Look at each fraction below. How are these fractions similar? How are they different?
The fractions above are similar since each one has a denominator of 4. Look at the circles below to see how these fractions are different.
1/4 (Onefourth) 4/4 (Fourfourths)
7/4 (Sevenfourths)
The fraction is called a proper fraction. The fractions and are improper fractions.
Definition: A proper fraction is a fraction in which the numerator is less than the denominator.
Definition: An improper fraction is a fraction in which the numerator is greater than or equal to the denominator.
In example 1, we will identify each fraction as proper or improper. We will also write each fraction using words.

What do the fractions in example 2 have in common?
Example 2
2/2 (Twohalves)
3/3 (Threethirds)
4/4 (Fourfourths)
5/5 (Fivefifths)
6/6 (Sixsixths)
In example 2, each fraction has a numerator that is equal to its denominator. Each of these fractions is an improper fraction, equal to one whole (1). An improper fraction can also be greater than one whole, as shown in example 3.
Example 3
7/4 (Seven Fourths)
In the improper fraction sevenfourths, the numerator (7) is greater than the denominator (4). We can write this improper fraction as a mixed number.
Definition: A mixed number consists of a wholenumber part and a fractional part.
In examples 4 through 6, we will write each improper fraction as a mixed number.
Example 4
In example 4, sevenfourths is an improper fraction. It is really the sum of fourfourths and threefourths. Sevenfourths is written as the mixed number one and threefourths, where one is the wholenumber part, and threefourths is the fractional part.
Example 5  


In example 5, the improper fraction seventhirds is written as the mixed number two and onethird, where two is the wholenumber part, and onethird is the fractional part.
Example 6  


In example 6, the improper fraction seventeenfifths is written as the mixed number three and twofifths, where three is the wholenumber part, and twofifths is the fractional part.
In example 7, we will write each number using words. We will then classify each number as a proper fraction, an improper fraction, or a mixed number. Place your mouse over the answer text see if you got it right.
Example 7  
Number  Words  Type of Fraction 
threeeighths  answer 1  
one and threesevenths  answer 2  
fivefifths  answer 3  
elevenfourths  answer 4  
three and fivesixths  answer 5  
eightthirds  answer 6  
fourteensevenths  answer 7 
The last number in example 7 can be written as a whole number: fourteensevenths is equal to two wholes (2).
How can you tell if a fraction is less than 1, equal to 1, or greater than 1?
Compare the numerator and denominator  Example  Type of Fraction  Write As 
If the numerator < denominator, then the fraction < 1.  proper fraction  proper fraction  
If the numerator = denominator, then the fraction = 1.  improper fraction  whole number  
If the numerator > denominator, then the fraction > 1.  improper fraction  mixed number 
Summary: A number can be classified as a proper fraction, an improper fraction, or as a mixed number. Any number divided by itself is equal to one. A mixed number consists of a wholenumber part and a fractional part.
Exercises
In Exercises 1 through 5, click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. Note: To write the fraction twothirds, enter 2/3 into the form.
1.  Write a proper fraction using only the digits 7 and 2. 
2.  Write an improper fraction using only the digits 5 and 8. 
3.  Write twelvesixths as a whole number. 
4.  Write one and twothirds as an improper fraction. 
5.  Write one and onefourth as an improper fraction. 