Number (Intermediate) - Fractions
 
Equivalent Fractions | Other Types of Fractions | Cancelling a Fraction | Adding and Subtracting
Multiplying | Dividing | Changing a Fraction to a Decimal
 

Equivalent Fractions

This rectangle has been divided into eight equal parts.

Each part is one eighth of the rectangle (1/8).

  1/8

  1/8

  1/8

  1/8

  1/8

  1/8

  1/8

  1/8

Shading fractions of diagrams.
We can shade three quarters of the rectangle as below.

  1
  4

  1
  4

  1
  4

  1
  4

Note that each quarter is made up of two eighths. So 3/4 is the same as 6/8

These are called Equivalent Fractions. We say 3/4 = 6/8

If we multiply the top and the bottom of a fraction by the same number, we get an equivalent fraction.

1/2 = 3/6 (multiply top and bottom by 3)

2/3 = 8/12 (multiply top and bottom by 4)


 

Other Types of Fractions

1. Improper or top-heavy fractions:
Where the top number is bigger than the bottom. For example, 3/2 is a top-heavy fraction.

2. Mixed numbers:
This is a mixture of whole numbers and fractions. For example, 1½.

This can be changed to a top-heavy fraction as follows.

1½
=
2 x 1+1
2
=
 3
2		  

We multiply the whole number by the bottom number of the fraction and add the top of the

fraction, i.e. 2 x 1 + 1 = 3, over the bottom, gives 3
2		  


e.g 5 2
3
=
3 x 5 + 2
3
=
17
3
    


Changing a top-heavy to a mixed fraction:
 
17
3
= 17 ÷ 3 = 5, remainder 2
  


This is written as 5 and 2/3 (5 2
3		 )

  

Cancelling a Fraction

This involves converting a fraction to its simplest form (or lowest terms).

We do this by dividing the top and bottom by the same number.

Example: 5/10 = 1/2 (divide top and bottom by 5)

6/8 = 3/4 (divide by 2)

12/20 = 3/5 (divide by 4)
 		  
Note: 1/2, 3/4 and 3/5 cannot be simplified further.  

Writing as a fraction

Example: I have 20 sweets and I eat 15. What fraction have I eaten?

The fraction eaten is 15 out of 20. This is written as a fraction, i.e. 15/20.

We must always simplify so 15/20 = 3/4 (divide by 5)

 
Finding ‘a fraction of'

Example 1: What is 3/4 of 8?

Rule: Divide by the bottom and multiply (times) by the top.

8 ÷ 4 x 3 = 6
 
So 3/4 of 8 = 6

  

Example 2: 2/5 of 15 = 15 ÷ 5 x 2 = 6

 
Adding and Subtracting Fractions
Examples: 3/5 + 1/5 = 4/5
 
4/5 – 1/5 = 3/5
 
2/7 + 3/7 = 5/7
 
7/8 – 3/8 = 4/8 = 1/2		  

When the bottom numbers are the same, we add or subtract the top numbers.

When the bottom numbers are different, we must make them the same by multiplying.

Example 1:

 		 1/2 + 1/3 = ?

      2 x 3 = 6		  

By making the bottom numbers 6, we must multiply the top numbers by the same amount.

1/2 + 1/3 = 3
6
+
 2
6
=
 5
6		  

In other words, 1/2 = 3/6 and 1/3 = 2/6 and these can now be added to get 5/6.

Example 2:

4/5 – 2/3 = ?
 
(5 x 3 =15)
 
4/5 = 12/15 and 2/3 = 10/15
 
So, 12/15 – 10/15 = 2/15

  

Mixed numbers
We can add or subtract the whole numbers and then the fractions.

Example:

  3 1
4
+
 2 1
3
=
 5
 7 
12
  

  5
7
12
–
 2 1
3
=
 3
3 
12
=
 3 1
4		  

 
 
Multiplying Fractions

When multiplying fractions, we multiply the top numbers and multiply the bottom numbers. Then we simplify the answer, if possible.

Example 1: 3/4 x 2/3
=

3 x 2
4 x 3
=
 6/12 = 1/2        

If the numbers are large, we can cancel diagonally.


Example 2: 15/16 x 24/35 = ?

15 and 35 cancel by 5 giving 3/16 x 24/7

16 and 24 cancel by 8 giving 3/2 and 3/7

So 15/16 x 24/35 = 3/2 x 3/7 = 9/14

 
For mixed number

Example 3: 3 1
4
x
2
 1
3		 = ?  

Change to a top-heavy fraction, so 13/4 x 7/3
=
  91/12 =7 7/12

  

Division of Fractions

Rule: change the sign from ÷ to x and turn over the second fraction. Then use the same method as multiplying.

Example:

3/4 ÷ 1/2 = 3/4 x 2/1 = 6/4 = 1

 2
4
=
 1½  

For mixed numbers

3 1
4
÷
 2 1
3
= 13/4 x 3/7 = 39/28 =
 1 11
28		  
 
Change to a top-heavy fraction first, then use the above rule.

 

Changing a Fraction to a Decimal

Rule: divide the top by the bottom.

Example 1: 3/4 = 3 ÷ 4 = 0. 75  
     
Or
 3/4 = 75/100 = 0.75 (multiply top and bottom by 25)
     
     
Example 2 : 3/5 = 3 ÷ 5 = 0.6  
     
Or
 3/5 = 6/10 = 0.6 (multiply top and bottom by 2)
     
Note: the second method is very useful for the non-calculator paper.

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