In this unit we shall see how to add and subtract
fractions. We shall also see how to add and
subtract mixed fractions by turning them into improper fractions.
In order to master the techniques explained here it is
vital that you undertake plenty of practice
exercises so that they become second nature.
After reading this text, and/or viewing the video tutorial
on this topic, you should be able to:
• add and subtract fractions;
• add and subtract mixed fractions.
Contents
1. Introduction 2
2. Fractions with different denominators 3
3. Mixed fractions 5
1. Introduction
Here is a simple example of adding fractions: calculate
To understand this, suppose we have a cake and divide it
into five equal pieces. Each piece is a
fifth, or 1/5 , of the cake. If we take one fifth , and then a further two
fifths , we have taken a total
of three fifths:
Example
Calculate 
Solution
In both of these examples we were adding ‘ like ’ things. In
the first example we were adding
fifths, and in the second we were adding eighths. So in both cases the
denominators were the
same. So to add ‘like’ fractions we just add the numerators.
The process is similar for subtraction, but we take away
instead of adding.
Example
Calculate 
Solution
Example
Calculate 
Solution
| Key Point When
adding or subtracting ‘like’ fractions, when the denominators are the
same, just add or
subtract the numerators. |
Exercises
1. Add the following fractions:
2. Subtract the following fractions:
2. Fractions with different denominators
What happens when we want to add or subtract fractions
where the denominators are not the
same? Let us look at a simple case. What is 
If we think of a pizza cut in half and then into quarters,
we can see that if we take a half and
then a quarter we will have taken a total of three quarters.
So 1/2 is equivalent to 2/4 , and then adding 1/4 gives us
3/4 in total:
Example
Calculate 
Solution
If we change the quarters into eighths, it becomes
straightforward. The fraction 3/4 is equivalent
to the fraction 6/8 , and since the fractions now have the same denominator, we
can just add the
numerators:
So far our examples have used fractions within the same
family, where it is easy to see a
connection between the fractions. For instance, quarters fit exactly into a
half, and eighths fit
exactly into a quarter. We shall now look at what happens when we add 1/2 and
1/3 .
This time 1/2 will not fit exactly into 1/3 , nor will 1/3
fit exactly into a 1/2 . So we need to find a number
that can be divided exactly by both 2 and 3, and then split each whole into that
number of
pieces . Now 6 can be divided by both 2 and 3, so if we split each whole into 6
pieces then we
can see that 1/2 is 3/6 and 1/3 is 2/6 .
So we have
Example
Calculate 
Solution
Again quarters and fifths are different sizes of fraction,
and we cannot exactly fit quarters into
fifths or fifths into quarters. So we need to find a size of fraction that will
fit into both quarters
and fifths.
Let us start by listing some numbers that can be divided
by 4:
4, 8, 12, 16, 20, 24, . . . .
And here are some numbers that can be divided by 5:
5, 10, 15, 20 —
and now we see that 20 can be divided by both 4 and 5. It
is the smallest number in both the
lists, so we shall split both wholes into 20 equal pieces.
If we look at this numerically , what we are doing is
finding the smallest number that can be
divided by the two denominators. The denominators are 4 and 5, so the number we
take is
20. Then we convert the two fractions into equivalent fractions with the same
denominator, 20,
before adding them. We say that 20 is the common denominator . To find the first
equivalent
fraction we see how many times 4 goes into 20. It goes 5 times, so we multiply
both the
numerator, 1, and the denominator, 4, by 5. To find the second equivalent
fraction, we see how
many times 5 goes into 20. It goes 4 times, so we multiply both the numerator,
2, and the
denominator, 5, by 4:
Example
Calculate 
Solution
To carry out this calculation , we must find the smallest
number that can be divided by both 4
and 6. That number is 12, so we need to convert both our fractions in to
twelfths:
In all these cases we have been changing the fractions
into equivalent fractions before adding or
subtracting. The denominator of the equivalent fraction is chosen so that it is
the lowest number
that can be divided by the other denominators, and it is called the lowest
common denominator,
or l.c.d. In some cases the l.c.d. can easily be found by multiplying together
the denominators
of the fractions to be added or subtracted. But, as our last example shows,
doing this does not
always result in the l.c.d. As you can see, if we had taken 4 × 6 and used 24 as
our common
denominator, the result would have been 14/24 and we would then have needed to
find the lowest
form of the fraction by dividing both numerator and denominator by any common
factors of 14
and 24.
3. Mixed fractions
Now let us look at how to add and subtract mixed
fractions. Take, for example, 
To add or subtract mixed fractions, we turn them into improper fractions first.
So
Now the improper fractions are treated just the same as
before. We find the lowest common
denominator of 4 and 5. The l.c.d. is 20, so
Example
Calculate 
Solution
First of all, write all the mixed fractions as improper
fractions:
We now want the lowest common denominator of 2, 4 and 5.
An easy way of finding this
is to count up in multiples of the largest denominator, in this case 5, see
whether the other
denominators, 2 and 4, are factors. So 5 is no good, 10 is no good, 15 is no
good, but 20 fits
our requirements. So
We can then turn the answer back into a mixed fraction by
dividing by the denominator and
finding the remainder: 213 ÷ 20 equals 10 remainder 13, so the answer is
| Key Point Turn
mixed fractions to improper fractions before adding or subtracting them. |
Exercises
3. Perform the following calculations:
Answers
