CONNECT: Fractions
FRACTIONS 4
FRACTIONS, DECIMALS, PERCENTAGES – how do they relate?
If you are unsure about what a fraction is, please refer to CONNECT:
Fractions 1 – Manipulating fractions.
To relate Fractions and Decimals, let’s first think about our Decimal System of
Numbers (or Place Value). This system gives us a very efficient way to write
any number we please, just by using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 in
columns, where each column represents a different size.
Here are the headings for the columns, which you are familiar with:
millions hundred ten
thousands hundreds tens units
thousands thousands
Unlike English language, we read numbers from right to left. Each time we
move from a column on the right to one directly on its left, we have multiplied
by 10 (so that for example, 3 x 10 = 30, or 58 x 10 = 580) – we attach a 0 on
the end because it tells us we have moved across a column. In the same
way, if we divide by 10, we are moving across a column from left to right – so,
for example, 4 930 ÷ 10 = 493 – we’ve picked up our number and moved it
across a column to the right.
← x 10
→ ÷10
millions hundred ten
thousands hundreds tens units
thousands thousands
All the columns listed so far refer to whole numbers. But what happens if we
have a whole number where the last digit is not 0 and we want to divide it by
10? How can we pick the number up and move across a column?
We need to be able to show that we are no longer dealing with a whole
number and will only have a fraction as our answer. This is why we have a
1
decimal point. The decimal point separates whole numbers from parts of a
whole, and parts if a whole are fractions. So:
millions hundred ten
thousands hundreds tens units .
thousands thousands
Let’s begin by dividing 1 by 10. We can write this as 1 ÷ 10. You probably
1
realise we can also write this as 1�10 or . Using the fact that when we
10
divide by 10 we move our number across a column from left to right, we would
have:
→ ÷10
units .
?
1
.
1
So what do we call our new column? 0.1, (which is the same as .1), must be
the same as
1
10
and so we call the new column tenths. (The “th” on the end
tells us that we are not dealing with a whole number any more, but part of a
number, that is, a fraction.)
We now know that 0.1 (meaning 0 units and 1 tenth) is exactly the same as
and in maths we would write 0.1 =
1
.
10
1
10
Our columns continue on the right. For example, if we divide each tenth into
1
10 bits (that is if we calculate 10 ÷ 10) we would get 100 bits and so we call
the next column across hundredths.
1
1
So, 10 ÷ 10 = 100, or 0.1 ÷ 10 = 0.01.
2
This is what we have if we put it all together in our decimal system:
millions hundred
ten
thousands hundreds
thousands thousands
tens
units . tenths hundredths thousandths
𝟏
𝟏𝟎
1
So 10 = 0.1,
Also,
3
= 0.3,
10
1
100
3
100
= 0.01,
23
1
1000
𝟏
𝟏𝟎𝟎
0
.
1
0
.
0
1
0
.
0
0
= 0.001.
= 0.03, 100 = 0.23,
23
1000
= 0.023,
594
1000
= 0.594,
1234
1000
= 1.234.
Remember, we can only put one digit in each column of our decimal system.
The number of 0s in the denominator tells us which column the last digit of the
numerator goes into.
Here are some for you to try. Match the decimals and fractions in the
following table (one is done for you). You can check with the table at the end
of this resource.
Fraction
Decimal
41
100
0.41
41
4.1
12
1000
0.12
1000
41
10
12
100
3
0.041
0.012
𝟏
𝟏𝟎𝟎𝟎
1
We have just looked at the relationship between fractions and decimals, as
long as the denominator of the fraction is 10, 100, 1000. But what about
fractions such as ½, ⅔, ¾, and so on? In these cases where the
denominator is not a power of 10, we need to consider a different method to
work out their decimal equivalents.
For this, we go back to the definition of a fraction, where the top number is
divided by the bottom number. For ½, we need to calculate 1 ÷ 2.
Set up a short division:
2)1
We see that 2 will not divide into 1, but we know that 1 is the same as 1.0, so
our division becomes
2)1.0
There are no 2s in 1, so we put a 0 above the 1, followed by a decimal point
above the decimal point in the 1.0.
0.
2)1.0
Now ask how many 2s in 10? There are 5, so we put 5 above the line.
0.5
2)1.0
There is no remainder. So, ½ = 0.5
For ⅔ we calculate 2 ÷ 3,
3)2
3 will not divide into 2 so put a 0 above the 2, and attach a decimal point and
a 0 after the 2.
0.
3)2. 0
4
Now ask how many 3s in 20? There are 6, so:
but there is 2 left over, so
0.6
3)2. 0
0.6
3)2. 02 0
Again ask how many 3s in 20? 6 again.
0.66
3)2. 02 0
There is a remainder of 2 again. We can keep attaching 0 at the end and
obtain 0.666666… for as long as we wish to keep writing 6s!
So, ⅔ is the same as 0.666… (the “…” means “continues the same”). You
can also write this as 0. 6̇ .
This method works for any fraction but for a denominator of 10 or 100 or 1000
(and so on) it is always easier to use the place value method from above.
Here are some for you to try. One has been done for you. You can check
with the solution at the end.
