7
Number
Decimals
You should feel comfortable working with decimal fractions because
they are used everywhere. That mountain bike you want costs $259.95,
or you need a desk 0.75 metres wide to fit between your bed and your
wardrobe. The building, business and banking world could not
function without the use of decimals.
In this chapter you will:
■
■
■
■
■
■
■
■
■
revise place value and order for decimals
write a (terminating) decimal as a common fraction
add and subtract decimals
multiply and divide a decimal by a power of 10, a whole number or
another decimal
use estimation and the calculator to find decimal answers
convert common fractions to terminating or recurring decimals
use the notation for recurring decimals
round decimals to a given number of places
solve a variety of real-life problems involving decimals and money.
Wordbank
■
■
■
■
■
■
decimal A number that includes a decimal point and place value to
indicate parts of a whole.
number of decimal places The number of digits after the decimal point.
estimate An educated guess concerning the answer to a calculation.
recurring decimal A decimal fraction that has one or more digits that
repeat forever.
terminating decimal A decimal fraction that is not recurring, but comes
to an end.
round To write a number to a given number of decimal places.
Think!
Why do you think people now find decimals more convenient to use than
common fractions when describing parts of a whole number?
1
■ How is writing 0.5 better than writing --- ?
2
■
How is writing 0.2 easier than writing 1--5- ?
■
How is 0.75 better than 3--4- ?
DECIMALS
199
CHAPTER 7
Start up
Worksheet
7-01
Brainstarters 7
1 Arrange the numbers in each of these sets in ascending order:
a 21, 32, 34, 17, 8
b 23, 26, 54, 66, 33
c 101, 100, 110, 99, 102, 111
d 251, 247, 256, 254, 244
2 Arrange the numbers in each of these sets in descending order:
a 44, 39, 42, 45, 38, 40
b 55, 56, 53, 50, 49, 52
c 1001, 1000, 1100, 1010, 1111
d 301, 310, 299, 318, 295
Skillsheet
7-01
Rounding
whole numbers
3 a Write 27 to the nearest ten.
c Write 9079 to the nearest thousand.
e Write 29 537 to the nearest hundred.
b Write 752 to the nearest hundred.
d Write 16 837 to the nearest thousand.
f Write 40 219 to the nearest ten.
4 Evaluate:
a 123 + 32 + 456
c 6 + 78 + 32 + 9 + 199
e 234 − 45
g 894 − 35
i 58 × 3
k 589 × 13
m 2920 ÷ 8
o 10 836 ÷ 21
b
d
f
h
j
l
n
p
432 + 45 + 2341 + 7
47 852 + 365 + 1458 + 81
1138 − 445
114 782 − 36 989
126 × 8
789 × 26
2163 ÷ 7
625 ÷ 25
5 Evaluate:
a 32 × 10
c 325 × 1000
e 400 ÷ 10
g 1 000 000 ÷ 1000
b
d
f
h
578 × 100
400 × 10
1200 ÷ 100
81 000 ÷ 100
Place value
Skillsheet
7-02
Decimal
fractions
You are familiar with numbers that have decimal points, for example:
• money
$3.52
• measurements
6.2 m.
The digits after the decimal point indicate a part of a whole.
The position of a digit in a number shows its size. This is known as place value.
Example 1
What does the number 532.81 mean?
Solution
5
hundreds
3
tens
2
units
.
8
1
tenths hundredths
decimal
point
So 532.81 is 5 × 100 + 3 × 10 + 2 × 1 + 8 ×
200
NEW CENTURY MATHS 7
1
-----10
+1×
1
--------- .
100
Example 2
Write each of these decimals in expanded form:
a 604.75
b 29.04
Solution
a 604.75 = 6 × 100 + 0 × 10 + 4 × 1 + 7 ×
b 29.04 = 2 × 10 + 9 × 1 + 0 ×
1
-----10
+4×
1
-----10
+5×
1
--------- .
100
1
--------- .
100
Example 3
What is the place value of the digit 8 in the number 612.87?
Solution
The 8 in 612.87 is in the tenths column, so it has a value of 8 tenths (or
8
------ ).
10
Example 4
Write in decimal form:
a 6+
1
-----10
+
8
--------100
b 2 × 10 + 4 ×
1
--------100
Solution
a 6.18
b 20.04
Exercise 7-01
1 Copy this place-value table. Put the 12 given numbers in the table, with the digits in
their correct columns.
hundreds
a
d
g
j
14.82
70.8
503.92
200.047
tens
units
b
e
h
k
6.014
297.86
8.3
4.025
tenths
hundredths
c
f
i
l
Example 1
thousandths
931.02
11.14
0.375
0.81
2 Write each of the following (a to p) as a decimal:
a five and four-tenths
b six and fifteen-hundredths
c eleven and thirty-eight hundredths
d eight and three tenths
e fourteen and six-hundredths
f four hundred and two and three-thousandths
g 6 + 2 tenths + 3 hundredths
h 3 + 7 tenths
i nineteen and nine-tenths
j 14 + 3 tenths + 9 hundredths
l 2 + 6 hundredths
k 2 tens + 4 + 0 tenths + 9 hundredths
n seventy-three hundredths
m 8 + 7 tenths + 5 hundredths
o nine-thousandths
p 8 tenths + 5 hundredths + 9 thousandths
DECIMALS
201
CHAPTER 7
Example 2
Example 3
3 Write in expanded form:
a 1.234
b 102.34
f 12.71
g 8.003
c 30.12
h 4.509
d 0.751
i 0.04
e 2.09
j 0.386
4 What is the place value of the digit 4 in each of these numbers?
a 431.70
b 31.047
c 761.04
e 3.734
f 907.431
g 0.064 75
i 72.314
j 26.74
k 100.0407
d 114.37
h 42 376
l 94.071
5 What is the place value of the digit 7 in each number listed in Question 4?
Example 4
6 Write in decimal form:
1
a 4 + ----10
c 4+
e 5+
3
-----10
2
-----10
g 19 +
i
11 +
+
+
5
--------100
7
--------100
d
+
9
-----------1000
m 3×
o 2×
q 9+
f
9
-----------1000
6
2
------ + ---------------10
10 000
1
------ + 1
10
1
1
1
------ + 4 × --------- + 1 × -----------100
1000
10
1
1
- + 3 × --------10 + 7 + 2 × ----100
10
1
1
- + 1 × -----------4 × -------100
1000
k 2 × 100 + 7 × 10 + 6 ×
8
-----10
9
3
- + --------12 + ----10
100
4
6
- + -----------0 + -------100
1000
7
3
- + -----------2 + ----10
1000
1
3 × 1 + 2 × ----10
b 3+
h
j
×
1
--------100
l
+
1
---------------10 000
+7×
7 × 10 + 6 × 1 + 5 ×
n 9×
p 4×
r 5×
1
-----10
1
-----------1000
1
10 + 9 × -------100
1
- +
100 + 4 × ----10
1
--------100
1
--------100
+7×
3×
1
-----------1000
Understanding the point
Example 5
Where would you place the decimal point in this weather report?
The day was fine and warm, with a maximum temperature of 245˚C.
Solution
Since a warm day has a temperature in the mid-twenties, we would place the decimal point
after the 4 so it reads 24.5˚C.
Exercise 7-02
Example 5
Read the following story carefully. List the numbers appearing in parts a to q and place a
decimal point in each so that the story makes sense.
a Maria filled her Holden car, which was nearly empty, with 434 litres of petrol
b at 849 cents per litre
c and handed the attendant a $5000 note.
d After she received her change of $1315
e she picked up her friend Fred, a tall man of 188 metres in height.
202
NEW CENTURY MATHS 7
f
g
h
i
j
k
l
m
n
o
p
q
Into the car jumped Charlie the dog, weighing 144 kilograms,
happily wagging his 150 centimetres of tail.
Maria and Fred had each packed a 200 kg backpack.
They stopped for a snack at McDougall’s and bought two beefburgers each, so that they
could have their daily intake of 250 kg of meat.
After they had driven for several hours, averaging 850 kilometres per hour, they were
within sight of Mt Kosciuszko, the highest mountain in Australia,
about 23000 m above sea level.
They enjoyed seeing the small eucalyptus trees about 105 m high.
They also saw a kangaroo, which jumped about 80 m,
110 times the Olympic record for the long jump that was set in Mexico City.
After a pleasant walk of 134 km they arrived back at their campsite.
They slept soundly and, next morning, drove the 2500 km home
in about 300 hours.
Ordering decimals
The number of digits after the decimal point tells us the number of decimal places in the
decimal number.
