Decimal Fractions
Decimal
Fractions
Curriculum Ready
www.mathletics.com
Decimal fractions 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 fraction system
is based around the number 10!
So we can think of decimal fractions as being fractions with powers of 10 in the denominator.
Write in this space EVERYTHING you already know about decimal fractions.
is
Give th
Q
a go!
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
Decimal Fractions
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How does it work?
Decimal Fractions
Place value of decimal fractions
L
' 10 000 000
' 1 000 000
' 100 000
#
1 = one tenth
10
2nd decimal place: ' 10 = # 1 = one hundredth
100
3rd decimal place: ' 10 = # 1 = one thousandth
1000
4th decimal place: ' 10 = # 1
= one ten thousandth etc...
10 000
1st decimal place: ' 10 =
I M A
' 10 000
' 1000
Te
nt
h
Hu s
nd
Th red
ou ths
Te sand
n
th th
Hu ou s
nd san
M red dth
illi th s
o
o
Te nth usa
n
s
nd
M
th
illi
s
on
th
s
C
' 100
1
#
#
Decimal point
#
D E
' 10
10
•
#
E
100
L
10 000
W H O
1000
Te
ns
o
Th f th
ou ou
s
Hu and san
nd s ds
Te red
ns s
On
es
Decimal fractions represent parts of a whole number or object.
#
Add ‘th’ to the
name for decimal
place values
Write the place value of each digit in the number 465.2703
465.2703
Multiply by multiples of 10
Expanded forms
2
Divide by multiples of 10
Place values
4......4 # 100
= 400
= 4 hundred
6......6 # 10
= 60
= 6 tens (or sixty)
5......5 # 1
= 5
= 5 ones (or five)
2......2 ' 10 `or 2 # 1 j
10
7......7 ' 100 `or 7 # 1 j
100
0......0 ' 1000 `or 0 # 1 j
1000
3......3 ' 10 000 `or 3 # 1 j
10 000
= 2
10
= 7
100
= 0
1000
3
=
10 000
= 2 tenths
1st decimal place
= 7 hundredths
2nd decimal place
= 0 thousandths
3rd decimal place
= 3 ten thousandths
4th decimal place
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Decimal Fractions
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Integer parts
How does it work?
Your Turn
Decimal Fractions
Place value of decimal fractions
1
Write the decimal fraction that represents these:
a
2 hundredths
b
9 tenths
c
1 ten thousandth
e
6 hundred thousandths
f
8 millionths
0.02
Always put a zero in front
(called a leading zero) when
there are no whole numbers
d
2
3
4
3 thousandths
Write the fraction that represents these:
a
3 tenths
b
7 thousandths
c
1 hundredth
d
9 ten thousandths
e
51 hundredths
f
11 ten thousandths
Write the place value of the digit written in square brackets for each of these decimal fractions:
a
63 @ 1.325
b
61 @ 0.231
c
64 @ 15.046
d
65 @ 0.05043
e
66 @ 0.79264
f
60 @ 8.56309
c
[hundred thousandths]
Circle the digit found in the place value given in square brackets:
a
[tenths]
b
8.171615
d
[hundredths]
9.12421
[thousandths]
4.321230
e
[ten thousandths]
16.123210
Decimal Fractions
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100.1001001
f
[millionths]
3.120619
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How does it work?
Your Turn
Decimal Fractions
Place value of decimal fractions
Each
digit is multiplied by the place value and then added together when writing a number in expanded form.
Write the decimal fraction 23.401 in expanded form
23.401 = 2 # 10 + 3 # 1 + 4 # 1 + 0 # 1 + 1 # 1
10
100
1000
= 2 # 10 + 3 # 1 + 4 # 1 + 1 # 1
10
1000
Write these decimal fractions in expanded form:
4.19 =
b
29.281 =
c
40.2685 =
d
3.74932 =
e
0.2306 =
f
0.0085 =
VALUE OF
DE
ACE
L
P
Simplify these numbers written in expanded form:
..../.
a
1 # 1+ 4 # 1 + 6 # 1 =
10
100
b
4 # 10 + 9 # 1 + 0 # 1 + 7 # 1 =
10
100
c
5 # 100 + 2 # 10 + 0 # 1 + 2 # 1 + 1 # 1 + 8 # 1 =
10
100
1000
d
1
6 # 1+ 8 # 1 + 5 # 1 + 0 # 1 + 2 # 1 + 9 #
=
10
100
1000
10 000
100 000
Psst: Remember to include a leading zero for these ones.
4
e
2 # 1 +0 # 1 +3 # 1 =
10
100
1000
f
1
6 # 1 +7 # 1 +0 # 1
+ 1#
=
100
1000
10 000
100 000
g
1
3 # 1 + 4 # 1 +1 # 1 + 0 # 1 + 8 #
=
10
100
1000
10 000
100 000
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Decimal Fractions
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FRACTIONS
MAL
I
C
a
..../2
0...
FRACTIONS
MAL
CI
6
Zero digits can be removed to simplify
VALUE OF
DE
ACE
PL
5
Multiply each digit by its place value
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How does it work?
Decimal Fractions
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
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.
Next digit
0
1
2
3
4
5
6
7
8
9
Closer to lower value, so round down
Closer to higher value, so round up
Leave the place value unchanged
Add 1 to the place value
Here are some examples to see how we round off numbers.
