UNIVERSITY OF GLOUCESTERSHIRE

FRACTIONS

For Numeracy Workshop and SLE 103 'Confidence Counts' module

At the end of this unit you should have • Revised the meanings of fraction, improper fraction, mixed

number and equivalent fraction.

• Learned to reduce a fraction to its lowest terms.

• Learned to convert mixed numbers to improper fractions and vice-

versa.

• Practised methods of adding, subtracting, multiplying and dividing

fractions WITHOUT using a calculator.

1

WHAT IS A FRACTION?

A fraction is a part of a whole This shaded area is one part out of 2. It is written 1 2 This shaded area is one part out of 3. It is written 1 3 This shaded area is two parts out of 3. It is written 2 3 These figures are all divided into EQUAL parts. If all the parts are not equal, as in the example below, then the shaded area is not a third. NOT

THIRDS

2

A fraction is shown as one number divided by another number.

The top number is called the NUMERATOR.

The bottom number is called the DENOMINATOR.

So a fraction = numerator denominator There are TWO types of fractions:

1. A fraction with a smaller numerator than denominator is called a COMMON or PROPER fraction.

Eg 1 3 5 2 4 8 2. A fraction with a bigger numerator than denominator is called an

IMPROPER or ‘TOP HEAVY’ fraction. Eg 5 7 19 2 3 7 You can also have MIXED NUMBERS which consist of whole numbers and parts of whole numbers (fractions).

Eg 1221

34594

271211

EQUIVALENT FRACTIONS Every fraction belongs to a ‘family’ or set of equivalent fractions.

1 = 2 = 3 = 4 = 5 ….. = …….50 …… 2 4 6 8 10 100

In this set of fractions the simplest form of the fraction is 21 and for all

other members of the set, the numerator is half of the denominator.

They are all said to be equivalent to 21

So 500 belongs to this set, but 99 does not. 1000 200

3

Another set of fractions is equivalent to 51 . Can you fill in the next 3

fractions in the series? 1 = 2 = 3 = 4 = 5 = = = 5 10 15 20 25 Here is another set. Try filling in the gaps. 2 = 4 = 6 = = 103 6 12 The important thing to remember is that the value of a fraction is unchanged if the numerator and the denominator are BOTH multiplied by the same number:

Eg 4030 is equivalent to

43 because both the top and bottom

numbers have been multiplied by 10.

Eg 426 is equivalent to

71 because both the top and bottom

numbers have been multiplied by 6. What is the numerator in the next example?

43 =

28?

What have you multiplied 4 by to get 28? Answer : 7 Then you need to multiply 3 by 7 to find the numerator. 3 x 7 = 21

43 =

2821

4

What is the denominator in the next example?

276 =

?2

What have you divided 6 by to get 2? Answer : 3 Then you need to divide 27 by 3 to find the unknown denominator. 27 ÷ 3 = 9

So 276 =

92

Now fill in these gaps: EXERCISE 1 1. 2 = ? 2. 4 = ? 3. 3 = 6 4. 3 = ? 7 14 5 25 8 ? 10 40 5. 1 = 5 6. 4 = 1 7. 10 = ? 8. 9 = 3 7 ? 12 ? 25 5 15 ? 9. 8 = ? 10. 18 = 3 64 8 48 ? Answers are given on page 22.

5

REDUCING A FRACTION TO ITS LOWEST TERMS (often called “cancelling”)

Eg Reduce 1210 to its lowest terms

Look for a number which divides both 10 and 12 with no remainder. The answer is 2. So divide both 10 and 12 by 2. 10 ÷ 2 = 5 12 ÷ 2 = 6

Then 1210 =

65

There is no number which divides both 5 and 6 with no remainder so the fraction is now reduced to its lowest terms. Eg Reduce 132 to its lowest terms 198 First try dividing top and bottom by 2

198132 =

9966

Divide the new fraction by 11

9966 =

96

Now divide this by 3

96 =

32

This will reduce no more, so it has been reduced to its lowest terms. It is the final answer. Now try some yourself.

6

EXERCISE 2 Reduce the following fractions to their lowest terms:

1. 189 =

2. 2515 =

3. 4842 =

4. 240180 =

5. 315210 =

Changing MIXED NUMBERS to ‘TOP HEAVY’ FRACTIONS

Eg 241

is a mixed number.

How can you change this into a ‘top heavy’ fraction, so that you can add, subtract, multiply or divide it? Process: Multiply the whole number by the denominator 2 x 4 = 8

Add on the numerator 8 + 1 = 9 This becomes the new numerator 9 The denominator does not change 4

Place the numerator over the denominator 49

So 241

= 49

7

Eg Change 573

to an improper fraction

Multiply the whole number by the denominator 5 x 7 = 35 Add on the numerator 35 + 3 = 38 This becomes the new numerator 38 The denominator does not change 7 Place the numerator over the denominator 38

7

So 573

= 7

38

Eg Change 4103

= 1043

Go through the process to check how this answer was calculated, then try the following examples EXERCISE 3

1. 321

=

2. 452

=

3. 387

=

4. 1521

=

5. 395

=

Answers can be found on page 22.

