Decimals MTH 3 – 03a, MTH 3 – 03b, MTH 3 – 01a, MNU 3 - 10a , MNU 3 - 08a
Place Value
It is vital to understand place value in decimal fractions. The decimal point is always between
the units (ones) and the tenths column. We have:
Thousands Hundreds
2
5
Tens
7
Units
1
Tenths
8
Hundredths Thousandths
9
3
In the number 2571.893, the 7 is worth 7 tens while the 8 is worth 8 tenths.
When adding or subtracting decimals it is important that we line up the columns so that the
place value is correct (i.e. we line up the decimal points)
So, 2.7 + 83.62 becomes:
2.7
+ 83.62
but it is better to have the same amount of decimal places in each number so we write:
2.70
+ 83.62
This is particularly important in subtraction where 5 – 2.71 becomes:
5.00
- 2.71
We also may have applications of decimal addition and subtraction in contexts such as time or
money. For instance:
John ran a race in a time of 54.6 seconds.
6
Terry ran
second faster. What was Terry’s time?
100
6
sec = 0.06 sec. He ran faster to so took less time.
100
i.e. subtract 0.06 sec.
54.60
- 0.06
54.54 secs
Examples:
1.
State the value of the 5 in each of the following:
a) 24.56
2.
b) 509.12
c) 0.0567
d) 5312.6
e) 2.105
c) 3.1 – 1.87
d) 14 – 2.613
Set out and evaluate:
a) 6.84 + 2.39
b) 6.8 + 5.47
e) 5.8 add
3.
a)
b)
2
10
f) 4.1 -
6
10
g) 5.8 +
18
100
h) 3.14 -
9
1000
Gary ran a race in 38.2 secs.
7
If Bob ran
secs faster, what was his time?
100
8
If Neil ran
secs slower, what was his time?
10
Ordering Decimals
So that we can compare the value of different decimals, it is important that we have all of the
numbers in the same format e.g. all with 3 decimal places.
For example, place these in order, starting with the smallest:
0.03
0.3
0.301
0.013
0.31
0.003
0.01
0.310
0.003
0.010
0.300
0.301
0.310
Writing each with 3 decimal places gives:
0.030
0.300
0.301
0.013
Now we can compare each value so in order we have:
0.003
0.010
0.013
0.030
Examples:
Write each of these lists in order of size, starting with the smallest:
a)
0.5
0.25
0.02
0.005
0.52
0.002
b)
1.16
1.006
1.6
1.061
1.1
1.06
c)
5.321
5.02
5.031
5.13
5.2
5.32
Multiplying and Dividing by 10, 100, 100
When we multiply a number by 10, the value of each digit increases by a factor of 10. The
result of this is that the digits move left one place as shown below:
H
T
U
1
10
1
100
6
5
5
7
7
1
1
6
So 65.71 x 10 = 657.1
1
1000
Again, 0.24 x 10 becomes 2.4 as below.
H
T
U
1
10
1
100
0
2
2
4
4
1
1000
Equally, multiplying by 100 moves digits 2 places to the left:
H
5
T
U
8
5
0
1
10
1
100
1
1000
8
So 5.8 x 100 = 580
Here there is a need to fill in the units column with a zero to maintain the other digits in the
correct place.
And also multiplying by 1000 will move the digits 3 places to the left, hence:
H
T
U
4
0
0
1
10
1
100
0
4
1
1000
So 0.04 x 1000 = 40 (again there is the need to fill in the units column with a zero in this case).
Dividing by 10, 100 and 1000 works in a similar manner, with the numbers this time getting
smaller and so moving to the right.
H
T
U
1
10
1
100
2
0
8
2
8
1
1000
So 2.8 ÷ 10 = 0.28
Examples:
1.
2.
Evaluate each of the following multiplications:
a) 0.32 x 10
b) 2.14 x 100
c) 3.1 x 1000
d) 0.04 x 10
e) 0.092 x 100
f) 0.0862 x 1000
g) 125.1 x 100
h) 27 x 10
Evaluate each of the following divisions:
a) 32.6 ÷ 10
b) 128 ÷ 100
c) 56.31 ÷ 100
d) 261.5 ÷ 1000
e) 94.56 ÷ 1000
f) 0.02 ÷ 10
g) 125400 ÷ 100
h) 3.16 ÷ 1000
Multiplying and Dividing Decimals
This works in a similar manner to multiplication and division of whole numbers. It is important to
keep the digits lined up with the decimal point in place, and to remember to include the decimal
point in the answer.
e.g.1 A roll of wallpaper is 2.4 metres long. Find the length of 7 rolls.
2.4
x 7
16 . 8 metres
e.g. 2 One fifth of the weight in a packet of crisps is fat content. If I eat a large portion of crisps
weighing 43.25 grams, how many grams of fat have I eaten?
5
8.65
43.25
Multiplying by Multiples of 10, 100, 1000
To multiply by a multiple of 10 we split the calculation up into two portions, just as we did with
whole numbers.
e.g.1
6.12 x 200 becomes 6.12 x 2 x 100
= 12.24 x 100
= 1224
Remember we can rearrange the order of our multiplication – always “split up” the multiple of
10.
e.g. 2
3000 x 5.07 becomes 5.07 x 3 x 1000
= 15.21 x 1000
= 15210
Examples:
1.
Evaluate:
a) 2.31 x 200
b) 1.4 x 50
c) 7.04 x 6000
d) 400 x 0.28
e) 200 x 78.1
Dividing by Multiples of 10, 100, 1000
As with multiplying, we split up the number we are dividing by.
e.g.1
24.8 ÷ 200 becomes 24.8 ÷ 2 ÷ 100
= 12.4 ÷ 100
= 0.124
e.g. 2
1.605 ÷ 50 becomes 1.605 ÷ 5 ÷ 10
= 0.321 ÷ 10
= 0.0321
Examples:
1.
Evaluate:
a) 50.2 ÷ 20
b) 31.2 ÷ 300
c) 8.545 ÷ 50
d) 264 ÷ 3000
e) 140.4 ÷ 60
Rounding and Estimating
To round a number to 1 decimal place the answer must have exactly one figure to the right of
the decimal point. To decide if the number is closer to the value above or below, we need to
look at the value of the hundredths column (the second decimal place).
e.g. Round to 1 d.p:
5.872
5.9
Look at this number – 7 is
more than halfway so we
round up.
5.849
Look at this number – 4 is
less than halfway so we
round to the value below.
5.8
Note: if the next number is a 5 we always round up.
The same process applies for rounding to 2 or 3 decimal places as shown:
Round 3.5671 to 2 d.p.
3.5671
3.57
Round 3.1998 to 3 dp
3.1998
3.200
Examples:
1.
Round these values to 1 decimal place:
a) 6.13
2.
c) 6.762
d) 21.209
e) 3.986
d) 5.6957
e) 2.0031
Round these values to 2 decimal places:
a) 2.301
3.
b) 41.25
b) 5.6192
c) 41.225
Round these values to 3 decimal places:
a) 3.1246
b) 3.1691
c) 4.0565
d) 29.9999 e) 1.333333
Foreign Currency
To exchange into an amount of money into a foreign currency we multiply by the exchange
rate.
To exchange into an amount of money out of a foreign currency we divide by the exchange
rate.
e.g. 1
The exchange rate is £1 = 4.3 Polish zloty.
How many zlotys will I get in exchange for £200.
Number of zlotys
e.g. 2
= 4.3 x 200
= 4.3 x 2 x 100
= 8.6x 100
= 860 zloty
The exchange rate is £1 = 300 Hungarian Forints.
How many pounds will I get if I exchange 33600 Forints?
Number of pounds = 33600 ÷ 300
= 33600 ÷ 3 ÷ 100
= 11200 ÷ 100
= £112