Mixed Numbers & Fractions: Convert, Examples

This article is about mixed numbers. You will learn what mixed numbers are, how to calculate with them and convert them to fractions.

This article belongs in the subject math and there in the area algebra – fractions and decimals.

  • Definition of a mixed number
  • Converting fractions to mixed numbers and vice versa
  • Localization of mixed numbers on the number line
  • Addition, subtraction, multiplication and division of mixed numbers in two ways
  • Mixed number problems with practice solutions at the end of the article

What is a mixed number?

A mixed number is a rational number made up of an integer and a fraction.

This notation is then called mixed notation. The mathematicians have been a bit lazy over time and it has become normal to simply omit an arithmetic sign (actually there should be a + between the number and the fraction (you will see why in the next two sections, in which it about converting mixed numbers into fractions and vice versa)). So the numbers in the example are the same as «Two wholes and a third» and «Three wholes and two fifths». This is important to know if you want to do arithmetic with mixed numbers.

The absolute value of mixed numbers is always greater than 1. They can also be converted into an improper fraction, i.e. a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator. Mixed numbers are therefore fractional numbers and belong – just like all fractions – to the rational numbers.

Note: Mixed numbers and mixed fractions are synonyms, so they mean the same thing.

Convert mixed numbers to fractions

Since mixed notation is sometimes impractical, you should be able to convert between mixed notation and fractional notation.

The first thing to do is convert the mixed number to a fraction. First of all, it is important to know that the + in the mixed spelling has been lost over time! So the following applies:

First, the whole number must now be written as a fraction with the denominator that also occurs in the mixed number. In this case, in thirds.

Note: If you don’t understand this conversion, you should take a look at the fractions chapter!

Now all you have to do is add the two fractions and you get the result:

Now it’s the turn of the mixed number.

With the addition of the +, the conversion of the 3 into fifths and the subsequent addition of the fractions, the following calculation results:

Convert fractions to mixed numbers

For the conversion in the other direction we take two new fractions – otherwise it would be too easy 🙂

The first thing to do is convert the fraction to a mixed number.

To do this, you first look at the numerator of the fraction and convert it into a sum so that the first summand can be divided by the number in the denominator without a remainder.

Here that would be 20=18+2, because 18 can be divided evenly by 6.

This sum is then written into the numerator:

Next, pull the fracture apart.

Finally, the first fraction is converted into a whole number and shortened if necessary:

As a second example, we take the fraction.

The appropriate division of the numerator is 45=42+3.

After the steps described in example 1, the calculation then looks like this:

Mixed numbers on the number line

One advantage of mixed numbers is that you can easily locate them on the number line. Because you will be shown directly next to which whole number the mixed number is.

We will again use the two numbers from the first example and locate them on the number line:

The mixed number can be entered as follows:

It is certainly greater than 2 and smaller than 3. It is therefore in the range between 2 and 3. Now the range between 2 and 3 is broken down into three equal parts, which are then the thirds. After the first third lies the desired number.

For the mixed number it works the same way:

It is between 3 and 4. This range is now divided into five equal parts, because the fraction in the mixed number has the 5 in the denominator. The number is after two of these five fifths.

Note: If you had major trouble with this section, you should check out the Comparing and arranging fractions article again. You can find this in the chapter «Fractions and decimals», where you also found this article.

Calculate with mixed numbers

Now it’s time to get down to business! We want to understand how to calculate with mixed numbers.

Adding mixed numbers

Adding mixed numbers is comparatively easy – you can add whole numbers and fractions separately.

Case 1: The denominators of the fractions are the same

This is the simplest case. You can just add the whole numbers and then the fractions.

Don’t forget to shorten completely!

You just have to make sure that the fraction in the mixed number of the result is a real fraction and not an improper one, i.e. the denominator is larger than the numerator. If this is not the case, you still have to extract a whole from the fraction, i.e. convert it again into a mixed number:

Case 2: The denominators of the fractions are not equal

In this case, the fractions for addition must be made to have the same name, i.e. in the same way as when adding fractions. Otherwise, the procedure is as in case 1:

Subtract mixed numbers

Even when subtracting, you can calculate wholes and fractions separately. Again, the fractions may need to be made of the same name.

Here you should think about the shortening again!

However, if the fraction of the subtrahend (i.e. of the number that is subtracted!) is greater than the fraction of the minu ending, then a whole of the minu ending (the number that is subtracted from) must be converted into a fraction.

Adding and subtracting mixed numbers using fractions

When subtracting, it is sometimes difficult to check whether the fraction in the subtrahend is larger than that in the minute end. When adding, check whether the result contains an improper fraction.

To get around these two problems, there is an alternative way of calculating: you can also simply convert the mixed numbers into fractions and then carry out the fraction calculation as normal.

You can always use this alternative. However, sometimes it is more clever if you calculate with the mixed numbers because it saves you a few calculation steps. But if you want to play it safe, you’ll always get the result.

Multiply and divide mixed numbers

If you want to multiply or divide two mixed numbers, you have to convert them to fractions. There is no way, like addition and subtraction, to calculate wholes and fractions separately. But then the calculation method is nothing new, because it works in the same way as with the multiplication and division of fractions.

Note: You can find out how fractions are multiplied or divided in the Fractions chapter.

exercises

Task 1

calculate For which tasks is it more adept to calculate with the mixed numbers?

exercise 2

calculate

solutions

Task 1:

Exercise 2:

Mixed numbers and fractions – the most important things at a glance