Proper and Improper Fractions
Learning Objective(s)
· Identify proper and improper fractions.
· Change improper fractions to mixed numbers.
· Change mixed numbers to improper fractions.
Introduction
Mathematicians use three categories to describe fractions: proper, improper, and mixed.
Fractions that are greater than 0 but less than 1 are called proper fractions. In proper fractions, the numerator is less than the denominator. When a fraction has a numerator that is greater than or equal to the denominator, the fraction is an improper fraction. An improper fraction is always 1 or greater than 1. And, finally, a mixed number is a combination of a whole number and a proper fraction.
In a proper fraction, the numerator is always less than the denominator. Examples of proper fractions include _{} and _{}.
In an improper fraction, the numerator is always greater than or equal to the denominator. Examples of improper fractions include _{} and _{}.
Identify _{} as a proper or improper fraction.
A) proper
B) improper

An improper fraction can also be written as a mixed number. Mixed numbers contain both a whole number and a proper fraction. Examples of mixed numbers include _{} and _{}.
Let’s look at a quick example. Below are three whole pizzas that are each cut into four pieces. A fourth pizza is there as well, but someone has taken one piece, leaving only three pieces.





You can use fractions to compare the number of pieces you have to the number of pieces that make up a whole. In this picture, the denominator is the total number of pieces that make up one whole pizza, which is 4. The total number of all pieces of pizza, which is 15, represents the numerator.
You can use the improper fraction _{} to represent the total amount of pizza here. Think: “Each whole pizza is cut into 4 equal pieces, and there are 15 pieces total. So, the total amount of whole pizzas is _{}.”
As you looked at the image of the pizzas, however, you probably noticed right away that there were 3 full pizzas and one pizza with a piece missing. While you can use the improper fraction _{} to represent the total amount of pizza, it makes more sense here to use a mixed number – a fraction that includes both a whole number and a fractional part. For this pizza scenario, you can use the fraction _{}.
The mixed number _{} can be easier to understand than the improper fraction _{}. However, both forms are legitimate ways to represent the number of pizzas.
Rewriting an improper fraction as a mixed number can be helpful, because it helps you see more easily about how many whole items you have.
Let’s look again at the pizzas above.
The improper fraction _{} means there are 15 total pieces, and 4 pieces makes a whole pizza. If you didn’t have the picture, you could change _{} into a mixed fraction by determining:
– How many groups of 4 pieces are there in 15 pieces? Since 15 ÷ 4 = 3 with a remainder, there are 3 whole pizzas.
– What is the remainder? The remainder is 3. So, there are 3 pieces of the last pizza left, out of the 4 that would make a whole pizza. So, _{} of a pizza is left.
Now, put the number of whole pizzas with the fraction of a pizza that is left over. The mixed number is _{}.
Writing Improper Fractions as Mixed Numbers
Step 1: Divide the denominator into the numerator.
Step 2: The quotient is the whole number part of the mixed number.
Step 3: The remainder is the numerator of the fractional part of the mixed number.
Step 4: The divisor is the denominator of the fractional part of the mixed number.

Example  
Problem  Write the improper fraction _{}as a mixed number.  
47 ÷ 7 = 6, remainder 5

Divide the denominator into the numerator.
The quotient, 6, becomes the whole number.
The remainder, 5, becomes the numerator.
The denominator, which is also used as the divisor, remains as 7.
 
Answer _{} = _{}  
Change _{} from an improper fraction to a mixed number.
A) _{}
B) _{}
C) _{}
D) _{}

Mixed numbers can also be changed to improper fractions. This is sometimes helpful when doing calculations with mixed numbers, especially multiplication.
Let’s start by considering the idea of one whole as an improper fraction. If you divide a cake into five equal slices, and keep all the slices, the one whole cake is equal to the 5 slices. So, 1 cake is the same as _{} cake.
Had you cut the cake into 4 pieces or 3 pieces, as shown below, you could have used the fractions _{} or _{} to represent the whole cake. The fractions may change depending on the number of cuts you make to the cake, but you are still dealing with only one cake.




Let’s explore how to write a simple mixed number, _{}, as an improper fraction. The mixed number is represented below. Each full circle represents one whole.



To write an improper fraction, you need to know how many equal sized pieces make one whole. You also need to know how many of those pieces you have. Since you have _{}, you should divide up all of the circles into 3 pieces.



Each whole circle has 3 pieces. You can multiply the number of whole circles, 2, by 3 to find how many onethird pieces are in the two whole circles. Then you add 1 for the onethird piece in the final, incomplete circle. As you can see from the diagram, there are 7 individual onethird pieces. The improper fraction for _{} is _{}.
Writing Mixed Numbers as Improper Fractions
Step 1. Multiply the denominator of the fraction by the whole number.
Step 2. Add this product to the numerator of the fraction.
Step 3. The sum is the numerator of the improper fraction.
Step 4. The denominator of the improper fraction is the same as the denominator of the fractional part of the mixed number.

Example  
Problem  Write _{} as an improper fraction.  
_{}
4 • 4 = 16
16 + 3 = 19
_{}

Multiply the denominator of the fraction by the whole number.
Add this result to the numerator of the fraction.
This answer becomes the numerator of the improper fraction.
Notice that the denominator of the improper fraction is the same as the denominator that was in the fractional part of the mixed number.
 
Answer _{}= _{}  
Change _{} from a mixed number to an improper fraction.
A) _{}
B) _{}
C) _{}
D) _{}

Summary
A fraction can be identified as proper or improper by comparing the numerator and the denominator. Fractions that are less than one are known as proper fractions, and the numerator (the top number) is less than the denominator (the bottom number). A fraction with a numerator that is greater than or equal to the denominator is known as an improper fraction. It represents a number greater than or equal to one. Numbers that are not whole numbers, but are greater than one, can be written as improper fractions or mixed numbers. A mixed number has a whole number part and a fraction part.