The topic of this guide is the multiplication of fractions. 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 calculation. 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.
Content on the topic "Multiplying Fractions"
How to Multiply Fractions?
Fractions are multiplied by multiplying all the numerators above the fraction bars and also by 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 this example, the numerator was multiplied by the other numerator, and the denominator was multiplied by the other 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 with examples first how to cleverly shorten fractions before multiplying them, so that we can then comfortably continue calculating with numbers that are as small as possible. Then we multiply whole numbers by fractions, multiply mixed fractions, and finally present you with a video on multiplying fractions.
How Do You Truncate Fractions Before Multiplication?
Early shortening, i.e., shortening the fractions before multiplying all the numerators and all denominators, subsequently avoids complicated calculations with large numbers. Meanwhile, 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 out more on the subject of reduction 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.
|Example 1: Reduce individual fractions before multiplication|
420 × 721 = 4 × 720 × 21 = 28420 = 115
shorten both fractions beforehand
420 × 721 = 15 × 13 = 1 × 15 × 3 = 115
As you can see, we have saved ourselves a lot of work by shortening the two fractions before multiplication (the left fraction is shortened by 5 and the right fraction is shortened by 7). While the first calculation can partly only be solved with a 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..
|Example 2: Truncate crosswise before multiplication|
421 × 720 = 4 × 720 × 21 = 28420 = 115
cut crosswise beforehand.
We start as before:
421 × 720 = 4 × 721 × 20
Now shorten the left numerator and right denominator by 5
4 × 721 × 20 = 1 × 721 × 5
Now shorten the right numerator and left denominator by 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 shorten these large numerators and denominators again at the end of the calculation, it makes a lot of sense to shorten 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.
|Example: Multiply whole number by fraction|
|5 × 23 = 51 × 23 = 5 × 21 × 3 = 103|
As described earlier, 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|
From the example below, the integer part of the mixed fraction, i.e., 2, was converted here into eight-quarter and added to the associated fraction one-quarter. The mixed fraction was thus converted into an improper fraction. Fractions are called improper if the numerator is greater than the denominator.
Converting mixed fractions into improper fractions
A mixed fraction or number is converted into an improper fraction by multiplying the integer part by the denominator and then adding the numerator to it. Meanwhile, the denominator remains unchanged.
Example of conversion
The mixed fraction from the above example is thus converted into an improper fraction as follows.
The integer 2 is multiplied by the denominator 4 and added to the previous numerator 1.
Multiplication of the two fractions
Now the two fractions shown in the example can be multiplied together.
Video on Multiplying Fractions
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, including "cross-shortening".
More online calculators
As source for the information in the 'Fractions' category, we have used in particular:
Last update on February 20, 2023
The pages of the 'Fractions' category were last editorially reviewed by Michael Mühl on February 20, 2023. They all correspond to the current status.
Previous changes on November 22, 2022
- 22.11.2022: Publication of the topic Fraction together with the corresponding texts.
- Editorial revision of all texts in this category