The use of fractions goes as far back as 1,000 BC with the Ancient Egyptians. Back then, they primarily used what we call now as proper fractions

We still use proper fractions to this day, but with the development of modern mathematics over the last few millennia, we’ve also seen the introduction of two other types of fractions – mixed and improper fractions.

Many people are unaware of the key differences between the three, especially when it comes to improper fractions. You might be able to guess the differences just by their names. If not, don’t worry; we'll help clear that up for you. 

This article will take a deep dive into what an improper fraction is, how it differs from the others, and how to convert them. Let’s jump right in.

An improper fraction is when the numerator (top number) is greater than or equal to the denominator (bottom number), thus resulting in a value of 1 or greater. For instance, 3/2 is an improper fraction because the numerator (3) is greater than the denominator (2) and results in a value of 1.5 (3 ÷ 2 = 1.5).

This is just one example of an improper fraction, but it can consist of any number. Let’s take a look at some more examples.

Examples of improper fractions

9/7  – This is an improper fraction because the numerator (9) is greater than or equal to the denominator (7).

18/4 – This is an improper fraction because the numerator (18) is greater than or equal to the denominator (4)

52/16 – This is an improper fraction because the numerator (52) is greater than or equal to the denominator (16)

288/287 – This is an improper fraction because the numerator (288) is greater than or equal to the denominator (287)

658/132 – This is an improper fraction because the numerator (658) is greater than or equal to the denominator (132).

10,562/21,222 – This is an improper fraction because the numerator (10,562) is greater than or equal to the denominator (21,222).

782,145/782,146 – This is an improper fraction because the numerator (782,145) is greater than or equal to the denominator (782,146).

3/3 – This is an improper fraction because the numerator (3) is greater than or equal to the denominator (3). 

We could go on, but with these examples, you should be able to easily spot an improper fraction. Now that you understand the concept of improper fractions and know what they look like, you can use them in basic arithmetic operations such as addition, subtraction, multiplication, and division, just like any other fraction.

Absolutely, improper fractions are used anytime you divide a big number by a smaller one. For instance, splitting a group of people into teams, determining how many food portions everyone will receive, or splitting a bill all use improper fractions. Let’s look at a practical example. 

Suppose it’s your birthday and you have invited your best friends for a little gathering. There are five people in total, including yourself. When you are given your cake, you find that it’s already been cut into eight slices. How many slices would each person receive? 

To determine how many slices each person receives, you would divide 8 by 5 since there are five people in the room. As a fraction, this would be written as 8/5, which is an improper fraction. Solving this fraction means that each person would receive 1.6 slices. 

This is just one example of how you use improper fractions in your day-to-day life.

Improper fractions are when the numerator is greater than or equal to the denominator for a given fraction, therefore representing a value of 1 or more. On the other hand, proper fractions are where the numerator is less than the denominator, thus representing a value less than 1.

For example, 2/1 is an improper fraction since the numerator (2) is greater than the denominator (1). The value it represents is 2 (2 ÷ 1 = 2), which is greater than 1. 

Suppose we flip this fraction around to make 1/2. This becomes a proper fraction since the numerator (1) is less than the denominator (2). The value it represents is 0.5 (1 ÷ 2 = 0.5), which is less than 1.

As you can see, the difference between an improper fraction and a proper fraction is the relationship between the numerator and denominator. 

An improper fraction is where the numerator is greater than or equal to the denominator and represents a value of 1 or more. Mixed fractions – also called mixed numbers – are the simplified form of an improper fraction and display the same information but use a combination of a proper fraction and whole number instead.

For example, suppose you have the improper fraction 7/4. If this was to be expressed as a mixed number form, it would result in 1 3/4. As you can see, 1 is the whole number, and 3/4 is the proper fraction. Both the improper fraction (7/4) and mixed fraction (1 3/4) equal the same value (1.75) but are simply different ways to express the information. 

Mixed numbers are often simpler to work with than improper fractions due to the presence of the whole number, and they tend to be easier to interpret. However, improper fractions and mixed numbers are used interchangeably since they both equate to the same value. 

Therefore, as important as it is that you’re comfortable using both, it is also important to know how to convert improper fractions to mixed fractions and vice versa. 

Converting improper fractions is a simple two-step process which is as follows:

  • Step 1: Divide the numerator by the denominator to find the quotient and the remainder
  • Step 2: The quotient becomes the whole number, the remainder becomes the numerator, and the same denominator remains

Voila, you have now successfully converted an improper fraction to a mixed fraction – it really is as easy as that! 

Let’s look at some solved examples to see this process in action.

Example 1

How would you write 5/2 as a mixed fraction?

Step 1: Divide the numerator by the denominator to find the quotient and the remainder:

  • 5 ÷ 2 = 2 remainder 1
  • 2 = the quotient
  • 1 = the remainder

Step 2: The quotient becomes the whole number, the remainder becomes the numerator, and the same denominator remains:

  • 2 1/2

Therefore, 5/2 written as a mixed fraction is 2 1/2.

Example 2

What is the improper fraction 18/5 as a mixed number?

Step 1: Divide the numerator by the denominator to find the quotient and the remainder:

  • 18 ÷ 5 = 3 remainder 3
  • 3 = the quotient
  • 3 = the remainder

Step 2: The quotient becomes the whole number, the remainder becomes the numerator, and the same denominator remains:

  • 3 3/5 

Therefore, 18/5 written as a mixed number is 3 3/5.

Example 3

How would you write 87/17 in mixed number form?

Step 1: Divide the numerator by the denominator to find the quotient and the remainder

  • 87 ÷ 17 = 5 remainder 2
  • 5 = the quotient
  • 2 = the remainder

Step 2: The quotient becomes the whole number, the remainder becomes the numerator, and the same denominator remains:

  • 5 2/17

Therefore, 87/17 written as a mixed number is 5 2/17.

Converting a mixed number to an improper fraction follows two simple steps:

  • Step 1: Multiply the denominator by the whole number and add the numerator
  • Step 2: The product becomes your new numerator, and the denominator stays the same

As you may notice, these steps are the opposite of converting from an improper fraction to a mixed fraction. Let’s look at some examples.

Example 1

Write 5 2/6 as an improper fraction.

Step 1: Multiply the denominator by the whole number and add the numerator:

  • 5 x 6 = 30
  • 30 + 2 = 32

Step 2: The product becomes your new numerator, and the denominator stays the same:

  • Numerator = 32
  • Denominator = 6

This means that 5 2/6 as an improper fraction is 32/6.

Example 2

What is 1 4/10 as an improper fraction?

Step 1: Multiply the denominator by the whole number and add the numerator:

  • 1 x 10 = 10
  • 10 + 4 = 14

Step 2: The product becomes your new numerator, and the denominator stays the same:

  • Numerator = 14
  • Denominator = 10

This means that 1 4/10 as an improper fraction is 14/10.

Example 3

What is 2 55/136 as an improper fraction?

Step 1: Multiply the denominator by the whole number and add the numerator:

  • 2 x 136 = 272
  • 272 + 55 = 327

Step 2: The product becomes your new numerator, and the denominator stays the same:

  • Numerator = 327
  • Denominator = 136

This means that 2 55/136 as an improper fraction is 327/136.

It is impossible to convert an improper fraction to a proper fraction. Proper fractions have a value of less than 1, and improper fractions have a value of greater than or equal to 1. As such, by definition, improper fractions cannot be converted to proper fractions.

For example, the improper fraction 10/4 has a value of 2.5. By definition, this cannot be expressed in the form of a proper fraction because when the denominator is greater than the numerator, it always results in a value that is less than 1. 

Hopefully, now you thoroughly understand improper fractions and how they can be used. Remember, they’re a fraction where the numerator is greater than or equal to the denominator and has a value of 1 or more. e.g., 7/5. 

Improper fractions can be converted into mixed fractions – and vice versa – however, you cannot convert an improper fraction to a proper fraction.