Round these numbers
(i) 2462 to the nearest hundred
2 4 6 2
The digit ‘4’ is in the hundreds position 2 4 6 2
The next digit is a 6, so round up by adding 1 to 4
2 5 0 0
Change the other smaller place value digits to 0’s
` 2462 . 2500 rounded to the nearest hundred
(ii) 0.3145 to one decimal place (or to the nearest tenth)
0 . 3 1 4 5
The digit ‘3’ is in the first decimal place 0 . 3 1 4 5
The next digit is a 1, so round down
0 . 3
Write decimal fraction with one decimal place only
` 0.3145 . 0.3 rounded to one decimal place
(iii) 26.35819 to four decimal places (or to the nearest ten thousandth)
2 6 . 3 5 8 1 9
The digit ‘1’ is in the fourth decimal place 2 6 . 3 5 8 1 9
The next digit is a 9, so round up by adding 1 to 1
2 6 . 3 5 8 2
Write decimal fraction with four decimal places only
` 26.35819 . 26.3582 rounded to four decimal places
Decimal Fractions
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Decimal Fractions
Round these whole numbers to the place value given in square brackets.
a
[nearest ten]
b
[nearest hundred]
c
34
44
5
...
[nearest thousand]
(i) 536 .
(i) 14 302 .
(i) 98 542 .
(ii) 8514 .
(ii) 4764 .
(ii) 18 401 .
(iii) 93025 .
(iii) 80 048 .
(iii) 120 510 .
2 Round these decimal fractions to the decimal places given in the square brackets.
a
3
[nearest tenth]
b
[nearest hundredth]
c
[nearest thousandth]
(i) 0.73 .
(i) 2.406 .
(i) 10.4762 .
(ii) 3.47 .
(ii) 0.007 .
(ii) 0.3856 .
(iii) 11.85 .
(iii) 1.003 .
(iii) 0.048640 .
Approximate the following distance measurements: a
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? .
b
Under a microscope the length of a dust mite was 0.000194 m
(i) A
pproximate 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. .
c
If Lichen City is 3 458 532 m away from Moss City: (i) W
hat 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.
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0...
../2
./...
N
IO T
Approximations through rounding numbers
1
. APPROXIMAT
RS
NUMBE
Your Turn
H ROUNDIN
OUG
G
HR
How does it work?
How does it work?
Your Turn
Decimal Fractions
Approximations through rounding numbers
Rounding up can affect more than one digit when the number 9 is involved.
Round 0.95 to one decimal place
9 rounds up to 10, so
the 9 becomes 0 and 1
is added to the digit in front. 0 . 9 5
The digit ‘9’ is in the tenths position 0 . 9 5
The next digit is a 5, so round up by adding 1 to 9
1 . 0
Change the other smaller place value digits to 0s
` 0.95 . 1.0 rounded to one decimal place
4
R
ound off these numbers according to the square brackets. a
1.98 .
d
5
b
[three decimal places]
[two decimal places]
11.899 .
f
[three decimal places]
0.1398 .
h
c
398 .
e
[nearest thousand]
49798 .
[nearest ten]
[nearest ones]
79.9 .
g
[one decimal place]
[nearest ones]
2.1995 .
i
[four decimal places]
199.9 .
9.89999 .
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 =
(iii) Is the digit 6 important when approximating to the nearest whole litre?
Briefly explain here why/why not.
Decimal Fractions
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.
1 L?
1000
.
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How does it work?
Decimal Fractions
Decimal fractions on the number line
The smallest place value in a decimal fraction is used to position points accurately on a number line.
• D
ecimal fractions 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:
6
3.0 3.6
4.0
Six tenths of
the way from
3.0 to 4.0
• The major intervals on the number line are marked according to the second last decimal place value
8
1.240
1.2481.250
So its eight thousandths of
the way from 1.240 to 1.250
Here are some more examples involving number lines:
(i) What value do the plotted points represent on the number lines below?
4
a)
0.10.2
Point is four steps from 0.1 towards 0.2, so the plotted point is: 0.14
9
b)
10.0610.07
Point is nine steps from 10.06 towards 10.07, so the plotted point is: 10.069
(ii) Round the value of the plotted points below to the nearest hundredth.
3
a)
2.142.15
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
5
b)
8.798.80
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
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Decimal Fractions
b
0.7
0.0
c
e
d
f
2
3
D EC
E
IN
2.0
3.0
9.1
9.2
5.21
5.22
9.15
0.2
2.34
2.3
2.1
1.0
0.13
0.1
NU M BE R L
a
./2
....
/
....
0...
S ON TH
D
isplay these decimal fractions on the number lines below: L FR A CT
I
4
ON
Decimal fractions on the number line
1
MA
I
Your Turn
E
How does it work?
5.212
2.4
Label these number lines and then display the given decimal fraction on them:
a
1.6
b
4.2
c
0.94
d
7.07
e
2.053
f
9.538
Round the value of the plotted points below to the nearest place value given in square brackets.
a
[tenth]
b
0.2
[hundredth]
0.3
1.03
` the value .
c
` the value .
[tenth]
d
0.8
[hundredth]
0.9
0.08
` the value .
e
f
[thousandth]
1.995
8.103
` the value .
g
8.104
` the value .
[thousandth]
2.902
0.09 ` the value .
[thousandth]
1.994
1.04
h
[thousandth]
2.903
0.989
` the value .
0.990
` the value .
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How does it work?
Decimal Fractions
Multiplying and dividing by powers of ten
Move the decimal point depending on the number of zeros
= decimal point moves right , = decimal point moves left Calculate these multiplication and division questions involving powers of 10:
(i) 5 # 1000
5 # 1000 = 5.0 # 1000 The whole number in decimal fraction form
1 2 3
We can simply add the same
number of zeros to the end
of the whole number
= 5.0
.
Fill the empty bounces with 0s
= 5 000
If the decimal point is on the left after dividing, an extra 0 is placed in front.
(ii) 8 ' 100
The whole number in decimal fraction form
8 ' 100 = 8.0 ' 100
21
Remember to include
the leading zero 
' 100 has 2 zeros, so move decimal point 2 spaces left
= . 8.0
= 0.08
(iii) 1.25893 # 10 000
Fill the empty bounces with 0s and put a zero in front
1234
Move decimal point 4 spaces right
1.25893 # 10 000 = 1 . 2 5 8 9 . 3
No empty bounces to fill, so this is the answer
= 12 5 8 9 . 3
(iv) 24.905 ' 100 000
54321
2 4 . 905 Move decimal point 5 spaces left
24.905 ' 100 000 = .
= 0.00024905
(v) 260.15 #
Fill empty bounces with 0s and put a zero in front
1
1000
260.15 #
1 = 260.15 ' 1000
1000
#
1 is the same as ' 1000
1000
3 2 1
10
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= . 2 6 0 . 15
Move decimal point 3 spaces left
= 0.2 6 015
Place a leading zero in front of the decimal point
Decimal Fractions
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How does it work?
Your Turn
Decimal Fractions
Multiplying and dividing by powers of ten
1
2
Calculate these multiplications. Remember, multiply means move decimal point to the right:
a
8 # 100
b
3.4 # 10
c
29 # 1000
d
12.45 # 10 000
e
0.512 # 100
f
0.0000469 # 1000 000
Calculate these divisions. Remember, divide means move decimal point to the left:
a
2 ' 100
b
d
70.80 ' 10 000
e
4590 ' 1000
1367.512 ' 1000
c
0.014 ' 10
f
421 900 ' 100 000 000
Here are some of the powers of 10 in exponent form. The power = the number of zeros.
101 = 10
104 = 10 000
3
102 = 100
105 = 100 000
103 = 1000
106 = 1000 000
Calculate these mixed problems written in exponent form:
a
31 # 102
b
2400 ' 105
c
0.0027 # 106
d
90.008 # 104
e
3.45 ' 103
f
2159 951 ' 107
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Decimal Fractions
MU
Multiplying and dividing by powers of ten
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
4
2830.3920 # 100
2 8 3 0 3 9 2 0
I9 and 2
b
23 857 ' 1000
2 3 8 5 7
N
c
0.4763892 # 105
0 4 7 6 3 8 9 2
A
d
382 961 ' 10 000
3 8 2 9 6 2
O
e
19 238.07 # 101
1 9 2 3 8 0 7
X
f
8.9236701 # 10 000
8 9 2 3 6 7 0 1
T
g
20 917 983 #
2 0 9 1 7 9 8 3
R
h
83 917 ' 105
8 3 9 1 7
I
i
902 873.021 # 1 2
10
9 0 2 8 7 3 2 0 1
D
j
0.08390 # 103
0 0 8 3 9 0
P
This is another mathematical name for a decimal point: I
0 and 9 8 and 98 and 79 and 20 and 73 and 98 and 20 and 83 and 86 and 7
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20...
..../
..../.
a
1
1000 000
DIVIDIN
G
ND
IPLYING A
LT
TEN
OF
Your Turn
POWER
S
BY
How does it work?
How does it work?
Decimal Fractions
Terminating decimal fractions to fractions
These have decimal fraction 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.
Decimal fraction
Write 0.3 as a fraction: Fraction
= 3
10
0.3
Decimal fraction digits in the numerator
Last digit is in tenths position
Integers in front of the decimal fraction values are simply written in front of the fraction.
Write 1.07 as a fraction: 1. 0 7
=1 7
100
Decimal fraction digits in the numerator
Last digit is in hundredths position
07 is just 7
Always simplify the fraction parts if possible. These two examples show you how.
Write each of these decimal fraction as an equivalent (equal) fraction in simplest form
(i) 0.25
0.25
= 25
100
Equivalent, un-simplified fraction
= 25 ' 25
100 ' 25
Divide numerator and denominator by HCF
= 1
4
Equivalent fraction in simplest form
= 2 105
1000
Equivalent, un-simplified mixed number
= 2 105 ' 5
1000 ' 5
Divide numerator and denominator by HCF
= 2 21
200
Equivalent mixed number in simplest form
Last digit is in hundredths position
(ii) 2.105
2 . 10 5
Last digit is in thousandths position
Decimal Fractions
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How does it work?
Your Turn
Decimal Fractions
Terminating decimal fractions to fractions
1
Write each of these decimal fractions as equivalent fractions:
a
0.1 =
b
0.7 =
c
0.09 =
d
0.03 =
e
0.001 =
f
0.007 =
g
0.013 =
h
0.049 =
i
0.129 =
j
0.081 =
k
0.1007 =
l
0.0601 =
2 Write each of these decimal fractions as equivalent fractions and then simplify:
a
0.5 =
b
=
0.6 =
Simplest form
d
0.08 =
=
c
=
Simplest form
e
0.004 =
=
Simplest form
g
0.12 =
=
h
0.25 =
0.045 =
=
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0.005 =
k
0.0028 =
=
Simplest form
Decimal Fractions
Mathletics Passport
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=
Simplest form
0.022 =
Simplest form
Simplest form
14
f
i
=
=
Simplest form
Simplest form
Simplest form
j
0.02 =
=
Simplest form
l
0.0605 =
=
Simplest form
Where does it work?
Decimal Fractions
G DECI
ATIN
M
IN
M
R
4
0...
../2
/...
.5 =
2
* TE
NS
0
Write each of these decimal fractions as equivalent mixed numbers:
a
2.3 =
b
1.1 =
c
3.07 =
d
1.03 =
e
4.001 =
f
2.009 =
....
TO FRACTI
NS
O
IO
3
FRAC
T
Terminating decimal fractions to fractions
AL
1
Your Turn
Write each of these decimal fractions as equivalent mixed numbers and then simplify:
a
b
2.8 =
=
c
1.4 =
=
=
Simplest form
d
Simplest form
e
3.05 =
=
2.75 =
Simplest form
f
=
Simplest form
g
=
5.005 =
=
Simplest form
h
1.004 =
4.06 =
2.025 =
i
=
Simplest form
Simplest form
3.144 =
=
Simplest form
Decimal Fractions
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Simplest form
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How does it work?
Decimal Fractions
Fractions to terminating decimal fractions
Where possible, just write as an equivalent fraction with a power of 10 in the denominator first.
numerator
denominator
3 = 3#2
5
5#2
= 6
10
Multiply numerator and denominator by the same value
Equivalent fraction with a power of 10 in the denominator
` = 0.6
Three fifths = six tenths = zero point six
Sometimes it is easier to first simplify the fraction before changing to a decimal fraction.
Write these as an equivalent decimal fraction
(i) 3
12
3'3 = 1
12 ' 3
4
Simplify fraction
1 = 1 # 25
4
4 # 25
= 25
100
Equivalent fraction with a power of 10 in the denominator
` = 0.25
Three twelfths = one quarter = twenty five hundredths = zero point two five
(ii) 2 3
15
2 3'3 = 21
15 ' 3
5
Simplify fraction part
21# 2 = 2 2
10
5#2
= 2.2
Equivalent fraction with a power of 10 in the denominator
Two and three fifteenths = two and one fifth = two and two tenths = two point two
16
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Decimal Fractions
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How does it work?
Your Turn
Decimal Fractions
include a
leading zero
Fractions to terminating decimal fractions
1
Write each of these fractions as equivalent decimal fractions.
a
2
3
9 =
10
b
3 =
100
c
11 =
100
7 =
1000
d
Write each of these as equivalent fractions with a power of 10 in the denominator.
a
1 =
2
b
2 =
5
c
3 =
4
d
9 =
20
e
8 =
25
f
3 =
250
g
11 =
200
h
2 =
125
i
14 =
5
j
3 1 =
25
k
6 7 =
20
(i) Write each of these as equivalent fractions with a power of 10 in the denominator.
(ii) Change to equivalent decimal fractions.
a
1 =
5
b
=
d
c
=
4 =
25
e
2 9 =
25
=
h
11 =
25
=
1 =
200
f
=
=
g
1 =
4
=
1 1 =
200
i
8 7 =
50
=
=
Decimal Fractions
Mathletics Passport
6 =
125
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How does it work?
Your Turn
Decimal Fractions
Fractions to terminating decimal fractions
4
Change each of these fractions to equivalent decimal fractions after first simplifying.
Show all your working.
a
12
20
b
20
25
ON
CTI S TO
RA
e
9
75
f
12
3 40
h
12
2 150
g
18
36
1 600
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Decimal Fractions
Mathletics Passport
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ONS
CTI
F
RA
22
40
..../...
IMAL
DEC
F
d
G
18
24
RMINATIN
TE
c
1 = 5
2 ../20...
How does it work?
Your Turn
Decimal Fractions
Fractions to terminating decimal fractions
When changing the denominator to a power of 10 is not easy, you can write the numerator as a decimal
fraction and then divide it by the denominator.
Write this fraction as an equivalent decimal fraction
5 = 5.000 ' 8
8
Write numerator as a decimal fraction and divide by the denominator
= 8g 5 . 0 0 0
If you need more
decimal place 0s, you
can add them in later!
0. 6 2 5
= 8 g 5 . 50 2 0 4 0
Complete division, keeping the decimal point in the same place
`= 0. 6 2 5
Five eighths = zero point six two five
5
Complete these divisions to find the equivalent decimal fraction:
a
d
2 = 2.000 ' 5
5
b
1 = 1.000 ' 4
4
c
3 = 3.000 ' 8
8
= 5g 2 . 0 0 0
= 4g1 . 0 0 0
= 8g 3 . 0 0 0
=
=
=
8 = 8.000 ' 5
5
e
11 = 11.000 ' 8
8
f
27 = 27.000 ' 4
4
= 5g 8 . 0 0 0
= 8g1 1 . 0 0 0
= 4g 2 7 . 0 0 0
=
=
=
Decimal Fractions
Mathletics Passport
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6
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19
How does it work?
Your Turn
Decimal Fractions
Fractions to terminating decimal fractions
6
20
Simplify these fractions and then write as an equivalent decimal fraction using the division method.
Show all your working. a
12
15
b
9
12
c
49
56
d
18
8
e
81
24
f
26
16
H
6
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TOPIC
Decimal Fractions
Mathletics Passport
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Where does it work?
Decimal Fractions
Adding and subtracting decimal fractions
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.
• Add 2.45 to 6.31 (i.e. 2.45 + 6.31)
2.4 5+
6.3 1
Decimal points lined up vertically
8.7 6
Add matching place values together
• Subtract 5.18 from 11.89 (i.e. 11.89 - 5.18)
1 1 . 8 9 5.18
Decimal points lined up vertically
6. 71
Subtract matching place values
Calculate each of these further additions and subtractions
(i) 24.105 + 11.06 + 6.5902
Any place value spaces
are treated as 0s
2 4 . 1 0 5 +
11 . 06
6 . 519 0 2
1
Decimal points lined up vertically
41 . 7552
Add matching place values together
` 24.105 + 11.06 + 6.5902 = 41.7552
Rounding decimal fractions before adding is sometimes used to quickly approximate the size of
the answer.
(ii) Round each value in question (i) to the nearest whole number before adding. ` 24.105 + 11.06 + 6.5902 . 24 + 11 + 7
. 42
Values rounded to nearest ones
Approximate value for addition
Note: Rounding values before adding/subtracting is not as accurate as rounding after adding/subtracting. (iii) 80.09 - 72.6081
8 10.10910 10-
Decimal points lined up vertically
7121. 6 0181 1
Subtract matching place values
7.4 8 1 9
Fill place value spaces
in the top number with
a ‘0’ when subtracting
` 80.09 - 72.6081 = 7.4819
Decimal Fractions
Mathletics Passport
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21
Where does it work?
Your Turn
Decimal Fractions
Adding and subtracting decimal fractions
Complete these additions and subtractions: 0 . 1 4 +
b
0 . 7 3
e
0 . 9 9 0 . 2 6
c
5 . 3 0
f
0 . 2 4 6+
1 2 . 1 9 4+
d
0 . 8 3 2
5 . 0 7 4-
g
1 . 0 6 4
5 . 2 4-
9 . 0 5 7
h
24 . 1 5 8-
0 . 8 3
13 . 694
AND SUBT
R
NG
-
b
Subtract 3.15 from 4.79
c
Add 0.936 to 0.865
d
Add 2.19, 5.6 and 0.13
e
Subtract 0.9356 from 8.6012
f
Add 10.206, 4.64 and 8.0159
+
-+
Add 8.75 to 1.24
22
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Decimal Fractions
Mathletics Passport
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..
/
....
FRACTIONS
a
..
0.
.. ./2
.
Calculate these additions and subtractions, showing all working:
G DECI
TIN
MA
AC
2
1 . 6 8 +
ADD
I
a
L
1
Where does it work?
Your Turn
Decimal Fractions
Adding and subtracting decimal fractions
3
a
Approximate these calculations by rounding each value to the nearest whole number first. (i) 5.7 + 6.2.
.
(iii) 8.3 - 1.9 .
.
+
+
.
-
.
(vi) 2.71 + 3.80 + 1.92 .
.
+
+
.
Calculate parts (v) and (vi) again, this time rounding after adding the numbers to get a more
accurate approximate value.
(ii) 2.71 + 3.80 + 1.92
(i) 8.34 + 1.61 + 0.54
4
+
(iv) 11.3 - 0.2 .
-
(v) 8.34 + 1.61 + 0.54 .
b
(ii) 0.9 + 9.4.
+
Calculate these subtractions, showing all your working: a
7.8 - 2.56
b
13.09 - 8.4621
Decimal Fractions
Mathletics Passport
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c
0.52 - 0.12532
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Where does it work?
Decimal Fractions
Multiplying with decimal fractions
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 4 # 1.2
Multiply both terms as whole numbers
4 # 12 = 4 8
1
1 decimal place in question = 1 decimal place in answer
48
` 4 # 1.2 = 4.8
2
Calculate 0.02 # 1.45
Multiply both terms as whole numbers
2 # 145 = 2 9 0
4321
290
` 0.02 # 1.45 = 0 . 0 2 9 0
4 decimal places in question = 4 decimal places in answer
How does this work when multiplying with decimal fractions? Excellent question! Very glad you asked!
Let’s do the second one again but this time change the decimal fractions to equivalent fractions first
0.02 # 1.45 = 2 # 145
100 100
Changing the decimal fractions to fractions
= 2 # 145
100 # 100
Multiply numerators and denominators together
= 290
10 000
Number of zeros in denominator = total of decimal places
in question
= 290 ' 10 000
4 3 2 1
= 0. 2 9 0
Dividing by 10 000 moves decimal point four places to the left
= 0.0290
` 4 decimal places in question = 4 decimal places in answer
Try this method for yourself on the first example above, remembering that 4 = 4 as a fraction.
1
24
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Decimal Fractions
Mathletics Passport
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Where does it work?
Your Turn
Decimal Fractions
Multiplying with decimal fractions
1
Calculate these whole number and decimal fraction multiplications, showing all you working:
a
0.8 # 2
b
d
0.62 # 4
e
5 # 1.5
3 # 0.032
c
0.14 # 6
f
1.134 # 2
a
3.8 # 0.2
b
1.09 # 0.08
c
2.7 # 2.5
d
7.1 # 1.4
e
3.21 # 2.1
f
17.2 # 9.3
Decimal Fractions
Mathletics Passport
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..
0.
... ../2
..../
MUL
TI
IONS
ACT
FR
Calculate these decimal fraction multiplications, showing all your working: ITH DEC
IMA
NG W
YI
L
PL
2
IONS
ACT
FR
MUL
TI
G WITH DECIM
YIN
AL
L
P
H
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25
Where does it work?
Decimal Fractions
Dividing with decimal fractions
Opposite to multiplying, we move the decimal point before dividing if needed. To find the quotient involving decimal fractions, the question must be changed so the divisor is a whole
number.
dividend ' divisor = quotient
• Calculate 4.28 ' 4
1. 0 7
4 g 4 . 2 28
Divisor already a whole number so no change needed
` 4.28 ' 4 = 1.07
• Calculate 0.0456 ' 0.006
0.0456 ' 0.006 = 0 0 4 5.6 ' 0 0 0 6 Move both decimal points right until divisor is a whole number
= 4 5.6 ' 6
0 7. 6
6 g 4 45 . 3 6
Quotient 2 Dividend
if divisor 1 1
` 0.0456 ' 0.006 = 7.6
Drop off any 0s at the front of the answer
Here’s another example showing how to treat remainders
Calculate 1.26 ' 0.8
Move both decimal points right until divisor is a whole number
1.26 ' 0.8 = 1.2 6 ' 0.8
= 12.6 ' 8
0 1. 5 7 5
1
= 8 g 1 2 . 4 6 60 4 0
` 1.26 ' 0.8 = 1.575
26
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TOPIC
Add 0s on the end of the dividend for each new remainder
Drop off any 0s at the front
Decimal Fractions
Mathletics Passport
© 3P Learning
Decimal Fractions
Calculate these decimal fraction and whole number divisions:
..../...
H
1
3.6 ' 4
g
b
0.63 ' 3
g
e
` 0.63 ' 3 =
2
g
16.2 ' 9
g
` 16.2 ' 9 =
` 17.5 ' 5 =
` 3.6 ' 4 =
d
c
17.5 ' 5
DIVIDING W
IT
a
÷
../20...
IMAL FRAC
TI
DEC
Dividing with decimal fractions
DIVIDING
WI
TH
NS
IMAL FRACT
DEC
IO
Your Turn
S
ON
Where does it work?
0.489 ' 5
f
g
10.976 ' 7
g
` 10.976 ' 7 =
` 0.489 ' 5 =
Calculate these decimal fraction divisions, showing all your working: a
5.2 ' 0.4
g
b
` 5.2 ' 0.4 =
d
1.58 ' 0.4
g
c
9.6 ' 0.6
g
g
` 0.56 ' 0.8 =
` 9.6 ' 0.6 =
e
0.56 ' 0.8
0.8125 ' 0.05
g
f
5.3682 ' 0.006
g
` 1.58 ' 0.4
` 0.8125 ' 0.05
` 5.3682 ' 0.006
=
=
=
Decimal Fractions
Mathletics Passport
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Where does it work?
Decimal Fractions
Recurring decimal fractions
Non-terminating decimal fractions have decimal parts that do not stop. They keep going on and on. 0.3582942049 ...
Three dots means it keeps going
If the decimal parts have a repeating number pattern, they are called recurring decimal fractions.
The pattern 21 keeps repeating in the decimal parts
5.212121 ...
Here are some examples involving recurring decimal fractions
A dot above the start and end digit of the repeating pattern is used to show it is a recurring
decimal fraction.
(i) Write these recurring decimal fractions using the dot notation
a) 10.81818...
Identify the start and end of the repeating pattern
10.81818 ...
Start End
oo
= 10.81
Dot above start and end of the repeating pattern
b) 0.2052052...
0.2052052 ...
Identify the start and end of the repeating pattern
Start End
o o
= 0.205
Dot above start and end of the repeating pattern
1047777 ...
Identify the start and end of the repeating pattern
c) 1.047777...
o o = 0.205
0.205
A bar over the whole
pattern can also be used
instead of dots
Start and End
= 1.047o
Dot above start and end of the repeating pattern
or
= 1.047r
(ii) Calculate 0.1 ' 0.6
0.1 ' 0.6 = 1 ' 6
= 1.0000 ' 6
Write 1 as a decimal fraction with a few 0s
0.1 6 6 6 ...
6g 1 .4 04 04 04 0
Repeats the same remainder when dividing
` 1 ' 6 = 0.1666 ... = 0.16o
28
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Recurring decimal fraction in simplest notation
Decimal Fractions
Mathletics Passport
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Where does it work?
Your Turn
Decimal Fractions
Recurring decimal fractions
1
What is the name of the horizontal line above the repeated numbers in a recurring decimal fraction?
Highlight the boxes that match the recurring decimal fractions 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!
4.1414 ...
0.14o
C z
0.4r
F h
41.1o
N d
0.144
W c
o o
0.141
D b
0.41o
A a
4.14
U n
o o
0.401
P t
4.1o
L f
oo
0.41
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
c
z
h
m
n
a
f
b
Calculate these divisions which have recurring decimal fractions as a result.
Write answers using dot notation.
2
a
1'3
g
b
1.6 ' 6
g
` 1.6 ' 6 =
c
g
`4'9 =
`1' 3 =
d
4'9
e
g
`5'6 =
f
2.5 ' 9
g
` 2.5 ' 9 =
0.34 ' 3
g
` 0.34 ' 3 =
Decimal Fractions
Mathletics Passport
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Your Turn
Decimal Fractions
ONS...RE
CU
CTI
RA
Recurring decimal fractions
(i) Complete the following divisions to five decimal places.
(ii) Determine whether the answer is a recurring decimal fraction or not.
3
a
d
g
b
2'3
g
c
g
1' 7
`1' 6 =
`1 ' 7 =
Recurring
decimal fraction?
Recurring
decimal fraction?
Recurring
decimal fraction?
Yes No
e
1.6 ' 7
g
Yes No
f
2.9 ' 3
g
` 0.33 ' 0.8 =
Recurring
decimal fraction?
Recurring
decimal fraction?
Recurring
decimal fraction?
h
0.68 ' 0.3
g
Yes No
0.019 ' 0.06
i
g
Yes No
g
` 2.9 ' 3 =
Yes No
..../
0.33 ' 0.8
` 1.6 ' 7 =
Yes No
0.00644 ' 0.002
g
` 0.68 ' 0.3
` 0.019 ' 0.06
` 0.00644 ' 0.002
=
=
=
H
6
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Yes No
Recurring
decimal fraction?
Yes No
Decimal Fractions
Mathletics Passport
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Recurring
decimal fraction?
..
0.
... ../2
g
`2'3 =
Recurring
decimal fraction?
30
1' 6
ONS... RE
CU
CTI
RA
DECIMAL
ING
F
RR
DECIMAL
ING
F
RR
Where does it work?
Yes No
What else can you do?
Decimal Fractions
Simple recurring decimal fractions into single fractions
Only recurring, non-terminating decimal fractions can be written in fraction form. Here is a quick way for simple decimal fractions with the pattern starting right after the decimal point.
0.111... = 0.1o = 1
9
One digit in repeating pattern, so that digit over 9
o o = 12
0.1212... = 0.12
99
Two digits in repeating pattern, so those two digits over 99
Always simplify
fractions
= 12 ' 3
99 ' 3
= 4
33
o o = 301
0.301301... = 0.301
999
Three digits in repeating pattern, so those three digits over 999
Here are some other examples including mixed numbers.
Write each of these recurring decimal fractions as mixed numbers in simplest form
(i) 3.777...
3.7777... = 3.7o
One digit in repeating pattern, so that digit over 9
= 37
9
(ii) 16.345345...
Digits in front of decimal point form the whole number part
o o
16.345345... = 16.345
Three digits in repeating pattern, so those digits over 999
= 16 345
999
Digits in front of decimal point form the whole number
= 16 345 ' 3
999 ' 3
Simplify the fraction part
= 16 115
333
Decimal Fractions
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What else can you do?
Your Turn
Decimal Fractions
Simple recurring decimal fractions into single fractions
2
a
0.4o
b
0.8r
c
0.6o
d
oo
0.11
e
oo
0.27
f
oo
0.57
Use the shortcut method to write each of these recurring decimal fractions as mixed numbers in
simplest form.
a
1.5o
b
2.7r
c
4.3r
d
3.6r
e
5.12
f
o o
0.117
g
0.162
h
5.1485
i
o o
0.4896
= 9
.
0
..=
0...
2
/
0. .
.
.
..
ONS
TI
(i) Write 0.9o as a fraction in simplest form.
32
(Ii) Does anything unusual seem to be happening with your answer? Explain.
H
6
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Decimal Fractions
Mathletics Passport
© 3P Learning
.
..../
INGLE F
RA
O S
C
NT
CURRING DE
CI
RE
RACTION
S
L F
I
MA
3
Use the shortcut method to write each of these recurring decimal fractions as a fraction in simplest form:
SIM
PL
E
1
What else can you do?
Decimal Fractions
Combining decimal fraction techniques to solve problems
All the techniques in this booklet can be used to solve problems.
These examples show different ways decimal fractions pop up in every-day life
(i) These rainfall measurements were taken during three days of rain from a small weather gauge:
36.1 mm
13.8 mm
27.6 mm
What was the total rainfall for the three days, to the nearest whole mm?
13.8 +
Add the decimal fraction values together
36.1
27.6
77.5
. 78 mm
Round to nearest whole mm
` The total rainfall over the three days was approximately 78 mm
Answer with a statement
(ii) The results for five runners in a 100 m race were plotted on the number line below.
11.2211.23
seconds
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
= (11.221 + 11.223 + 11.223 + 11.226 + 11.228) ' 5 Read off all the times
Add, then divide by 5
= 56.121 ' 5
1 1. 2 2 4 2
= 5g 5 6. 11 12 21 10
The average time ran by all the runners in the race = 11.2242 seconds
Decimal Fractions
Mathletics Passport
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Answer with a statement
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What else can you do?
Your Turn
Decimal Fractions
Combining decimal fraction 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
Use multiplication to show how much yellow paint you will need if you use all
of the 12.46 mL of blue paint you have.
e?
er m
Rememb
b
2
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.
