CONNECT: Fractions
FRACTIONS 1 – MANIPULATING FRACTIONS
Firstly, let’s think about what a fraction is.
1.
One way to look at a fraction is as “part of a whole”.
Fractions
•
consist of a numerator and a denominator:
•
the denominator represents the kind of pieces the whole has been
divided into
•
the numerator represents the number of pieces we are interested in
Example:
Looking at the shaded section, we write this fraction as
1
.
2
The denominator
is 2 because the whole strip has been divided into 2 pieces. The numerator is
1 because we are interested in one of those pieces (the shaded piece).
2.
We can also think of fractions in a different way: the line between the
numerator and denominator is another way of writing the symbol ÷, that is,
the division sign. This means we can think of a fraction as the top number
(numerator) being divided by the bottom number (denominator), or “how
many times does the denominator go into the numerator?”.
So, for
example,
1
1
2
is the same as 1÷ 2.
Note that we can write fractions either with the numerator vertically above the
denominator, such as
denominator, like 1�2.
1
,
2
or with the numerator followed by a slash then the
Proper or improper?
Proper fractions
Think about the first definition of a fraction. We have a number of pieces out
of all the pieces that a whole has been divided into. So, if the fraction we are
1
looking at has the numerator smaller than the denominator, for example, or
2
2
, then we have fewer pieces than the whole is made up of.
3
(1 piece is shaded out of 2 so the fraction is
1
)
2
(2 pieces are shaded out of 3 so the fraction is
2
)
3
A fraction where the numerator is smaller than the denominator is called a
proper fraction.
Another way we can tell that a fraction is proper is if we try to divide the
2
numerator by the denominator and get 0. For example with , we would say
3
2
“how many times will 3 go into 2?” It will not go at all and so is a proper
3
fraction.
2
Improper fractions
If the numerator is bigger than the denominator, (for example, 4�3), then the
denominator will divide into the numerator at least once. This type of fraction
is called an improper fraction.
A fraction where the numerator is larger than the denominator is called an
improper fraction.
4
1
is the same as 1 3, because 3 divides into 4 once (1 whole) and there is 1
piece left over, which is one third piece. You can see this from the diagram.
3
1
We call 1 3 a mixed number, because it is made up of a whole number and a
fraction. We can only make mixed numbers from improper fractions, not from
proper fractions.
Here are some for you to try.
Are these fractions proper or improper? For the improper fractions, make
them into a mixed number. You might like to draw the fractions to help you.
(Answers are just below, written upside down.)
3
10
3. Proper
4
3.
4.
1
5
4
4. Improper, 2
6
1
8
2.
13
6
2. Improper, 1
3
5
Answers: 1. Proper
1.
Whole numbers (Integers)
Just for a moment, we need to consider our original definition of a fraction.
Whole numbers, that is numbers like 1, 2, 3, 4, 5, and so on, have not been
split into parts – they are still whole. So there is just one “part” of them. This
means that if we want to write a whole number as a fraction then its
denominator is always 1.
1
2
3
So, 1 is the same as , 2 is the same as , 3 is the same as , 100 is the
same as
100
1
1
1
0
1
and so on, even 0 is the same as . This is helpful when we
1
want to apply operations like +, –, x, and ÷ and fractions are involved.
EQUIVALENT FRACTIONS
1
Can you name any fractions that are the same as (equivalent to) ?
2
We can split the fraction into any number of pieces and obtain other fractions
1
that are the same as . For example:
2
The diagram tells us that
1
2
is the same as
2
.
4
2
We could also obtain by multiplying both the numerator and denominator of
4
1
1
1×2
by 2. So, =
2
2
2×2
=
4
2
4
We can also obtain
3
6
as an equivalent fraction for
1
.
2
Here is the model:
For the algorithm (procedure), you would multiply both the numerator and
1
denominator of by 3:
2
1 1×3
=
2 2×3
=
3
6
In fact, there is an infinite set of fractions that are equivalent to
fraction for that matter!
For example,
2
3
1
2
is equivalent to
is equivalent to
4 6
, ,
8
, …,
6 9 12
2 3 4
, , ,
5
, …,
4 6 8 10
50
,…
25
1
,
2
or to any
,…
50
75
3
We can also work backwards, that is we could start from , say, and divide
6
3÷3
1
both numerator and denominator by 3 to get
= . For the model, we
6÷3
2
1
3
combine the parts into the :
6
2
5
Going backwards (dividing the numerator and denominator by the same
number) is the way to write a fraction in its simplest form, that is, you obtain
the lowest possible numerator and denominator for that particular fraction.
This is called simplifying a fraction.
Calculating Equivalent Fractions
1. Sometimes you are given a starting numerator or denominator and you can
work out the other value depending on what has been done to the original
fraction.
Example: If we were given that
2
2
3
=

12
, we could complete the blank. Look at
the complete fraction ( ). Compare the denominators of both fractions. 3 has
3
been multiplied by 4 to make 12, so we need to multiply the numerator (2) by
4 as well to make the equivalent fraction.
So
2
3
=
=
2×4
3×4
8
12
The model is over the page:
6
2. When you “simplify” a fraction, that is, express it in its “lowest terms”
(working backwards), you are again making equivalent fractions.
Example: Simplify
Using the model:
6
10
From the diagrams, we can see that
6
10
3
is equivalent to .
5
Using the algorithm, we can see that we can divide both 6 and 10 by 2 (in fact
2 is the only number we can divide both 6 and 10 by!), so we would write
6
10
7
=
=
6÷2
10÷2
3
5
This time, simplify
8
20
Now, we could do this in two steps, or we can do it in one. Both will work, but
obviously using only one step is quicker (and more efficient), however using
two steps is just as clever!
For one step, we look for the highest number that we can divide both 8 and 20
by, and find that that number is 4. So our solution looks like this:
8
20
=
=
8÷4
20÷4
2
5
4
We could also have divided both 8 and 20 by 2, (and got 10) then divided both
4 and 10 by 2 again and we would have found the same result.
Using the model:
8
Here are some for you to try. You will need to multiply sometimes and divide
sometimes; the trick is for you to work out which is which. You can check with
the solutions at the end of this resource.
1. Complete the blanks in each of the following:
(a)
5
8
=

16
(b)
9
12
=
3

(c)
7
10
=

100
(d)
35
100
=
7

2. Simplify the following fractions by putting them into their lowest form:
(a)
9
90
100
(b)
24
32
(c)
25
75
(d)
12
36
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.
Worked solutions (page 9).
(There are other methods to achieve the correct answers.)
1.
(a)
5
8
=
8
=
8
=
5
5

9
12
=
8×2
12
=
16
12
=
(b)
16
5 ×2
9
10
9
3

(c)
9 ÷3
7
10
=
10
=
7
12 ÷3
3
7
4
10

100
(d)
7 ×10
70
100
100
=
100 ÷5
100
=
20
35
100
7
=
35
10 ×10
=
35

35 ÷5
7
2.
(a)
90
100
90
100
10
=
=
90 ÷10
100 ÷10
9
10
(b)
24
32
24
32
=
=
24 ÷8
32 ÷8
3
4
(c)
25
75
25
75
=
=
25 ÷25
75 ÷25
1
3
(d)
12
36
12
36
=
=
12 ÷12
36 ÷12
1
3