year 9
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Example
16a
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i
ii
Show your working to estimate the quotient.
Use your calculator to find the answer, then round sensibly.
a
d
g
Example
16b
6
i
ii
7
i
ii
4 L 0.25
15 m 0.12
14 cm 0.71
b
e
h
1 m 0.6
7 tonnes 0.7
8 Hz 0.7
c
f
i
9 cm2 0.4
12 L 0.9
6 km 0.8
b
e
h
8 m 0.9
7 km 0.37
8 L 0.48
c
f
i
3 t 0.14
6 m2 0.19
5 kg 0.89
Show your working to estimate the quotient.
Use your calculator to find the answer, then round sensibly.
7 kg 0.251
15 cm 0.176
10 kg 0.534
b
e
h
8 L 0.193
4 t 0.232
12 cm 0.639
c
f
i
4 m3 0.114
3 g 0.027
5 km 0.228
M
PL
a
d
g
8
Sally buys 15 m of ribbon for giftwrapping small parcels. She uses 0.37 m of ribbon
for each parcel. How many parcels can she wrap? (Show your working.)
9
Tinesia is making bookshelves, and her local timber merchant sells timber shelving in
6 m lengths. If her shelves are 0.55 m long, how many can Tinesia make from each
length of timber? (Show your working.)
10
Hone cuts firewood into 0.375 m lengths.
a
b
How many pieces does he get from a tree trunk 8 m long?
How long is the short leftover piece that can be used for kindling?
Enrichment: Outdoor camp
11
Rimu College is running an outdoor camp for the students. Find the maximum
number of students who could attend this camp.
Food for each student has been calculated as follows:
SA
16c
4 kg 0.7
18 L 0.1
5 g 0.6
Show your working to estimate the quotient.
Use your calculator to find the answer, then round sensibly.
a
d
g
Example
Page 115
E
Chapter 03.qxd
Meat:
Potatoes:
Vegetables:
Fruit:
0.125 kg
0.12 kg
0.235 kg
0.345 kg
The cook buys 20 kg sausages, 45 kg potatoes,
32 kg fruit, and 18 kg of cabbages.
Four manuka tent pegs, 0.375 m long, are required for
each three-person tent. They are cut from 2 m long manuka stakes. Liz provides 65
stakes. There can be fewer than three students in a tent.
The number of students permitted to attend the camp is restricted by the area of
the bathrooms, which are 42 m2. There must be 0.72 m2 per student at the camp.
Chapter 3 — Decimals
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Dividing decimals
Division of a decimal number by another decimal number simply means that we are using
numbers with more digits to carry out the division. We need to know what we are doing and
if the answer obtained is sensible. Calculators help speed up the process, but we need to
understand what is happening in the division problem.
E
Key ideas
Glide the last place value of both numbers to the left by the same number of places so
that the divisor becomes a whole number.
Estimate the quotient.
M
PL
Use the calculator and check the answer against your estimate.
Example 17
Calculate: 185.4 1.06
Solution
T
O
H
T
1
8
O
t
h
1
0
6
h
1
0
6
H
T
O
t
1
8
5
4
5
4
0
SA
T
O
Explanation
185.4 1.06 18 540 106
⬇ 20 000 100
200
Calculator quotient 174.9056604
Sensible quotient 174.91 (2 dp)
116
Digits of the divisor glide two
place values to the left.
Digits of the other number also
glide two place values to the left.
Round 18 540; round 106.
Divide 20 000 by 100.
Calculator: quotient is close
to 200.
Round sensibly (2 dp).
Mathematics and Statistics Year 9
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Exercise 3H
1
Calculate using gliding place values and estimation before using your calculator.
Round the quotient sensibly.
a
e
i
2
b
12.34 1.2
f
1.03 0.05
468.06 2.482 j
c
27.3 3.4
g
143 3.56
53.471 1.509 k
567 12.6
63.8 4.93
47 2.008
d
h
l
68.23 7.5
687 9.51
2.1 1.072
Calculate using gliding place values and estimation before using your calculator.