8

Changing ‘TOP HEAVY’ FRACTIONS to MIXED NUMBERS Eg 22 is a ‘top heavy fraction 7 How do you change it to a mixed number? Process: Divide the numerator by the denominator

22 ÷ 7 = 3, remainder 1 The 3 is the whole number 3 The remainder 1 becomes the new numerator 1 The denominator does not change 7 Place the numerator over the denominator and place the

whole number in front 371

So 722 = 3

71

Eg Change 47 to a mixed number

Divide the numerator by the denominator

7 ÷ 4 = 1, remainder 3 The 1 is the whole number 1 The remainder 3 becomes the new numerator 3 The denominator does not change 4 Place the numerator over the denominator and place the

whole number in front 143

So 47 = 1

43

9

Here are some for you to try: EXERCISE 4 1. 13 = 2 2. 23 = 7 3. 18 = 5 4. 29 = 11 5. 53 = 12 Answers can be found on page 22. We will now go on to look at how you multiply, divide, add and subtract,

fractions.

MULTIPLICATION OF FRACTIONS

MULTIPLICATION is very straightforward and works logically exactly as

you would expect

Process: multiply numerators

multiply denominators

place new numerator over new denominator

10

Eg 43 x

107

numerators 3 x 7 = 21 denominators 4 x 10 = 40

43 x

107 =

4021

Eg 32 x

54

numerators 2 x 4 = 8 denominators 3 x 5 = 15

32 x

54 =

158

The same method works for mixed numbers, but first convert them to

‘top heavy’ fractions.

Eg 121 x 2

51 =

23 x

511

= 1033

= 3103

That is all there really is to multiplication, except for the useful technique

of CANCELLING which can simplify the numbers you are working with.

You can do this when you can find a number which divides one of the

numerators and one of the denominators with no remainder.

11

Eg 112 x

2522 CANCEL by 11

2

112 x

2522 =

12 x

252 =

254

1

Eg 145 x

107 CANCEL by 5 and by 7

1 1

145 x

107 =

21 x

21 =

41

2 2 The main point to remember is that when you are cancelling you can

cancel ANY NUMERATOR with ANY DENOMINATOR:

1 1 1 1 Eg 2 x 3 x 3 x 5 = 1 x 1 x 1 x 1 = 1 3 4 10 6 1 2 2 2 8

1 2 2 2

1

Eg 132

x 254

= 35 x

514 =

314 = 4

32

1 Do not worry if you have not always cancelled where possible – at the

end of your calculation you will be able to reduce the fraction to its

lowest terms and finish with the same answer as if you had cancelled at

the beginning.

12

Eg 1 1

21

x 32

x 43

= 41

1 1

or 21

x 32

x 43

= 246

1

= 246

4

= 41

Final example: What is two thirds of four and a half?

You know that ‘of’ means multiply, so this problem can be written 1 3

32

x 421

= 32

x 29

1 1

= 13

= 3

Now try the following exercise, using the examples given above to help

you.

EXERCISE 5

1. 53

x 74

2. 53

x 95

13

3. 53

x 31

4. 221

x 54

5. 321

x 471

6. 3 x 72

7. 53

x 5 8. 2 x 121

x 31

9. 343

x 153

x 181

10. 54

of 121

Answers can be found on page 22. DIVISION OF FRACTIONS Division can be thought of as the inverse (opposite) operation to

multiplication. There is a little more to do

Process:

• The first fraction remains the same.

• Change the division sign to a multiplication sign.

• Invert the second fraction (turn it upside down).

• Then continue as you would for multiplication of fractions.

Eg 32

÷ 75

= 32

x 57

= 1514

14

If there are mixed numbers, change them to ‘top heavy’ fractions before

starting the division procedure.

Eg 154 ÷

32 =

59 x

23

= 1027

= 2 107

Eg 154 ÷ 2

31 =

59 ÷

37

= 59 x

73

= 3527

Eg 4 divided by 43

becomes

14 ÷

43

= 14 x

34

= 3

16 = 531

When you reach the multiplication stage (but not before) you can cancel

any numerator with any denominator.

15

Eg 221 divided by 10 becomes

221 ÷

110 =

25 ÷

110

1

= 25 x

101 CANCEL by 5

2

= 41

Now try the following exercise: EXERCISE 6

1. 43 ÷

21 = 2.

