Algebra (Higher) - Simplifying Fractions
 
Fractions | Other Types of Fractions | Cancelling a Fraction
Adding and Subtracting | Multiplication and Division
Fractions

1/8 1/8 1/8 1/8
1/8 1/8 1/8 1/8

This rectangle has been divided into eight equal parts.
Each part is one eighth of the rectangle ( 1/8 ).
Shading fractions of diagrams
Shade three quarters of the rectangle
1 ////////////////////////////// 1 ////////////////////////////// 1 ////////////////////////////// 1
4 ////////////////////////////// 4 ////////////////////////////// 4 ////////////////////////////// 4
 
Note each quarter is made up of two eighths. So ¾ is the same as 6/8

These are called Equivalent Fractions. We say ¾ = 6/8
If we multiply the top and the bottom of a fraction by the same number we get an equivalent fraction.
  ½ = 3/6 ( x3 top and bottom )
   
  2/3 = 8/12 ( x4 top and bottom )

Other Types of Fractions

1) Improper or top heavy fractions:

The top number is bigger than the bottom. For example 3/2 is a top heavy fraction.
 
2) Mixed numbers:

A mixture of whole numbers and fractions. For example 1½.
 
 This can be changed to a top heavy:
1½ =                                                        2x1+1 = 3
  2 2

We multiply the whole number by the bottom number of the fraction and add the top of the fraction:
2 x 1 + 1 = 3 , over the bottom , gives 3/2.

For Example
5   2 = 3x5+2 = 17  
    3       3           3  

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

This is written as 5 and 2/3   (52)		  
3     
 
Cancelling a Fraction

  
This is converting a fraction to its simplest form (Lowest terms).
We do this by dividing the top and bottom by the same number.
For Example
  5/10 = ½ ( divide top and bottom by 5)
  6/8 = ¾ (divide by 2)
  12/20 = 3/5 (divide by 4)
Note: ½ , ¾ , and 3/5 cannot be simplified further.
     
Writing as a fraction.
For example; I have 20 sweets and I eat 15, what fraction have I eaten?
Fraction eaten is 15 out of 20, this is written as a fraction 15/20.
We must always simplify: 15/20 = ¾ ( divide by 5)
 
Finding ‘a fraction of’:
For example: What is ¾ of 8 ?
Rule: Divide by the bottom and times by the top.
  8 ÷ 4 x 3 = 6
So, ¾ of 8 = 6

For Example
  2/5 of 15 = 15 ÷ 5 x 2 = 6
 
Adding and Subtracting Fractions
For Example
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: For example ½ + 1/3 = ?
We must make them the same by multiplying: 2 x 3 = 6. By making the bottom numbers 6, we must multiply the top numbers by the same amount:
½ + 1/3 =               3 + 2 =  5
     6     6    6
In other words ½ = 3/6 and 1/3 = 2/6 and these can now be added to get 5/6
 
For example
     4/5 - 2/3 = ?  
     (5 x 3 =15)  
     4/5 = 12/15 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.
For example
                             3 1 + 2 1 + 5 7
  4   3    12
 
                               5 7 - 2 1 = 3 3 = 3 1
  12    3  12        4
 
Multiplying Fractions
   ¾ x 2/3 = 3x2 = 6/12 = ½
4x3

We multiply the top numbers and multiply the bottom. Then simplify the answer, if possible.
If the numbers are large: 15/16 x 24/35 = ?
We can cancel diagonally:
  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
 
Mixed numbers 3 1 x 2 1 = ?
    4  3
Change to top heavy:  
13/4 x 7/3 = 91/12 =7
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