Round the quotient sensibly.
a
c
e
12.78 m 1.3
3.456 tonne 2.113
98.32 L 5.608
b
d
f
E
17
45.8 km 0.55
207.6 m2 4.03
516.1 kg 21.752
One dress takes 2.56 m of material. How many dresses could be made from 120.45 m
of material?
4
Jerry travels 456.78 km in 8.06 hours. What is his average speed?
5
Star Hospital allows $2.017 for food per patient each day. If the budget allows $2508
per day, how many patients can be provided with food?
6
Petrol costs $1.8694 per litre. William spends $87.04 to fill his car’s petrol tank. How
much petrol did he buy?
M
PL
3
Enrichment
7
Jason is interested in the profit he will receive when he sells some cattle from his farm.
He sells them for $2679.89. The beef schedule pays him $0.301 per kilogram. He
knows that feed costs $3.072 per 100.75 kg of beef produced. Other production costs
are shown in this chart.
SA
Example
Item
Cost per x kg of beef produced
Wages
Fencing
Vet, medicines, drenches
$4.208 per 47.9 kg
$1.008 per 25.607 kg
$0.157 per 60.23 kg
Use this information to calculate his profit.
Chapter 3 — Decimals
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Applications of decimals using
a calculator
M
PL
Key ideas
E
Decimals are used widely in everyday life, as seen in previous exercises. When performing
operations with decimals in everyday situations, we often get answers that have no real
meaning.
For example, consider an answer of $18.987. We normally round this to $18.99. Then we
round it to $19.00 if we are calculating the cost of something, because the smallest coin we
use is 10 cents. So, when calculating problems involving money, we always need to check
how sensible our solution is.
When solving more difficult problems, it helps to break them into steps.
Steps for solving problems:
1
2
3
4
5
6
Understand the problem. What am I given and what am I asked to find?
Decide on a method.
Write a mathematical statement.
Estimate the answer if necessary.
Determine your answer.
Check that the answer is sensible and round off if necessary.
Example 18
SA
Fran orders 26 packs of 33 mini pizzas for a fundraising
event. She purchases each pack for $17.58 and sells the
pizzas individually. She wishes to raise $300.
a
b
What is the total price for the packs of pizzas?
For how much should each mini pizza be sold?
Solution
a
b
118
$17.58 26 $457.08
The total cost is $457.08.
26 33 858
$457.08 $300 $757.08
$757.08 858 $0.882377622
Each mini pizza would sell for 90
cents.
Explanation
Cost per pack number of packs
Number of packs number of pizzas in a pack
This is the total revenue to be raised.
Total revenue number of pizzas
Round to the nearest 10 cents.
Mathematics and Statistics Year 9
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Exercise 3I
18
1
What is the cost for each of these orders?
a
b
c
d
e
A small Hawaiian pizza with extra green olives
A family size Volcano pizza and a small Americana pizza
A large Napoli pizza with extra mushrooms and a small Kiwi pizza
A family size Mushroom pizza with extra bacon and a large Marguerita pizza
A small Rocky’s Special pizza with pineapple, and a garlic pizza pastry
2
The Scott family orders a main spaghetti marinara for Kaylene, lasagne for Matthew, a
small Rocky’s Special pizza with pineapple for David, a large Kiwi pizza for
Christopher and a small Hawaiian pizza for Samantha, with a side order of garlic
pizza pastry. How much will the meal cost and what is the average price per person?
3
Rocky purchases ham at $15.00 a bag. Each bag contains 6 kg of sliced ham pieces.
On average he can use this ham on 30 Kiwi pizzas. How much ham is used on each
pizza and what is the cost per pizza for the ham only?
SA
Example
M
PL
E
Everyone has ordered a meal at one time or another. But have you ever thought about how
much mathematics is involved? Look at the copy of the menu from Rocky’s Restaurant below
and use it to help you answer the questions that follow.