Worksheet
7-02
Dewey decimals
Example 6
How many decimal places are there in:
a 3.6567?
b 15.801?
Solution
a 3.6567 has four decimal places.
1 23 4
b 15.801 has three decimal places.
1 23
Example 7
Arrange these numbers in ascending order (smallest to largest):
67.41
67.14
6.714
67.04
Solution
To compare decimals more easily, insert zeros at the end to give all the numbers the same
amount of decimal places.
67.41
67.14
6.714
67.04
become: 67.410 67.140 6.714
67.040
In ascending order: 6.714, 67.04, 67.14, 67.41
Example 8
Arrange these numbers in descending order (largest to smallest):
0.5
0.08
1.7
0.85
Solution
0.5
0.08
1.7
0.85
become: 0.50
0.08
1.70
0.85
In descending order: 1.7, 0.85, 0.5, 0.08
DECIMALS
203
CHAPTER 7
Exercise 7-03
Example 6
1 How many decimal places does each of these numbers have?
a 1.65
b 3.881
c 15.3062
d 0.005
e 7.045 73
f 814.3
g 9.100 001
h 203.222 22
i 0.040 400 4
Example 7
2 Arrange these numbers in ascending order (smallest first):
a 43.89, 56.324, 9.998, 80.879, 400, 23.89, 56.314
b 0.568, 0.684, 0.099, 1.002, 0.586, 5.608, 0.0586
c 1.23, 0.891, 1.814, 0.222, 7.007, 0.89
d 0.5, 0.05, 0.005
e 3.441, 3.404, 3.4, 3.44, 3.004, 3.044
f 0.2, 0.202, 0.22, 0.022
g 1.01, 1.002, 1.012, 1.21
h 0.07, 0.67, 0.71, 0.007, 7
Example 8
3 Arrange these numbers in descending order (largest first):
a 570.25, 125.63, 0.9899, 4000.99, 1256.3, 400.099
b 5.37, 6.539, 5.639, 5.367, 3.659, 3.66, 5.369
c 1.6, 1.61, 1.599, 1.601, 1.509
d 6, 0.06, 0.6, 6.6
e 0.7, 0.07, 0.707, 0.77, 0.007, 7.07
f 0.4004, 0.044, 0.404, 0.44
g 0.1, 0.08, 0.65, 0.029
h 0.92, 0.921, 0.09, 0.099
4 Insert or to make each of these true:
a 0.2
0.25
c 0.035
0.305
d 1.59
1.059
g 44.44
4.444
b
d
f
h
0.731
0.007
0.099
0.7932
0.73
0.070
0.99
0.7239
5 Copy each of those number lines carefully and fill in the values of the points marked with
purple dots:
a
1.9
1.6
2.3
b
4.07
4.09
4.11
c
0.67
0.65
0.71
d
8.0
8.1
8.16
8.2
e
0.23
0.07
f
2.6
2.0
g
4.3
204
4.4
NEW CENTURY MATHS 7
4.5
4.6
4.7
6 Look at the following diagram.
a Find a path from the START circle to the TOP circle. You can make your first move in
either direction, but then you can only move to a circle with a larger number.
TOP
0.029
0.12
0.09
0.2
0.121
0.071
0.081
0.17
0.11
0.3
0.05
0.005
0.14
0.015
START
b Now write out the number sequence for:
i the longest path
ii a path requiring seven moves
iii the route requiring nine moves
c There are several paths that use the smallest number of moves.
i Find and write the number sequence for all the shortest paths you can find.
ii How many moves are in the shortest path?
Using technology
Ordering decimals
A spreadsheet can be used for ordering a group of numbers, from smallest to largest or
vice versa.
Instructions
Step 1: Type your numbers into the green column, like the green box below.
(You can overtype Spreadsheet 7-01.)
Step 2: Hold down the left mouse key and drag, to highlight the numbers that you typed.
Step 3: Click on the Data menu, and choose ‘Sort’.
Step 4: You must choose this column to sort, but you can decide whether to choose to sort
in ascending order or descending order.
Step 5: When you press ‘OK’, the spreadsheet will automatically order the data for you.
Example:
Spreadsheet
7-01
Ordering
decimals
0.2
0.23
0.232
0.233
0.3
0.32
0.322
DECIMALS
205
CHAPTER 7
Worksheet
7-03
Decimals
squaresaw 1
Decimals are special fractions
A decimal number usually includes a part of a whole unit. The unit is divided into 10 parts
(tenths), 100 parts (hundredths), 1000 parts (thousandths), or any ‘power of 10’ parts.
This shape has been divided into 10 equal parts.
Nine out of the 10 parts are coloured blue.
9
- or nine-tenths
Writing this as a common fraction, ----10
of the shape is coloured.
Writing it as a decimal fraction, 0.9 or ‘zero-pointnine’ of the shape is coloured.
A decimal number has a decimal point to mark where the whole part is separated from
the fraction part of the number.
Example 9
Write each of these as a decimal:
a
22
--------100
b
7
3 ----10
c eighty-five thousandths
Solution
a
22
--------100
= 0.22
7
- = 3.7
b 3 ----10
c eighty-five thousandths =
85
-----------1000
= 0.085
Note:
• tenths have one decimal place
• hundredths have two decimal places
• thousandths have three decimal places, and so on.
Example 10
Write each of these as a common fraction:
a 0.09
b 0.273
c 8.1
Solution
a 0.09 =
9
--------100
b 0.273 =
273
-----------1000
c 8.1 = 8 +
1
-----10
(two decimal places means hundredths)
(three decimal places means thousandths)
1
- (one decimal place means tenths)
= 8 ----10
Note that the number of zeros in the denominator (bottom number) of the common fraction
is the same as the number of decimal places in the decimal.
206
NEW CENTURY MATHS 7
Exercise 7-04
1 What part of each of these shapes has been shaded? (Give your answers as decimals.)
a
b
c
d
2 Write each of these as a decimal:
9
15
a ----b -------10
100
e four-tenths
i
l
p
704
-----------1000
14
--------100
17
--------100
Example 9
c
79
--------100
6
-----10
d
f
23
--------100
j
eighty-seven hundredths
k
m
8
-----10
368
-----------1000
o
q
g
n
r
h
235
-----------1000
9345
---------------10 000
s
60
--------100
411
-----------1000
7
--------100
247
-----------1000
493
-----------1000
3 Write each of these as a decimal:
a
d
g
j
m
67
--------100
11
-----------1000
1
--------100
17
---------------10 000
33
---------------10 000
b
e
3
-----10
8
--------100
h thirty-two hundredths
k
n
2
--------100
46
------------------100 000
c
f
4
--------100
6
-----------1000
i
five-thousandths
l
57
-----------1000
o seven-millionths
4 Write each of these as a decimal:
67
a 1 -------100
47
b 4 -------100
8
d 6 ----10
e forty-five and twenty-three thousandths
f
3
2 ----10
11
g 6 ----------1000
i
0
42 ----10
j
0
5 -------100
5 Write each of these as a common fraction:
a 0.7
b 0.4
d 0.572
e 0.003
g 0.11
h 0.309
j 0.999
k 0.013
m 0.0471
n 0.3333
p 0.91
q 0.087
s 27.33
t 2.007
v 7.41
w 101.3
9
c 23 ----10
143
h 89 ----------1000
6
k 4 ----------1000
c
f
i
l
o
r
u
x
Example 10
0.39
0.05
0.9
0.0004
0.5001
1.9
10.349
6.0102
DECIMALS
207
CHAPTER 7
Working mathematically
Applying strategies and reflecting: Decimal distances
Equipment: A tape measure and some coloured chalk.
Step 1: Form a group of three or four students and find a flat area such as a path,
basketball court, or the classroom floor.
Step 2: Mark a starting point on your flat area.
Step 3: Choose a distance between 1 and 4 metres, for example 3.25 m.
Step 4: Using chalk, take turns to mark your estimate of 3.25 m from the starting
point. Measure the exact length and mark it with an X.
Step 5: Give points to each person. Score 5 points for best estimate, 3 points for next
best, 2 points for next and 1 point for next.
Take turns to choose distances, and play the game until someone reaches 20 points
and wins.
Sample score sheet for one person:
Trial
Distance
Guess
1
3.25
2.57
2
4.50
3.16
3
3.50
3.41
4
1.75
1.82
5
2.20
2.38
Points
Total
Skillbank 7
SkillTest
7-01
Estimation
Estimating answers
A quick way of estimating the answer to a calculation is to round each number in the
calculation.
