Classifying Fractions

rectangle three fourths blue nonroutine

Look at each fraction below. How are these fractions similar? How are they different?

classify_intro.gif

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 (One-fourth)      4/4 (Four-fourths)

circle one fourth blue 1  circle_fourths_blue_2.gif

                 7/4 (Seven-fourths)

circle fourths blue  circle_three_fourths_blue_1.gif

The fraction 1over4.gif is called a proper fraction. The fractions 4over4_1.gif and 7over4.gif 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.

Example 1
Fraction Type Words
3over2.gif improper three-halves
2over5_4.gif proper two-fifths
3over3.gif improper three-thirds
5over6_1.gif proper five-sixths
11over8.gif improper eleven-eighths
8over8.gif improper eight-eighths

What do the fractions in example 2 have in common?

Example 2

circle two halves red

  2/2 (Two-halves)

circle three thirds pink

 3/3 (Three-thirds)

circle four fourths blue

 4/4 (Four-fourths)

circle five fifths green

   5/5 (Five-fifths)

circle six sixths yellow

   6/6 (Six-sixths)

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

circle four fourths blue circle_three_fourths_blue_1.gif

                7/4 (Seven Fourths)

In the improper fraction seven-fourths, the numerator (7) is greater than the denominator (4). We can write this improper fraction as a mixed number.

Definition: mixed number consists of a whole-number part and a fractional part.

In examples 4 through 6, we will write each improper fraction as a mixed number.

Example 4

write7over4_mixed.gif

circle four fourths blue circle three fourths blue

4over4.gif

3over4.gif

7over4equals.gif
7over4equals_step2.gif

In example 4, seven-fourths is an improper fraction. It is really the sum of four-fourths and three-fourths. Seven-fourths is written as the mixed number one and three-fourths, where one is the whole-number part, and three-fourths is the fractional part.

Example 5
write7over3_mixed.gif
  rectangle_three_thirds_small_vertical.gif     rectangle_one_third_small_vertical.gif  
  3over3.gif   3over3.gif   1over3.gif  
  7over3_equals.gif
  7over3_equals_step2.gif

In example 5, the improper fraction seven-thirds is written as the mixed number two and one-third, where two is the whole-number part, and one-third is the fractional part.

Example 6
write17over5_mixed.gif

circle five fifths green

circle five fifths green

circle five fifths green

circle five fifths green

5over5.gif

5over5.gif

5over5.gif

2over5_1.gif

17over5_equals.gif

17over5_equals_step2.gif

In example 6, the improper fraction seventeen-fifths is written as the mixed number three and two-fifths, where three is the whole-number part, and two-fifths 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
3over8_1.gif three-eighths answer 1
1_and_3over7.gif one and three-sevenths answer 2
5over5.gif five-fifths answer 3
11over4.gif eleven-fourths answer 4
3_and_5over6.gif three and five-sixths answer 5
8over3.gif eight-thirds answer 6
14over7.gif fourteen-sevenths answer 7

The last number in example 7 can be written as a whole number: fourteen-sevenths 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. 1over4.gif proper fraction proper fraction
If the numerator = denominator, then the fraction = 1. 4over4_1.gif improper fraction whole number
If the numerator > denominator, then the fraction > 1.

7over4.gif

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 whole-number 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 two-thirds, enter 2/3 into the form.

1. Write a proper fraction using only the digits 7 and 2.
 
ANSWER BOX:   

RESULTS BOX: 

2. Write an improper fraction using only the digits 5 and 8.
 
ANSWER BOX:   

RESULTS BOX: 

3. Write twelve-sixths as a whole number.
 
ANSWER BOX:   

RESULTS BOX: 

4. Write one and two-thirds as an improper fraction.
 
ANSWER BOX:   

RESULTS BOX: 

5. Write one and one-fourth as an improper fraction.
 
ANSWER BOX:   

RESULTS BOX: