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Calculating fractions - that is, calculating with fractions - we show you here. Let's look at this:

  • A Explanationwhat a fraction is and how to calculate with it.
  • Lots Examples for adding, subtracting, multiplying and dividing fractions as well as reducing and expanding.
  • tasks / Exercises for fractions
  • A Video on the subject.
  • A Question and answer area for fractions.

A little tip to start with: We'll see in a moment that a fraction is nothing more than a division. So if you don't know dividing - such as 3: 4 - you can take a look at this under Divide / Divide.

Fractional arithmetic: what is a fraction?

First of all, we should first cover the basics of fractions: What is a fraction? Let's say you eat rolls. Then you can eat 1 roll or 2 rolls or even 3. But sometimes you don't eat a whole roll, but only part of a roll. Or you can only eat part of a pizza. Of course, you have to be able to write something down in mathematics somehow. This is exactly what you do with a fraction.

How can you easily represent the fractions for children or beginners in this topic? Let's take a pizza. We cut this pizza into 8 equal pieces:

Suppose we now eat 5 out of 8 pieces. Then you express this in mathematics with a fraction. In this case you write this down

Let's mark the 5 pieces once in red (and the rest in green):

If we eat the 5 pieces (marked in red) there are 3 pieces left. These 3 of 8 pieces can of course also be written as a fraction.

Let's get to the terms in fractions: We have a number that is at the top. Then a line and below that a number. The number above is called the numerator, the number below is the denominator and there is a fraction line in between.

The definition of fractions:


Fractions in mathematics are concerned with representing parts of something as a whole. These proportions can be used to perform the basic arithmetic operations of addition (+), subaction (-), multiplication (·) and division (:).


Calculate with fractions, abbreviate and expand

In the basics above, it was already mentioned that fractions can also be used for arithmetic. This is exactly what is supposed to happen here. We look at all the basic arithmetic operations. These are examples with calculation methods and solutions.

Example 1: Adding fractions

We cut a cake into 9 equal pieces. Anna eats 3 pieces and Marc eats 4 pieces. How do you write this as a fraction? How many pieces were eaten in the end?


We first write both entries as a fraction. These are 3 out of 9 pieces and 4 out of 9 pieces. We add these up, i.e. there is a plus sign between the two fractions. Since the denominators - i.e. the numbers below - are the same, we can simply add the numerators.

In the end, 7 out of 9 pieces were eaten with it.

Note: We will learn how to deal with different denominators below.

Example 2: subtracting fractions

The reverse of addition is subtraction. We will now look at the "minus calculation" for fractions. We assume that a cake was cut into 12 pieces. All parts are the same size and were offered to the guests. The guests eat 7 pieces. How many pieces of cake are left?

Solution to the example:

At the beginning we have a whole cake with 12 pieces that was given to the guests. 12 of 12 pieces got this in front of them. 7 out of 12 pieces were eaten. We therefore have to subtract 7 pieces from the 12 pieces.

There are 5 of 12 pieces left.

Here, too, the hint: We'll look at how to deal with different denominators below.

Example 3: multiplying fractions

The next basic arithmetic is multiplication, which is the multiplication of two (or more) fractions. Some people also refer to this as "fractions times". The calculation is actually quite simple. Numerator is multiplied by numerator and denominator is multiplied by denominator.

When multiplying fractions, there are often times that fractions can be shortened. What it is and how it works is covered below.

Example 4: dividing fractions

The last basic arithmetic is division. It's about dividing fractions. We have two fractions between which there is a divided sign. The calculation looks like this, that one reverses the second fraction (numerator and denominator exchanged, marked in red). This is called forming the reciprocal. After forming the reciprocal value, the two fractions are multiplied together (numerator times numerator and denominator times denominator).

In the next section you will learn how to shorten such a result.

Example 5: Shorten fractions

This section is about truncating fractions. This means simply dividing the numerator and denominator by the same number. This is normally done when you have natural numbers in the numerator and denominator and both are divisible by another natural number without a remainder. This simplifies the fraction.

You can either try out whether this is the case by dividing by 2, 3, 4 etc. in your head, or you can also use the divisibility rules. An example should clarify this. So the next fraction was shortened by 3, because the numerator and denominator were each divided by 3.

The next fraction was shortened with 2 and then shortened again (with 3). This means that first the numerator and denominator were each divided by 2 and then divided again by 3.

Example 6: expanding fractions

When truncating fractions, the numerator and denominator were each divided by a number. When expanding fractions, the opposite happens: fractions are expanded by multiplying the numerator and denominator by a number (or variable). As an example, we want to expand a fraction with 5. Therefore we multiply the numerator and denominator by 5 each.

Example 7: Find the main denominator

In fractions, a distinction is made between eponymous and unlike fractions. Fractions with the same name all have the same denominator (for example, they all have the denominator 3). Fractions of the same name have different denominators. For the additions and subtractions of fractions above, we had the same denominators. Therefore we were allowed to simply add or subtract the counters here.

However, if you have different denominators, you first have to look for a common denominator, this is called the main denominator. So we have to make sure that all fractions have the same denominator. There are several ways to do this. The easiest way to understand is to be shown here. To be calculated:

We have 3 and 4 as the denominator here. A simple way to find a common denominator is to simply multiply these two numbers. 3 4 = 12.

What do we do with the counters now? Well, on the first fraction, we had to multiply the denominator by 4 to get 12. We also do this with the numerator: 2 · 4 = 8. For the second denominator, we had to multiply the 4 by 3 to get the 12. We now do the same with the numerator: 3 · 3 = 9.

Now we can just add the numerators (and take the denominator).

Exercises / exercises with fractions


Videos Fractional Calculation

Explanations for calculating fractions

The next video covers the following content about fractions:

  • What is a break
  • Why do you need fractions in mathematics?
  • How do you represent a break on the Internet or on paper?

Next video »

Questions with Answers Fractional Calculation

This section deals with typical questions with answers about fractions.

Q: What rules do you have to follow when calculating fractions?

A: Of course, the usual calculation rules also apply to fractions. Calculations must take place in this order:

  1. Brackets
  2. Potencies
  3. Point calculation (multiplication and division)
  4. Line calculation (addition and subtraction)
  5. Left to right

More on this in the calculation rules.

Q: Can you also work out a fraction?

A: Of course. You can also calculate a fraction. For example, the fraction 3: 4 can be calculated with the written division with decimal numbers.

Q: When is the fractions calculated in school?

A: In elementary school, fractions are usually not dealt with yet. But from the 5th grade onwards, it often starts with the basics of fractions. First we will deal with what a fracture actually is. Then it is about how to deal with fractions, i.e. the basic arithmetic operations as well as expanding and shortening fractions. Sometimes different types of fractions are also covered in class. Fractions are dealt with again in the 6th and 7th grades. Fractional arithmetic is taught in both grammar school and secondary school.

Q: How can you compare fractions?

A: Bring all fractions to the same denominator (and adjust the numerators accordingly). Then all you have to do is look at the counter. The smaller the counter, the smaller the number. And vice versa: the larger the counter, the larger the number.