Three Kinds of Fractions Study Guide (page 2)

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Updated on Oct 5, 2011

Three Kinds of Fractions

There are three kinds of fractions - proper, improper, and mixed - explained in this Stud Guide.

Proper Fractions

In a proper fraction, the top number is less than the bottom number:

The value of a proper fraction is less than 1. 


Suppose you eat 3 slices of a pizza that's cut into 8 slices. Each slice is of the pizza. You've eaten of the pizza.

Working with Fractions

Improper Fractions

In an improper fraction, the top number is greater than or equal to the bottom number:

The value of an improper fraction is 1 or more.

  • When the top and bottom numbers are the same, the value of the fraction is 1. For example, all of these fractions are equal to , etc.
  • Any whole number can be written as an improper fraction by writing that number as the top number of a fraction whose bottom number is 1, for example, .


Suppose you're very hungry and eat all 8 slices of that pizza. You could say you ate of the pizza, or 1 entire pizza. If you were still hungry and then ate 1 slice of your best friend's pizza, which was also cut into 8 slices, you'd have eaten of a pizza. However, you would probably use a mixed number, rather than an improper fraction, to tell someone how much pizza you ate. (If you dare!)

Working with Fractions


If a shape is divided into pieces of different sizes, you cannot just add up all the sections. Break the shape up into equal sections of the smaller pieces and use the total number of smaller pieces as your denominator. For example, break this box into 16 of the smaller squares instead of counting this as just six sections. The fraction that represents the shaded area would then be .

Working with Fractions

Mixed Numbers

When a fraction is written to the right of a whole number, the whole number and fraction together constitute a mixed number:

The value of a mixed number is greater than 1: It is the sum of the whole number plus the fraction. 


Remember those 9 slices you ate? You could also say that you ate pizzas because you ate one entire pizza and one out of eight slices of your best friend's pizza.

Changing Improper Fractions into Mixed or Whole Numbers

Fractions are easier to add and subtract as mixed numbers rather than as improper fractions. To change an improper fraction into a mixed number or a whole number:

  1. Divide the bottom number into the top number.
  2. If there is a remainder, change it into a fraction by writing it as the top number over the bottom number of the improper fraction. Write it next to the whole number.

Example: Change into a mixed number.

1. Divide the bottom number (2) into the top number (13) to get the whole number portion (6) of the mixed number:
2. Write the remainder of the division (1) over the original bottom number (2):      
3. Write the two numbers together:    
4. Check: Change the mixed number back into an improper fraction (see steps starting on page 20). If you get the original improper fraction, your answer is correct.


Example: Change into a mixed number.

1. Divide the bottom number (4) into the top number (12) to get the whole number portion (3) of the mixed number:
2. Since the remainder of the division is zero, you're done. The improper fraction is actually a whole number:    3
3. Check: Multiply 3 by the original bottom number (4) to make sure you get the original top number (12) as the answer.
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