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In this pdf, we will do multiplication and division on fractions. A fraction represents a part of a whole or, more generally, any number of equal parts. A fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters.

1.
Fractions: multiplying

and dividing

mc-TY-fracmult-2009-1

In this unit we shall see how to multiply fractions. We shall also see how to divide fractions by

turning the second fraction upside down.

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:

• multiply fractions, including improper fractions and mixed fractions;

• divide fractions, including improper fractions and mixed fractions.

Contents

1. Multiplying fractions 2

2. Dividing fractions 5

www.mathcentre.ac.uk 1 c mathcentre 2009

and dividing

mc-TY-fracmult-2009-1

In this unit we shall see how to multiply fractions. We shall also see how to divide fractions by

turning the second fraction upside down.

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:

• multiply fractions, including improper fractions and mixed fractions;

• divide fractions, including improper fractions and mixed fractions.

Contents

1. Multiplying fractions 2

2. Dividing fractions 5

www.mathcentre.ac.uk 1 c mathcentre 2009

2.
1. Multiplying fractions

How do we multiply fractions? Let us start with an example. What is 4 × 13 ? This means 4 lots

of one third.

Numerically, we perform the calculation like this:

1 1 1 1 1 4

4× = + + + = = 1 13 .

3 3 3 3 3 3

The concept is the same as when we multiply whole numbers, for instance

4 × 5 = 5 + 5 + 5 + 5 = 20 .

Now when we multiply numbers, we often use the fact that multiplication is ‘commutative’. This

means that, for example, 4 × 5 = 5 × 4, so that

5 + 5 + 5 + 5 = 4 + 4 + 4 + 4 + 4,

20 = 20 .

1 1

Let us see what this means when we are using fractions. We expect to find that 6 × 3

= 3

× 6.

1 1 1 1 1 1 1

6× = + + + + + ,

3 3 3 3 3 3 3

and ( 13 + 31 + 13 ) + ( 13 + 31 + 13 ) equals 1 + 1 = 2.

On the other hand, 31 × 6 means one-third of 6, and that is just 6 ÷ 3, so that we have 6 wholes

split into 3 equal parts, giving us 2 again.

So taking a fraction a whole number of times is the same as taking a fraction of a whole number.

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How do we multiply fractions? Let us start with an example. What is 4 × 13 ? This means 4 lots

of one third.

Numerically, we perform the calculation like this:

1 1 1 1 1 4

4× = + + + = = 1 13 .

3 3 3 3 3 3

The concept is the same as when we multiply whole numbers, for instance

4 × 5 = 5 + 5 + 5 + 5 = 20 .

Now when we multiply numbers, we often use the fact that multiplication is ‘commutative’. This

means that, for example, 4 × 5 = 5 × 4, so that

5 + 5 + 5 + 5 = 4 + 4 + 4 + 4 + 4,

20 = 20 .

1 1

Let us see what this means when we are using fractions. We expect to find that 6 × 3

= 3

× 6.

1 1 1 1 1 1 1

6× = + + + + + ,

3 3 3 3 3 3 3

and ( 13 + 31 + 13 ) + ( 13 + 31 + 13 ) equals 1 + 1 = 2.

On the other hand, 31 × 6 means one-third of 6, and that is just 6 ÷ 3, so that we have 6 wholes

split into 3 equal parts, giving us 2 again.

So taking a fraction a whole number of times is the same as taking a fraction of a whole number.

www.mathcentre.ac.uk 2 c mathcentre 2009

3.
Calculate 5 × 32 .

We obtain

2 2 2 2 2 2 10

5× = + + + + = = 3 13 .

3 3 3 3 3 3 3

This is just the same as two-thirds of 5. Now one-third of 5 is 35 , so two-thirds of 5 is 2 × 5

3

which is 10

3

.

Any whole number can be written as a fraction. For example, 2 can be written as 12 , 42 , 63 , and

so on. Any numbers can be used, as long as the numerator is twice the denominator. So, for

3 2 3 2×3 6 3

2× = × = = = = 1 21 .

4 1 4 1×4 4 2

Calculate 7 × 59 .

We obtain

5 7 5 7×5 35

7× = × = = = 3 98 .

9 1 9 1×9 9

1

Now let us look at multiplying fractions by other fractions. For example, what is 3

× 12 ? This

means one third of one half.

If we take a half and split it into 3, we have 61 of a whole. We can see this if we also divide the

other half into 3 pieces. Numerically, we have

1 1 1×1 1

× = = .

3 2 3×2 6

As you can see, we have obtained the answer 16 by first multiplying together the two numerators

to give the numerator of the answer. We have then multiplied together the two denominators

to give the denominator of the answer. This is the general technique we shall use.