Fraction
Decimal
4
5
1
6
0.1666… (or 0.16̇ )
2
9
0.41666… (or 0.416̇ )
3
8
5
12
5
0.8
0.222… (or 0.2̇ )
0.375
We have found ways in which to convert fractions into their decimal form.
Now, to work back the other way. (For people who do not like division, this is
much easier!)
Let’s say we want to know what the decimal 0.59 is as a fraction. From our
decimal system (or place value) above, we know that this number consists of
5 tenths and 9 hundredths and we can write it as
fraction any further so 0.59 =
59
100
.
59
100
. We cannot simplify this
Now, what about 0.155? We know that this is the same as
155 and 1000 can be divided by 5, so we get
155÷5
1000÷5
=
31
200
155
1000
and that both
Here are some for you to try. Change these decimals to fractions and write
your answer as the simplest fraction possible. (You can check your answers
at the end.)
(1) 0.7
(2) 0.89
(3) 0.123
(4) 0.2
(7) 0.5
(8) 0.25
(9) 0.125
(10) 1.4
6
(5) 0.1
(6) 0.10
PERCENTAGES
Percentages relate to fractions and decimals because per cent means out of
80
100. So, for example, 80% means 80 out of 100, or 100.
We can simplify this both as a fraction and as a decimal.
Firstly as a fraction:
80
80÷20
=
100
100÷20
(Try to divide by the highest number that will go into both numerator and
denominator here.)
So
80
100
=
4
As a decimal:
5
80
By place value, we know 100 is the same as 0.80, which is the same as 0.8.
So the relationship is 80% =
4
5
= 0.8. Notice that all of these indicate the same
relatively high proportion. If you got 80%, or 4 out of 5, or 0.8 for an
assessment task, you should be pretty happy!
We have changed a percentage into a fraction and a decimal, now let’s go the
other way.
1
Example: Change 2 into a percentage. We’ll do this as you probably already
know the answer. But we can then use the same method for all other cases.
To change fractions – or decimals – into percentages, we just multiply by 100
and call the answer %.
So,
1
2
=
1
2
1
× 100(%)
=2×
=
100
2
100
(%)
(%)
= 50%
7
1
3
Let’s try a fraction we do not know: change 8 into a percentage.
3
3
= × 100(%)
8
8
3
100
= 8 × 1 (%)
300
= 8
Here we can go two ways, either divide 8 into 300 with a short division, or
divide both numerator and denominator by 4.
3 7. 5
6
or
4
8)30 0. 0
300
8
=
=
300÷4
8÷4
75
2
1
= 372
Either way, we end up with 37.5%. This seems right, especially because
37.5% is halfway between 25% and 50%.
Now let’s try a decimal – which is actually easier than a fraction!
Change 0.5 to a percentage: just times by 100, so 0.5 x 100(%) gives 50%.
What about changing 1.5 to a percentage? 1.5 = 1.5 x 100% = 150%. Does
this make sense? It does if we remember that 100% is the same as 1 whole!
Here are some for you to try. Fill in the blanks in the table.
Fraction
Decimal
2
5
0.23
Percent
75%
30%
0.10
2
3
8
If you need help with any of the Maths covered in this resource (or any other Maths
topics), you can make an appointment with Learning Development through
Reception: phone (02) 4221 3977, or Level 3 (top floor), Building 11, or through your
campus.
Solutions
Solution to table exercise (from page 3):
Fraction
Decimal
41
0.41
100
41
4.1
1000
12
0.12
1000
41
10
0.041
12
0.012
100
Solution to table exercise (from page 5):
Fraction
Decimal
4
0.1666… (or 0.16̇ )
5
1
6
3
8
2
9
5
12
9
0.8
0.222… (or 0.2̇ )
0.41666… (or 0.416̇ )
0.375
Solutions to questions from page 6
7
(1) 0.7 = 10 (cannot be simplified further)
89
(2) 0.89 = 100 (cannot be simplified further)
123
(3) 0.123 = 1000 (cannot be simplified further)
(4) 0.2 =
2 ÷2
10 ÷2
1
1
=5
(5) 0.1 = 10 (cannot be simplified further)
(6) 0.10 =
10 ÷10
=
100 ÷ 10
1
(Note that 0.1 and 0.10 are the same, and they are also
the same as 0.100000, etc!)
5 ÷5
10
1
(7) 0.5 = 10 ÷5 = 2
25 ÷25
(8) 0.25 = 100 ÷25 =
125 ÷5
(9) 0.125 = 1000 ÷5 =
1
4
25 ÷25
1
=8
200 ÷ 25
(You could also divide numerator and denominator both by 125 at the first step,
or you could divide both numerator and denominator by 5 and by 5 again instead
of by 25 at the second step and you would obtain the same result.)
14 ÷2
7
7
2
(10) 1.4 = 10 ÷2 = 5. (You can also make 5 into 1 5.)
10
Solutions to Table (page 8)
11
Fraction
Decimal
Percent
2
5
23
100
75
3
=
100 4
30
3
=
100 10
1
10
2
3
0.4
40%
0.23
23%
0.75
75%
0.3
30%
0.10
10%
0.666…
2
66.666…% (or 663%)