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PASSPORT
PASSPORT
Deci
mal
s DECIMALSDECIMALSDECIMALS
www.mathletics.co.uk
6HSERIES TOPIC
1Decimals
Mathletics Passport © 3P Learning
����To�make�dark-green�coloured�paint,�you�can�mix�yellow�and�blue�together,�using�exactly�0.5�(half)�as�much�yellow�as�you�do�blue.�
How�much�dark-green�paint�will�you�make�if�you�use�all�of�the�12.5 mL�of�blue�paint�you�have?
Work through the book for a great way to do this
Give this a go!Give this a go!
Decimals�allow�us�to�be�more�accurate�with�our�calculations�and�measurements.
Because�most�of�us�have�ten�fingers,�it�is�thought�that�this�is�the�reason�the�decimal�system�is�based�around�the�number�10!
So�we�can�think�of�decimals�as�being�fractions�with�powers�of�10 in�the�denominator.
Write�in�this�space�EVERYTHING�you�already�know�about�decimals.
Q
2 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
DecimalsHow does it work?
1st�decimal�place:� 10101
#' = = �one�tenth
2nd�decimal�place:� 101001
#' = = �one�hundredth
3rd�decimal�place:� 1010001
#' = = �one�thousandth
4th�decimal�place:� 1010000
1#' = = �one�ten�thousandth etc...
Decimal�point
Add�‘th’�to�the�name�for�decimal�place�values
or
or
or
or
......
......
......
......
2 10 2101
7 100 71001
0 0 1000 010001
3 3 10 000 310000
1
2
7
#
#
#
#
'
'
'
'
`
`
`
`
j
j
j
j
......
......
......
4 100
6 10
5 1
4
6
5
#
#
#
102
1007
10000
100003
=
=
=
=
400
60
5
=
=
=
Multiply�by�multiples�of 10 Divide�by�multiples�of 10
1st�decimal�place
2nd�decimal�place
3rd�decimal�place
4th�decimal�place
= 2 tenths
= 7 hundredths
= 0 thousandths
= 3 ten�thousandths
= 4�hundred
= 6�tens�(or�sixty)
= 5�ones�(or�five)
# 1
0 0
00
# 1
000
# 1
00
# 1
0
# 1
' 1
0
' 1
00
' 1
000
' 1
0 0
00
' 1
00 0
00
' 1
000 0
00
' 1
0 0
00 0
00
•
Tens�of�thousands
Tenths
Hundredths
Thousandths
Ten�thousandths
Hundred�thousandths
Millionths
Ten�M
illionths
Thousands
Hundreds
Tens
Ones
Place value of decimals
Decimals�represent�parts�of�a�whole�number�or�object.
W H O L E D E C I AM L
Write�the�place�value�of�each�digit�in�the�number�465.2703
4 6 5 . 2 7 0 3
Expanded forms Place values
Integer�parts
6HSERIES TOPIC
3Decimals
Mathletics Passport © 3P Learning
How does it work? Your Turn Decimals
Place value of decimals
Write�the�decimal�that�represents�these:
b ca
e fd
Write�the�fraction�that�represents�these:
Write�the�place�value�of�the�digit�written�in�square�brackets�for�each�of�these�decimals:
2�hundredths 9�tenths 1�ten�thousandth
3�thousandths 6�hundred�thousandths
b ca
e fd
3�tenths 7�thousandths 1�hundredth
9�ten�thousandths 51�hundredths 11�ten�thousandths
b ca
e fd
Circle�the�digit�found�in�the�place�value�given�in�square�brackets:
.3 1 3256 @ .1 0 2316 @ 1 .4 5 0466 @
.5 0 050436 @ .6 0 792646 @ .0 8 563096 @
b ca
e fd
[tenths]
8 . 1 7 1 6 1 5
[thousandths]
4 . 3 2 1 2 3 0
[millionths]
3 . 1 2 0 6 1 9
[hundredths]
9 . 1 2 4 2 1
[ten�thousandths]
1 6 . 1 2 3 2 1 0
[hundred�thousandths]
1 0 0 . 1 0 0 1 0 0 1
Always�put�a�zero�in�front�(called�a�leading zero)�when�there�are�no�whole�numbers
0.02
8�millionths
2
3
4
1
4 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
How does it work? Your Turn Decimals
Write�the�decimal�23.401�in�expanded�form
Place value of decimals
Each�digit�is�multiplied�by�the�place�value�and�then�added�together�when�writing�a�number�in�expanded�form.
PLACE VALUE OF DECIMALS PLACE VALUE
OF DECIMALS
..../...../20...
a
b
c
d
e
f
.
.
.
.
.
.
4 19
29 281
40 2685
3 74932
0 2306
0 0085
=
=
=
=
=
=
5
.23 401 2 10 3 1 4101 0
1001 1
10001
2 10 3 1 4101 1
10001
# # # # #
# # # #
= + + + +
= + + +
Multiply�each�digit�by�its�place�value
Zero�digits�can�be�removed�to�simplify
6
Write�these�decimals�in�expanded�form:
Simplify�these�numbers�written�in�expanded�form:
1 1 4101 6
1001
4 10 9 1 0101 7
1001
5 100 2 10 0 1 2101 1
1001 8
10001
6 1 8101 5
1001 0
10001 2
100001 9
1000001
# # #
# # # #
# # # # # #
# # # # # #
+ + =
+ + + =
+ + + + + =
+ + + + + =
a
b
c
d
Psst:�Remember�to�include�a�leading zero�for�these�ones.
e
f
g
2101 0
1001 3
10001
61001 7
10001 0
10 0001 1
1000001
3101 4
1001 1
10001 0
100001 8
1000001
# # #
# # #
# # # # #
#
+ + =
+ + + =
+ + + + =
6HSERIES TOPIC
5Decimals
Mathletics Passport © 3P Learning
How does it work? Decimals
Approximations through rounding numbers
Look�at�these�two�statements�made�about�a�team�of�snowboarders:
� •� They�have�attempted�4937�tricks�since�starting���= Accurate statement
� •� They�have�attempted�nearly�5000�tricks�since�starting���= Rounded off approximation
Closer�to�lower�value,�so�round down
Leave�the�place�value�unchanged
Closer�to�higher�value,�so�round up
Add�1�to�the�place�value
Round�these�numbers
The�digit�‘4’�is�in�the�hundreds�position��
The�next�digit�is�a�6,�so�round up�by�adding�1�to�4
Change�the�other�smaller�place�value�digits�to�0’s�
The�digit�‘3’�is�in�the�first�decimal�place��
The�next�digit�is�a�1,�so�round down
Write�decimal�with�one�decimal�place�only
The�digit�‘1’�is�in�the�fourth�decimal�place��
The�next�digit�is�a�9,�so�round�up�by�adding�1�to�1
Write�decimal�with�four�decimal�places�only�
Here�are�some�examples�to�see�how�we�round�off�numbers.