Let us take another example. What is 13 × 25 ? This means that we want to start with two-fifths

of a whole, and then take one third of that.

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We obtain

2 2 2 2 2 2 10

5× = + + + + = = 3 13 .

3 3 3 3 3 3 3

This is just the same as two-thirds of 5. Now one-third of 5 is 35 , so two-thirds of 5 is 2 × 5

3

which is 10

3

.

Any whole number can be written as a fraction. For example, 2 can be written as 12 , 42 , 63 , and

so on. Any numbers can be used, as long as the numerator is twice the denominator. So, for

3 2 3 2×3 6 3

2× = × = = = = 1 21 .

4 1 4 1×4 4 2

Calculate 7 × 59 .

We obtain

5 7 5 7×5 35

7× = × = = = 3 98 .

9 1 9 1×9 9

1

Now let us look at multiplying fractions by other fractions. For example, what is 3

× 12 ? This

means one third of one half.

If we take a half and split it into 3, we have 61 of a whole. We can see this if we also divide the

other half into 3 pieces. Numerically, we have

1 1 1×1 1

× = = .

3 2 3×2 6

As you can see, we have obtained the answer 16 by first multiplying together the two numerators

to give the numerator of the answer. We have then multiplied together the two denominators

to give the denominator of the answer. This is the general technique we shall use.

Let us take another example. What is 13 × 25 ? This means that we want to start with two-fifths

of a whole, and then take one third of that.

www.mathcentre.ac.uk 3 c mathcentre 2009

4.
To split 25 into 3 parts, it is easier to split each fifth into 3 parts and take one from each. Splitting

1 2

each fifth into 3 gives us 15 pieces, so we have 15 from each one fifth section, giving 15 :

1 2 1×2 2

× = = .

3 5 3×5 15

So, as before, we are multiplying the numerators together and then multiplying the denominators

2

Calculate 5

× 49 .

We obtain

2 4 2×4 8

× = = .

5 9 5×9 45

2

Calculate 3

× 45 .

We obtain

2 4 2×4 8

× = = .

3 5 3×5 15

2 9

Calculate 3

× 10

.

We obtain

2 9 2×9 18

× = = .

3 10 3 × 10 30

3

Now in this example we can simplify the result by cancelling, and we would get 5

as our final

result. But often it is easier to cancel as we go along. If we do that, we get

2 9 2×9 1×3 3

× = = = .

3 10 3 × 10 1×5 5

The process is exactly the same if we wish to multiply three fractions rather than two.

1

Calculate 2

× 34 × 32 .

We obtain

1 3 2 1×3×2 1×1×1 1

× × = = = .

2 4 3 2×4×3 1×4×1 4

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1 2

each fifth into 3 gives us 15 pieces, so we have 15 from each one fifth section, giving 15 :

1 2 1×2 2

× = = .

3 5 3×5 15

So, as before, we are multiplying the numerators together and then multiplying the denominators

2

Calculate 5

× 49 .

We obtain

2 4 2×4 8

× = = .

5 9 5×9 45

2

Calculate 3

× 45 .

We obtain

2 4 2×4 8

× = = .

3 5 3×5 15

2 9

Calculate 3

× 10

.

We obtain

2 9 2×9 18

× = = .

3 10 3 × 10 30

3

Now in this example we can simplify the result by cancelling, and we would get 5

as our final

result. But often it is easier to cancel as we go along. If we do that, we get

2 9 2×9 1×3 3

× = = = .

3 10 3 × 10 1×5 5

The process is exactly the same if we wish to multiply three fractions rather than two.

1

Calculate 2

× 34 × 32 .

We obtain

1 3 2 1×3×2 1×1×1 1

× × = = = .

2 4 3 2×4×3 1×4×1 4

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5.
What happens if we have mixed fractions?

Just as before when dealing with mixed fractions, we turn them into improper fractions first.

Calculate 2 31 × 34 .

We obtain

2×3+1 3 7×3 7×1 7

2 13 × 3

4

= × = = = = 1 43 .

3 4 3×4 1×4 4

Calculate 1 52 × 2 56 .

We obtain

1×5+2 2×6+5 7 × 17 119

1 25 × 2 65 = × = = = 3 29

30

.

5 6 5×6 30

Key Point

To multiply fractions, multiply the numerators together and multiply the denominators together

separately. To multiply mixed fractions, turn them into improper fractions first.

1. Perform the following multiplications:

2 5 3 3 7

(a) 6 × 5

(b) 2 × 9

(c) 4 × 10

(d) 4

× 11

3 10

(e) 5

× 12

(f) 2 51 × 3 23

2. Dividing fractions

1

We shall now look at what happens when we divide fractions. Let us take 4

and divide it by 2.

