Download Fractions MTH 3-07 a, b, c Fraction Terms Equal Fractions

Fractions
MTH 3-07 a, b, c
Fraction Terms
3
4
numerator
denominator
The denominator describes how many equal parts the shape has been cut into. The numerator
describes how many of these parts are shaded.
Equal Fractions
The pairs of fractions shown in the diagram below are equal.
1 3

2 6
1 2

3 6
3 6

4 8
To create equal fractions numerically we use the following method:
1

4 12
1

4 12
x3
Work out what the denominator has
been multiplied by
(i.e. how many 4’s go into 12?)
x3
1 3

4 12
Now multiply the numerator by the
same number.
x3
Similarly we have :
5

6 24
which gives
x4
5

6 24
x4
and so
5 20

6 24
x4
And:
x8
7

9 72
which gives
7

9 72
and so
x8
7 56

9 72
x8
Exercise
Copy and complete these equal fractions:
a)
2

3 12
b)
4

5 30
c)
6

7 28
d)
3

8 40
e)
5

11 66
f)
7

12 36
g)
6

7 56
Simplifying Fractions
This works in a similar way to equal fractions, except this time we are dividing the numerator
and denominator of the fractions by the same number to change the fraction into its simplest
form.
e.g. Reduce
15
to its simplest form.
18
First we find the largest number which divides into 15 and 18. This is the highest common
factor of the two numbers. In this case, 15 and 18 both divide by 3.
÷3
15 5

18 6
So we have:
÷3
Similarly we have:
27
45
÷9
27 3

45 5
and the highest common factor of 27 and 45 is 9 so
÷9
Sometimes the highest common factor is not obvious, so we may have to repeat the process if
we have not found the largest number, for example:
Simplify
÷2
48
.
72
÷2 ÷6
48 24 12 2



72 36 18 3
÷2
÷2 ÷6
This could be simplified in any of these ways:
÷6 ÷4
or
48 8 2


or
72 12 3
÷6
÷4
÷8 ÷3
48 6 2
 
72 9 3
÷8 ÷3
÷ 24
or
48 2

72 3
÷ 24
Obviously, it is preferable to find the highest common factor in the first place if we can since
this takes less time.
Exercise
Reduce each of these fractions to their simplest form:
a)
15
20
b)
12
16
c)
18
24
d)
14
21
e)
30
36
f)
18
45
g)
42
48
Comparing Fractions
Knowing how to change the denominator of a fraction is important so that fractions can be
ordered, compared, added and subtracted.
5
1
1
For instance, to work out whether
or
is bigger, we can convert
into twelfths as follows:
12
4
4
x3
1 3

4 12
x3
Now we can compare the two fractions:
1 3
5
5

and
so we can see that
is larger.
4 12
12
12
e.g.1 Which is bigger,
7
5
or ?
9
6
Here we have to change both fractions so that their denominators are the same – here the first
number that 6 and 9 both divide into is 18 so we change them into eighteenths as follows:
7

9 18
and
x3
x2
So we have:
x2
7 14

9 18
x3
and
x2
And now we can see that
5

6 18
5 15

6 18
x3
5
is slightly bigger.
6
e.g. 2 Put this list of fractions in order of size, starting with the smallest.
2 5
7
3
, ,
,
3 6 12 4
In order to compare the size of the fractions we must change each one so that all of the
denominators are the same. We need to find the lowest common denominator of our
fractions i.e. the lowest common multiple of 3, 6, 12 and 4 = 12
2 8

3 12
5 10

6 12
x4
x2
3 9

4 12
7
12
x3
So now we can order the fractions:
7 2 3 5
, ,
,
12 3 4 6
Exercise
1.
By using equal fractions, decide which of these fractions is bigger:
a)
5 17
or
6 18
2.
Put these fractions in order of size, starting with the smallest:
a)
b)
3 7
13
,
,
5 10 20
b)
2
5
or
9
27
c)
7
8
or
10 15
5 19 11
,
,
6 24 12
c)
d)
3
7
or
4 9
4 11 5
9
,
,
,
5 15 6 10
Fraction of a Quantity
A reminder: To find a fraction of an amount we divide by the denominator and then multiply the
answer by the numerator as follows:
4
of 35
7
= 35 ÷ 7 x 4
=5x4
= 20
Note that this calculation could also be written as 35 x
4
.
7
For example, which is bigger?
5
of 84
12
5
of 84
12
= 84 ÷ 12 x 5
=6 x 5
= 30
or
72 x
and
72 x
4
9
4
9
= 72 ÷ 9 x 4
=8x4
= 32
So 72 x
4
is bigger.
9
Examples
1.
a)
Evaluate:
2
of 36
3
f) 60 x
3
4
b)
4
of 72
9
g) 48 x
c)
5
6
7
of 56
8
h) 81 x
7
9
d)
i)
3
of 265
5
3
x 104
8
e)
j)
4
of 343
7
4
x 612
9
Adding and Subtracting Fractions
If the fractions have the same denominator then simply add/subtract the numerators of the
fractions as follows:
2 3 5
 
