# Multiplying Fractions

Fractions ﹣ Multiply Fractions

The multiplication of fractions is the topic of this guide. After an explanation of the rules for multiplying simple fractions, the multiplication of mixed fractions is shown. With the help of the calculator for multiplying fractions, you can perform any calculations. Each step of the multiplication together with the clever shortening of the entered fractions is derived in detail in the calculator.

On the general page on fractions you will find a lot of basic information on fractions and their transformations. If you want to learn how to do the other arithmetic operations on fractions, visit our guides on Dividing fractions, Adding fractions or Subtracting fractions.

## How to multiply fractions?

Fractions are multiplied by multiplying all the numerators above the fraction bars and also multiplying all the denominators below the fraction bars. The result of multiplying fractions is the product of the fractions.

Example: Multiplication of fractions
34 × 12 = 3 × 14 × 2 = 38

In the example, numerator was multiplied by numerator and denominator was multiplied by denominator. The multiplication of fractions is therefore simpler than the addition of fractions or the subtraction of fractions: While you first have to calculate a common denominator for the addition and subtraction of fractions, this is not necessary for multiplication. When multiplying fractions, only the numerators and the denominators have to be multiplied.

In the following, we will show step by step, using examples, first how to cleverly shorten fractions before multiplication in order to be able to continue calculating comfortably with numbers that are as small as possible afterwards. Then we multiply whole numbers with fractions, multiply mixed fractions and finally present you with a video on multiplying fractions.

## How do you truncate fractions before multiplication?

Early shortening, as shortening the fractions before the multiplication of all numerators as well as the multiplication of all denominators, subsequently avoids complicated calculations with large numbers. On the one hand, the individual fractions involved in the multiplication can be shortened if necessary. In addition, when multiplying fractions, you can also shorten them "crosswise", i.e. shorten the numerator of one fraction with the denominator of the other fraction, as we would like to illustrate with the following examples. By the way, you can find more on the subject of reducing on our overview page on fractions.

### Truncate individual fractions before multiplying

The following example shows the advantage of truncating the fractions involved in multiplication before multiplication.

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Example 1: Reduce individual fractions before multiplication

Statt

420 × 721 = 4 × 720 × 21 = 28420 = 115

shorten both fractions beforehand

420 × 721 = 15 × 13 = 1 × 15 × 3 = 115

As you can easily see, we have saved ourselves a lot of work by shortening the two fractions before multiplication (left fraction shortened by 5 and right fraction shortened by 7). While the first calculation can partly only be solved by pocket calculator, the second multiplication is much easier to calculate by shortening beforehand.

### Cross-shortening fractions before multiplying

The following example shows the advantage of cross-shortening when multiplying fractions, i.e. shortening the numerator of one fraction with the denominator of the other fraction and vice versa.

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Example 2: Truncate crosswise before multiplication

Statt

421 × 720 = 4 × 720 × 21 = 28420 = 115

cut crosswise beforehand.

We start as before:

421 × 720 = 4 × 721 × 20

Now shorten left numerator and right denominator with 5

4 × 721 × 20 = 1 × 721 × 5

Now shorten right numerator and left denominator with 7

1 × 721 × 5 = 1 × 13 × 5 = 115

Here, too, you can see the benefit of shortening beforehand. Instead of making the numerator and denominator very large by multiplying them and then having to trim these large numerators and denominators again at the end of the calculation, it makes a lot of sense to trim before multiplying the fractions. Not only can you shorten the individual fractions, but, as we have seen, you can also intelligently shorten them crosswise.

## How to multiply whole numbers by fractions?

When we want to multiply whole numbers by a fraction, we take advantage of the fact that whole numbers can easily be converted into a fraction: Every whole number can be represented as a "unit", so the whole number 5, for example, forms the fraction 5 units, as we can see in the following example.

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Example: Multiply whole number by fraction
5 × 23 = 51 × 23 = 5 × 21 × 3 = 103

As described at the beginning, the integer 5 was converted into a fraction and then the multiplication of this fraction with the other fraction of the task was carried out.

## How to multiply mixed fractions?

Mixed fractions, also called mixed numbers, are made up of a whole number and an ordinary fraction. These two are added together even though there is no plus sign between them. To multiply mixed fractions, for each mixed fraction you first convert the whole number into the corresponding fraction so that the resulting fraction can then be multiplied by the other fraction in the task.

Example: Multiplication of mixed fractions
214 × 13 = 94 × 13 = 9 × 14 × 3 = 912 = 34

The integer part of the mixed fraction, i.e. the two, was converted here into 8 quarters and added to the associated fraction. The mixed fraction was thus converted into a non-genuine fraction. Fractions are called improper if the numerator is greater than the denominator.

### Umwandlung gemischter in unechte Brüche

A mixed fraction or number is converted into a non-genuine fraction by multiplying the integer part by the denominator and then adding the numerator to it. The denominator remains unchanged.

### Example of conversion

The mixed fraction from the above example is thus converted into a non-genuine fraction as follows.

The integer 2 is multiplied by the denominator 4 and added to the previous numerator 1.

214 = 2 × 4 + 14 = 94

### Multiplication of the two fractions

Now the two fractions of the example can be multiplied together.

94 × 13 = 9 × 14 × 3 = 912 = 34

### Video on multiplying simple fractions

Here we present a video on multiplying fractions by Math Antic. The multiplication of fractions is explained using some examples. Later the advantages of shortening individual fractions before multiplication are shown and the "cross-shortening".

## Source information

As source for the information in the 'Fractions' category, we have used in particular:

## Last update on November 22, 2022

The last changes in the 'Fractions' category were implemented by Michael Mühl on November 22, 2022. The main changes were:

• 22.11.2022: Publication of the topic Fraction together with the corresponding texts.
• Editorial revision of all texts in this category