(i)� 2462��to�the�nearest�hundred
(ii)�� 0.3145�to�one�decimal�place�(or�to�the�nearest�tenth)
(iii)� 26.35819 to�four�decimal�places�(or�to�the�nearest�ten�thousandth)
2462 2500` . rounded�to�the�nearest�hundred
. .0 3145 0 3` . rounded�to�one�decimal�place
.3 .26 5819 26 3582` . rounded�to�four�decimal�places
2 6 . 3 5 8 1 9
2 6 . 3 5 8 1 9
2 6 . 3 5 8 2
0 . 3 1 4 5
0 . 3 1 4 5
0 . 3
2 4 6 2
2 4 6 2
2 5 0 0
0 1 2 3 4 5 6 7 8 9
Next�digit
Rounding�off�values�is�used�when�a�great�deal�of�accuracy�is�not�needed.
The�next�digit�following�the�place�value�where�a�number�is�being�rounded�off�to�is�the�important�part.
6 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
How does it work? Your Turn Decimals
Approximations through rounding numbers
1 � Round�these�whole�numbers�to�the�place�value�given�in�square�brackets.
Round�these�decimals�to�the�decimal�places�given�in�the�square�brackets.2
Approximate�the�following�distance�measurements:����3
A�group�of�people�form�an�8.82 m�long�line�when�they�stand�together.
(i)� How�long�is�this�line�to�the�nearest�10�cm�(i.e.�1�decimal�place)?
(ii)��What�is�the�approximate�length�of�this�line�to�the�nearest�10�metres?�
Under�a�microscope�the�length�of�a�dust�mite�was�0.000194 m
(i)� �Approximate�the�length�of�this�dust�mite�to�the�nearest�ten�thousandth� of�a�metre.
(ii)���Approximate�the�length�of�this�dust�mite�to�the�nearest�hundredth�of�a�metre.
If�Lichen�City�is�3 458 532 m�away�from�Moss�City:��
(i)� �What�is�this�distance�approximated�to�the�nearest�km?� (i.e.�nearest�thousand)
(ii)���What�is�the�approximate�distance�between�the�cities�to�the�nearest�100 km?
(iii)�����Are�the�digits�2, 3�or�even�5�important�to�include�when�describing�the�total�distance�between�the�two�cities?�Briefly�explain�here�why/why�not.
4
544
3
APPROXIMATION THROUGH ROUNDING
NUMBE
RS.
..../...../2
0...
a
a
[nearest�ten]
[nearest�tenth]
b
b
c
c
[nearest�hundred]
[nearest�hundredth]
[nearest�thousand]
[nearest�thousandth]
(i)
(i)
(ii)
(ii)
(iii)
(iii)
536 .
8514 .
93025 .
(i)
(i)
(ii)
(ii)
(iii)
(iii)
14302 .
4764 .
80048 .
(i)
(i)
(ii)
(ii)
(iii)
(iii)
98542 .
18401 .
120510 .
.0 73 .
.3 47 .
.11 85 .
.2 406 .
.0 007 .
.1 003 .
.10 4762 .
.0 3856 .
.0 048640 .
.
.
.
.
.
.
a
b
c
6HSERIES TOPIC
7Decimals
Mathletics Passport © 3P Learning
How does it work? Your Turn Decimals
Approximations through rounding numbers
4 � �Round�off�these�numbers�according�to�the�square�brackets.����
Rounding�up�can�affect�more�than�one�digit�when�the�number�9�is�involved.
Round�0.95��to�one�decimal�place The�digit�‘9’�is�in�the�tenths�position��
The�next�digit�is�a�5,�so�round up�by�adding�1�to�9
Change�the�other�smaller�place�value�digits�to�0s�
. .0 95 1 0` . rounded�to�one�decimal�place
9�rounds�up�to�10,�so�the 9�becomes�0�and�1 is�added�to�the�digit�in�front.��
a � [one�decimal�place]
.1 98 .
d � [nearest�ones]
.79 9 .
g � [nearest�thousand]
49798 .
b � [nearest�ten]
398 .
e � [three�decimal�places]
.0 1398 .
h � [nearest�ones]
.199 9 .
c � [two�decimal�places]
.11 899 .
f � [three�decimal�places]
.2 1995 .
i � [four�decimal�places]
.9 89999 .
5 � Approximate�these�values:
a � A�call�centre�receives�an�average�of�2495.9�calls�each�day�during�one�month.� � (i)� Approximate�the�number�of�calls�received�to�the�nearest�hundreds.
� (ii)� Approximately�how�many�thousands�of�calls�did�they�receive?� � (iii)� Estimate�the�number�of�calls�received�daily�throughout�the�month.
b � A�swimming�pool�had�a�slow�leak,�causing�it�to�empty�9599.5896�L�in�one�week.��� � (i)� How�much�water�was�lost�to�the�nearest�10�litres?
� (ii)� How�much�water�was�lost�to�the�nearest�mL�if�1mL = 10001 L?
� (iii)� �Is�the�digit�6�important�when�approximating�to�the�nearest�whole�litre?
Briefly�explain�here�why/why�not.
.
.
.
.
.
0 . 9 5
0 . 9 5
1 . 0
8 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
How does it work? Decimals
Decimals on the number line
The�smallest�place�value�in�a�decimal�is�used�to�position�points�accurately�on�a�number�line.
•� �Decimals�are�based�on�the�number�10,�so�there�are�always�ten�divisions�between�values Eg:�Here�is�the�value�3.6�on�a�number�line:
So�its�eight�thousandths�of�the�way�from�1.240�to�1.250
Six�tenths�of�the�way�from�3.0�to�4.0
Here�are�some�more�examples�involving�number�lines:
1.240 1.248 1.250
8
3.0 3.6 4.0
6
0.1 0.2
4a)
2.14 2.15
a)
•� The�major�intervals�on�the�number�line�are�marked�according��to�the�second last�decimal�place�value
(i)� What�value�do�the�plotted�points�represent�on�the�number�lines�below?