7 7 7
Note that the denominators do not change.
Likewise:
8 5 3


11 11 11
However, in order to either add or subtract fractions, all fractions must have the same
denominator. In this case, the method of equal fractions must be used to convert fractions
where necessary.
For example:
2 3

5 10
Here we have to change
2 4
2
into tenths. We have 
5
5 10
x2
2 3

5 10
4
3


10 10
7

10
Sometimes we have to convert both fractions, such as:
2 1

3 8
Look at the denominators – the first number that 3 and 8 both divide into is 24.
This is known as the lowest common denominator and is the lowest common multiple of the two
denominators.
2 16
1 3

and 
3 24
8 24
x8
2 1

3 8
16 3


24 24
19

24
so
x3
The same process applies to subtraction, so:
5 2

9 5
The lowest common denominator of 5 and 9 is 45.
5 25
2 18

and 
9 45
5 45
x5
5 2

9 5
25 18


45 45
7

45
so
x9
Exercise
1. Copy and complete the calculations below:
a)
2 1

5 5
b)
h)
1 1

2 3
i)
8 7

9 9
c)
2 1

5 15
d)
7 1

9 3
e)
3 2

10 5
f)
2 1

5 3
j)
2 1

3 4
k)
4 1

7 8
l)
2 5

3 7
m)
17 4

18 9
9 3

11 4
g)
4 2

7 21
n)
3 2

8 5
Mixed Numbers
Mixed numbers are numbers which contain a mixture of whole parts and fractions e.g. 4
3
5
Some fractions have a numerator that is larger than the denominator – these are called
improper or top heavy fractions. It is useful to be able to convert between mixed numbers and
improper fractions.
Note: One whole can be written as : 1 =
2 3 4 5
    .....
2 3 4 5
Mixed Numbers to Improper:
3
Convert 4 to an improper fraction.
5
5
For every whole part we have
so here we could write :
5
3 5 5 5 5 3 23
4 = + + + + =
5 5 5 5 5 5
5
23
5
It is quicker to realise that in 4 whole parts there will be 4 x 5 = 20 fifths, giving a total of
when we add on the extra
3
.
5
So, for example,
2 6 x 7  2 44
6 =

7
7
7
and
9
3 9  8  3 75


.
8
8
8
Examples:
1.
Convert these mixed numbers to improper fractions:
a) 3
2
5
b) 2
1
7
c) 8
3
7
d) 6
5
6
e) 10
4
9
f) 5
7
8
g) 12
5
11
Improper to Mixed Number:
To convert back to a mixed number we need to know how many wholes can be made with the
given fractions:
i.e.
7 3 3 1
1
   2
3 3 3 3
3
or
7÷3=2r1
so
7
1
2
3
3
Here we can see that there are 2 wholes since 3 can divide into 7 twice. There would be a
remainder of 1 third.
19 4 4 4 4 3
3
     4
4 4 4 4 4 4
4
or
19 ÷ 4 = 4 r 3
so
19
3
4
4
4
So to convert an improper fraction into a mixed number we divide the numerator by the
denominator. The remainder of this is the fractional part of the mixed number.
Examples:
1.
a)
Convert each improper fraction back into a mixed fraction:
11
2
b)
8
3
c)
13
4
d)
21
5
e)
36
7
f)
50
8
g)
45
6
Adding and Subtracting with Mixed Numbers – Extension Work
If we want to add or subtract mixed numbers we always deal with the whole number part first,
and then add/subtract the fractions as before, remembering that they must have the same
denominators first.
For example:
2
1
6 3
5
2
4
5
9 
10 10
9
9
10
7
2
5 2
8
3
21 16
3

24 24
5
3
24
Note: 6 + 3 = 9
Note: 5 – 2 = 3
With both addition and subtraction there can occasionally be an extra complication as follows:
1
3
4 2
2
5
6
5
6 
10 10
11
6
10
1
 6 1
10
1
7
10
1
1
8 5
3
2
2 3
3 
6 6
6 2 3
2  
6 6 6
5
2
6
Convert the
improper fraction
here
We can’t take 3 from 2
here. We need to
break down one of the
wholes into sixths.
Examples:
1.
Copy and complete these fraction calculations:
a)
2 1
2 1
7 7
b)
2
2
9 3
5
5
c)
7
8
10  4
9
9
d)
5
1
3
2
2
11
e)
1
5
7 4
4
8
f)
2 2
9 1
9 3
g)
4
3
8 4
7
5
h)
2 2
5 -3
5 3