DECIMALS
Place values
When working with decimals we need to insert a decimal point after the unit to separate
the whole numbers from the fractions and extend the place values to include tenths,
hundredths, then thousandths etc.
Hundreds
Tens
Units
Point
Tenths
Hundredths Thousandths
100
10
1

1
10
1
100
1
1000
4
7
2

6
5
3
400
70
2

6
10
5
100
3
1000
= 472.653
Converting decimals to fractions
To convert a decimal fraction such as 0.73 to a fraction you divide it by 1 with as many zeros
after the decimal point as you have numbers after the decimal point.
 0.73 =
 0.812 =
 0.4 =
 0.015 =
73
100
(2 digits, 2 zeroes)
812
1000
(3 digits, 3 zeroes)
4
10
(1 digit, 1 zero)
015
15

1000
1000
 0.030 = 0.03 =
(3 digits, 3 zeroes)
03
3

100
100
(2 digits, 2 zeroes)
(the zero between the decimal point and the 1 is included as a number)
1
Updated Feb 2013
Converting fractions to decimals
Count the number of zeroes on the bottom (the denominator).
There must be the same number of digits after the decimal point, inserting zeroes if
necessary.




= 0.7
(1 zero, 1 number after the decimal point)
= 0.024
= 0.09
(3 zeroes, 3 numbers after the decimal point)
(2 zeroes, 2 numbers after the decimal point)
= 6.0003 (4 zeroes, 4 numbers after the decimal point)
The relative sizes of decimals
The sign  means “is greater than” i.e. 6  3
The sign  means “is less than”
i.e. 2  3
When comparing decimals, convert them to fractions with the same denominator, then
compare the numbers on top.



0.72 =
72
100
0.71 =
71
100
3.4
=
340
100
3.04 =
304
100
5.3
=
530
100
5.33 =
533
100
72
71

100
100
340
304

100
100
530
533

100
100
2
Updated Feb 2013
Addition and subtraction of decimals
Addition and subtraction are performed in the same way as for whole numbers, with extra
care being taken to ensure that the decimal points and the place values are lined up.
 62.1 + 3.04
(fill up spaces with zeros)
62.10
3.04
65.14
 423.015 - 10.3
(fill up spaces with zeros)
423.015
10.300
412.715
 13.24 + 2.153 + 21 + 3.6 + 0.0005
13.2400
2.15 3 0
21.0000
3.6000
0.0005
39.9935
(fill up spaces with zeros)
Multiplication
To multiply decimals you go through the following steps
1) Count the total number of digits after the decimal points.
2) Ignore the decimal points and multiply as for whole numbers.
3) Count the number of decimal places from the right and insert the decimal point.
 21.2  0.5 ( 2 decimal places)
Count 2 decimal places from the right,
giving 10.60
 4.53  2.14 ( 4 decimal places)
Count 4 decimal places from the right,
Giving 9.6942
3
Updated Feb 2013
212
5
1060
453
214
1812
453
906
96942
 0.09  0.004 ( 5 decimal places)
4  9 = 36
Counting from the right, we need to insert zeros so that we have 5 places after the
decimal point.
giving 0.00036
Multiplication by powers of 10
 When multiplying by 10 move the decimal point one place to the right.
 When multiplying by 100 move the decimal point two places to the right.
 When multiplying by 1000 move the decimal point three places to the right.
 The general rule is that you move the decimal point one place to the right for every zero
in the multiplier, adding zeros if necessary.
 6.3  10 = 63
 4.2  1000 = 4200 ( adding 2 zeros)
 0.015  100 = 1.5
 If petrol costs $1.075 per litre, what would be the cost of:
10 L, 100 L and 1000 L in?
10 L

10  $1.075 = $10.75
100 L

100  $1.075 = $107.50
1000 L

1000  $1.075 = $1075
4
Updated Feb 2013
Dividing a decimal by a small whole number
Divide as with whole numbers, but insert a decimal point in the answer directly above the
decimal point in the number being divided into.
 43.65  5
0.141  3
8.73
5
0.047
43.65
3
40
0.141
12
36
21
35
21
15
15
 3.04 ÷6
0.5066…
6 3.04000
3.04 ÷6 = 0.5067
30
040
In this case we added zeros to
continue the division, which would
keep giving the same answer of 6, so
we rounded it off.
36
40
36
5
Updated Feb 2013
Dividing decimals by powers of 10
 When dividing by 10 move the decimal point one place to the left.
 When dividing by 100 move the decimal point two places to the left.
 When dividing by 1000 move the decimal point three places to the left.
 The general rule is that you move the decimal point one place to the left for every
zero in the divisor, adding zeros to the right of the decimal point if necessary.
 6.3  10 = 0.63
 4.2  1000 = 0.0042 (inserting 2 zeros after the decimal point)
 0.015  100 = 0.00015 (inserting 2 zeros after the decimal point)
Dividing a decimal by another decimal
When dividing a decimal by another decimal you need to move the decimal point the
same number of places, in the same direction, in both numbers, till the divisor becomes
a whole number. Then you divide as before.
 46.2  0.6 
46.2
0.6

462
6
0.45
0.9

4.5
9
77
6
462
42
42
42
 0.45  0.9
0.5
9 4.5
4.5
2

 2.35 ÷ 0.005 
2.35
0.005

2350
5
470
5 2350
20
35
35
0
Decimal Exercises
1.
Convert the following decimals to fractions:
a) 0.04
b) 0.75
c) 0.93
d) 0.50
e) 0.09
f) 0.45
g) 3.14
h) 7.02
i) 7.20
j) 0.006
k) 0.010
l) 0.023
2.
a)
Convert the following fractions to decimals:
3
100
d) 3
g)
13
100
123
1000
j) 11
1
10
b)
11
100
c)
45
1000
e)
4
10
f)
7
1000
h) 9
k)
3
1000
1
100
i) 5
l)
77
1000
75
100
3
3.
Compare these decimals from right to left using the signs >, <, =
a) 0.42 ____ 0.452
b) 0.82 ____0.723
c) 0.361 ____ 0.316
d) 0.110 ____ 0.101
e) 0.22 ____0.219
f) 0.242 ____0.24
g) 0.4 ____0.400
h) 0.876 ____0.87
i) 0.105 ____0.11
j) 0.91 ____ 0.929
k) 0.75 ___0.750
l) 0.4 ___ 0.399
4. Library books are numbered according to the Dewey decimal system and are put on
shelves in numerical order. Sort the following books into numerical order.
5.
A 843.1
B 729.32
C 843.10
D 620.9
E 540.82
F 729.4
G 843.101
H 620.101
I 729.04
Calculate the following:
a) 194.25 + 23.6 + 4.501
b) 2.005 + 71.26 + 0.5
c) 32.060 - 4.999
d) 4.68 + 5.27 + 3.191
e) 9.5 - 2.020
f) 11.111 - 9.2
g) 0.7 + 0.009 + 0.222 + 0.08
h) 0.1 - 0.08
i) 99.01 - 89.99
j) 7.09 + 4.36 - 2.9
k) 6.003 + 2.008 - 1.9
l) 1.2 + 2.03 + 3.004 - 4.0005
4
6.
Calculate the following:
a) 3.2  0.4
b) 4.65  1.1
c) 0.007  0.02
d) 4.005  0.2
e) 1.3  1.20
f) 9.01  0.02
g) 46.2  3
h) 3.333  1.1
i) 5.25  0.005
j) 8.08  0.0004
k) 0.0092  2
l) 0.003645  0.03
7.
Calculate the following :
a) 4.3  1000
b) 0.25  100
c) 2.123  10
c) 8.805  10 000
d) 0.23  1000
e) 1.066  100
f) 5.2  100
g) 0.06  10
h) 0.046  1000
i) 5267.45  1000
j) 1.050  100
k) 9.9632  10 000
8.
If petrol costs 107.9 cents per litre, how much would it cost to fill a car with 30
litres of petrol?
9.
How much would 4.6 metres of fabric cost at $16.50 per metre?
10.
If decking timber costs 80 cents per metre, how much would it cost for a deck that
needs 35 slats each 1.5 metres long?
11.
If picture rail costs $2.25 per metre, how much would it cost to erect picture rail in
a room that is 3.25 m long and 2.60 metres wide?
12.
If a map is drawn that uses 1 cm to represent a distance of 15.2 km, what distance
would 3.8 cm represent?
13.
If a lamp uses 0.2 kilowatt of electricity per hour, how many kilowatts would it use
for 5.5 hours?
5
14.
A carpenter does a small renovation job where the materials cost $192.50 and the
labour costs are 20.5 hours at $25 per hour. How much is the total cost of the job?
15.
A doctor orders 0.15 grams of a drug for a patient. The drug is only available in
0.05 gram tablets. How many tablets should the patient receive?
16.
If John works from 8am to 4.30pm, with a 45 minute lunch break, how much
would he be paid for a 5 day week at $22.75 per hour?
17.
Employees in a certain company who use their cars for travel can claim 35 cents
per km travel allowance. If Allan's odometer reading at the beginning of the day
reads 18 642.8 km and at the end of the day reads 18 726.9 km, how much should
he claim?
18.
If I spent $28.50 filling my car with petrol, costing 106.2 cents per litre, how many
litres did I put in?
19.
A dollar coin is about 0.25 cm thick. How high would a stack of 100 coins be? How
many coins would make a pile 1 m high?
20.
Fuel economies of cars are quoted in litres/100 km. If a car averages 9.2 L/ 100
km, how much fuel would it use to travel 1000 km? How far could the car travel on
1 L of petrol?
Decimal exercises solutions
1.
a)
4
100
b)
75
100
c)
93
100
d)
5
10
e)
9
100
f)
45
100
g) 3
j)
2.
3.
6
14
100
6
1000
h) 7
k)
2
100
1
100
i) 7
l)
2
10
23
1000
a) 0.03
b) 0.11
c) 0.045
d) 3.13
e) 0.4
f) 0.007
g) 0.123
h) 9.003
i) 5.007
j) 11.1
k) 0.01
l) 0.75
a) 0.42  0.452
b) 0.82  0.723
c) 0.361  0.316
d) 0.110  0.101
e) 0.22  0.219
f) 0.242 0.24
h) 0.876 0.87
k) 0.75 = 0.750
g) 0.4 = 0.400
j) 0.91  0.929
4.
E
5.
6.
7.
H
D
I B
F
A&C
i) 0.105  0.11
l) 0.4  0.399
G
a) 222.351
b) 73.765
c) 27.061
d) 13.141
e) 7.48
f) 1.911
g) 1.011
h) 0.02
i) 9.02
j) 8.55
k) 6.111
l) 2.2335
a) 1.28
b) 5.115
c) 0.00014
d) 0.801
e) 1.56
f) 0.1802
g) 15.4
h) 3.03
i) 1050
j) 20 200
k) 0.0046
l) 0.1215
a) 4300
b) 25
c) 21.23
d) 88 050
e) 230
f) 106.6
g) 0.052
h) 0.006
i) 0.000046
j) 5.26745
k) 0.01050
l) 0.00099632
8.
$32.37
18.
26.8 L
9.
$75.90
19.
25 cm, 400 coins
10.
$42
20.
92 L, 10.9 km
11.
$26.33
12.
57.76 km
13.
1.1 kilowatts
14.
$705
15.
3
16.
$881.56
17.
$29.44
7