A proper fraction is the most commonly expressed type of fraction and is characterized as having a numerator smaller than the denominator. These types of fractions represent numbers smaller than a whole, or less than 1. Examples include 1/2, 3/4 and 3/8.
An improper fraction has a numerator that is greater than, or equal to, the denominator. Thus, an improper fraction identifies the presence of a whole or more parts than a whole when the numerator is equal to or greater than the denominator, respectively. Examples include 4/3, 8/5 and 3/3.
A mixed fraction is composed of a whole number and a proper fraction. It is possible to convert between a mixed fraction and an improper fraction; therefore, these two types may be used interchangeably to identify the same number. To convert a mixed fraction to a whole number, multiply the whole number by the denominator of the proper fraction, add the result to the numerator and write that value on top of the denominator. To convert back to the mixed number, divide the numerator by the denominator and write down the whole number with any remainder above the denominator to form the proper fraction part.
Both proper and improper fractions are used extensively in all levels of math, ranging from algebra to differential equations. Mixed fractions, however, are normally not used in equations and higher levels of math because of their added complexity. They are most readily used in everyday situations to describe amounts. For example, it would be better received in saying that someone ate one and a half (1 1/2) slices of pizza than to say they ate three halves (3/2) of pizza.