Fractions

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.

 

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.

 

Example 

 

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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

(1)   

(2)   

 

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.

 

Example

Work out .

 

Solution

 

 

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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  

 

Solution

 

 

 

 

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  

 

Solution

 

 

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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  

So,

 

 

Example 2:  Find .

 

Solution: