Dividing fractions is similar to multiplying fractions. However, one fraction is multiplied by the reciprocal of the other fraction, which we will see below. After explaining the rules for dividing simple fractions, we will show you how to divide mixed fractions. The calculator for dividing fractions allows you to perform any calculation. All steps of the division as well as the intelligent shortening of the entered fractions are comprehensively derived in the calculator.
The general page on fractions provides you with a lot of basic information on fractions and the transformation of fractions. Would you like to know how to perform the other arithmetic operations on fractions? Then visit our guides on the topics Multiplying fractions, Adding fractions or Subtracting fractions.
Contents on the topic "Dividing Fractions"
Contents
How to Divide Fractions
Fractions are divided by multiplying one fraction by the reciprocal of the other fraction. After that, fractions are multiplied by multiplying all the numerators above the fraction bars and also by multiplying all the denominators below the fraction bars.
| Example: Division of fractions |
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| 12 ÷ 34 = 12 × 43 = 1 × 42 × 3 = 46 |
In the example above, the reciprocal fraction was formed first, i.e., the reciprocal of the right-hand fraction with numerator 3 and denominator 4. The numerator and denominator were therefore swapped so that the left-hand fraction is now multiplied by the right-hand reciprocal fraction. In contrast to addition and subtraction of fractions, the new numerators and denominators can now be multiplied together.
In the following section, we will show step by step, using examples, how to cleverly shorten fractions before division, so that you can then continue to calculate with the smallest possible numbers without any problems. Then we divide whole numbers with fractions, divide mixed fractions and finally present you with a video on dividing fractions.
How Do You Truncate Fractions Before Division?
Early shortening, i.e., shortening the fractions before dividing the left fraction by the right fraction, subsequently avoids complicated calculations with large numbers. For one thing, the individual fractions involved in the division can be shortened if necessary. Furthermore, when dividing fractions, you can also shorten them "crosswise", i.e., you can shorten the numerator of one fraction with the denominator of the reciprocal fraction, or the denominator of one fraction with the numerator of the reciprocal fraction, as we will show in the following examples. By the way, you can find more about shortening on our overview page on fractions.
Shorten individual fractions before dividing
The following example shows the advantage of shortening the fractions before dividing.
| Example 1: Shortening individual fractions before division |
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Instead of 420 ÷ 721 = 420 × 217 = 4 × 2120 × 7 = 84140 = 35 shorten both fractions beforehand 420 ÷ 721 = 15 ÷ 13 = 15 × 31 = 1 × 35 × 1 = 35 |
As you can easily see, we have saved ourselves a lot of work by shortening the two fractions before division (the left fraction is shortened by 5 and the right fraction is shortened by 7). While the first calculation can only be solved with a pocket calculator, the second division is much easier to calculate by shortening beforehand.
Cross-cut fractions before dividing
The following example shows the advantage of being able to cross-cut when dividing fractions, i.e., to cut the numerator of one fraction with the denominator of the reciprocal fraction to be multiplied and vice versa.
| Example 2: Cross-cut before division |
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Instead of 421 ÷ 207 = 421 × 720 = 4 × 721 × 20 = 28420 = 115 Cross-shorten beforehand We start as before: . 421 ÷ 207 = 421 × 720 = 4 × 721 × 20 Now shorten left numerator and right denominator by 5 4 × 721 × 20 = 1 × 721 × 5 Now shorten right numerator and left denominator by 7 1 × 721 × 5 = 1 × 13 × 5 = 115 |
Here, too, the benefit of shortening beforehand becomes clear. Instead of making the untruncated values of the numerator and denominator very large by multiplying them by the reciprocal fraction following the division and then awkwardly truncating these large numerators and denominators again at the end of the calculation, it makes a lot of sense to carry out the truncation before multiplying the fraction and the reciprocal fraction. Not only can you shorten the individual fractions but, as we have seen, you can also intelligently cross-shorten them after forming the reciprocal fraction.
How to Divide Whole Numbers by Fractions
When we want to divide 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". The whole number 4 can thus be represented by the fraction 4 ones, as we can see in the following example.
| Example: Multiply whole number by fraction |
|---|
| 4 ÷ 32 = 4 × 23 = 41 × 23 = 4 × 21 × 3 = 83 |
As described earlier, the integer 4 was converted into a fraction, and then the division of this fraction with the other fraction of the task was carried out.
How to Divide Mixed Fractions?
Mixed fractions or mixed numbers are composed of a whole number and an ordinary fraction that are added together, even though there is no plus sign between them. To divide mixed fractions, first convert the whole number for each mixed fraction into the corresponding fraction so that the resulting fraction can then be divided with the other fraction of the task.
| Example: Division of mixed fractions |
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| 214 ÷ 13 = 94 ÷ 13 = 94 × 31 = 9 × 34 × 1 = 274 = 634 |
From the example below, the integer part of the mixed fraction, i.e., the 2, was converted here into an eight-quarter and added to the corresponding one-quarter. The mixed fraction was thus converted into a non-genuine fraction. Fractions are called improper if the numerator is greater than the denominator.
Conversion of mixed fractions into improper fractions
A mixed fraction or a mixed 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 a non-genuine fraction as follows.
The integer 2 is multiplied by the denominator 4 and added to the previous numerator 1.
Dividing the two fractions
Now the two fractions of the example can be divided.
Video on “Dividing Fractions”
Finally, a video on “dividing fractions” by Math Antic. The first example on dividing fractions is followed by a slightly more difficult example. Later, Math Antics explains the division of mixed fractions.
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Source information
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