1 Examine these examples:
a 631 + 280 + 51 + 43 + 96 ≈ 600 + 300 + 50 + 40 + 100
= (600 + 300 + 100) + (50 + 40)
= 1000 + 90
= 1090
(Actual answer = 1101)
b 55 + 132 − 34 + 17 − 78 ≈ 60 + 130 − 30 + 20 − 80
= (60 + 20 − 80) + (130 − 30)
= 0 + 100
= 100
(Actual answer = 92)
c 67 × 13 ≈ 70 × 12
= 840
(Actual answer = 871)
d 78 × 7 ≈ 80 × 7
= 560
(Actual answer = 546)
208
NEW CENTURY MATHS 7
e 929 ÷ 5 ≈ 1000 ÷ 5
= 200
f 510 ÷ 24 ≈ 500 ÷ 20
≈ 50 ÷ 2
= 25
(Actual answer = 185.8)
(Actual answer = 21.25)
2 Now estimate the answers to these:
a 27 + 11 + 87 + 142 + 64
c 684 + 903
e 517 − 96
g 766 − 353
i 83 × 81
k 828 ÷ 3
b
d
f
h
j
l
55 + 34 − 22 − 46 + 136
35 + 81 + 110 + 22 + 7
210 − 38 − 71 + 151 − 49
367 × 2
984 × 16
507 ÷ 7
Adding and subtracting decimals
Careful setting out is essential when adding and subtracting decimal numbers.
When adding or subtracting whole numbers, we must write them under one another in their
correct place-value columns:
67
Not: 67
486
486
9
9
302
302
+ 59
+ 59
923
???
The same is true when adding and subtracting numbers that have decimal points.
Example 11
Add 16.27, 10.92, 4.03, 0.89, 32, 0.6
Solution
Estimate: 16 + 11 + 4 + 1 + 32 + 1 = 65.
The answer should be about 65.
16.27
10.92
4.03
0.89
32.0
+ 0.6
Remember that 32 is the same as 32.0 or 32.00.
64.71
When adding or subtracting decimals, keep decimal points under one another.
DECIMALS
209
CHAPTER 7
Example 12
1 Subtract 8.914 from 46.029.
Solution
Estimate: 46 − 9 = 37.
The answer should be about 37.
46.029
− 8.914
37.115
2 Find the answer to 4.31 − 2.183.
Solution
Estimate: 4 − 2 = 2.
4.310
− 2.183
Hint: Fill the spaces with zeros.
2.127
Spreadsheet
7-02
Counting with
decimals
Example 11
To really understand decimals, we need to be able to count up and down by decimal fraction
steps. Use the link to go to an activity enabling you to practise this skill.
Exercise 7-05
1 Copy and complete these addition calculations:
a 5.3
b 4.723
c 43.5
6.2
0.01
116.29
+ 0.5
+ 12.2
7.3
+ 0.227
e
Example 12
0.0076
1.23
+ 0.9
f
37
3.45
1.98
+ 548.7
g
6.83
0.5
12.387
+ 5.001
2 Copy and complete these subtraction calculations:
a
8.936
b 57.703
c 6.1
− 3.513
− 16.21
− 0.2
e
22.6
− 13.54
f
30.001
− 1.73
3 Find the answers to these:
a 103.67 + 9.81 + 0.24 + 3.7
c 98.64 − 41.09
e 38.624 + 1.109 + 23.7 + 0.65
g 2.91 + 32.5 − 18.9
i 58.94 − 2.31 − 46.13
g
b
d
f
h
j
6.03
− 4.091
d 12.755
1.35
18.675
+ 49
h
0.057
0.1324
5.6731
+ 38.009
d
23.57
− 16.88
h
48.6
− 9.951
4.6 + 2.9 + 15 + 0.16 + 32.32
7.41 − 3.95
81.91 + 3.254 + 681.7 + 458
1.57 − 0.89 + 2.34
14.62 − 0.195 + 6.37
4 An electrician needed these lengths of cable to complete a wiring job: 12.3 m, 4.8 m,
18.7 m, 7.98 m, 13.65 m and 23.6 m.
a How many metres of cable did the electrician use?
b If the full spool of cable was 100 m long, how many metres of cable were left in the
spool after the electrician completed the job.
210
NEW CENTURY MATHS 7
5 Add each set of prices. Calculate the exact change from the amount shown in brackets.
a $7.01
$0.34
$2.19
($10)
b $2.14
$1.34
39c
$0.45
($5)
c $5.26
$10.09
$14.31
$11.36
($50)
d $0.85
$4.34
$1.17
$8.79
($20)
e $11.34
$9.15
$3.95
$7.92
$2.36
($50)
f $1.46
$13.74
$42.99
$7.46
$29.05
($100)
6 To keep fit, Angela runs each day. Last week she ran 3.8 km, 4.1 km, 2.3 km, 2.6 km,
3.0 km, 0.9 km and 1.8 km. How far did she run last week?
7 A truck carrying sand had a total mass of 13 248.56 kg. If the truck alone had a mass of
5210.8kg, what is the mass of the sand?
8 Five runners in the school’s 100 m race recorded the following times: 13.5 s, 13.81 s,
12.7 s, 14.62 s, 12.45 s
a Place these times in order, from fastest to slowest.
b What is the time difference between the first runner to finish and the last?
c If the runner in second place had run 0.3 seconds faster, would she have won the race?
Explain your answer.
9 Krysten’s expenses for one week are shown in
the table on the right.
a How much did she spend?
b How much did she have left out of $620.80?
Item
Food
Cost ($)
128.80
Clothing
88.45
Car
58.35
Rent
185.00
Entertainment
78.95
Savings
66.00
10 A block of wood 11.27 cm thick has 0.34 cm shaved off one side and 0.55 cm shaved
off the other side. How thick is the block of wood then?
11 Wendy was making teddy bears. She needed these amounts of material for five bears:
2.6 m, 0.8 m, 1.2 m, 0.75 m and 0.88 m. How much material did she need altogether?
12 The odometer in a car measures the total distance the car has travelled. The odometer
below reads 21 456.9 km. The blue digit shows tenths of a kilometre.
2
1
4
5
6
9
Find the distance travelled during a holiday if the odometers below give the readings at
the start of the holiday and at the end of the holiday.
2
1
5
7
6
4
2
2
3
1
5
3
DECIMALS
211
CHAPTER 7
Working mathematically
Applying strategies: Comparing heights
1 Use the clues below to find each girl’s name and height.
• Mandy is taller than Sarah.
• Kelly is taller than Sarah but shorter than Mandy.
• Sarah is shorter than Yin.
• Mandy is not the tallest.
• The heights of the girls are 168.5 cm, 166.3 cm, 164.2 cm and 160.7 cm.
A
B
C
D
2 Use the clues below to find each boy’s name and height.
• Steve is 164.7 cm tall.
• Mike is 14.7 cm taller than Milof.
• Steve is 3.9 cm shorter than Milof.
• Ganesh is 1.6 cm taller than Mike.
A
B
C
D
3 Use the clues below to find each student’s name and height.
• Yollande is 15.1 cm taller than Peter.
• Jade is 13.7 cm shorter than Yollande.
• Karl is 20.6 cm taller than Jade.
• Yollande is 6.9 cm shorter than Karl.
• Peter is 163 cm tall.
A
B
C
D
4 Devise your own problem using four to six class members. Estimate their heights and
height differences, and write a set of clues. (Don’t forget to change their names!)
212
NEW CENTURY MATHS 7
Working mathematically
Applying strategies and reasoning: Contents of envelopes
The objects listed below are typical of what we put in envelopes.
• pin 0.06 g
• stamp 0.13 g
• lined notepaper 3.69 g
• staple 0.07 g
• cheque 0.97 g
• paper clip 0.22 g
The total mass (in grams) is printed on each
envelope on the right. If a standard envelope
has a mass of 2.23 g, identify the contents of
each of these envelopes from its total mass.
Each envelope can contain more than one of
each object listed above.
a Contents: 1
and 1
b Contents: 1
and 1
and 5
c Contents: 2
,1
d Contents: 1
,1
,1
and 3
e Contents: 2
,1
and 1
f Contents: 3
,2
,1
, 1
a
b
3.42 g
6.14 g
c
d
8.16 g
7.32 g
e
f
9.9 g
15.66 g
and 1
Multiplying and dividing by powers of 10
3
7
3
6
7
0
6
7
0
0
th
ou ten
sa
nd
th
s
s
nd
sa
ou
th
7
2
3
2
3
5
2
3
5
3
5
0
2.35 × 10
th
s
th
re
d
nd
hu
6
2
hs
te
3
2.35 × 100
nt
ts
ni
6
2.35
2.35 × 1000
U
3
36.7 × 10
36.7 × 1000
Te
n
dr
ed
H
un
sa
ou
36.7
36.7 × 100
s
s
s
nd
s
Th
ou Ten
sa
nd
th
C
al
cu
la
tio
n
This place-value chart shows what happens when you multiply and divide a decimal by
powers of 10. Check it using your calculator.
5
36.7 ÷ 10
3
6
7
36.7 ÷ 100
0
3
6
7
36.7 ÷ 1000
0
0
3
6
2.35 ÷ 10
0
2
3
5
2.35 ÷ 100
0
0
2
3
7
5
Note how the decimal point moves to the right when multiplying and to the left when
dividing. The number of places moved is the same as the number of zeros in the power of 10
you are multiplying or dividing by.
DECIMALS
213
CHAPTER 7
Exercise 7-06
1 Copy and complete each of these statements:
a To multiply by 10, move the decimal point
place to the
.
b To multiply by 100, move the decimal point
places to the
.
c To multiply by 1000, move the decimal point
places to the
.
d To divide by 10, move the decimal point
places to the
.
e To divide by 100, move the decimal point
places to the
.
f To divide by 1000, move the decimal point
places to the
.
2 Copy and complete:
a 46.3 ÷ 10 =
d 36.4 ÷ 100 =
g 6.43 ÷ 1000 =
j 66 ÷ 100 =
m 24.9 × 100 =
p 0.416 × 100 =
s 60 451 × 100 =
Worksheet
7-04
Where’s the
point?
507 ÷ 100 =
705 ÷ 1000 =
64.3 ÷ 1000 =
0.31 ÷ 1000 =
0.81 × 10 =
0.81 × 100 =
6.02 × 100 =
c
f
i
l
o
r
u
1203 ÷ 1000 =
3102 ÷ 10 =
4.28 ÷ 1000
0.02 ÷ 10
37.42 × 1000 =
2.192 × 1000 =
0.031 × 10 =
3 Evaluate, using the rules you have found:
a 45.213 × 100
b 10.64 × 1000
d 5.98 ÷ 1000
e 847.612 × 100
g 36.28 ÷ 100
h 519.4 × 1000
j 81.348 ÷ 1000
k 502 × 100
m 0.4 ÷ 1000
n 17.01 × 100
p 66 ÷ 10 000
q 5.2 × 10 ÷ 100
c
f
i
l
o
r
6304 ÷ 100
0.0592 × 10
40.075 ÷ 10
0.61 ÷ 100
12.3 × 10 000
14.71 ÷ 100 × 10
4 Evaluate:
a 469 187 ÷ 100 000
d 1 200 000 ÷ 1 000 000
c 27.43 ÷ 1 million
f 137.429 × 1000 ÷ 10 000
b
e
h
k
n
q
t
b 437.421 ÷ 1000
e 235 000 137 ÷ 10 000
Multiplying decimals
Estimation and ‘fixing the point’
If you know the result of a whole number multiplication, you can use estimation to work out
the answer to similar decimal multiplications and correctly position the decimal point.
Example 13
1 Given that 17 × 12 = 204, find:
a 1.7 × 12
b 1.7 × 120
Solution
Multiplication
Estimate
17 × 12
204
a
1.7 × 12
2 × 10 = 20
b
1.7 × 120
2 × 100 = 200
214
Answer
NEW CENTURY MATHS 7
20.4
204
(given)
use the digits 204 to make a number near 20
use the digits 204 to make a number near 200
2 Given that 23 × 47 = 1081, find:
b 230 × 4.7
a 2.3 × 47
c 23 × 0.47
Solution
Multiplication
Estimate
23 × 47
a
2.3 × 4.7
b
230 × 4.7
c
Answer
1081
2 × 5 = 10
10.81
200 × 5 = 1000 1081
23 × 0.47
20 × 0.5 = 10
10.81
(given)
use the digits 1081 to make a number near 10
use the digits 1081 to make a number near 1000
use the digits 1081 to make a number near 10
Exercise 7-07
Example 13
1 Copy and complete each of the following tables:
a
Multiplication
Estimate
69 × 18
Answer
1242
690 × 180
6.9 × 180
6.9 × 1.8
690 × 1.8
b
Multiplication
Estimate
104 × 42
Answer
4368
10.4 × 42
1.04 × 4.2
104 × 4.2
0.104 × 4.2
c
Multiplication
38 × 92
Estimate
Answer
3496
3.8 × 92
38 × 920
38 × 0.92
380 × 0.92
2 Given that 63 × 34 = 2142, use estimates to find:
b 0.63 × 3.4
a 630 × 340
e 6.3 × 34 000
d 3.4 × 630
c 0.63 × 3400
f 63 × 340
DECIMALS
215
CHAPTER 7
3 Given that 1.7 × 1.2 = 2.04, use estimates to find:
b 17 × 12
a 1.7 × 12
e 0.12 × 17
d 120 × 170
c 0.17 × 1.2
f 17 000 × 0.12
4 Given that 7.2 × 3.4 = 24.48, use estimates to find:
b 72 × 3.4
a 7.2 × 34
e 7.2 × 0.34
d 72 × 34
c 0.72 × 3.4
f 720 × 340
5 Given that 1.26 × 6 = 7.56, use estimates to find:
b 126 × 6
a 12.6 × 6
e 0.126 × 0.6
d 0.126 × 6
c 1.26 × 0.6
f 126 000 × 600
6 Use the fact that 0.3 × 0.24 = 0.072 to find:
b 0.3 × 2.4
a 3 × 0.24
e 300 × 2.4
d 30 × 0.24
c 0.3 × 24
f 0.03 × 0.24
Working mathematically
Reflecting and reasoning: Decimal places in multiplication answers
1 What happens when you multiply by a number less than one? As a group activity,
consider the product of 12 × 0.8.
a Is the answer more or less than 12? Why?
b Estimate the answer to 12 × 0.8.
c How many decimal places have 12 and 0.8?
d Use a calculator to evaluate 12 × 0.8. How many decimal places has the answer?
2 a
b
c
d
Is the answer to 0.7 × 0.3 more or less than 0.7? Why?
Estimate the answer to 0.7 × 0.3.
How many decimal places have 0.7 and 0.3?
Use a calculator to evaluate 0.7 × 0.3. How many decimal places has the answer?
3 a
b
c
d
e
Is the answer to 2.5 × 4.1 more or less than 2.5? Why?
Estimate the answer to 2.5 × 4.1.
How many decimal places have 2.5 and 4.1?
Use a calculator to evaluate 2.5 × 4.1. How many decimal places has the answer?
What is the relationship between the number of decimal places in the question
and the number of decimal places in the answer?
4 a
b
c
d
If 82 × 6 = 492, what do you think is the answer to 82 × 0.6? Why?
If 4 × 17 = 68, what do you think is the answer to 0.4 × 1.7? Why?
If 367 × 51 = 18 717, what do you think is the answer to 3.67 × 5.1? Why?
What is the answer to 0.5 × 0.9?
When multiplying decimals, the number of decimal places in the answer
equals the total number of decimal places in the question.
5 Make up a question about multiplying decimals. Swap questions with all members
of the group. If you disagree about the correct answers, check with your teacher.
216
NEW CENTURY MATHS 7
Using the multiplication rule
Animation
7-01
Multiplying
decimals
Example 14
1 Evaluate 3.05 × 4.8.
Solution
Step 1: Do the multiplication without decimal points.
305
× 48
2440
12 200
Worksheet
7-05
Which
decimals?
14 640
Step 2: Decide where to place the decimal point. There are two decimal places in 3.05
and one decimal place in 4.8. So there are a total of three decimal places in the
question. We write the answer, 14 640, with three decimal places: 14.640 = 14.64.
2 Evaluate 0.6 × 4.1.
Solution
41
×6
246
Because there is a total of two decimal places in the question, we need to write the
answer, 246, with two decimal places. The answer is 2.46.
Exercise 7-08
1 Use the method shown above to calculate:
a 3.05 × 4
b 1.02 × 7
d 17.1 × 2
e 10 × 2.25
g 6.95 × 5
h 1.0004 × 8
j 0.4 × 12
k 6 × 0.002
m 10.2 × 4
n 3 × 4.2
p 0.71 × 2
q 7 × 2.193
s 0.63 × 5
t 492 × 0.12
c
f
i
l
o
r
u
2.001 × 9
3 × 4.20
0.18 × 5
0.25 × 11
0.5 × 10
10.941 × 3
0.11 × 365
2 Calculate:
a 0.4 × 0.8
d 0.3 × 0.24
g 12.6 × 0.06
j 2.93 × 0.3
m 0.9 × 0.75
p 2.45 × 0.1
b
e
h
k
n
q
3.9 × 0.5
0.08 × 0.04
0.28 × 3
6.80 × 4
8.06 × 4.1
0.02 × 3.5
c
f
i
l
o
r
0.8 × 0.6
3.1 × 0.4
0.39 × 9
0.54 × 20
0.11 × 1.01
6.3 × 0.04
3 Find:
a 47.9 × 23
d 2.95 × 5.13
g 0.237 × 1.2
j 0.023 × 0.042
b
e
h
k
6.43 × 7.2
2.45 × 3.9
9.2 × 7.9
0.7521 × 3.6
c
f
i
l
83.4 × 6.3
94.6 × 5.1
11.1 × 0.621
321.2 × 8.1
Example 14
DECIMALS
217
CHAPTER 7
Working mathematically
Applying strategies and reflecting: Multiplication estimation game
This is either a game for two players, or a class team game.
Players should copy this table into their books or print it from the CD-ROM, but
should not commence calculating.
Multiplication
a
42 × 3.5
b
5.1 × 2.4
c
1.6 × 7.3
d
3.8 × 6.2
e
2.2 × 8.8
f
7.5 × 3.2
g
5.6 × 5.7
h
2.6 × 5.1
i
9.3 × 3.9
j
4.8 × 3.3
Estimation
Correct answer
from calculator
Points
Total
Rules
Step 1: You each have a two-minute time limit in which to estimate all the answers for
the 10 multiplication questions in the table, correct to one decimal place.
Step 2: On the instruction ‘Go’ given by your partner, you begin to write the estimates
for the multiplication questions in your table.
Step 3: After two minutes, stop. Then the other person completes Step 2.
Step 4: Use a calculator to find the exact answers, and score yourself for each
multiplication.
• 5 points for each estimate within 0.1 of the correct answer.
• 4 points for each estimate within 0.2 of the correct answer.
• 3 points for each estimate within 0.3 of the correct answer.
• 2 points for each estimate within 0.4 of the correct answer.
• 1 point for each estimate within 0.5 of the correct answer.
• 0 points if the estimate is more than 0.5 from the correct answer.
Step 5: Total your points, and compare your total with that of your partner.
Investigate further
1 Did you tend to overestimate or underestimate?
Spreadsheet
7-03
Multiplication
game
2 Now make up and try your own set of 10 decimal multiplications. See if you perform
better with practice.
3 Use this link to try an Excel version of this game.
218
NEW CENTURY MATHS 7
Calculating change
Example 15
Find the change from $50 if the following items are bought:
• 3 kg of butter at $2.50 per kilogram
• 500 g of cheese at $12.90 per kilogram
• 5 kg of meat at $5.75 per kilogram
• 2 dozen eggs at $1.77 per dozen.
Round the total to the nearest 5 cents.
Solution
3
0.5
5
2
× 2.50
× 12.90
× 5.75
× 1.77
= 7.50
= 6.45
= 28.75
= + 3.54
Total 46.24
Rounded to the nearest 5 cents, this total is $46.25.
Change from $50:
50.00
− 46.25
SALESPERSON 31
MILK 1LT
BUTTER
BREAD
1.13
1.86
1.64
RND TOTAL
CASH
CHANGE
4.65
5.00
0.35
10:52
Worksheet
7-05
Which decimals?
Worksheet
7-06
Shopping and
change
12/12/2004
3.75
Exact change will be $3.75.
Exercise 7-09
1 Calculate the change from $50 for each of the following purchases. Round totals to the
nearest 5 cents.
a • meat $25.36
• 2 dozen eggs at $1.36 per dozen
• 5.5 kg of apples at $1.49 per kilogram
b • 3 kg of tomatoes at $2.45 per kilogram
• 10 kg of potatoes at $1.29 per kilogram
• 4 L of milk at $1.37 per litre
• 2 kg of butter at $3.08 per kilogram
c • 3.5 kg of chops at $3.89 per kilogram
• 2.5 kg of T-bone steak at $8.10
per kilogram
• 5 L of soft drink at $1.26 per litre
d • toothpaste $2.56
• jam $1.13
• cheese $3.08
• 4 kg of oranges at $1.74 per kilogram
• 3 kg of sausages at $1.85 per kilogram
e • 2 kg of nails at $2.51 per kilogram
• 2 m of wood at $13.48 per metre
• 2.5 L of glue at $6.74 per litre
DECIMALS
219
Example 15
SkillBuilder
1-09
Rounding
money
CHAPTER 7
• 7 pens at $2.15 each
• 12 glue sticks at $1.99 each
• 4 erasers at $0.35 each
g • 2 tubes of toothpaste at $3.15 each
• 2 bottles of drink at $1.37 each
• 4 boxes of tissues at $3.29 each
• 1.5 kg of tomatoes at $2.65 per kilogram
h • 46.5 litres of petrol at 93.6 cents per litre
• 0.5 litres of premium oil at $6.60 per litre
f
2 Find as many words as you can beginning with ‘deci’, meaning ‘ten’.
Using technology
Spreadsheet
7-04
Calculating
change
Calculating change
Your task is to solve all of the parts of Question 1 in Exercise 7-09 using a spreadsheet
similar to the one below.
A
6
B
C
D
E
Item
Quantity
Price/Unit
Cost
F
7
8
Meat
9
Eggs
10
Apples
1 $
2 $
5.5 $
25.36
1.36
1.49
11
12
TOTAL:
13
14
15
Amount given: $
CHANGE:
50.00
1 What formula will you use to calculate the cost?
2 What formula will you use to calculate the total?
3 What formula will you use to calculate the change?
Working mathematically
Applying strategies and reflecting: Multiplication maze
1 Can you work out how to get through the maze on the facing page? Start with 1 in
your calculator display and, as you travel along each path, multiply the number in
the display by the number you pass.
You may travel along each path only once, but it can be in any direction. You must
finish with a 5 in the display.
2 How many steps are needed to complete the maze?
3 Is there only one solution to this maze?
220
NEW CENTURY MATHS 7
× 0.1
× 2.5
× 0.5
× 0.2
× 10
1 START
× 1.5
×5
×2
×4
FINISH 5
×2
× 0.4
× 0.25
Dividing decimals
Dividing decimals by whole numbers
When dividing by a whole number, keep decimal points under one another.
Example 16
Evaluate each of the following:
a 10 ÷ 4
b 13.62 ÷ 3
c 0.018 ÷ 6
d 2.63 ÷ 4
0.003
c 6 0.018
0.6575
d 4 2.6300
Solution
2.5
a 4 10.0
4.54
b 3 13.62
Always check by estimation that your answers sound reasonable.
Exercise 7-10
1 Evaluate each of the following:
b
a 4.8 ÷ 2
e
d 32.8 ÷ 8
h
g 0.056 ÷ 7
k
j 7.35 ÷ 2
m 13.42 ÷ 8
n
18.6 ÷ 3
29.3 ÷ 2
10.71 ÷ 4
4.15 ÷ 8
22.47 ÷ 7
c
f
i
l
o
20.8 ÷ 5
8.79 ÷ 4
195.6 ÷ 8
19.23 ÷ 5
0.318 ÷ 3
2 Evaluate each of the following:
b
a 12 ÷ 5
e
d 88.88 ÷ 11
h
g 18.48 ÷ 15
k
j 205 ÷ 50
m 58 ÷ 8
n
13.56 ÷ 12
107.1 ÷ 9
177 ÷ 12
165.2 ÷ 20
0.732 ÷ 6
c
f
i
l
o
23 ÷ 4
82.5 ÷ 6
0.044 ÷ 11
2075.6 ÷ 8
6.23 ÷ 50
Example 16
DECIMALS
221
CHAPTER 7
3 Find:
a 0.651 ÷ 3
d 675 ÷ 12
g 2.75 ÷ 4
j 79.86 ÷ 33
m 0.471 ÷ 3
b
e
h
k
n
37 ÷ 8
89.341 ÷ 7
0.093 ÷ 11
31.98 ÷ 13
256.84 ÷ 4
c
f
i
l
o
0.078 ÷ 6
4.275 ÷ 5
117.44 ÷ 16
130.76 ÷ 28
0.696 ÷ 12
Dividing decimals by decimals
Look at this pattern:
18 ÷ 3 = 6
180 ÷ 30 = 6
1800 ÷ 300 = 6
When dividing two numbers, if we multiply both numbers by the same power of 10 first, the
answer stays the same. We can use this property to help us divide decimals. For example:
9.8 ÷ 0.08 = 980 ÷ 8 (multiplying both numbers by 100)
= 122.5
The answer to 980 ÷ 8 is easier to find because it involves division by a whole number.
