CONNECT: Fractions
FRACTIONS 3 – OPERATIONS WITH FRACTIONS:
+ and –
If you have not worked with Fractions for a while, you might like to begin by
looking at CONNECT: FRACTIONS 1 – MANIPULATING FRACTIONS.
For information on multiplying and dividing fractions, please refer to
CONNECT: FRACTIONS 2 – OPERATIONS WITH FRACTIONS: x and ÷.
ADDITION
Imagine you had a chocolate bar and had eaten
1
4
1
2
of it. Then you ate another
of it. You wonder what fraction of the chocolate bar you’ve eaten.
1
1
2
4
We would need to use equivalent fractions here and replace the
1
4
In other words, we replace
1
2
4
4
with an equivalent fraction which we can obtain
by multiplying both numerator and denominator of
this case, 2.
So our algorithm would look like this:
1
2
2
with :
1
4
1
1
1
2
by the same number, in
1 1
1 ×2 1
+ =
+
2 4
2 ×2 4
=
2
=
4
+
3
1
4
4
When adding fractions, we need to make sure that we are combining the
same kinds of pieces. With the model, we can show this by replacing at least
one kind of piece with another kind. With the algorithm, we replace at least
one of the fractions with an equivalent fraction, depending on denominators.
The denominator tells us what kind of pieces we are dealing with. We cannot
add 2 fractions whose denominators are not the same and so we have to
make them the same by choosing appropriate equivalent fractions.
To work out the equivalent fraction(s) involved, using a model, we need to
think about the original denominators and try to picture if we could cut them
up in any way to obtain the other denominator. This is reasonably
straightforward with the example above (where the 2 quarter pieces fit neatly
onto the 1 half piece - we are used to halves and quarters on the face of a
clock, for example) but with other fractions perhaps it might not be so easy.
Let’s try to add
3
4
1
+ .
3
plus
It doesn’t look like we can cut either fraction up to fit neatly onto the other, so
we need to work slightly differently here. How about we cut the quarters (or
fourths) into 3 pieces and each third into 4 pieces:
2
plus
Now we have exactly the same size pieces and can combine them to obtain:
So our result is
13
, which is the same as 1
12
1
.
12
Using the algorithm, we look at the denominators of each fraction. Can either
denominator divide exactly into the other one? Not in this case. So we need
to find another number that both 4 and 3 divide exactly into. The lowest
number (always the most efficient to use) in this case is 12 because both 4
and 3 divide exactly into 12.
So we replace each of our fractions with their equivalent fractions where the
denominators are 12. We can see that we need to multiply 4 by 3 to get 12
and 3 by 4 to get 12, so we do the same to their numerators as well.
3 1  
+ =
+
4 3 12 12
=
=
3× 3
4 ×3
+
12
12
9
13
= 12
=1
3
1
12
+
4
1×4
3 ×4
We can also model this in a slightly different way:
3
We represent 4 like this:
1
and 3 like this:
3
4
can be split horizontally, so:
and
1
3
can be split vertically, so:
We still have the correct shape and size pieces so we can combine them
again to get (over):
4
13
1
1
Again, we have 12 or 1 whole and 12, that is, 1 12.
Let’s try
3
5
+
2
3
plus
We need to cut the fifths into 3 pieces and the thirds into 5 pieces:
plus
So now we have
5
9
15
and
10
.
15
Putting all together, we would obtain
19
15
which is the same as 1
Using the algorithm:

3 2 
+ =
+
5 3 15 15
=
=
=
3 ×3
5×3
+
15
15
9
19
+
10
4
.
15
2×5
3 ×5
15
4
= 1 15
Using the second method of modelling:
plus
6
gives:
plus
Putting as many of the shaded pieces of the second fraction as possible into
the unshaded pieces of the first fraction gives:
So we have
19
15
4
altogether, which is the same as 1 15.
Did you notice that we never add denominators?
