FRACTIONS
INTRODUCTION
A fraction consists of two numbers. One is known as the NUMERATOR and the
other is the DENOMINATOR. It is basically a ratio of the numerator to the
denominator.
Consider a circle that is divided into four equal parts. Then each part relative to the
1
complete circle is a quarter of the circle. This is written as:
. The “1” represents
4
the part and is the numerator, whilst the “4” is the total number of parts in the
complete figure and is the denominator. To represent three parts of the circle, it is
3
written as: .
4
If the numerator is less than the denominator, then the fraction is called a PROPER
FRACTION whilst if the numerator is greater than the denominator, it is called an
IMPROPER FRACTION. An improper fraction can also be converted to a mixed
number which is a whole number and a proper fraction.
Ex 1:
3
8
is a PROPER FRACTION, whilst is an IMPROPER FRACTION.
4
5
8
3
converted to a mixed number is 1 .
5
5
ADDITION & SUBTRACTION
To carry out addition and subtraction, it is necessary to change all the denominators to
be the same. This is achieved by selecting a number that all the denominators will
divide into. Usually we select the lowest common denominator (or multiple) but it is
not absolutely necessary. In fact we could multiply out all the denominators to
produce a common denominator with the knowledge that all the initial denominators
would divide into. In some cases, the product of these denominators will produce the
lowest common denominator.
1
Ex 2:
3 7 2
+ − =?
10 12 3
Now the lowest common denominator is 60, therefore all fractions have to be
converted so their denominators are 60. This is achieved by dividing each of the
denominators into the lowest common denominator and then multiplying by their
respective numerators.
3 7 2
+ −
10 12 3
=
( 6 × 3) + (5 × 7 ) − ( 20 × 2 )
60
=
18 + 35 − 40
60
=
13
60
or
( 6 × 3) + (5 × 7 ) − ( 20 × 2 )
or
18 35 40
+ −
60 60 60
60
60
60
1
5
1
Ex 3: 5 + 4 − 7 = ?
8
12
3
Firstly, convert all the mixed numbers to improper fractions.
1
5
1
5 +4 −7
8
12
3
=
41 53 22
+ −
8 12 3
Secondly, calculate the lowest common denominator or any suitable common
denominator and convert these improper fractions so they have the same denominator.
=
( 3 × 41) + ( 2 × 53) − (8 × 22 )
24
123 + 106 − 176
=
24
53
5
=
=2
24
24
2
MULTIPLICATION
To carry out multiplication, there is no need to change all the denominators to a
common denominator, however if any of the numbers are mixed, then it will be
necessary to convert them to improper fractions.
One method that can be used is to multiply out all the numerators, then do the same
again with denominators and then express as a fraction ideally in its lowest terms.
Ex 4:
3
1
1 5
×2 ×
10
8 34
=
31 17 5
× ×
10 8 34
=
31× 17 × 5
10 × 8 × 34
=
2635 31
=
2720 32
An alternative method is to see if any of the numbers in the numerators and
denominators can be reduced by dividing by a series of common numbers. Note that
the number of numerators that have been divided must equal the number of
denominators that have been divided. Using the same example as above:
3
1
1 5
×2 ×
10
8 34
1
1
=
31 17
5
×
×
2
2
8
10
34
=
31
31
=
2 × 8 × 2 32
3
DIVISION
The first step is to change each division sign into a multiplication sign and invert each
fraction that follows it. After that, the process is exactly the same as multiplication.
Ex 5:
1
1 3
4 ÷2 ÷
8
16 7
=
33 33 3
÷ ÷
8 16 7
=
33 16
7
×
×
1
1
8
33 3
=
2 × 7 14
=
3
3
1
=4
2
2
3
4