Pizza menu
Small
Large
Family
Pasta menu
Entree
Main
Marguerita
$4.90
$8.20
$10.90
Spaghetti marinara
$5.50
$8.90
Kiwi
$5.80
$9.40
$12.30
Chicken carbonara
$6.50
$9.50
Hawaiian
$5.80
$9.40
$12.30
Lasagne
$7.90
Volcano
$5.80
$9.40
$12.30
Napoli
$5.80
$9.40
$12.30
Extra pizza toppings will be charged for
Usual
$5.50
$8.80
$11.80
60c small
80c large $1.00 family
Melton Special
$6.30
$9.90
$12.70
Americana
$5.80
$9.40
$12.30
Garlic bread
Mushroom
$5.80
$9.40
$12.30
$4.30 small $6.30 large $7.50 family
Rocky’s Special
$6.90
$10.60
$13.90
Garlic pizza pastry (One size only) $3.80
4
A Melton Special is the same as a Usual pizza with two extra toppings, but a Melton Special
costs less. How much cheaper is a small Melton Special than the equivalent Usual pizza?
5
If 15 people each gave you $5.00 to purchase as many large pizzas as you could and
receive the least charge possible, what would you order?
Enrichment: Pizza and pasta
6
The Year 7 students at Rimu College have a ‘pizza and pasta’ day on the last day of
term. Rocky charges $6.00 for chicken carbonara, $6.50 for lasagne and $5.00 for
any small pizza. The local supermarket sells drinks for $8.40 per dozen.
a
b
If 80 students choose pizza, 20 choose lasagne and 29 choose chicken
carbonara, and each student has one drink, what is the overall cost of the day?
Each student is to be charged the same amount. How much will each student pay?
Chapter 3 — Decimals
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W O R K
I
Mathematically
N G
Decimals
Building
Andrea wished to build a wardrobe for her bedroom and wondered if she could afford to do
it with the $100 she had saved. The panelling costs $14.60 per square metre and comes in
sheets of different sizes. The glue, nails and hinges cost $15.50 in total. The design is
shown below. The two doors are the same size and there is no back on the wardrobe.
20 cm
E
220 cm
200 cm
Calculating
Complete the table below.
M
PL
1
40 cm
180 cm
partition
Item
Size (cm)
Area (m2)
Side
Side
Partition
Base
Top
Door
Door
Shelf
Shelf
Shelf
Nails, glue and hinges
20 200
20 200
0.2 2.0 0.4
Cost
0.4 $14.60 $5.84
$15.50
Total cost
SA
22 Can Andrea afford to build the wardrobe?
Modifying
Andrea decided to modify her wardrobe so that the partition and
the shelves are only 15 cm wide.
How much money will she save using this new design?
Can she afford to build it?
Improving and comparing
1
2
1
2
3
120
3
40 cm
15 cm
Can you use the same guidelines to design a wardrobe that
has more space than Andrea’s?
Find out the sizes of panelling sheets and decide which sheet sizes have minimal
wastage, and so improve your costing calculated in Question 1.
Compare the cost of Andrea’s wardrobe to the cost of some readymade wardrobes.
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Using technology to set up spreadsheets
Spreadsheets are very useful when you are doing repetitive
calculations or wish to vary values and not calculate the answer
each time. Fionna wishes to buy figurines. Some are made of
lead and cost $15.40 each and some are made of plastic, and
cost $6.47 each. She needs to purchase eight figurines of any
type to complete her set and can spend no more than $100.
3
To help Fionna find out what combinations of the numbers of figurines to buy, set up a
spreadsheet.
1
Complete columns A and B, ensuring that the total number of figurines is eight.
2
Determine the rule you will use to calculate the total in:
a C2
b D2
c E2
3
Enter these in the table and then use the Fill Down operation to complete columns
C, D and E.
M
PL
2
E
Setting up the spreadsheet
FPO
A
No. of
lead
figurines
B
No. of
plastic
figurines
2
0
8
3
1
7
4
2
6
5
3
1
C
Total cost
of lead
figurines
D
Total cost
of plastic
figurines
E
Total cost
of
figurines
6
7
SA
Using the spreadsheet
Which combinations of the numbers of lead and plastic figurines are possible for Fionna to buy?
Modifying the spreadsheet
1
1
2
2
Suppose Fionna’s friend Tomika has $50 to spend. Set up a spreadsheet to determine
how many different combinations of figurines she could afford to buy.
If Fionna and Tomika combined their resources, set up a spreadsheet to determine:
a the maximum number of figurines they could buy
b the minimum number of figurines they could buy if they spent most of the
money.
c If a new set consisted of a minimum of two lead and a minimum of seven plastic
figurines, could they each buy a set?
d What are the possible combinations of each set if both Fionna and Tomika buy the
same sets?
ary
Chapter 3 — Decimals
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Decimals
7.346 means 7 units, 3 tenths, 4 hundredths and 6 thousandths.
To convert a fraction to a decimal, divide the numerator by the denominator.
E
Rounding
Round down if the next number is less than 5: 3.131 3.1 to one decimal place.
Round up if the next number is 5 or more: 32.1356 32.14 to two decimal places.
M
PL
Addition and subtraction of decimals
Use place value houses.
Add or subtract whole numbers.
Write decimal fraction as fraction, and write in place value house.
Add or subtract the same place value digits.
Change the fraction back to a decimal.
Multiplication and division of decimals
To multiply by 10 or 100 or 1000, glide the place value to the left the same number of
places as there are zeros.
To divide a decimal by 10 or 100 or 1000, glide the place value to the right the same
number of places as there are zeros.
To multiply decimals by decimals:
1 Change decimals to whole numbers.
2 Use multiplication strategies to solve.
3 Divide through by the multiples of 10 used to convert to the decimals.
To divide a whole number by a decimal, glide the place value to the left the same
number of places for both numbers, to make the divisor a whole number. Carry out the
whole number division.
To divide a decimal number by another decimal, glide the place value to the right for
both numbers the same number of places, to make the divisor a whole number.
Estimate your answer. Carry out the calculation on the calculator and round sensibly.
SA
Review
Chapter summaryDecimals
Short-answer questions
1
2
3
4
5
6
7
8
122
What is the value of 4 in 3.042?
1
Express
as a decimal.
100
Write the numbers 0.023, 2.358, 5.23, 2.3 in order from smallest to largest.
Find the answer to 36.45 1000.
Find the answer to 6 0.2.
Find the answer to 0.4 0.02.
Find the answer to 28 0.04.
Write this decimal fraction in words: 34.703
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M
PL
E
Review
9 Estimate the total cost, using leading digit estimation:
one dog roll at $2.98 and 3 kg of washing powder at $1.57 per kg
10 Find the answer to:
a 0.245 4000
b 50 0.5
11 Find the answer to:
a 12.32 6.45
b 48.37 26.016
c 2.94 13.7 6.23
12 Find the answer to:
a 2 0.32
b 3 0.004
c 4 0.26
13 Find the answer to:
a 3 0.01
b 123 0.3
c 18 0.04
14 At a sale DVDs cost $19.50 each and CDs cost $14.95 each. What is the total cost of
three DVDs and three CDs?
15 A cardboard box has a mass of 0.37 kg. When filled with drink bottles it has a mass of
21.25 kg. How many bottles, each weighing 0.87 kg, are in the carton?
16 Mark saved $20 to go to the grand final of his District League. His return fare cost
$6.35, his ticket was $8.00, a football record was $2.50 and his food cost $1.55.
a How much did it cost him for the day?
b How much money did he have left from his $20?
1
SA
Extended-response questions
1
2
A bottle contains 250 mL of medicine. You are required to take 0.8 mL three times per day.
a How many equal doses will you get from a bottle?
b How long will the bottle last before you need a new one?
c How much will be left in the bottle at the end?
Pauline has $300 to spend at the shopping mall. She purchases five photo frames at
$29.55 each and six CDs for $18.35 each.
a How much did she spend on photo frames?
b How much did she spend on CDs?
c How much did she spend altogether?
d How much did Pauline have left after these purchases?
Chapter 3 — Decimals
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