Derek types his essays at an average speed of 93.45 words every minute. How many words does he type in
five minutes (to the nearest whole word)?
126.23
126.24
126.25
MS
LE
B
O
COM
BI
126.22
126.26
a
What was the slowest time recorded to 3 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?
34
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Decimal Fractions
Mathletics Passport
© 3P Learning
..
/20.
.
.
.
.
.
/
.
...
O
TO S LVE P
R
Nine people were trying out for a speed roller skating team around an oval flat track.
The shortest time to complete six full laps of the track for each person were recorded on
the number line below:
126.27
ES
3
ON TECHNI
QU
TI
AC
DECIMAL
FR
NG
NI
seconds
What else can you do?
Your Turn
Decimal Fractions
Combining decimal fraction techniques to solve problems
4 T he 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 =
3 bars =
2 bars =
1 bar =
0 bars =
0.81 to 1.0 signal strength
0.61 to 0.8 signal strength
0.41 to 0.6 signal strength
0.21 to 0.4 signal strength
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?
5 R
uofan 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.
Decimal Fractions
Mathletics Passport
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What else can you do?
Your Turn
Decimal Fractions
Combining decimal fraction 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.
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
8
36
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Decimal Fractions
Mathletics Passport
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What else can you do?
Your Turn
Decimal Fractions
Reflection Time
Reflecting on the work covered within this booklet:
1
2
3
What useful skills have you gained by learning about decimal fractions?
Write about one or two ways you think you could apply decimal fractions to a real life situation.
If you discovered or learnt about any shortcuts to help with decimal fractions or some other cool facts,
jot them down here:
Decimal Fractions
Mathletics Passport
© 3P Learning
H
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37
Cheat Sheet
Decimal Fractions
Here is a summary of the things you need to remember for decimal fractions
L
' 10 000 000
' 1 000 000
I M A
' 100 000
#
C
' 1000
#
1
' 10
' 100
D E
•
#
10
100
E
#
1000
L
#
10 000
W H O
' 10 000
Te
ns
o
Th f th
ou ou
s
Hu and san
nd s ds
Te red
ns s
On
es
Te
nt
h
Hu s
nd
Th red
ou ths
Te sand
n
th ths
Hu ou
nd san
M red dth
illi th s
o
o
Te nth usa
n
s
nd
M
th
illi
s
on
th
s
Place value of decimal fractions
Approximations through rounding numbers
The next digit following the place value where a number is being rounded off to is the important part.
Next digit
0
1
2
3
4
5
Closer to lower value, so round down
Leave the place value unchanged
6
7
8
9
Closer to higher value, so round up
Add 1 to the place value
Decimal fractions on the number line
The smallest place value in a decimal fraction is used to position points accurately on a number line.
6
3.0
8
3.64.0
1.240
1.2481.250
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
= decimal point moves right , = decimal point moves left 5 # 1000 = 5.0 # 1000
8 ' 100 = 8.0 ' 100
1 2 3
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= 5.0
= . 8.0
= 5000
= 0.08
Decimal Fractions
Mathletics Passport
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Cheat Sheet
Decimal Fractions
Terminating decimal fractions to fractions
The place value of the last digit on the right helps us to write it as a fraction.
Decimal fraction Fraction
Decimal fraction Fraction
3
0.3
=
Write 0.3 as a fraction:
Write 1.07 as a fraction:
1.07 = 1 7
10
100
Last digit is in tenths position
Last digit is in hundredths position
Fractions to terminating decimal fractions
Where possible, just write as an equivalent fraction with a power of 10 in the denominator first.
Eg: 3 = 3 # 2 Multiply numerator and denominator by the same value
5
5#2
Equivalent fraction with a power of 10 in the denominator
= 6
10
Three fifths = six tenths = zero point six
` = 0.6
When this method is not easy, write the numerator as a decimal fraction and then divide it by the denominator.
Adding and subtracting decimal fractions
Line up the decimal points and matching place values vertically before adding or subtracting.
Multiplying and dividing decimal fractions
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!
: 4 # 1.2 = 4.8
Eg:
: 0.02 # 1.45 = 0.0290
Dividing with decimal fractions
The question must be changed so the divisor is a whole number first.
Eg:
: 13.5 ' 0.4 = 135 ' 4
dividend ' divisor = quotient
: 89.25 ' 0.003 = 89250 ' 3
Recurring decimal fractions
These have decimal parts with a repeating number pattern.
Eg:
o o = 5.21
: 5.212121... = 5.21
Start
End
o o = 0.3698
: 0.3698698... = 0.3698
Start
End
Simple recurring decimal fractions into single fractions
Always simplify fractions
Only recurring, non-terminating decimal fractions can be written in fraction form.
This is the method for simple decimal fractions with the pattern starting right after
the decimal point.
: 0.111... = 0.1o = 1
9
o o = 12 = 4
: 0.1212... = 0.12
99
33
o o = 8 301
: 8.301301... = 8.301
999
One digit in repeating
pattern, so that digit over 9
Two digits in repeating pattern,
so those two digits over 99
Three digits in repeating pattern, so
those three digits over 999, Keep
whole number out the front.
Decimal Fractions
Mathletics Passport
© 3P Learning
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Decimal Fractions
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Notes
Decimal Fractions
Mathletics Passport
© 3P Learning
Decimal Fractions
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