When students are given mixed fractions, their first reaction is to panic. After all, there are so many numbers floating about. You have a whole number and a fraction together – which goes against everything you’ve faced up until now – and can look quite intimidating.

Luckily, mixed numbers are a piece of cake once you know what to do. All it requires is knowledge of a few key facts. 

To help you understand what they are, this article will look at all things mixed number-related, such as what it is, how it relates to improper and proper fractions, how to convert between them, and how to solve problems.

Let’s jump right in.

A mixed number is a number that consists of an integer (whole number) and a proper fraction. Mixed numbers precisely describe a value without using decimal points, which can be cumbersome to write out and lose accuracy due to rounding.

By definition, the fraction part of a mixed number can only be a proper fraction – where the numerator is less than the denominator. If the numerator is larger than the denominator, the mixed number has not been simplified correctly.

Examples of a mixed number

Suppose you had one and a half avocados. That can be represented as 1 1/2. This would be a mixed number since it consists of a whole number (1) and a proper fraction (2/3). Let's look at a few more examples.

1 2/3 is a mixed number consisting of a whole number (1) and a proper fraction (2/3). 

4 5/7 is a mixed number since it consists of a whole number (1) and a proper fraction (5/7)

2 6/4 is not a mixed number. Although it consists of a whole number (2), it does not have a proper fraction. The fraction’s numerator (6) is greater than the denominator (4).

2 4/6 is a mixed number consisting of a whole number (2) and a proper fraction (4/6).

7 3/8 is a mixed number since it consists of a whole number (7) and a proper fraction (3/8).

18 11/4 is not a mixed number. Although it consists of a whole number (18), it does not have a proper fraction.

6 5/3 is not a mixed number. Although it consists of a whole number (6), it does not have a proper fraction.

What is a mixed fraction?

A mixed fraction is the same as a mixed number; the two terms are used synonymously. For example, 3/2 can be described as a mixed number or a mixed fraction. It’s important to know this little fact since both terms are used interchangeably – as we will in this article – and you may come across either during your tests and exams.

Mixed numbers vs. improper fractions

Mixed numbers and improper fractions both express the same number but simply do it differently. Solving an improper fraction will yield the answer in a mixed number form – i.e., mixed numbers are a simplified version of improper fractions. To see this relationship, it’s important to first understand what an improper fraction is. 

An improper fraction is when the numerator is greater than or equal to the denominator. As such, solving for the fraction equates to a value of 1 or more. For instance, 5/2 is an improper fraction since the numerator (5) is greater than the denominator (2). When solving for the fraction, you would get the answer 2.5 (5 divided by 2 equals 2.5). Since 2.5 is greater than or equal to 1, we can confirm that 5/2 is an improper fraction.

As discussed earlier, a mixed fraction is when there is a presence of a whole number and a proper fraction. Suppose we take the example of 5/2 again. Expressing this as a mixed number would result in 2 1/2. How, you may ask? Let’s find out.

How to convert an improper fraction to a mixed number?

You don’t need to solve any complicated calculations to convert an improper fraction to a mixed number, you just need to follow two easy steps:

  • Step 1: Divide the numerator by the denominator. This will provide you with the quotient and remainder
  • Step 2: Arrange the answers to Step 1 in the mixed number format
    • Whole number = Quotient
    • Numerator = Remainder
    • Denominator remains the same

Let’s see a few examples.

Example 1: 9/4

Suppose you wanted to convert 9/4 to a mixed number. Let’s follow the steps outlined above.

Step 1: Divide the numerator by the denominator. This will provide you with the quotient and remainder:

  • 9 ÷ 4 = 2 remainder 1

Step 2: Arrange the answers to Step 1 in the mixed number format:

  • Whole number = Quotient: The quotient is 2
  • Numerator = Remainder: The remainder is 1
  • Denominator: Remains the same at 4

Therefore, after combining this all together, you can conclude that 9/4 as a mixed number is 2 1/4.

Example 2: 59/13

Suppose you wanted to convert the improper fraction 59/13 into a mixed number. Again, follow the steps we did in the previous example.

Step 1: Divide the numerator by the denominator. This will provide you with the quotient and remainder:

  • 59 ÷ 13 = 4 remainder 7

Step 2: Arrange the answers to Step 1 in the mixed number format:

  • Whole number = Quotient: The quotient is 4
  • Numerator = Remainder: The remainder is 7
  • Denominator: Remains the same at 13
  • Arranging these into their correct places gives you 4 7/13.

Therefore, you can conclude that 9/4 as a mixed number is 4 7/13.

We’ve just looked at examples of converting an improper fraction to a mixed fraction. However, knowing how to do the opposite calculation is equally important – converting a mixed fraction to an improper fraction. 

How to convert a mixed number to an improper fraction?

You must work backwards from the steps above to convert mixed numbers into improper fractions. The process is as follows:

  • Step 1: Multiply the denominator by the whole number
  • Step 2: Add the numerator
  • Step 3: The final numerator is the answer to step 2, and the denominator stays the same

Sounds easy enough, right? Let’s look at it in action.

Example 1: 6 8/12

Suppose you wanted to convert the mixed fraction 6 8/12 into an improper fraction. 

Step 1: Multiply the denominator by the whole number:

  • The denominator is 12, and the whole number is 6. Therefore, 12 x 6 = 72

Step 2: Add the numerator:

  • 72 + 8 = 80

Step 3: The final numerator is the answer to step 2, and the denominator stays the same:

  • Numerator = 80
  • Denominator = 12
  • This means that 6 8/12 as an improper fraction is 80/12

Example 2: 4 3/5

Converting 4 3/5 into an improper fraction would be as follows.

Step 1: Multiply the denominator by the whole number:

  • The denominator is 5, and the whole number is 4. Therefore, 5 x 4 = 20

Step 2: Add the numerator:

  • 20 + 3 = 23

Step 3: The final numerator is the answer to step 2, and the denominator stays the same:

  • Numerator = 23
  • Denominator = 5
  • This means that 4 3/5 as an improper fraction is 23/5

Can you convert a mixed fraction to a proper fraction and vice versa?

No. By definition, proper fractions have a value of less than 1. In contrast, mixed fractions have a value of greater than or equal to 1. Therefore, this conversion is not mathematically possible.

How to add mixed fractions?

Adding mixed fractions together is easier than you think. As with many things in math, there’s a step-by-step process you must follow:

  • Step 1: Convert all mixed fractions to an improper fraction
  • Step 2: Determine if the fractions have the same denominator. If they are the same, skip to step 4
  • Step 3: If the denominators are different, find the lowest common multiple of all denominators and make them the same 
  • Step 4: Add the numerators together and keep the denominator the same
  • Step 5: Simplify if possible

Example 1: 1 3/5 + 2 2/5

Solve 1 3/5 + 2 2/5.

Step 1: Convert all mixed fractions to improper fractions:

  • This step can be accomplished by doing the conversion process we mentioned above. 1 3/5 would be converted to 8/5, and 2 2/5 would be converted to 12/5
  • Therefore, the addition equation can be rewritten as 8/5 + 12/5

Step 2: Determine if the fractions have the same denominator. If they are the same, skip to step 4:

  • Since both fractions have the same denominator of 5, you can skip to the fourth step

Step 4: Add the numerators together and keep the denominator the same:

  • Numerator: 8 + 12 = 20
  • Denominator: Remains as 5
  • This results in the fraction 20/5

Step 5: Simplify if possible:

  • This fraction can be simplified from 20/5 to 4/1, which you simplify again to 4

As such, 1 3/5 + 2 2/5 = 4

Example 2: 2 4/7 + 7 3/4

Solve 2 4/7 + 7 3/4.

Step 1: Convert all mixed fractions to an improper fraction:

  • 2 4/7 as an improper fraction is 18/7, and 7 3/4 as an improper fraction is 31/4
  • Therefore, the addition equation can be rewritten as 18/7 + 31/4

Step 2: Determine if the fractions have the same denominator. If they are the same, skip to step 4:

  • Both fractions have different denominators. This means that you can move on to step 3

Step 3: If the denominators are different, find the lowest common multiple of all denominators and make them the same:

  • The lowest common multiple of 7 and 4 is 28. To make the denominators the same, you must multiply the first fraction by 4 (7 x 4 = 28) and the second fraction by 7 (4 x 7 = 28)
    • 18/7 x 4 = 72/28
    • 31/4 x 7 = 217/28
  • Therefore, the equation can be rewritten as 72/28 + 217/28

Step 4: Add the numerators together and keep the denominator the same:

  • Numerator: 72 + 217 = 289
  • Denominator: Remains as 28
  • This results in the fraction 289/28

Step 5: Simplify if possible:

  • 289/28 is an improper fraction, but it can be simplified into a mixed number
  • 28 can go 10 times into 289, leaving a remainder of 9
  • Writing this as a mixed fraction results in 10 9/28

As such, 2 4/7 + 7 3/4 = 10 9/28.

How to subtract mixed fractions?

Subtracting fractions is incredibly easy. You can follow the exact process outlined above but simply subtract in Step 4 instead of adding. Let’s look at a quick example.

Example 1: 5 2/6 - 3 5/6

Solve 5 2/6 - 3 5/6.

Step 1: Convert all mixed fractions to an improper fraction:

  • 5 2/6 as an improper fraction is 32/6, and 3 5/6 as an improper fraction is 23/6
  • Therefore, the addition equation can be rewritten as 32/6 + 23/6

Step 2: Determine if the fractions have the same denominator. If they are the same, skip to step 4:

  • Both fractions are the same, which means you can go straight to step 4

Step 4: Subtract the numerators together and keep the denominator the same:

  • Numerator: 32 - 23 = 9
  • Denominator: Remains as 6
  • This results in the fraction 9/6

Step 5: Simplify if possible:

  • The greatest common factor of 9 and 6 is 3. Therefore, the fraction can be simplified to 3/2 and then converted into the mixed fraction 1 1/2

As such, 5 2/6 - 3 5/6 = 1 1/2.