To divide a decimal by a decimal:
Step 1: Make the second decimal a whole number by moving the decimal point the
appropriate number of places to the right.
Step 2: Move the decimal point in the first decimal the same number of places to
the right.
Step 3: Divide the new first number by the new second number.
This works because in Steps 1 and 2 we multiply both decimals by the same power of 10.
Animation
7-02
Dividing
decimals
Example 17
Evaluate each of the following:
a 0.4 ÷ 0.2
b 1.75 ÷ 0.5
c 122.4 ÷ 7.65
Solution
a 0.4 ÷ 0.2
Move the decimal points one place to the right (multiplying both decimals by 10).
=4÷2=2
b 1.75 ÷ 0.5
Move the decimal points one place to the right (multiplying both decimals by 10).
= 17.5 ÷ 5 = 3.5
c 122.4 ÷ 7.65
Move the decimal points two places to the right (multiplying both decimals by 100).
= 12 240 ÷ 765 = 16
Always check by estimation that your answers sound reasonable.
222
NEW CENTURY MATHS 7
Exercise 7-11
1 What happens when you divide by a number less than 1?
a 18 ÷ 0.5 means ‘how many times does 0.5 go into 18’. Is the answer more or less
than 18? Why?
b Estimate the answer to 18 ÷ 0.5.
c Use the method from Example 17 to evaluate 18 ÷ 0.5.
Example 17
2 a 20.4 ÷ 0.3 means ‘how many times does 0.3 go into 20.4’. Is the answer more or less
than 20.4?
b Estimate the answer to 20.4 ÷ 0.3.
c Find the exact answer to 20.4 ÷ 0.3.
3 Rewrite each of the following divisions so that the second decimal is a whole number.
b 17.82 ÷ 0.11
c 333 ÷ 4.5
a 508.8 ÷ 1.2
e 129.2 ÷ 0.38
f 49.5 ÷ 1.5
d 1.725 ÷ 2.5
h 14.823 ÷ 0.61
i 0.66 ÷ 0.022
g 168 ÷ 0.75
4 Evaluate each of these, and check that your answers seem reasonable by estimating:
b 7.32 ÷ 0.2
c 2.94 ÷ 0.6
a 3.48 ÷ 0.4
e 27 ÷ 0.25
f 10.08 ÷ 2.4
d 16.28 ÷ 2.2
h 5.6 ÷ 0.07
i 1.71 ÷ 0.3
g 10.4 ÷ 0.05
k 532 ÷ 3.5
l 78.12 ÷ 6.2
j 40.82 ÷ 5.2
n 98.4 ÷ 0.08
o 465 ÷ 7.5
m 0.272 ÷ 0.08
Working mathematically
Reflecting and applying strategies: Practising your skills
The cards for this set of questions have been printed with the decimal points in the
wrong places. Insert the decimal points so that the numbers on the cards fit the clues.
1 The product of these two numbers is 134.64.
The sum of the numbers is 58.5.
2 The difference between the numbers is 178.8.
When you divide the greater number by the smaller
number, the quotient is between 43 and 44.
561
2.4
183
42
3 The sum of the three numbers is 5.36.
The product of the numbers is 1.2096.
8
42
36
4 The sum of the three numbers is 4.61.
The product of the numbers is 0.9135.
15
29
21
31
14
14
5 The sum of the four numbers is 2.55.
The product of two numbers is 0.196.
The product of the other two numbers
is 0.217.
7
DECIMALS
223
CHAPTER 7
Decimals at work
Exercise 7-12
1 Find the cost of 352 units of electricity at 12.3 cents per unit.
2 A farmer wants to fence a rectangular paddock. The paddock is 35.6 metres long and
20.85 metres wide. How many metres of fencing will be needed?
3 Mark buys golf balls for $4.85 each and sells them for $5.15 each. How much money
does he make if he sells 30 golf balls?
4 A drink bottle holds 0.75 litres. How many drink bottles can be filled from a tub that
holds 4.5 litres?
5 A car travels 220.6 kilometres on 14 litres of petrol. How many kilometres would the car
travel on one litre of petrol? (Give the answer to one decimal place.)
6 Pamela runs 3.8 kilometres each day of the week. How far does she run in one week?
7 A long distance train is made
up of a diesel engine, two
dining cars and 15 passenger
carriages. The engine has a
mass of 20.2 tonnes, each
dining car has a mass of
14.35 tonnes and each
passenger carriage has a
mass of 13.96 tonnes. How
heavy is the entire train?
8 George is cutting shelves from a board which is 4.6 metres long. Each shelf needs to be
1.25 metres long. How many shelves can be cut?
9 Samantha ran 100 metres in 15.21 seconds. How long would Samantha take to run
400 metres at this pace?
10 Henry has a faulty calculator; it does not
show the decimal point. For each of the
calculations in the table shown on the right,
write the correct answer.
Calculation
Answer
3.42 × 12
4104
4.145 × 0.2
829
37.3 × 8.8
32824
0.03 × 157.64
47292
8.3902 × 0.3
251706
11 Mick walks to work and back each day. He works six days a week and, in one week,
walks 16.8 kilometres. How far is Mick’s apartment from work?
12 The following are calculator displays for amounts in dollars and cents. Give the answers
in dollars and cents.
a
b
c
d 1234.047126
15.236
6.8412
487.759318
13 A sheet of paper is 0.01 cm thick. How many sheets would be in a stack 4 cm high?
224
NEW CENTURY MATHS 7
Converting common fractions to decimals
We often use fractions in our everyday conversations:
Are we half-way there yet?
Nearly two-thirds of the crowd barracked for NSW.
It is useful to recognise fractions in their decimal form.
Example 18
Write each of these as a decimal:
a 3--5-
b
5
--8
b
5
--8
Solution
a
3
--5
means 3 ÷ 5
means 5 ÷ 8
0.6
5 3.0
0.625
2 4
8 5.000
3
--5
5
--8
= 0.6
= 0.625
Exercise 7-13
1 Copy and complete this table. Use a calculator if you need to.
Common fraction
Meaning as division
a
3
--5
3÷5
b
1
--2
1
--4
4
--5
2
--5
1÷2
c
d
e
Decimal
Example 18
Equivalent common
fraction
6
-----10
0.6
5
-----10
0.25
f
3÷4
g
1÷5
h
1÷8
0.75
2 Use your completed table from Question 1 to help you find decimals equal to the
following common fractions:
a
2
--5
b
3
--8
c
3
--4
d
2
--2
e
2
--4
f
6
--8
g
3
--5
h
2
--8
i
5
--8
j
7
--8
k
4
--8
l
5
--5
3 Explain why some of the fractions in Question 2 have the same decimal value.
DECIMALS
225
CHAPTER 7
4 Write as decimals:
8
a 4 ----10
b 23 3--4-
e 57 2--5-
f
j
6 1--5-
i
Spreadsheet
7-05
Common
fractions to
decimals
Worksheet
7-07
Fraction families
c 12 5--8-
d 6 3--5-
19 1--8-
g 110 7--8-
h 80 1--4-
13 1--2-
k 5 3--8-
l
6 2--4-
5 You can use Excel to change decimals into common fractions and vice versa. Use this
link to go to an activity that will show you how.
Recurring decimals
Sometimes when a common fraction is converted to a decimal, we get a repeating or
recurring decimal. One or more of the digits in the decimal repeat forever.
Example 19
Worksheet
7-08
Decimals
squaresaw 2
1 Change
1
--3
to a decimal.
Solution
1
--3
1
--3
0.333…
3 1.0000…
means 1 ÷ 3
= 0.333… = 0. 3 or 0. 3
2 Change
5
--6
to a decimal.
Solution
5
--6
5
--6
0.8333…
6 5.0000…
means 5 ÷ 6
= 0.8333… = 0.8 3 or 0.8 3
3 Change
2
-----11
to a decimal.
Solution
2
-----11
2
-----11
0.18181…
11 2.000000…
means 2 ÷ 11
= 0.181 818… = 0.18 or 0.18
Notice the repeating or recurring pattern in the numbers following the decimal point. To
show that the pattern goes on forever, we place a dot or a line above the recurring digits.