1 1

÷2 =

4 8

Dividing by 2 is the same as taking a half, so dividing by 2 and multiplying by a half are the same

thing. We could write

1 1 1 1

÷2 = × = .

4 4 2 8

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Just as before when dealing with mixed fractions, we turn them into improper fractions first.

Calculate 2 31 × 34 .

We obtain

2×3+1 3 7×3 7×1 7

2 13 × 3

4

= × = = = = 1 43 .

3 4 3×4 1×4 4

Calculate 1 52 × 2 56 .

We obtain

1×5+2 2×6+5 7 × 17 119

1 25 × 2 65 = × = = = 3 29

30

.

5 6 5×6 30

Key Point

To multiply fractions, multiply the numerators together and multiply the denominators together

separately. To multiply mixed fractions, turn them into improper fractions first.

1. Perform the following multiplications:

2 5 3 3 7

(a) 6 × 5

(b) 2 × 9

(c) 4 × 10

(d) 4

× 11

3 10

(e) 5

× 12

(f) 2 51 × 3 23

2. Dividing fractions

1

We shall now look at what happens when we divide fractions. Let us take 4

and divide it by 2.

1 1

÷2 =

4 8

Dividing by 2 is the same as taking a half, so dividing by 2 and multiplying by a half are the same

thing. We could write

1 1 1 1

÷2 = × = .

4 4 2 8

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6.
1

What about 3

divided by 4? We would have

1 1 1 1

÷4 = × = .

3 3 4 12

1

And again, for 2

÷ 2 we would have

1 1 1 1

÷2 = × = .

2 2 2 4

In all these cases it looks as if, to divide by a number, we can instead multiply the denominator

by that number. Another way of saying this is that we write the divisor, that is the number we

are dividing by, as a fraction. When we have done this, we turn that fraction upside-down and

multiply instead.

Now we can extend this approach to divide by numbers which are themselves fractions rather

than whole numbers. For example, what is 12 ÷ 14 ? In other words, how many times does a quarter

go into a half?

Following the rule, we obtain

1 1 1 4 1×4 4

÷ = × = = = 2

2 4 2 1 2×1 2

1 1

which makes sense since 4

+ 4

= 21 , so 1

4

goes into 1

2

twice.

1

Calculate 3

÷ 15 .

We obtain

1 1 1 5 1×5 5

÷ = × = = = 1 32 .

3 5 3 1 3×1 3

Calculate 2 ÷ 18 .

We obtain

1 8

2÷ = 2× = 2×8 = 16 .

8 1

Calculate 4 ÷ 31 .

We obtain

1 3 12

4÷ = 4× = = 12 .

3 1 1

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What about 3

divided by 4? We would have

1 1 1 1

÷4 = × = .

3 3 4 12

1

And again, for 2

÷ 2 we would have

1 1 1 1

÷2 = × = .

2 2 2 4

In all these cases it looks as if, to divide by a number, we can instead multiply the denominator

by that number. Another way of saying this is that we write the divisor, that is the number we

are dividing by, as a fraction. When we have done this, we turn that fraction upside-down and

multiply instead.

Now we can extend this approach to divide by numbers which are themselves fractions rather

than whole numbers. For example, what is 12 ÷ 14 ? In other words, how many times does a quarter

go into a half?

Following the rule, we obtain

1 1 1 4 1×4 4

÷ = × = = = 2

2 4 2 1 2×1 2

1 1

which makes sense since 4

+ 4

= 21 , so 1

4

goes into 1

2

twice.

1

Calculate 3

÷ 15 .

We obtain

1 1 1 5 1×5 5

÷ = × = = = 1 32 .

3 5 3 1 3×1 3

Calculate 2 ÷ 18 .

We obtain

1 8

2÷ = 2× = 2×8 = 16 .

8 1

Calculate 4 ÷ 31 .

We obtain

1 3 12

4÷ = 4× = = 12 .

3 1 1

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7.
So far, when dividing we have only looked at fractions with a numerator of 1. Let us now look

at other fractions. What is 34 ÷ 2? Using the rule, we obtain

3 3 2 3 1 3×1 3

÷2 = ÷ = × = = .

4 4 1 4 2 4×2 8

So again the process is turn the second fraction, the divisor, upside down, and then to multiply.

3

Calculate 5

÷ 4.

We obtain

3 3 4 3 1 3×1 3

÷4 = ÷ = × = = .

5 5 1 5 4 5×4 20

2

Calculate 3

÷ 34 .

How many times does 43 fit into 32 ? Since 3

4

is larger than 2

3

we should expect the answer to be

less than 1. Following the rule, we obtain

2 3 2 4 2×4 8

÷ = × = = .

3 4 3 3 3×3 9

Finally we need to look at how to deal with mixed fractions when dividing.

Calculate 1 32 ÷ 2 14 .

As before, we need to convert mixed fractions to improper fractions before dividing:

1×3+2 2×4+1 5 9

1 32 ÷ 2 14 = ÷ = ÷ .

3 4 3 4

Now we continue as before by turning the divisor upside down, and multiplying:

5 9 5 4 20

÷ = × = .

3 4 3 9 27

Calculate 2 54 ÷ 4 23 .

2×5+4 4×3+2 14 14 14 3 3

2 45 ÷ 4 32 = ÷ = ÷ = × = .

5 3 5 3 5 14 5

Key Point

To divide fractions, turn the second fraction upside-down and multiply. To divide mixed fractions,

turn them into improper fractions first.

www.mathcentre.ac.uk 7 c mathcentre 2009

at other fractions. What is 34 ÷ 2? Using the rule, we obtain

3 3 2 3 1 3×1 3

÷2 = ÷ = × = = .

4 4 1 4 2 4×2 8

So again the process is turn the second fraction, the divisor, upside down, and then to multiply.

3

Calculate 5

÷ 4.

We obtain

3 3 4 3 1 3×1 3

÷4 = ÷ = × = = .

5 5 1 5 4 5×4 20

2

Calculate 3

÷ 34 .

How many times does 43 fit into 32 ? Since 3

4

is larger than 2

3

we should expect the answer to be

less than 1. Following the rule, we obtain

2 3 2 4 2×4 8

÷ = × = = .

3 4 3 3 3×3 9

Finally we need to look at how to deal with mixed fractions when dividing.

Calculate 1 32 ÷ 2 14 .

As before, we need to convert mixed fractions to improper fractions before dividing:

1×3+2 2×4+1 5 9

1 32 ÷ 2 14 = ÷ = ÷ .

3 4 3 4

Now we continue as before by turning the divisor upside down, and multiplying:

5 9 5 4 20

÷ = × = .

3 4 3 9 27

Calculate 2 54 ÷ 4 23 .

2×5+4 4×3+2 14 14 14 3 3

2 45 ÷ 4 32 = ÷ = ÷ = × = .

5 3 5 3 5 14 5

Key Point

To divide fractions, turn the second fraction upside-down and multiply. To divide mixed fractions,

turn them into improper fractions first.

www.mathcentre.ac.uk 7 c mathcentre 2009

8.
2. Perform the following divisions:

1 1 1 1 1 1

(a) 5

÷2 (b) 6

÷4 (c) 4

÷ 2

(d) 3

÷ 3

1 1 2 3

(e) 2÷ 5

(f) 3÷ 4

(g) 5

÷3 (h) 4

÷2

3 2 3 1 2 2

(i) 4

÷ 3

(j) 5

÷ 2

(k) 4÷ 5

(l) 5÷ 3

3. Perform the following divisions:

1

(a) 2 10 ÷ 1 18 (b) 5 14 ÷ 3

8

(c) 1 13 ÷ 3 14

12

(a) 5

or 2 25 (b) 10

9

or 1 19 (c) 6

5

or 1 51 (d) 21

44

1 121 1

(e) 2

(f) 15

or 8 15

1 1 1

(a) 10

(b) 24

(c) 2

(d) 1

2 3

(e) 10 (f) 12 (g) 15

(h) 8

9

(i) 8

or 1 81 (j) 6

5

or 1 51 (k) 10 (l) 15

2

or 7 12

28 13 16

(a) 15

or 1 15 (b) 14 (c) 39

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1 1 1 1 1 1

(a) 5

÷2 (b) 6

÷4 (c) 4

÷ 2

(d) 3

÷ 3

1 1 2 3

(e) 2÷ 5

(f) 3÷ 4

(g) 5

÷3 (h) 4

÷2

3 2 3 1 2 2

(i) 4

÷ 3

(j) 5

÷ 2

(k) 4÷ 5

(l) 5÷ 3

3. Perform the following divisions:

1

(a) 2 10 ÷ 1 18 (b) 5 14 ÷ 3

8

(c) 1 13 ÷ 3 14

12

(a) 5

or 2 25 (b) 10

9

or 1 19 (c) 6

5

or 1 51 (d) 21

44

1 121 1

(e) 2

(f) 15

or 8 15

1 1 1

(a) 10

(b) 24

(c) 2

(d) 1

2 3

(e) 10 (f) 12 (g) 15

(h) 8

9

(i) 8

or 1 81 (j) 6

5

or 1 51 (k) 10 (l) 15

2

or 7 12

28 13 16

(a) 15

or 1 15 (b) 14 (c) 39

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