(ii)� Round�the�value�of�the�plotted�points�below�to�the�nearest�hundredth.�
Point�is�four�steps�from�0.1�towards�0.2,�so�the�plotted�point�is:�0.14
Point�is�nine�steps�from�10.06�towards�10.07,�so�the�plotted�point�is:�10.069
Point�is�three�steps�from�2.14�towards�2.15,�so�the�plotted�point�is�2.143
` the�value�of�the�plotted�point�to�the�nearest�hundredth�is:�2.14
10.06 10.07
9b)
Point�is�five�steps�from�8.79�towards�8.80,�so�the�plotted�point�is�8.795
` the�value�of�the�plotted�point�to�the�nearest�hundredth�is:�8.80
8.79 8.80
b)
3
5
6HSERIES TOPIC
9Decimals
Mathletics Passport © 3P Learning
How does it work? Your Turn Decimals
Decimals on the number line
1 � �Display�these�decimals�on�the�number�lines�below:���
2 � Label�these�number�lines�and�then�display�the�given�decimal�on�them:
3 � Round�the�value�of�the�plotted�points�below�to�the�nearest�place�value�given�in�square�brackets.
` the�value��.
` the�value��.
` the�value��.
` the�value��.
` the�value��.
` the�value��.
` the�value��.
` the�value��.
4
DECIMA
LS ON THE NUMBER LINE
DECIMALS ON THE NUMBER
LINE
..../...../2
0...
0.0 1.0
0.2 0.3
0.8 0.9
1.994 1.995
2.902 2.903
0.1 0.2
2.3 2.4
a
a
c
e
g
a
c
c
e
e
0.7
[tenth]
[tenth]
[thousandth]
[thousandth]
[hundredth]
[hundredth]
[thousandth]
[thousandth]
1.6
0.13
0.94
2.34
2.053
b
d
f
4.2
7.07
9.538
2.0 3.0
1.03 1.04
0.08 0.09
8.103 8.104
0.989 0.990
9.1 9.2
5.21 5.22
b
b
d
f
h
d
f
2.1
9.15
5.212
10 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
How does it work? Decimals
Multiplying and dividing by powers of ten
Move�the�decimal�point�depending�on�the�number�of�zeros
(i)� 5 1000#
(ii)� 8 100'
(iii)� . 10001 25893 0#
(iv)� . 10 0024 905 0 0'
(v)� . 1260 151000
#
� Calculate�these�multiplication�and�division�questions�involving�powers�of�10:
=��decimal�point�moves�right�����,������������������������� =��decimal�point�moves�left���
.
. .
8 100 8 0 100
8 0
' '=
=
0.08=
. . .
.
1 25893 10000 1 258 9 3
1258 9 3
# =
=
. . .
.
24 905 100000 24 905
0 00024905
' =
=
. .
. .
260 1510001 260 15 1000
2 60 15
# '=
=
The�whole�number�in�decimal�form�
'100�has�2�zeros,�so�move�decimal�point�2�spaces�left
Fill�the�empty�bounces�with�0s�and�put�a�zero�in�front
Move�decimal�point�4�spaces�right
No�empty�bounces�to�fill,�so�this�is�the�answer�
Move�decimal�point�5�spaces�left
Fill�empty�bounces�with�0s�and�put�a�zero�in�front
1000
1# is�the�same�as�' 1000
Move�decimal�point�3�spaces�left
Place�a�leading�zero�in�front�of�the�decimal�point
.
..
5 1000 5 0 1000
5 0
# #=
=
The�whole�number�in�decimal�form�
Fill�the�empty�bounces�with�0s�
We�can�simply�add�the�same�number�of�zeros�to�the�end�of�the�whole�number�
0.2 6015=
Remember�to�include�the�leading�zero�
5000=
1 2 3
2 1
1 2 3 4
5 4 3 2 1
3 2 1
If�the�decimal�point�is�on�the�left�after�dividing,�an�extra�0�is�placed�in�front.
6HSERIES TOPIC
11Decimals
Mathletics Passport © 3P Learning
How does it work? Your Turn Decimals
Multiplying and dividing by powers of ten
1 � ��Calculate�these�multiplications.�Remember,�multiply�means�move�decimal�point�to�the�right:�
2 � �Calculate�these�divisions.�Remember,�divide�means�move�decimal�point�to�the�left:
3 � Calculate�these�mixed�problems�written�in�index�form:
Here�are�some�of�the�powers�of�10�in�index�form.�The�power��= ��the�number�of�zeros.
10 10
10 10000
1
4
=
=
10 100
10 100000
2
5
=
=
10 1000
10 1000000
3
6
=
=
a
a
a
d
d
d
b
b
b
c
c
c
e
e
e
f
f
f
8 100# 29 1000#3.4 10#
12.45 10000# 0.512 100# 0.0000469 1000000#
1002 ' 4590 1000' .0 014 10'
70. 0 100008 ' .1367 512 1000' 421900 100000000'
31 102# 2400 105
' 0.0027 106#
90.008 104# .3 45 103
' 2159 951 107'
12 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
How does it work? Your Turn Decimals
Multiplying and dividing by powers of ten
4 � For�these�calculations:� (i)�Show�where�our�character�needs�to�spray�paint�a�new�decimal�point,�and� (ii)�write�down�the�two�numbers�the�new�decimal�point�is�between�to�solve�the�puzzle�
2 8 3 0 3 9 2 0
2 3 8 5 7
0 4 7 6 3 8 9 2
3 8 2 9 6 2
1 9 2 3 8 0 7
8 9 2 3 6 7 0 1
2 0 9 1 7 9 8 3
8 3 9 1 7
9 0 2 8 7 3 2 0 1
0 0 8 3 9 0
I 9 and 2
This�is�another�mathematical�name�for�a�decimal�point:��
0�and�9 8�and�9 8�and�7 9�and�2 0�and�7 3�and�9 8�and�2 0�and�8 3�and�8 6�and�7
MU
LTIPLYING AND DIVIDING BY POWERS
OF TEN
..../...../20.
..
a
b
c
d
e
f
g
h
i
j
2830.3920 100#
23857 1000'
0.4763892 105#
382 961 10000'
19238.07 101#
8.9236701 10000#
20 917 9831000000
1#
83917 105'
902873.02110
12
#
0.08390 103#
N
A
O
X
T
R
I
D
P
I
6HSERIES TOPIC
13Decimals
Mathletics Passport © 3P Learning
How does it work? Decimals
MU
LTIPLYING AND DIVIDING BY POWERS
OF TEN
..../...../20.
..
(i)�0.25
07�is�just 7
� Write�each�of�these�decimals�as�an�equivalent�(equal)�fraction�in�simplest�form
Write�1.07�as�a�fraction:�������
�Last�digit�is�in�hundredths�position
Decimal�digits�in�the�numerator
0.2510025= Equivalent,�un-simplified�fraction
Divide�numerator�and�denominator�by�HCF
Equivalent�fraction�in�simplest�form
Equivalent,�un-simplified�mixed�numeral
Divide�numerator�and�denominator�by�HCF
Equivalent�mixed�numeral�in�simplest�form
10025
2525
''=
41=
2.105 21000105=
100105
552
0 ''=
220021=
1.07 11007=
103=Write�0.3�as�a�fraction:�������
�Last�digit�is�in�tenths�position
Decimal�digits�in�the�numerator0.3
Decimal Fraction
�Last�digit�is�in�hundredths�position
Terminating decimals to fractions
These�have�decimal�parts�which�stop�(or�terminate)�at�a�particular�place�value.
