Levels 6-7


The objectives of this section are to

*  simplify fractions;

*  order fractions;

*  convert between improper and mixed fractions;

*  add and subtract fractions (including mixed numbers);

*  multiply fractions.



The top number of a fraction is called the numerator.

The bottom number is called the denominator.

The fraction  is called a mixed number.

The fraction  is called a top-heavy (or improper) fraction  its numerator is bigger than its denominator.



Converting between mixed numbers and improper fractions


Example:  Convert  to an improper fraction.

Solution:  There are 20 fifths in 4 whole numbers.  So,  


Example 2:  Convert  to an improper fraction.

Solution  (the numerator can be found as 6 × 3 + 2 = 20)


Example 3:  Convert  to a mixed number.

Solution:  6 goes into 25 four times, remainder 1.  So,  


Example 4:  Convert  to a mixed number.

Solution:  4 goes into 15 three times, remainder 3.  So  



Simplifying and ordering fractions


A fraction can be simplified if there is a whole number that divides exactly into the numerator and denominator.




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Ordering fractions


Example:  Which of these fractions is larger:  ?

Solution:  To decide which fraction is bigger, we write both fractions with the same denominator.

To find a common denominator we can multiply 5 and 24 together: 5 × 24 = 120.

We therefore write both fractions over 120:




Adding and subtracting fractions


Fractions can only be added or subtracted if they have the same denominator.


Simple cases




Note:  It is important to simplify your answers as much as possible.  It is usually preferable to give your answers as a mixed number rather than as an improper fraction.


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Adding/ subtracting fractions with different denominators


If the fractions have different denominators, the first step must be to find a common denominator.


Example 1:  Find .


Solution:  Both 8 and 6 divide into 24.  So we can use 24 as the common denominator.



Example 2:  Find .


Solution:  Both 3 and 7 divide into 21.  So we use 21 as the common denominator.



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Adding mixed numbers


Mixed numbers can be added by

(1)  adding the whole numbers together

(2)  adding the fractions together

(3)  combining both answers together.



Work out .





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Subtracting mixed numbers


When subtracting mixed numbers, it is usually simplest to begin by converting them to improper fractions.


Example:  Work out  







Multiplying Fractions


Multiplying fractions is simple.  You just multiply the top and bottom numbers together.  But remember to see if your answer can simplify!!


Example:  Find  





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Multiplying mixed numbers


Mixed numbers should be converted to improper fractions before multiplying.


Example 1:



Example 2:



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Multiplying a fraction by a whole number


A whole number can be written as a fraction with denominator 1.  We can then use the rules for multiplying fractions.


Example 1:  Find .


Solution:  10 can be thought of as the fraction  




Example 2:  Find .