SUBTRACTION
The procedure for subtraction is exactly the same: we cannot subtract 2
fractions unless their denominators are the same. And of course, rather than
adding the numerators in the final step, we subtract them but we do NOT
subtract denominators!
7
Example:
3
4
−
1
3
3 1 3×3 1×4
− =
−
4 3 4×3 3×4
=
9
12
5
−
= 12
4
12
Model:
minus
becomes
5
We have 12 left.
Over the page are some for you to try. You can check your answers with the
worked solutions at the end.
8
1.
1
5.
5
2
9
+
−
3
2.
1
2
6.
7
4
9
4
8
+
−
5
8
1
2
3.
2
7.
5
3
6
+
−
5
6
4
5
4.
3
8.
9
7
+
10
−
4
7
2
3
Adding and subtracting Mixed Numbers
Method 1:
One method used to add or subtract mixed numbers is to change them to
improper fractions before adding or subtracting.
2
3
For example, if we want to add 1 3 + 2 4, we could do the following:
2
3
5 11
1 +2 = +
3
4
3
4
=
=
=
5 ×4
3 ×4
20
12
53
+
+
12
=4
5
12
Model (over the page):
9
11 ×3
4 ×3
33
12
plus
is the same as:
10
plus
Cut the thirds into 4 pieces and the fourths (quarters) into 3 pieces, so (over):
11
plus
2
We can see that 1 3 is the same as
we have
53
20
3
, and 2 4 is the same as
12
33
. Altogether
12
. If we take 4 shaded pieces from the last diagram and fill up the
12
empty spaces in the first diagram, we would have 4 wholes and 5 shaded
pieces, which is the same as 4
12
5
. (See over).
12
Method 2:
2
3
The other method is to add wholes and fractions separately, so to do 1 3 + 2 4,
2
3
first add the 1 and 2 (to get 3) and separately add the 3 and 4:
2×4 3×3
2 3
+ =
+
3×4 4×3
3 4
8
9
= 12 + 12
17
= 12
5
= 1 12
5
5
Now combine the total of the wholes (3) and the 1 12 and get 4 12. To model
this, just combine the wholes and add the fraction parts separately.
13
For subtraction, generally it is easier to use Method 1 but you can use either.
3
1
Here is one for you to try: 1 8 + 2 2. You can check your solution with the one
at the end.
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.
14
SOLUTIONS
Adding and Subtracting (page 8)
1.
1
2
3
+
=
=
1
4
4
3
6.
7
8.
7
8
9
+
1×2
4×2
2
= +
8
7
4
=
7
1
7
7
=
8
7
2
3
5
3.
8
2
3
+
5
6
8
7
=1
1×4
2×4
4
=
9×3
3
10×3
−
30
30
27
−
2×2
=
=
3×2
4
=
−8= 8
8
− =
15
5
8
− = −
10
+
4
4
=
8
2
3
1
=
4.
+
5
5
+
2 ×2
2 3
+
4 4
=1
2.
1 ×2
=
4
20
2×10
3×10
=
7
30
5.
5
7.
5
9
6
−
2
9
4
=
− =
5
=
=
6
+
9
9÷3
5×5
6×5
1
30
6
6
3
6
3÷3
30
5
5
=1 =1
6
25
+
=
−
−
1
3
4×6
5×6
24
30
1
2
SOLUTIONS
Adding and Subtracting Mixed Numbers (page 13)
Method 1:
3
1
11 5
1 +2 =
+
8
2
8 2
=
=
=
8
11
8
31
+
+
5×4
2×4
20
8
8
=3
Method 2:
3
11
7
8
1
1 8 + 2 2. Add the wholes to get 3.
Add
3
8
1
+ 2:
3
1
1×4
3
+ 2 = 8 + 2×4
8
3
4
=8+8
𝟕
=𝟖
7
Combine wholes and fractions to get 3 8.
16
Model:
plus
gives:
17
plus
gives:
18
7
that is, 3 8.
19