65 ÷

32 =

3. 221 ÷ 1

41 = 4. 5

31 ÷

94 =

5. 125 ÷ 2

21 = 6. 1

21 ÷

73 =

7. 143 ÷

81 = 8. 1

61 ÷ 4

32 =

9. 53 ÷

209 = 10. 3

31 ÷

95 =

Answers can be found on page 22.

16

ADDING AND SUBTRACTING FRACTIONS The same procedure works for both adding and subtracting fractions. If the fractions have the same denominator this is a straightforward procedure: + =

53

51

= 54

Just as you can add 3 apples and 1 apple to get 4 apples, you can add 3 fifths and 1 fifth to get 4 fifths. Notice that you only add the NUMERATORS. The denominator is unchanged.

Eg 123

+ 124

= 127

Eg 2

103

+ 101

= 104

= 52

5 So you add the NUMERATORS and the denominator is not changed at all. Exactly the same procedure can be used for subtracting: Eg 3

109

- 103

= 106

= 53

5 But what if the denominators are different?

17

Eg 31

+ 41

Is the denominator of the answer going to be thirds or quarters?

Neither.

The fractions cannot be added until you alter them so that the

denominators are the same.

You have to look for a common denominator.

Now is the time to use your knowledge of EQUIVALENT FRACTIONS.

Consider the sets of fractions equivalent to the two given fractions:

31

= 62

= 93

= 124

= 155

= . . . . . . .

41

= 82

= 123

= 164

= . . . . . .

So instead of 31

+ 41

you can write 124

+ 123

Now that you have two fractions with the same denominator you can add them:

124

+ 123

= 127

This is how the process works, but writing out sets of equivalent

fractions is very time-consuming so in practice you look for the

LOWEST COMMON DENOMINATOR. This is the LOWEST number

into which both denominators will divide with no remainder.

18

Sometimes this is just the product of the two denominators, but

sometimes you can find a smaller number.

What is the lowest common denominator for each of the following pairs

of fractions?

a) 31

and 51

(Answer: 15

)

b) 101

and 52

(Answer: 10

)

c) 61

and 103

(Answer: 30

)

PROCESS for ADDING/SUBTRACTING fractions with different

denominators:

Eg 72

+ 43

28 is the lowest number into which 7 and 4 will divide

(7 goes into 28 4 times, 4 goes into 28 7 times)

72

= 28

2x4 ,

43

= 28

3x7

72

+ 43

= 28

3x7+x24

= 28

21+8 =

2829

= 1281

19

Eg 32

- 53

15 is the lowest number into which 3 and 5 will divide

32

- 53

= 15

33x- x25 =

15910 -

= 151

If mixed numbers are involved, then convert them to improper fractions

before doing the adding or subtracting.

OR

Add/subtract the whole number parts before doing the fraction

adding/subtraction if this seems easier.

Eg

141

+ 232

EITHER convert to mixed numbers and add:

45

+ 38

= 12

4x8+3x5 =

1232+15

= 1247

= 31211

OR Add the whole numbers 1 + 2 = 3 and then the fraction parts

41

+ 32

= 12

2x4+x13 =

128+3

= 1211

and then combine them to give 31211

The second method is often preferable when adding fractions with large

numerators and denominators.

20

Now try the following exercises: EXERCISE 7 Adding fractions

1. 43

+ 21

2. 71

+ 32

3. 43

+ 81

4. 54

+ 32

5. 43

+ 54

6. 521

+ 141

7. 453

+ 141

8. 731

+ 1121

9. 431

+ 32

10. 583

+ 143

+ 487

+ 3165

and EXERCISE 8 Subtracting fractions

1. 87

- 83

2. 72

- 141

3. 21

- 83

4. 54

- 103

5. 113

- 221

6. 121

- 32

7. 331

- 125

8. 2125

+ 1127

9. 343

- 187

10. 5 - 83

21

ANSWERS

EXERCISE 1

1. 144

2. 2520

3. 166

4. 4012

5. 355

6. 31

7. 52

8. 53

9. 81

10. 83

EXERCISE 2

1. 21

2. 53

3. 87

4. 43

5. 32

EXERCISE 3

1. 27

2. 522

3. 831

4. 231

5. 932

EXERCISE 4

1. 621

2. 372

3. 353

4. 2117

5. 4125

EXERCISE 5

1. 3512

2. 31

3. 51

4. 2 5. 1421

6. 76

7. 3 8. 1 9. 643

10. 151

EXERCISE 6

1. 121 2. 1

41 3. 2 4. 12 5.

61

6. 321 7. 14 8.

41 9. 1

31 10. 6

22

EXERCISE 7

1. 141 2.

2117 3.

87 4. 1

157 5. 1

2011

6. 643 7. 5

2017 8. 8

125 9. 5 10. 15

165

EXERCISE 8

1. 21 2.

143 3.

81 4.

21 5.

225

6. 65 7. 2

1211 8.

65 9. 1

87 10. 4

85

23