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.

 

Identifying Proper and Improper Fractions

 

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

 

Show/Hide Answer

A) proper

Incorrect. In the fraction, the numerator is greater than the denominator, so it is an improper fraction. The correct answer is improper.

 

B) improper

Correct. The fraction is greater than 1, and the numerator is greater than the denominator, so  is an improper fraction.

 

 

 

Changing Improper Fractions to Mixed Numbers

 

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.

 

pie-4-4.jpg

pie-4-4.jpg

pie-4-4.jpg

 

pie-3-4.jpg

 

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 .

 

3-and-3-4-pizza.png

 

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

 

6-and-5-7.png

 

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)

 

Show/Hide Answer

A)

Incorrect. You probably confused the numerator with the whole number. This is much greater than . The correct answer is .

 

B)

Correct. The improper fraction  can be thought of as 12 ÷ 5 = 2, with a remainder of 2. So,  is the correct answer.

 

C)

Incorrect. To find the mixed number, you need to divide the denominator into the numerator. The correct answer is .

 

D)

Incorrect. You probably mixed up the numerator and the denominator. The correct answer is .

 

 

 

Changing Mixed Numbers to Improper Fractions

 

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.

 

 

1-as-whole-number.png

 

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.

 

blue-full-quarters.png

blue-full-thirds.png

  

 

 

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.

 

complete_grape_juice_circle.png

complete_grape_juice_circle.png

one-third.png

 

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.

three-thirds.png

three-thirds.png

one-third.png

 

Each whole circle has 3 pieces. You can multiply the number of whole circles, 2, by 3 to find how many one-third pieces are in the two whole circles. Then you add 1 for the one-third piece in the final, incomplete circle. As you can see from the diagram, there are 7 individual one-third 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)

 

Show/Hide Answer

A)

Incorrect. You probably multiplied the whole number by the numerator of the fraction instead of the denominator, and then added it to the 5 that was initially at the top. The correct answer is .

 

B)

Incorrect. You probably put the whole number 3 in the tens place of the numerator without following the correct process. The correct answer is .

 

C)

Correct. . The denominator stays the same, so  is the improper form.

 

D)

Incorrect. You probably reversed the numerator and denominator after finding your answer. The correct answer is .

 

 

 

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.