The dot or line shows the repeating section: 0.259 259 259 2… = 0. 259 or 0. 259
Non-repeating decimals are called terminating decimals. ‘Terminate’ means ‘to stop’.
Exercise 7-14
Example 19
1 Change these to repeating decimals:
a
f
k
226
1
--9
2
--7
5
--7
b
g
l
NEW CENTURY MATHS 7
1
--3
2
--9
5
--9
c
h
m
1
--6
3
--7
6
--7
d
i
n
1
--7
4
--7
7
--9
e
j
o
2
--3
4
--9
8
--9
2 Copy and complete the following pattern:
Fraction:
1
--9
2
--9
Decimal:
0. 1
0. 2
3
--9
4
--9
5
--9
6
--9
7
--9
8
--9
3 Perform these calculations and write the answers as recurring decimals:
b 11.3 ÷ 7
c 58.43 ÷ 9
a 11 ÷ 3
d 1.9 ÷ 6
e 76 ÷ 9
f 0.67 ÷ 6
g 2÷7
h 13.4 ÷ 6
i 149 ÷ 7
j 13 ÷ 3
k 196 ÷ 11
l 12 ÷ 7
m 7 ÷ 11
n 17 ÷ 12
o 18.1 ÷ 9
Rounding decimals
Skillsheet
7-01
Rounding
whole numbers
Sometimes, to approximate an answer with many decimal places, we round to fewer
decimal places. We need to be able to round when working with money, measuring
quantities or writing answers to division calculations.
To round a decimal:
•
•
•
•
cut the number at the required decimal place
look at the digit immediately to the right of the specified place
if this digit is 0, 1, 2, 3 or 4, leave the number in the specified place unchanged
if the digit is 5, 6, 7, 8 or 9, add 1 to the number in the specified place.
Example 20
1 Round 86.246 correct to one decimal place.
Solution
86.2 46
cut the next digit is 4, so the number 2 does not change
So 86.245 is 86.2 (correct to one decimal place).
2 Round 86.246 correct to two decimal places.
Solution
86.24 6
cut the next digit is 6, so add 1 to 4 to give 5
So 86.246 is 86.25 (correct to two decimal places).
Example 21
The number 0.087 1245 has seven decimal places. Round it to:
a one decimal place
b two decimal places
c the nearest thousandth
Solution
a 0.1
b 0.09
c 0.087
DECIMALS
227
CHAPTER 7
Rounding to:
• the nearest tenth = one decimal place
• the nearest hundredth = two decimal places
• the nearest thousandth = three decimal places
Exercise 7-15
Example 20
1 Write each of the following correct to one decimal place:
a 0.35
b 0.47
c 0.81
e 2.55
f 0.32
g 0.90
d 0.69
h 2.88
2 Round each of the following to two decimal places:
a 0.481
b 0.736
c 0.069
e 0.309
f 0.655
g 2.096
d 0.293
h 3.995
3 Write each of the following correct to two decimal places:
a 25.3456
b 341.7675
c 321.3333
d 734.6541
e 27.757 575
f 1314.123 45
Example 21
4 Copy this table into your book. Use a calculator to help you complete it.
Question
12.19 ÷ 3
Calculator display
Rounded to
one decimal place
Rounded to
two decimal places
4.063 333 3333
4.1
4.06
12.32 ÷ 6
19.82 ÷ 9
56.85 ÷ 11
17.13 ÷ 4
12.65 ÷ 12
4.875 ÷ 21
27.45 ÷ 8
17 ÷ 12
254.678 ÷ 32
5 Use a calculator to find the value of each of the following divisions. Give your answers
to the nearest hundredths.
b 8.57 ÷ 6
c 0.812 ÷ 9
a 2.89 ÷ 3
d 7 ÷ 11
e 5.12 ÷ 6
f 11.71 ÷ 7
g 8÷6
h 12 ÷ 7
i 18.87 ÷ 11
j 12.62 ÷ 13
k 7 ÷ 12
l 9.38 ÷ 15
6 Round each of these numbers to four decimal places:
a 10.33374
b 431.543 27
d 3217.654 061
e 4.670 89
228
NEW CENTURY MATHS 7
c 1.444 95
f 0.888 88
7 The answers to the following are whole numbers but, for particular reasons, some need to
be rounded up and some need to be rounded down. Find the answers.
a A box of chocolates with 33 chocolates is shared among a family of five people.
How many chocolates does each person receive?
b A new bathroom requires 32 square metres of tiles. The tiles come in boxes
containing 1.5 square metres. How many boxes are needed to tile the bathroom?
c A team of four golfers wins 27 new golf balls in a competition. How many does each
person receive?
d Some timber comes in 1.8 m lengths. How many lengths are needed to build a chicken
house needing 23 m of timber?
e Each blouse requires 1.3 m of material. How many blouses can be made from a 5 m
length of material.
8 Give a reason for:
a rounding up
b rounding down.
Working mathematically
Questioning and reflecting: Rounding prices
Since the use of 1c coins and 2c coins stopped in 1990, prices have been rounded to
the nearest 5 cents. Find out:
a when amounts are rounded
b how amounts are rounded (check the major stores in your area)
c what happens with bills such as electricity, phone, etc.
d who decides how the rounding will work.
SkillBuilder
1-09
Rounding
money
Just for the record
The salami technique
When banks started using computers to keep track of customers’ accounts, they left
themselves open to a new type of crime: computer theft. One such crime employs the
salami technique, where computer hackers steal a cent or a fraction of a cent from
many bank accounts. They round down the decimal amount of an account balance (for
example $234.6523 would become $234.65) and the stolen fraction of a cent
($0.0023) is deposited into the hacker’s account, with no one noticing it missing. When
this is done to thousands of bank customers over a number of years, a considerable
amount of money can be accumulated.
If the salami technique is applied to $1723.35631, $456.3277, $6701.2315 and
$488.29891, how much will the computer criminal have in his or her account?
Why do you think this type of crime is difficult to detect?
Why do you think it is called the ‘salami technique’?
DECIMALS
229
CHAPTER 7
More decimals at work
Example 22
Use the table below to find the cost of a 6-minute phone call made over a distance of
236 km on Friday at 11:00am.
Cost of long-distance phone calls
Approximate cost per minute (rounded to the nearest cent) with a minimum charge of 21c per call.
Under
50 km
50 km to
85 km
86 km to
165 km
166 km to
745 km
Over
745 km
Day
8:00am–6:00pm
Monday to Saturday
$0.14
$0.28
$0.35
$0.43
$0.60
Night
6:00pm–10:00pm
Monday to Friday
$0.09
$0.17
$0.26
$0.29
$0.41
Economy
6:00pm Saturday
to 8:00am Monday
10:00pm–8:00am
everyday
$0.08
$0.13
$0.18
$0.21
$0.26
Distance
Solution
The distance is between 166 km and 745 km so we use the fourth column of costs. The time
is a weekday during daytime, so we use the first row.
The call is charged at $0.43 a minute.
Cost of call = 6 × $0.43
= $2.58
Exercise 7-16
Example 22
1 Long-distance phone calls
Use the table in Example 22 to help you answer these questions:
a Find the cost of a 5-minute call over a distance of 21 km made at 7:45pm on a Tuesday
night.
b Find the cost of an 11-minute phone call from Manly to Penrith (60 km) on a Sunday
morning at 9:30am.
c Janis called her boyfriend Mal, who lives 112 km away, on Saturday night at 6:10pm.
Find the cost of the call if they spoke for an hour-and-a-half.
d A phone call from Sydney to Coffs Harbour (400 km) appeared on the phone bill as a
charge of $2.73. If the call was made on Monday at 10:30pm, work out how long the
call took.
e Make up three questions about phone calls using the table of costs. Work out the
answers and give a classmate the three questions to complete.
230
NEW CENTURY MATHS 7
f
Linh is an interstate truck driver. Her company gave her a $20 phone card before she
left Melbourne on an interstate trip to Moree in New South Wales. After making
deliveries at each of the towns marked on the strip map below, she telephoned her
truck base in Bendigo. The phone calls were recorded in her log book.
80 km
230 km
Bendigo Echuca
230 km
Narrandera
110 km
340 km
Moree
Parkes Dubbo
LOG
Call from
Time
Length of call
Echuca
7:30am
6 min
Narrandera
11:15am
9 min
Parkes
6:50pm
15 min
Dubbo
8:00am
4 min
Moree
4:15pm
22 min
Did Linh’s phone card cover the cost of her phone calls? Explain your answer.