The�place�value�of�the�last digit on the right�helps�us�to�write�it�as�a�fraction.
Integers�in�front�of�the�decimal�values�are�simply�written�in�front�of�the�fraction.
Always�simplify�the�fraction�parts�if�possible.�These�two�examples�show�you�how.
(ii)�2.105
�Last�digit�is�in�thousandths�position
14 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
How does it work? Your Turn Decimals
Terminating decimals to fractions
1 Write�each�of�these�decimals�as�equivalent�fractions:
c da
Write�each�of�these�decimals�as�equivalent�fractions�and�then�simplify:2
Simplest�form
.0 1 = .0 09 = .0 03 =
b ca 0.5 = = 0.6 = = 0.02 = =
b .0 7 =
g he 0. 100 = .0 013 = .0 049 =f 0.007 =
k li 0.129 = .0 1007 = .0 0601 =j .0 081 =
e fd 0.08 = = 0.004 = = 0.005 = =
h ig 0.12 = = 0.25 = =
k lj 0.045 = = 0.0028 = = 0.0605 = =
Simplest�form Simplest�form
Simplest�formSimplest�form Simplest�form
Simplest�formSimplest�form Simplest�form
Simplest�formSimplest�form Simplest�form
0.022 = =
6HSERIES TOPIC
15Decimals
Mathletics Passport © 3P Learning
Where does it work? Your Turn Decimals
Terminating decimals to fractions
3 � Write�each�of�these�decimals�as�equivalent�mixed�numerals:�
b ca .2 3 = .1 1 = .03 7 =
e fd .01 3 = .4 001 = .002 9 =
b ca 2.8 = 1.4 = .04 6 =
e fd .03 5 = .2 75 = 5.005 =
h ig .1 004 = .0252 = .3 144 =
4 � Write�each�of�these�decimals�as�equivalent�mixed�numerals�and�then�simplify:
..../...
../20...
0. 5 =
1
2TE
RMINAT
ING DECIMALS TO FRACTIO
NS *
Simplest�form
=
Simplest�form
=
Simplest�form
=
Simplest�form
=
Simplest�form
=
Simplest�form
=
Simplest�form
=
Simplest�form
=
Simplest�form
=
16 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
How does it work? Decimals
Fractions to terminating decimals
Write�these�as�an�equivalent�decimal
(i)��123
Sometimes�it�is�easier�to�first�simplify�the�fraction�before�changing�to�a�decimal.
Where�possible,�just�write�as�an�equivalent�fraction�with�a�power�of�10�in�the�denominator�first.
.
123
41
41
41
10025
0 25
33
2525
#
#
`
'' =
=
=
=
.
2153 2
51
251 2 2
102
2 2
33
2#
#
'' =
=
=
.
53
53
106
0 6
22
#
#
`
=
=
=
numeratordenominator
Three�twelfths��= �one�quarter��= �twenty�five�hundredths��= �zero�point�two�five�
Two�and�three�fifteenths��= �two�and�one�fifth��= �two�and�two�tenths��= �two�point�two�
Three�fifths��= �six�tenths��= �zero�point�six
Multiply�numerator�and�denominator�by�the�same�value
Equivalent�fraction�with�a�power�of�10�in�the�denominator
(ii)� 2153
Simplify�fraction
Equivalent�fraction�with�a�power�of�10�in�the�denominator
Simplify�fraction�part
Equivalent�fraction�with�a�power�of�10�in�the�denominator
6HSERIES TOPIC
17Decimals
Mathletics Passport © 3P Learning
How does it work? Your Turn Decimals
include�a� leading�zero
Fractions to terminating decimals
1 � Write�each�of�these�fractions�as�equivalent�decimals.�
2 Write�each�of�these�as�equivalent�fractions�with�a�power�of�10�in�the�denominator.�
3 (i)� Write�each�of�these�as�equivalent�fractions�with�a�power�of�10�in�the�denominator.� (ii)� Change�to�equivalent�decimals.
c da 12
=43 =
209 =b
52 =
g he258 =
20011 =
1252 =f
2503 =
c da109 =
10011 =
10007 =b
1003 =
i kj
b ca51 =
=
e fd
h ig
154 = 3
251 = 6
207 =
41 =
=
2511 =
=
254 =
=
2001 =
=
1256 =
=
2259 =
=
12001 =
=
8507 =
=
18 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
How does it work? Your Turn Decimals
Fractions to terminating decimals
4 � ��Change�each�of�these�fractions�to�equivalent�decimals�after�first�simplifying.�Show�all�your�working.�
a 2012
c 2418
e 759
g 1 60036
b 2520
d 4022
f 34012
h 215012
FRACT
IONS TO TERMINATING DECIMAL
S
..../...../20...
1 = 0.52
6HSERIES TOPIC
19Decimals
Mathletics Passport © 3P Learning
How does it work? Your Turn Decimals
Fractions to terminating decimals
When�changing�the�denominator�to�a�power�of�10�is�not�easy,�you�can�write�the�numerator�as�a�decimal�and�then�divide�it�by�the�denominator.
5 Complete�these�divisions�to�find�the�equivalent�decimal:�
a b.52 2 000 5'=
d e f
Write�numerator�as�a�decimal�and�divide�by�the�denominator
Complete�division,�keeping�the�decimal�point�in�the�same�place
Five�eighths�= �zero�point�six�two�five
If�you�need�more�decimal�place�0s,�you�can�add�them�in�later!
.00041 1 4'= c .000
83 3 8'=
.000 558 8 '= 1.000
811 1 8'= .000
427 27 4'=
.
.
.
85 5 000 8
8 5 0 0 0
0 6 2 5
'
`
=
=
=
g
.
.
0 6 2 5
8 5 0 0 05 2 4= g
Write�this�fraction�as�an�equivalent�decimal
.5 2 0 0 0=
=
g
.5 8 0 0 0=
=
g .8 1 1 0 0 0=
=
g .4 2 7 0 0 0=
=
g
.4 1 0 0 0=
=
g .8 3 0 0 0=
=
g
20 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
How does it work? Your Turn Decimals
Fractions to terminating decimals
6 � ���Simplify�these�fractions�and�then�write�as�an�equivalent�decimal�using�the�division�method.� Show�all�your�working.��
a b
c
1512
129
5649 d
e f2481
818
1626
6HSERIES TOPIC
21Decimals
Mathletics Passport © 3P Learning
Decimals
(i)� . . .24 105 11 06 6 5902+ +
•� Add�2.45�to�6.31�(i.e.�2.45 + 6.31)
Decimal�points�lined�up�vertically
Decimal�points�lined�up�vertically
Rounding�decimal�values�before�adding�is�sometimes�used�to�quickly�approximate�the�size�of�the�answer.
24.105 11.06 6.5902 41.7552` + + =
•� Subtract�5.18�from�11.89�(i.e.�11.89 - 5.18)
(ii)�Round�each�value�in�question�(i)�to�the�nearest�whole�number�before�adding.������
�(iii)� . .80 09 72 6081-
. . .24 105 11 06 6 5902 24 11 7
42
` .
.
+ + + +
�Note:��Rounding�values�before�adding/subtracting�is�not�as�accurate�as�rounding�after�adding/subtracting.����
Any�place�value�spaces�are�treated�as�0s
Fill�place�value�spaces�in�the�top�number�with�a�‘0’�when�subtracting�
80.09 72.6081 7.4819` - =
Where does it work?
2 4 . 1 0 5 +
1 1 . 0 6
1 6 . 5
1 9 0 2
4 1 . 7 5 5 2
Decimal�points�lined�up�vertically
Add�matching�place�values�together
Values�rounded�to�nearest�ones
Approximate�value�for�addition
Decimal�points�lined�up�vertically
Subtract�matching�place�values
2 . 4 5 +
6 . 3 1
8 . 7 6
1 1 . 8 9 -
5 . 1 8
6 . 7 1
Add�matching�place�values�together
Subtract�matching�place�values
Adding and subtracting decimals
Just�add�or�subtract�the�digits�in�the�same�place�value.����
To�do�this,�line�up�the�decimal�points�and�matching�place�values�vertically�first.�
Calculate�each�of�these�further�additions�and�subtractions
8 10 .
10 9
10
10 -
71 21 . 6
01 81 1
7 . 4 8 1 9
22 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
Where does it work? Your Turn Decimals
Adding and subtracting decimals
1 � Complete�these�additions�and�subtractions:����
2 Calculate�these�additions�and�subtractions,�showing�all�working:
c da b
g he f
Add�8.75�to�1.24a Subtract�3.15�from�4.79b
Add�0.936�to�0.865c
Subtract�0.9356�from�8.6012e
Add�2.19, 5.6�and�0.13d
Add�10.206, 4.64�and�8.0159f
ADDING AND SUBTRACTING DECIMALS +
- +
- .
..../.....
/20...
0 . 1 4 +
0 . 7 3
0 . 9 9 -
0 . 2 6
1 . 6 8 +
5 . 3 0
0 . 2 4 6 +
0 . 8 3 2
5 . 2 4 -
0 . 8 3
5 . 0 7 4 -
1 . 0 6 4
1 2 . 1 9 4 +
9 . 0 5 7
2 4 . 1 5 8 -
1 3 . 6 9 4
6HSERIES TOPIC
23Decimals
Mathletics Passport © 3P Learning
Where does it work? Your Turn Decimals
Adding and subtracting decimals
3 a � Approximate�these�calculations�by�rounding�each�value�to�the�nearest�whole�number�first.����
b
c
4
a b
Calculate�parts�(v)�and�(vi)�again,�this�time�rounding�after�adding�the�numbers�to�get�a�moreaccurate�approximate�value.�
(i)� �(ii)��������. . .2 71 3 80 1 92+ +. . .8 34 1 61 0 54+ +
Calculate�these�subtractions,�showing�all�your�working:�����
7.8 2.56- . .13 09 8 4621- . .0 52 0 12532-
5.7 + 6.2 .
.
+
8.3 - 1.9 .
.
-
8.34 + 1.61 + 0.54 .
.
2.71 + 3.80 + 1.92 .
.
+ ++ +
11.3 - 0.2 .
.
-
0.9 + 9.4 .
.
+(i)
(iii)
(v) (vi)
(iv)
(ii)
24 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
Where does it work? Decimals
Multiplying with decimals
How�does�this�work�when�multiplying�with�decimals?�Excellent�question!�Very�glad�you�asked!�
Just�write�the�terms�as�whole�numbers�and�multiply.�Put�the�decimal�point�back�in�when�finished.�
The�number�of�decimal�places�in�the�answer�=��the�number�of�decimal�places�in�the�question!��
1 � Calculate��
2 Calculate
4 1.2#
0.02 1.45#
4 12 4 8# =
. .
4 8
4 1 2 4 8#` =
1
Multiply�both�terms�as�whole�numbers
1�decimal�place�in�question��= 1�decimal�place�in�answer
Multiply�both�terms�as�whole�numbers
4�decimal�places�in�question��= 4�decimal�places�in�answer
2 145 2 9 0# =
0.02 1.45 0 . 0 2 9 0` # =
2 9 0
4 3 2 1
Let’s�do�the�second�one�again�but�this�time�change�the�decimals�to�equivalent�fractions�first
Changing�the�decimals�to�fractions
Multiply�numerators�and�denominators�together
Number�of�zeros�in�denominator�=�total�of�decimal�places�in�question
Dividing�by�10 000�moves�decimal�point�four�places�to�the�left
` 4�decimal�places�in�question�= 4�decimal�places�in�answer
Try�this�method�for�yourself�on�the�first�example�above,�remembering�that�414= �as�a�fraction.�
. .
.
.
0 02 1 451002
100145
100 1002 145
10 000290
290 10 000
0 2 9 0
0 0290
# #
##
'
=
=
=
=
=
=
4 3 2 1
6HSERIES TOPIC
25Decimals
Mathletics Passport © 3P Learning
Where does it work? Your Turn Decimals
Multiplying with decimals
1 � Calculate�these�whole�number�and�decimal�multiplications,�showing�all�you�working:
a b c0.8 2# .5 1 5# 0.14 6#
d e f0.62 4# 3 .0 032# 1.134 2#
2 � Calculate�these�decimal�multiplications,�showing�all�your�working:�����
a b c.8 .23 0# . .1 09 0 08# . .2 7 2 5#
d e f. .47 1 1# 3. .21 2 1# . .17 2 9 3#
MULTIPLY
ING WITH DECIMALS MULTIPLYING WITH DE
CIMALS
..../...../2
0...
26 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
Where does it work? Decimals
Calculate� . .1 26 0 8'
•� Calculate� .4 28 4'
Divisor�already�a�whole�number�so�no�change�needed
•� Calculate� . .0 0456 0 006'
0.0456 0.006 .7 6` ' =
. . . .
.
0 0456 0 006 0 045 6 0 006
45 6 6
' '
'
=
=
dividend�' �divisor�= �quotient
. 2
.
4 4 8
1 0 72g
4.28 4 1.07` ' =
Move�both�decimal�points�right�until�divisor�is�a�whole�number
4 .
.
6 5 6
0 7 64 3g
Drop�off�any�0s�at�the�front�of�the�answer
Quotient�2 �Dividendif�divisor�1 1
Here’s�another�example�showing�how�to�treat�remainders
1.26 0.8 1.2 6 0.8
.12 6 8
' '
'
=
=
. . .1 26 0 8 1 575` ' =
8 .
.
1 2 6 0 0
0 1 5 7 51 4 6 4
= g
Move�both�decimal�points�right�until�divisor�is�a�whole�number
Add�0s�on�the�end�of�the�dividend�for�each�new�remainder
Drop�off�any�0s�at�the�front
Dividing with decimals
Opposite�to�multiplying,�we�move�the�decimal�point�before�dividing�if�needed.��
To�find�the�quotient�involving�decimals,�the�question�must�be�changed�so�the�divisor�is�a�whole�number.
6HSERIES TOPIC
27Decimals
Mathletics Passport © 3P Learning
Where does it work? Your Turn Decimals
Dividing with decimals
1 � Calculate�these�decimal�and�whole�number�divisions:
a b c3.6 4' 17.5 5' .16 2 9'
d e f0.63 3' .0 489 5' .10 976 7'
3.6 4` ' = 17.5 5` ' = 16.2 9` ' =
0.63 3` ' = 0.489 5` ' = 10.976 7` ' =
2 � Calculate�these�decimal�divisions,�showing�all�your�working:��
a b c. .45 2 0' . .9 6 0 6' . .0 56 0 8'
d e f. .1 58 0 4' 0. .8125 0 05' . .5 3682 0 006'
5.2 0.4` ' = 9.6 0.6` ' = 0.56 0.8` ' =
1.58 0.4` ' 0.8125 0.05` ' 5.3682 0.006` '
DIVIDING WITH DECIMALS DIVIDING
WITH DECIMAL
S
..../...../20...÷
g
g
g
g
g
g
g
g
g
g
g
g
= = =
28 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
Where does it work? Decimals
Recurring decimals
Identify�the�start�and�end�of�the�repeating�pattern
Dot�above�start�and�end�of�the�repeating�pattern
Identify�the�start�and�end�of�the�repeating�pattern
Dot�above�start�and�end�of�the�repeating�pattern
Identify�the�start�and�end�of�the�repeating�pattern
Dot�above�start�and�end�of�the�repeating�pattern
Write�1�as�a�decimal�with�a�few�0s
Repeats�the�same�remainder�when�dividing
Recurring�decimal�in�simplest�notation
If�the�decimal�parts�have�a�repeating�number�pattern,�they�are�called�recurring�decimals.
Non-terminating�decimals�have�decimal�parts�that�do�not�stop.�They�keep�going�on�and�on.���
. ...
.
0 2052052
0 205= o o
. ...0 3582942049
. ...5 212121
A�dot�above�the�start�and�end�digit�of�the�repeating�pattern�is�used�to�show�it�is�a�recurring�decimal.
(i)� Write�these�recurring�decimals�using�the�dot�notation
. ...
.
10 81818
10 81= o o
Three�dots�means�it�keeps�going
Start���End
Start�����End
or
.
.. ...
1 0000 6
6 1 0 0 0 00 16 6 6
4 4 4 4
'=
g
...
.
1047777
1 047= o
Start�and�End
1.047= r
0.1 0.6 1 6' '=
(ii)� Calculate�0.1 0.6'
1 6 0.1666 ... 0.16` ' = = o
A�bar�over�the�whole�pattern�can�also�be�used�instead�of�dots
The�pattern�21�keeps�repeating�in�the�decimal�parts
Here�are�some�examples�involving�recurring�decimals�
a)
b)
c)
10.81818...
0.2052052...
1.047777...
. .0 205 0 205=o o
6HSERIES TOPIC
29Decimals
Mathletics Passport © 3P Learning
Where does it work? Your Turn Decimals
Recurring decimals
1 � What�is�the�name�of�the�horizontal�line�above�the�repeated�numbers�in�a�recurring�decimal?
� �Highlight�the�boxes�that�match�the�recurring�decimals�in�each�row�with�the�correct�simplified�notation�in�each�column�to�find�the�answer.�
� Not�all�of�the�matches�form�part�of�the�answer!
2 � �Calculate�these�divisions�which�have�recurring�decimals�as�a�result.�������� Write�answers�using�dot�notation.
a b c1 3' 4 9' 5 6'
d e f1.6 6' .2 5 9' .0 34 3'
1 3` ' = 4 9` ' = 5 6` ' =
.1 6 6` ' = .2 5 9` ' = .0 34 3` ' =
c z h m n a f b
g
g
g
g
g
g
.0 14o .0 4r .41 1o .0 144 0.141o o .0 41o .414 .0 401o o .4 1o .0 41o o
4.1414 ... C z F h N��d W c D b A a U n P t L f O m
0.144144 ... Y��n A m R��f T t K z E��h R��d I��c U b S a
0.1444 ... L a D b A m I��h M t B f S c A d U z Q n
0.401401 ... R��h Z d A n E��z A c N��t 0 a M b A h G��f
4.111 ... A f T z P c H d T a Y��n A t A h C m A b
0.4111... I��d Y��t A b U n H m I��z E��f S m I��t T a
0.4141 ... A b L a D t E��f A d N��c L m E��z O d N��h
41.111 ... W c J f B d A a X h M m A b U n A A z
0.444 ... P m V��c E��a F b A n B d T Y��f E��c I��t
0.1411411 ... H t A n A m A m U f A b A h A a D d R��c
30 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
Where does it work? Your Turn Decimals
Recurring decimals
3 � (i)� Complete�the�following�divisions�to�five�decimal�places.������ (ii)� Determine�whether�the�answer�is�a�recurring�decimal�or�not.
a b c2 3' 1 6' 1 7'
d e f.1 6 7' .2 9 3' . .0 33 0 8'
2 3` ' = 1 6` ' = 1 7` ' =
.1 6 7` ' = .2 9 3` ' =
Recurring�decimal?Yes� ���No
Recurring�decimal?Yes� ���No
Recurring�decimal?Yes� ���No
Recurring�decimal?Yes� ���No
Recurring�decimal?Yes� ���No
Recurring�decimal?Yes� ���No
Recurring�decimal?Yes� ���No
Recurring�decimal?Yes� ���No
Recurring�decimal?Yes� ���No
g h i0.6 .38 0' 0. .019 0 06' 0. 0.00644 002'
0.68 0.3` ' 0.019 0.06` ' 0.00644 0.002` '
RECU
RRING DECIMALS... RECURRING DECIMALS... RECURRIN
G DECIMALS...
..../...../20.
..
g
g
g
g
g
g
g
g
g
= = =
0.33 .80` ' =
6HSERIES TOPIC
31Decimals
Mathletics Passport © 3P Learning
DecimalsWhat else can you do?
Simple recurring decimals into single fractions
Only�recurring,�non-terminating�decimals�can�be�written�in�fraction�form.����
Here�is�a�quick�way�for�simple�decimals�with�the�pattern�starting�right�after�the�decimal�point.
(i)� . ...3 777
Three�digits�in�repeating�pattern,�so�those�three�digits�over�999
One�digit�in�repeating�pattern,�so�that�digit�over�9
Two�digits�in�repeating�pattern,�so�those�two�digits�over�99
One�digit�in�repeating�decimal�pattern,�so�that�digit�over�9
Digits�in�front�of�decimal�point�form�the�whole�number�part
Three�digits�in�repeating�decimal�pattern,�so�those�digits�over�999
Digits�in�front�of�decimal�point�form�the�whole�number
Simplify�the�fraction�part
. ... .3 7777 3 7
397
=
=
o
9912
334
33
''=
=
�(ii)�� . ...16 345345
0.111... 0.1
. ... .
91
0 1212 0 129912
= =
= =
o
o o
0.301301... 0.301999301= =o o
Here�are�some�other�examples�including�mixed�numerals.
. ... .16 345345 16 345
16999345
16999345
16333115
33
''
=
=
=
=
o o
Always�simplify fractions
Write�each�of�these�recurring�decimals�as�mixed�numerals�in�simplest�form
32 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
What else can you do? Your Turn Decimals
Simple recurring decimals into single fractions
1 � Use�the�shortcut�method�to�write�each�of�these�recurring�decimals�as�a�fraction�in�simplest�form:
a b c.0 4o .0 8r .0 6o
d e f.0 11o o .0 27o o .0 57o o
g h i0.162 5.1485 0.4896o o
Use�the�shortcut�method�to�write�each�of�these�recurring�decimals�as�mixed�numerals�in�simplest�form.
2
a b c.1 5o .2 7r .4 3r
d e f.3 6r 5.12 0.117o o
3 � (i)� Write� .0 9o �as�a�fraction�in�simplest�form.
� (Ii)� Does�anything�unusual�seem�to�be�happening�with�your�answer?�Explain...../..
.../20...
= 9
= 0. ...
0. SIMPLE R
ECURRING DECIMALS INTO SINGLE FRACTIO
NS
6HSERIES TOPIC
33Decimals
Mathletics Passport © 3P Learning
What else can you do? Decimals
Combining decimal techniques to solve problems
All�the�techniques�in�this�booklet�can�be�used�to�solve�problems.
(i)� These�rainfall�measurements�were�taken�during�three�days�of�rain�from�a�small�weather�gauge:
Add�the�decimal�values�together
(ii)�� The�results�for�five�runners�in�a�100�m�race�were�plotted�on�the�number�line�below.
13.8
36.1
27.6
77.5
+
78. mm
Read�off�all�the�times
a) What�was�the�fastest�time�run�(to�the�nearest�thousandth�of�a�second)?
� Fastest�time��= ��left-most�plotted�point��= 11.221�seconds
b) What�time�did�two�runners�finish�the�race�together�on?
� Two�runners�with�the�same�time��= ��two�dots�at�the�same�point��= 11.223�seconds
c) What�was�the�average�time�ran�by�all�runners�in�this�race?
� Average�time��=�The�sum�of�all�the�times�ran�divided�by�the�number�of�runners
The�average�time�ran�by�all�the�runners�in�the�race�� .11 2242= seconds
( . . . . . )
.
11 221 11 223 11 223 11 226 11 228 5
56 121 5
'
'
= + + + +
=
5 5 6. 1 2 1 0
.11 2 2 4 21 11 2
= g
11.22 11.23seconds
These�examples�show�different�ways�decimals�pop�up�in�every-day�life
What�was�the�total�rainfall�for�the�three�days,�to�the�nearest�whole�mm?
13.8 mm 36.1 mm 27.6 mm
Round�to�nearest�whole�mm
Answer�with�a�statement
Add,�then�divide�by�5
Answer�with�a�statement`��The�total�rainfall�over�the�three�days�was�approximately�78 mm
34 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
What else can you do? Your Turn Decimals
Remember me?
Combining decimal techniques to solve problems
1 To make dark-green coloured paint, you can mix yellow and blue together, using exactly 0.5 (half) as much yellow as you do blue.
a Usemultiplicationtoshowhowmuchyellowpaintyouwillneedifyouuseall of the 12.46 mL of blue paint you have.
b How many millilitres of dark-green paint can you make with 18.45 mL of yellow paint in the mix? Round your answer to the nearest tenth of a mL.
2 Derek types his essays at an average speed of 93.45 words every minute. How many words does he type in fiveminutes(tothenearestwholeword)?
3 Ninepeopleweretryingoutforaspeedrollerskatingteamaroundanovalflattrack. Theshortesttimetocompletesixfulllapsofthetrackforeachpersonwererecordedon the number line below:
a Whatwastheslowesttimerecordedto3 decimal places?
b To make the team, a skater had to complete the six laps in less than 126.245 seconds. How many skaters made it into the team?
c How many skaters missed out making the team by less than 0.01 seconds?
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COMBIN
ING DECIMAL TECHNIQUES TO SOLVE PRO
BLEMS
126.22 126.23 126.24 126.25 126.26 126.27seconds
6HSERIES TOPIC
35Decimals
Mathletics Passport © 3P Learning
What else can you do? Your Turn Decimals
Combining decimal techniques to solve problems
� �The�wireless�transmitter�in�Laura’s�house�reduces�in�signal�strength�by�0.024�for�every�1�metre�of�distance�she�moves�her�computer�away�from�the�transmitters�antenna.�Her�computer�displays�signal�strength�using�bars�as�shown�below:
4�bars� .0 81= �to�1.0�signal�strength3�bars��� 0. 16= �to�0.8�signal�strength2�bars�� 0. 14= �to�0.6�signal�strength1�bar��� 0. 12= �to�0.4�signal�strength0�bars�� 0.2= �or�below�signal�strength����
� �How�many�bars�of�signal�strength�would�Laura�have�if�using�her�computer�16.25m�away�from� the�antenna?�
� �Ruofan�is�putting�together�a�video�of�a�recent�karaoke�party�with�her�friends.�She�will�be�using�five�of�her�favourite�music�tracks�for�the�video.�
� The�length�of�time�each�of�the�tracks�play�for�is:
3.55 min,�5.14 min, 2.27 min, 3.18 min�and 4.86 min
� �If�she�uses�the�entire�length�of�the�tracks�with�a�0.15�min�break�in�each�of�the�four�gaps�between�songs,�how�long�will�her�video�run�for�(to�the�nearest�whole�minute)?�Show�all�your�working.