How to multiply mixed fractions?

Multiplying mixed fractions can be done via a three-step process:

  • Step 1: Convert all mixed fractions to an improper fraction
  • Step 2: Multiply the numerators together, and multiply the denominators together
  • Step 3: Simplify if possible

Example 1: 4 7/9 x 2 1/3

Solve 4 7/9 x 4 7/9 x 2 1/3.

Step 1: Convert all mixed fractions to an improper fraction:

  • 4 7/9 converted to an improper fraction is 43/9, and 2 1/3 is 7/3
  • Thus, the equation becomes 43/9 x 7/3

Step 2: Multiply the numerators together, and multiply the denominators together:

  • Numerators: 43 x 7 = 301
  • Denominators: 9 x 3 = 27
  • The resulting fraction is 301/27

Step 3: Simplify if possible:

  • 27 goes into 301 at least 11 times, leaving a remainder of 4 (301 ÷ 27 = 11 remainder 4)
  • Converting this into a mixed fraction results in 11 4/27

As such, 4 7/9 x 2 1/3 = 11 4/27.

Example 2: 6 1/2 x 3 4/5

Solve 6 1/2 x 3 4/5.

Step 1: Convert all mixed fractions to an improper fraction:

  • 6 1/2 converted to an improper fraction is 13/2, and 3 4/5 is 19/5
  • Thus, the equation becomes 13/2 x 19/5

Step 2: Multiply the numerators together, and multiply the denominators together:

  • Numerators: 13 x 19 = 247
  • Denominators: 2 x 5 = 10

The resulting fraction is 247/10

Step 3: Simplify if possible:

  • 10 goes into 247 at least 24 times, leaving a remainder of 7 (247 ÷ 10 = 24 remainder 7)
  • Converting this into a mixed fraction results in 10 7/10

As such, 6 1/2 x 3 4/5 = 10 7/10.

How to divide mixed fractions?

Dividing fractions is a little different than the other mathematical operations. After converting the fractions into improper form, you must utilize the ‘keep it, change it, flip it’ process. The full steps are as follows:

  • Step 1: Convert all mixed fractions to an improper fraction
  • Step 2: Keep the first fraction as is
  • Step 3: Change the division sign into a multiplication sign
  • Step 4: Flip the numerator and denominator of the second fraction around
  • Step 5: Multiply across and simplify if possible.

If that sounds a little confusing, don't worry. It will all make sense once you see a couple of practice problems.

Example 1: 3 2/3 ÷ 1 1/5

Solve 3 2/3 ÷ 1 1/5.

Step 1: Convert all mixed fractions to an improper fraction:

  • 3 2/3 becomes 11/3, and 1 1/5 becomes 6/5.
  • Therefore, the new equation is 11/3 ÷ 6/5

Step 2: Keep the first fraction as is:

  • 11/3 remains the same

Step 3: Change the division sign into a multiplication sign:

  • By changing the division sign to a multiplication sign, the equation now becomes 11/3 x 6/5

Step 4: Flip the numerator and denominator of the second fraction around:

  • 6/5 turns into 5/6, which means that equation now looks like 11/3 x 5/6

Step 5: Multiply across and simplify if possible:

  • Numerator: 11 x 5 = 55
  • Denominator: 3 x 6 = 18
  • This gives you the fraction 55/18
  • To simplify, you can calculate that 18 goes into 55 at least 3 times, leaving a remainder of 1 (55 ÷ 18 = 3 remainder 1)
  • Converting this into a mixed fraction is 3 1/18

As such, 3 2/3 ÷ 1 1/5 =  3 1/18.

Example 2: 2 2/9 ÷ 4 2/6

Solve 2 2/9 ÷ 4 2/6.

Step 1: Convert all mixed fractions to an improper fraction:

  • 2 2/9 as an improper fraction is 20/9, and 4 2/6 is 26/6
  • Therefore, the new equation is 20/9 ÷ 26/6

Step 2: Keep the first fraction as is:

  • 20/9 remains the same

Step 3: Change the division sign into a multiplication sign:

  • By changing the division sign to a multiplication sign, the equation now becomes 20/9 x 26/6

Step 4: Flip the numerator and denominator of the second fraction around:

  • 26/6 turns into 6/26, which means that equation now looks like 20/9 x 6/26

Step 5: Multiply across and simplify if possible:

  • Numerator: 20 x 6 = 120
  • Denominator: 9 x 26 = 234
  • This gives you the fraction 120/234
  • The greatest common factor of both 120 and 234 is 6
  • Therefore, the numerator becomes 20 (120 ÷ 6), and the denominator becomes 39 (234 ÷ 6 = 39)

As such, 2 2/9 ÷ 4 2/6 =  20/39.

Summary

A mixed number – also known as a mixed fraction – is a number that consists of a whole number and a proper fraction. They are simplified versions of improper fractions; thus, you can convert between the two. 

Doing mathematical operations using mixed numbers is fairly straightforward. Simply follow the simple steps outlined in this article and you’ll be able to tackle any problem you face with ease. Good luck!