2 FM radio
In the radio and television guide you will find a list of
FM stations and their allocated frequencies measured in
megahertz (MHz).
a Locate the stations on a number line.
90
100
110
b What is the frequency difference in megahertz
between 2DAY and WS-FM?
c Find the smallest frequency difference between
adjacent stations. (‘Adjacent’ means side-by-side.)
d What is the largest difference in frequency between
adjacent stations?
Station
Frequency
(MHz)
ABC
92.9
The Edge
96.1
STAR-FM
99.7
WS-FM
101.7
2MBS
102.5
FM 103
103.2
2DAY
104.1
MMM
104.9
JJJ
105.7
MIX
106.5
2SER
107.3
3 Decimal currency
a Find out about the history of the decimal money system in Australia.
i When was it introduced?
ii What system did it replace?
iii Why was the change made?
iv When were the different coins and notes introduced?
v What changes have occurred since?
b Design a set of notes ($5, $10, $20, $50, $100) that blind and sighted people could
both use.
DECIMALS
231
CHAPTER 7
4 Decimal time
a Investigate the idea of decimal time. How would
you measure time in this system?
b Design a decimal time calendar.
c How would decimal time affect your birthday
and age?
d Do you think decimal time is possible? Explain.
9
10 1
8
2
7
3
6
5
4
Power plus
1 Code breaker
A long time ago, the warrior Wolf was trying to break a code that would open some
dungeon doors. He had to free the prisoners before midnight so that they would not be
turned into frogs by an evil spell. Each castle door was operated by a combination lock.
a Use the following clues to match the various combinations with the doors to the
different rooms in the castle.
Combinations
Rooms
8.262
Queen’s chamber
9.24
armoury
9.96
throne room
8.07
banquet hall
8.16
kitchen
8.79
dungeon
Clues
• Combination 9.24 opens a door to a room that deals with food.
• The combination to the armoury has a 6 in the hundredths place.
• The combinations of the throne room and the banquet hall add to 18.03.
• The Queen’s chamber has a combination that is bigger than 3.78 × 2.1 but smaller than
25.11 ÷ 3.1. (Hint: The doors in the dungeon and the throne room remained locked
when Wolf tried 9.96 and 8.262.)
• The kitchen combination is one of the three largest combination numbers.
b Invent another story that sets a code-breaking task involving decimals and ask your
classmates to solve it.
2 Calculate:
a 0.02 × 0.02
d
0.04
b (0.02)3
c (1.1)3
e
f
0.36
3
0.027
3 Decide where the decimal point should be so that the numbers in the ovals fit the clues.
a The product of the two numbers is 91.02.
246
37
The sum of the numbers is 28.3.
b The sum of the three numbers is 14.
57
6
23
The product of the three numbers is 78.66.
c The sum of the four numbers is 28.32.
72
51
The product of two of the numbers is 3.672.
18
45
The product of the other two is 81.
232
NEW CENTURY MATHS 7
Language of maths
ascending
descending
place value
tenth
decimal
estimate
recurring decimal
thousandth
decimal place
fraction
round
decimal point
hundredth
terminating decimal
Worksheet
7-09
Decimals
crossword
1 Use the words ‘ascending’ and ‘descending’ in a non-mathematical way.
2 ‘The bushfire decimated the possum population of the forest.’ Look up the
meaning of the word ‘decimate’.
3 What is a recurring dream or a recurring back pain? What does ‘recurring’ mean?
4 What happens when a train terminates at a station?
5 What is a ‘decimetre’?
6 Find examples of words beginning with ‘deci’ or ‘deca’ meaning ten of
something.
Topic overview
• What parts of this topic were new to you? What parts did you already know?
• Write any rules you have learnt about working with decimals.
• What parts of this topic did you not understand? Be specific. Talk to a friend or your
teacher about them.
• Give three examples of where decimals are used.
• Copy this overview into your workbook and complete it. Check it with a friend or your
teacher, to be sure nothing is left out.
Converting
common
fractions to
decimals
Place value
Operations
DECIMALS
Rounding
Problems
Recurring
decimals
DECIMALS
233
CHAPTER 7
Chapter 7
Ex 7-01
Review
Topic test
Chapter 7
1 Write each of these in decimal form:
a 4 × 10 + 3 ×
1
-----10
+6×
1
--------100
b 5 × 100 + 2 × 10 + 7 ×
d 8+6×
c twelve thousandths
1
--------100
+9×
1
-----10
+3×
1
-----------1000
1
-----------1000
Ex 7-02
2 Read the story below and write the number in each part with a decimal point so that
the story makes sense.
a Angelo and Loren decided to go to the motorbike races. They caught the train at
12:45pm after walking 123 km to the station.
b They gave the station master a $1000 note
c and received $460 change.
d The train trip took 15 hours.
e Angelo said that each bike weighed 1400 kg.
f There were 4500000 people at the race meeting
g who paid a total of $50850000 in entrance costs.
h The average price of a ticket was $113.
i A mechanic told Loren that the price of race fuel was 968 cents per litre.
Ex 7-03
3 Draw a decimal number line for each of these sets of decimals and mark them in their
correct locations.
a 1.6
1.7
1.9
2.0
2.2
2.3
2.4
b 4.00
4.02
4.03
4.06
4.08
4.11
c 0.63
0.64
0.67
0.69
0.71
d 8.0
8.02
8.06
8.1
8.12
8.16
8.2
Ex 7-03
4 Arrange these numbers in order, from largest to smallest:
a 34.98
56.86
3.998
50.141
340
34.89
b 0.136
0.86
0.652
0.662
0.23
1.006
c 0.556
0.559
0.595
0.565
d 1.015
1.293
1.1015 1.239
e 16.409 16.087 16.009 16.49
Ex 7-04
Ex 7-05
8.6
5 Write each of these as a decimal:
a
Ex 7-04
3.099
0.086
4
-----10
b
13
--------100
c
7
--------100
d
111
-----------1000
6 Write each of these as a common fraction or mixed number:
a 0.5
b 0.89
c 0.09
d 0.444
7 Find the value of each of these:
a 12.35 + 4.53 + 0.56 + 3.125 + 24.7 + 20.09
b 214.33 − 109.84
c 0.568 + 23 + 4.027 − 16.28 + 0.65
d 1600.8 − 562.9
e 1453.6 + 1287.31 − 2344.4
f 7 + 7.7 + 77.77 + 777.777 + 7777.7
234
NEW CENTURY MATHS 7
e
45
-----------1000
e 4.051
g
h
i
j
k
l
m
204.9 + 23.24 + 0.015
363.2 − 120.49
0.14 + 0.375 + 0.0042
16.95 + 28.2 − 12.4
9.23 − 6.851
16.51 + 9.003 + 125 + 0.9
16.821 − 5.84
8 Find the answer to each of these:
b 0.5 ×1.2
a 2.75 × 6
e 0.92 × 10
d 6.1 × 1.2
h 0.05 × 0.02
g 3.25 × 0.41
c 12.23 × 4
f 100 × 6.7
i 4.67 × 1.1
9 Find the answer to each of these:
b 12.5 ÷ 0.5
a 762.4 ÷ 2
e 12.72 ÷ 0.4
d 2.75 ÷ 100
c 97.6 ÷ 10
f 6.9 ÷ 0.03
Ex 7-08
Ex 7-11
10 The Liverpool Women’s Cricket Club was having a pizza night. They ordered
16 Super Supreme pizzas at $13.70 each and 10 Hawaiian pizzas at $12.10 each.
How much will the club need to spend?
Ex 7-12
11 Maria saved $90 to go to a rock concert. Her return fare cost $5.60, her concert
ticket cost $48.95, the program cost $11.00 and food cost $8.70. She did not have
enough to buy the band’s latest compact disc (priced $24.00) after the concert.
How much did she need to borrow from her friend Sam to buy the disc?
Ex 7-12
12 Rae bought 1200 bricks for $465.70. How much did one brick cost?
Ex 7-12
13 Change to decimals:
a
4
--5
Ex 7-13
b
3
--8
c
5
--9
d
6 2--3-
14 Round each of these numbers to the place value shown:
a 456.8 to the nearest ten
b 125.84 to the nearest unit
c 2345.876 to two decimal places
d 3.8967 to the nearest tenth
e 78 654.056 to the nearest thousand
f 4821.623 to the nearest tenth
g 678.439 to one decimal place
h 102.007 to the nearest hundred
i 102.007 to the nearest hundredth
j 26.046 to two decimal places
DECIMALS
Ex 7-15
235
CHAPTER 7