4
5
36 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
What else can you do? Your Turn Decimals
Combining decimal techniques to solve problems
6 � �����After�a�recent�study�by�a�city�council,�the�average�number�of�people�in�each�household�was�determined��to�be�3.4.�Explain�how�this�is�possible�if�a�household�cannot�actually�have�0.4�of�a�person?�
psst:�Check�example�on�page�33�to�see�how�average�calculations�are�made.
7 A�Mexican�chef�has�split�up�a�mystery�ingredient�“Sal-X”�into�four�exactly�identical�quantities�in�separate�jars.�He�then�distributes� .138 2o mL�of�the�secret�ingredient�“Sa-Y”�amongst�the�four�jars,�producing�in�total� .863 9o mL�of�the�special�sauce�“SalSa-XY”.�How�much�of�the�mystery�ingredient�“Sal-X”�is�there�in�each�jar�(to�the�nearest�mL)?�Show�all�your�working.
8 After�completely�flat�water�conditions�(waves�with�a�height�of�0.0m),�the�height�of�the�waves�at�a�local�beach�start�increasing�by�0.2 m�every� .0 3o �hours.
� �If�the�waves�need�to�be�at�least�1.4�metres�high�before�surfers�will�ride�them�at�this�beach,�how�long�will�it�be�until�people�start�surfing�there�to�the�nearest�minute?�Show�all�your�working.
psst:�1.0�hours��� 60= �minutes
6HSERIES TOPIC
37Decimals
Mathletics Passport © 3P Learning
What else can you do? Your Turn Decimals
Reflection Time
Reflecting�on�the�work�covered�within�this�booklet:
� �What�useful�skills�have�you�gained�by�learning�about�decimals?
2 � Write�about�one�or�two�ways�you�think�you�could�apply�decimals�to�a�real�life�situation.
3 � �If�you�discovered�or�learnt�about�any�shortcuts�to�help�with�decimals�or�some�other�cool�facts,�jot�them�down�here:
1
38 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
Cheat Sheet Decimals
1.240 1.248 1.250
8
=��decimal�point�moves�right�����,������������������������� =��decimal�point�moves�left����
.
.
5 1000 5 0 1000
5 0
# #=
=
5000=
.
. .
8 100 8 0 100
8 0
' '=
=
.0 08=
1 2 3 2 1
Closer�to�lower�value,�so�round downLeave�the�place�value�unchanged
Closer�to�higher�value,�so�round upAdd�1�to�the�place�value
0 1 2 3 4 5 6 7 8 9
Next�digit
•
# 1
0 0
00
# 1
000
# 1
00
# 1
0
# 1
Tens�of�thousands
Thousands
Hundreds
Tens
Ones
W H O L E
' 1
0
' 1
00
' 1
000
' 1
0 0
00
' 1
00 0
00
' 1
000 0
00
' 1
0 0
00 0
00
Tenths
Hundredths
Thousandths
Ten�thousandths
Hundred�thousandths
Millionths
Ten�M
illionths
D E C I AM L
Here is a summary of the things you need to remember for decimals
Place value of decimals
Approximations through rounding numbers
The�next�digit�following�the�place�value�where�a�number�is�being�rounded�off�to�is�the�important�part.�
Decimals on the number line
The�smallest�place�value�in�a�decimal�is�used�to�position�points�accurately�on�a�number�line.
3.0 3.6 4.0
6
Six�tenths�of�the�way�
from�3.0�to�4.0
Eight�thousandths�of�the�
way�from�1.240�to�1.250
Multiplying and dividing by powers of ten
Move�the�decimal�point�depending�on�the�number�of�zeros
6HSERIES TOPIC
39Decimals
Mathletics Passport © 3P Learning
Cheat Sheet Decimals
.1 07 11007=
Eg:
.
53
53
106
0 6
22#
#
`
=
=
=
Multiplynumeratoranddenominatorbythesamevalue
Equivalentfractionwithapowerof10 in the denominator
Threefifths= six tenths = zero point six
Terminating decimals to fractions
The place value of the last digit on the righthelpsustowriteitasafraction.
.0 3103=Write 0.3asafraction:
Lastdigitisintenthsposition
Decimal DecimalFraction Fraction
Fractions to terminating decimals
Wherepossible,justwriteasanequivalentfractionwithapowerof10inthedenominatorfirst.
When this method is not easy, write the numerator as a decimal and then divide it by the denominator.
Adding and subtracting decimals
Lineupthedecimalpointsandmatchingplacevaluesverticallybeforeaddingorsubtracting.
Multiplying and dividing decimals
Writethetermsaswholenumbersandmultiply.Putthedecimalpointbackinwhenfinished. The number of decimal places in the answer = thenumberofdecimalplacesinthequestion!
4 1.2 4.8#: = 0.02 1.45 0.0290#: =Eg:
Eg:
Eg:
Recurring decimals
Thesehavedecimalpartswitharepeatingnumberpattern.
Dividing with decimals
Thequestionmustbechangedsothedivisorisawholenumberfirst. dividend ' divisor = quotient
13.5 0.4 135 4: ' '= 89.25 0.003 89250 3: ' '=
AlwayssimplifyfractionsSimple recurring decimals into single fractions
Onlyrecurring,non-terminatingdecimalscanbewritteninfractionform.Thisisthemethodforsimpledecimalswiththepatternstartingrightafterthedecimalpoint.
0.111... 0.191: = =o 0.1212... 0.12
9912
334: = = =o o
Onedigitinrepeatingpattern,sothatdigitover9
Twodigitsinrepeatingpattern,so those two digits over 99
8.301301... 8.301 8999301: = =o o
Threedigitsinrepeatingpattern,sothose three digits over 999, Keep whole number out the front.
5.212121... 5.21 5.21: = =o o 0.3698698... 0.3698 0.3698: = =o o
Start StartEnd End
Write 1.07asafraction:
Lastdigitisinhundredthsposition
40 Decimals
Mathletics Passport © 3P Learning
6HSERIES TOPIC
Decimals Notes
4
DECIMA
LS ON
THE NUMBER LINE DECIMALS ON THE NUM
BE
R LINE
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/20...
MU
LTIPLYING AND DIVIDING BY POWERS
OF TEN
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..
MULTIPLY
ING WITH DECIMALS MULTIPLYING WITH DE
CIMALS
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/20...
DIVIDING WITH DECIMALS DIVIDING
WITH DECIMAL
S
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PLACE
VALUE OF DECIMALS PLACE VALUE O
F DECIMALS
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