fractions
TRANSCRIPT
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