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June 20th

June 20th

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 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

The process is similar for subtraction, but we take away instead of adding.

Example
Calculate

Solution

Example

Calculate

Solution

 Key PointWhen adding or subtracting ‘like’ fractions, when the denominators are the same, just add or subtract the numerators.

Exercises

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 PointTurn mixed fractions to improper fractions before adding or subtracting them.

Exercises

3. Perform the following calculations: